Vibration Monitoring, Testing, and Instrumentation (Mechanical Engineering Series)

Author: Clarence W. de Silva

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Vibration Monitoring, Testing, and Instrumentation

© 2007 by Taylor & Francis Group, LLC

53191—14/3/2007—18:49—VELU—246493— CRC – pp. 1–15

Mechanical Engineering Series Frank Kreith and Roop Mahajan - Series Editors Published Titles Computer Techniques in Vibration Clarence W. de Silva Distributed Generation: The Power Paradigm for the New Millennium Anne-Marie Borbely &Jan F. Kreider Elastoplasticity Theory Vlado A. Lubarda Energy Audit of Building Systems: An Engineering Approach MoncefKrarti Engineering Experimentation Euan Somerscales Entropy Generation Minimization Adrian Bejan Finite Element Method Using MATLAB®, 2 n d Edition Young W. Kwon & Hyochoong Bang Fluid Power Circuits and Controls: Fundamentals and Applications John S. Cundiff Fundamentals of Environmental Discharge Modeling Lorin R. Davis Heat Transfer in Single and Multiphase Systems GregF. Naterer Introductory Finite Element Method Chandrakant S. Desai & Tribikram Kundu Intelligent Transportation Systems: New Principles and Architectures Sumit Ghosh & Tony Lee Mathematical & Physical Modeling of Materials Processing Operations Olusegun Johnson Ilegbusi, Manabu Iguchi & Walter E. Wahnsiedler Mechanics of Composite Materials AutarK. Kaw

© 2007 by Taylor & Francis Group, LLC

Mechanics of Fatigue Vladimir V. Bolotin Mechanics of Solids and Shells: Theories and Approximations Gerald Wempner & Demosthenes Talaslidis Mechanism Design: Enumeration of Kinematic Structures According to Function Lung-Wen Tsai The MEMS Handbook, Second Edition MEMS: Introduction and Fundamentals MEMS: Design and Fabrication MEMS: Applications Mohamed Gad-el-Hak Nanotechnology: Understanding Small Systems Ben Rogers, Jesse Adams & Sumita Pennathur Nonlinear Analysis of Structures M. Sathyamoorthy Practical Inverse Analysis in Engineering David M. Trujillo & Henry R. Busby Pressure Vessels: Design and Practice Somnath Chattopadhyay Principles of Solid Mechanics Rowland Richards, Jr. Thermodynamics for Engineers Kau-Fui Wong Vibration Damping, Control, and Design Clarence W. de Silva Vibration Monitoring, Testing, and Instrumentation Clarence W. de Silva Vibration and Shock Handbook Clarence W. de Silva Viscoelastic Solids Roderic S. Lakes

Vibration Monitoring, Testing, and Instrumentation

Edited by

Clarence W. de Silva The University of British Columbia Vancouver, Canada

Ltfi) CRC Press VV^

J Taylor & Francis Group Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Croup, an informa business

© 2007 by Taylor & Francis Group, LLC

This material was previously published in Vibration and Shock Handbook © 2005 by CRC Press, LLC.

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2007 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed in the United States of America on acid-free paper 10987654321 International Standard Book Number-10:1-4200-5319-1 (Hardcover) International Standard Book Number-13:978-1-4200-5319-7 (Hardcover) This book contains information obtained from authentic and highly regarded sources. Reprinted material is quoted with permission, and sources are indicated. A wide variety of references are listed. Reasonable efforts have been made to publish reliable data and information, but the author and the publisher cannot assume responsibility for the validity of all materials or for the consequences of their use. No part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC) 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging-in-Publication Data Vibration monitoring, testing, and instrumentation / editor Clarence W. de Silva. p. cm. — (Mechanical engineering series) Includes bibliographical references and index. ISBN-13: 978-1-4200-5319-7 (alk. paper) ISBN-10:1-4200-5319-1 (alk. paper) 1. Vibration. 2. Shock (Mechanics) I. De Silva, Clarence W. II. Title. III. Series. TA355.V5228 2007 620.3028'7-dc22 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

© 2007 by Taylor & Francis Group, LLC

2006100139

Preface

Individual chapters authored by distinguished leaders and experienced professionals in their respective topics, this book provides for engineers, technicians, designers, researchers, educators, and students, a convenient, thorough, up-to-date and authoritative reference source on techniques, tools, and data for monitoring, testing, and instrumentation of mechanical vibration and shock. Included are shock and vibration methodologies particularly for civil and mechanical engineering systems; instrumentation and testing methods, including sensors, exciters, signal acquisition, conditioning, and recording, and LabVIEWw tools for virtual instrumentation; testing and design for seismic vibration, and related regulatory issues; and human response to vibration. Important information and results are summarized as windows, tables, graphs, and lists, throughout the chapters, for easy reference and information tracking. References are given at the end of each chapter, for further information and study. Cross-referencing is used throughout to indicate other places in the book where further information on a particular topic is provided. In the book, equal emphasis is given to theory and practical application. Analytical formulations, design approaches, monitoring and testing techniques, and commercial software tools are presented and illustrated. Commercial equipment, computer hardware, and instrumentation are described, analyzed, and demonstrated for ﬁeld application, practical implementation, and experimentation. Examples and case studies are given throughout the book to illustrate the use and application of the included information. The material is presented in a format that is convenient for easy reference and recollection. Mechanical vibration is a manifestation of the oscillatory behavior in mechanical systems as a result of either the repetitive interchange of kinetic and potential energies among components in the system, or a forcing excitation that is oscillatory. Such oscillatory responses are not limited to purely mechanical systems, and are found in electrical and ﬂuid systems as well. In purely thermal systems, however, free natural oscillations are not possible, and an oscillatory excitation is needed to obtain an oscillatory response. Sock is vibration caused by brief, abrupt, and typically highintensity excitations. Low levels of vibration mean reduced noise and improved work environment. Consequently, a considerable effort is devoted today to monitoring, studying, and modifying the vibration and shock generated by machinery components, machine tools, transit vehicles, impact processes, civil engineering structures, ﬂuid ﬂow systems, and aircraft. Before designing a system for good vibratory or shock performance, it is important to understand, analyze, and represent the dynamic characteristics of the system. This may be accomplished through monitoring, testing and analysis of test data, which is the emphasis of this book. In recent years, educators, researchers, and practitioners have devoted considerable effort towards studying, monitoring, and testing vibration and shock in a range of applications in various branches of engineering; particularly, civil, mechanical, aeronautical and aerospace, and production and

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vi

Preface

manufacturing. Speciﬁc applications are found in machine tools, transit vehicles, impact processes, civil engineering structures, construction machinery, industrial processes, product qualiﬁcation and quality control, ﬂuid ﬂow systems, ships, and aircraft. This book is a contribution towards these efforts.

Clarence W. de Silva

Editor-in-Chief Vancouver, Canada

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Acknowledgments

I wish to express my gratitude to the authors of the chapters for their valuable and highly professional contributions. I am very grateful to Michael Slaughter, Acquisitions Editor-Engineering, CRC Press, for his enthusiasm and support throughout the project. Editorial and production staff at CRC Press have done an excellent job in getting this volume out in print. Finally, I wish to lovingly acknowledge the patience and understanding of my family.

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Editor-in-Chief

Dr. Clarence W. de Silva, P.Eng., Fellow ASME, Fellow IEEE, Fellow Canadian Academy of Engineering, is Professor of Mechanical Engineering at the University of British Columbia, Vancouver, Canada, and has occupied the NSERC-BC Packers Research Chair in Industrial Automation, since 1988. He has earned Ph.D. degrees from the Massachusetts Institute of Technology and the University of Cambridge, England. De Silva has also occupied the Mobil Endowed Chair Professorship in the Department of Electrical and Computer Engineering at the National University of Singapore. He has served as a consultant to several companies including IBM and Westinghouse in U.S., and has led the development of eight industrial machines and devices. He is recipient of the Henry M. Paynter Outstanding Investigator Award from the Dynamic Systems and Control Division of the American Society of Mechanical Engineers (ASME), Killam Research Prize, Lifetime Achievement Award from the World Automation Congress, Outstanding Engineering Educator Award of IEEE Canada, Yasundo Takahashi Education Award of the Dynamic Systems and Control Division of ASME, IEEE Third Millennium Medal, Meritorious Achievement Award of the Association of Professional Engineers of BC, and the Outstanding Contribution Award of the Systems, Man, and Cybernetics Society of the Institute of Electrical and Electronics Engineers (IEEE). He has authored 16 technical books including Sensors and Actuators: Control System Instrumentation (Taylor & Francis, CRC Press, 2007); Mechatronics—An Integrated Approach (Taylor & Francis, CRC Press, Boca Raton, FL, 2005); Soft Computing and Intelligent Systems Design—Theory, Tools, and Applications (with F. Karry, Addison Wesley, New York, NY, 2004); VIBRATION Fundamentals and Practice (Taylor & Francis, CRC Press, 2nd edition, 2006); INTELLIGENT CONTROL Fuzzy Logic Applications (Taylor & Francis, CRC Press, 1995), Control Sensors and Actuators (Prentice Hall, 1989), 14 edited volumes, over 170 journal papers, 200 conference papers, and 12 book chapters. He has served on the editorial boards of 14 international journals, in particular as the Editor-in-Chief of the International Journal of Control and Intelligent Systems, Editor-in-Chief of the International Journal of Knowledge-Based Intelligent Engineering Systems, Senior Technical Editor of Measurements and Control, and Regional Editor, North America, of Engineering Applications of Artiﬁcial Intelligence – the International Journal of Intelligent Real-Time Automation. He is a Lilly Fellow at Carnegie Mellon University, NASA-ASEE Fellow, Senior Fulbright Fellow at Cambridge University, ASI Fellow, and a Killam Fellow. Research and development activities of Professor de Silva are primarily centered in the areas of process automation, robotics, mechatronics, intelligent control, and sensors and actuators, with cash funding of about $6 million, as principal investigator.

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Contributors

Kourosh Danai

University of Massachusetts Amherst, Massachusetts

T.H. Lee

National University of Singapore Singapore

Priyan Mendis

Clarence W. de Silva

Y.P. Leow

Tuan Ngo

The University of British Columbia Vancouver, British Columbia, Canada

P. S. Heyns

University of Pretoria Pretoria, South Africa

S. Huang

National University of Singapore Singapore

Hirokazu Iemura Kyoto University Kyoto, Japan

Sarvesh Kumar Jain

University of Melbourne Melbourne, Victoria, Australia

Singapore Institute of Manufacturing Technology Singapore

University of Melbourne Melbourne, Victoria, Australia

S.Y. Lim

Kyoto University Kyoto, Japan

Singapore Institute of Manufacturing Technology Singapore

Jiaohao Lin

Dalian University of Technology Liaoning, People’s Republic of China

W. Lin

Mulyo Harris Pradono

C. Scheffer

University of Stellenbosch Pretoria, South Africa

K.K. Tan

National University of Singapore Singapore

Madhav Institute of Technology and Science Madhya Pradesh, India

Singapore Institute of Manufacturing Technology Singapore

K.Z. Tang

Christian Lalanne

Chris K. Mechefske

Yahui Zhang

Engineering Consultant Jalles, France

Queen’s University Kingston, Ontario, Canada

National University of Singapore Singapore

Dalian University of Technology Liaoning, People’s Republic of China

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Contents

1

Vibration Instrumentation

2

Signal Conditioning and Modiﬁcation

3

Vibration Testing Clarence W. de Silva .......................................................................................... 3-1

4

Experimental Modal Analysis Clarence W. de Silva .................................................................. 4-1

1.1 1.2 1.3 1.4 1.5 1.6

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9

3.1 3.2 3.3 3.4 3.5

4.1 4.2 4.3 4.4 4.5 4.6

Clarence W. de Silva ....................................................................... 1-1 Introduction ................................................................................................................................ 1-1 Vibration Exciters ....................................................................................................................... 1-3 Control System ......................................................................................................................... 1-15 Performance Speciﬁcation ....................................................................................................... 1-21 Motion Sensors and Transducers ............................................................................................ 1-27 Torque, Force, and Other Sensors .......................................................................................... 1-50 Appendix 1A Virtual Instrumentation for Data Acquisition, Analysis, and Presentation ....................................................................................................................... 1-73 Clarence W. de Silva ................................................ 2-1 Introduction ................................................................................................................................ 2-2 Ampliﬁers .................................................................................................................................... 2-2 Analog Filters ............................................................................................................................ 2-15 Modulators and Demodulators ............................................................................................... 2-29 Analog –Digital Conversion ..................................................................................................... 2-37 Bridge Circuits .......................................................................................................................... 2-43 Linearizing Devices .................................................................................................................. 2-49 Miscellaneous Signal Modiﬁcation Circuitry ......................................................................... 2-56 Signal Analyzers and Display Devices .................................................................................... 2-62

Introduction ................................................................................................................................ 3-1 Representation of a Vibration Environment ............................................................................ 3-3 Pretest Procedures .................................................................................................................... 3-24 Testing Procedures .................................................................................................................... 3-37 Some Practical Information .................................................................................................... 3-52

Introduction ................................................................................................................................ 4-1 Frequency-Domain Formulation .............................................................................................. 4-2 Experimental Model Development ........................................................................................... 4-8 Curve Fitting of Transfer Functions ....................................................................................... 4-10 Laboratory Experiments .......................................................................................................... 4-18 Commercial EMA Systems ...................................................................................................... 4-24 xiii

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5

6

7

8

9

Contents

Mechanical Shock 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.9 5.10 5.11

Christian Lalanne ............................................................................................ 5-1 Deﬁnitions .................................................................................................................................. 5-2 Description in the Time Domain ............................................................................................. 5-3 Shock Response Spectrum ......................................................................................................... 5-4 Pyroshocks ................................................................................................................................ 5-17 Use of Shock Response Spectra ............................................................................................... 5-18 Standards ................................................................................................................................... 5-24 Damage Boundary Curve ........................................................................................................ 5-26 Shock Machines ........................................................................................................................ 5-28 Generation of Shock Using Shakers ....................................................................................... 5-44 Control by a Shock Response Spectrum ................................................................................ 5-52 Pyrotechnic Shock Simulation ................................................................................................ 5-58

Machine Condition Monitoring and Fault Diagnostics Chris K. Mechefske ................... 6-1 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11

Introduction ................................................................................................................................ 6-2 Machinery Failure ...................................................................................................................... 6-2 Basic Maintenance Strategies .................................................................................................... 6-4 Factors Which Inﬂuence Maintenance Strategy ...................................................................... 6-7 Machine Condition Monitoring ............................................................................................... 6-8 Transducer Selection ................................................................................................................ 6-10 Transducer Location ................................................................................................................. 6-14 Recording and Analysis Instrumentation ............................................................................... 6-14 Display Formats and Analysis Tools ....................................................................................... 6-16 Fault Detection ......................................................................................................................... 6-21 Fault Diagnostics ...................................................................................................................... 6-25

Vibration-Based Tool Condition Monitoring Systems C. Scheffer and P.S. Heyns .......... 7-1 7.1 7.2 7.3 7.4 7.5 7.6

Introduction ................................................................................................................................ 7-1 Mechanics of Turning ................................................................................................................ 7-2 Vibration Signal Recording ....................................................................................................... 7-7 Signal Processing for Sensor-Based Tool Condition Monitoring ........................................ 7-11 Wear Model/Decision-Making for Sensor-Based Tool Condition Monitoring .................. 7-15 Conclusion ................................................................................................................................ 7-20

Fault Diagnosis of Helicopter Gearboxes Kourosh Danai ..................................................... 8-1 8.1 8.2 8.3 8.4 8.5 8.6

Introduction ................................................................................................................................ 8-1 Abnormality Scaling ................................................................................................................... 8-5 The Structure-Based Connectionist Network .......................................................................... 8-8 Sensor Location Selection ........................................................................................................ 8-11 A Case Study ............................................................................................................................. 8-14 Conclusion ................................................................................................................................ 8-23

Vibration Suppression and Monitoring in Precision Motion Systems T.H. 9.1 9.2 9.3 9.4

K.K. Tan, Lee, K.Z. Tang, S. Huang, S.Y. Lim, W. Lin, and Y.P. Leow .......................................................... 9-1 Introduction ................................................................................................................................ 9-1 Mechanical Design to Minimize Vibration .............................................................................. 9-2 Adaptive Notch Filter ............................................................................................................... 9-10 Real-Time Vibration Analyzer ................................................................................................. 9-17

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Contents

xv

9.5 9.6

Practical Insights and Case Study ........................................................................................... 9-29 Conclusions ............................................................................................................................... 9-35

10

Vibration and Shock Problems of Civil Engineering Structures Priyan Mendis

11

Seismic Base Isolation and Vibration Control Hirokazu Iemura, Sarvesh Kumar Jain,

12

Seismic Random Vibration of Long-Span Structures

13

Seismic Qualiﬁcation of Equipment Clarence W. de Silva ................................................... 13-1

14

Human Response to Vibration

and Tuan Ngo ......................................................................................................................................... 10-1 10.1 Introduction .............................................................................................................................. 10-2 10.2 Earthquake-Induced Vibration of Structures ........................................................................ 10-3 10.3 Dynamic Effects of Wind Loading on Structures ................................................................ 10-22 10.4 Vibrations Due to Fluid – Structure Interaction .................................................................. 10-33 10.5 Blast Loading and Blast Effects on Structures ..................................................................... 10-34 10.6 Impact Loading ...................................................................................................................... 10-47 10.7 Floor Vibration ....................................................................................................................... 10-51

and Mulyo Harris Pradono .................................................................................................................... 11-1 11.1 Introduction .............................................................................................................................. 11-1 11.2 Seismic Base Isolation .............................................................................................................. 11-4 11.3 Seismic Vibration Control ..................................................................................................... 11-33

12.1 12.2 12.3 12.4 12.5 12.6

13.1 13.2 13.3

14.1 14.2 14.3 14.4

Jiahao Lin and Yahui Zhang ...... 12-1 Introduction .............................................................................................................................. 12-2 Seismic Random-Excitation Fields ....................................................................................... 12-11 Pseudoexcitation Method for Structural Random Vibration Analysis .............................. 12-16 Long-Span Structures Subjected to Stationary Random Ground Excitations .................. 12-27 Long-Span Structures Subjected to Nonstationary Random Ground Excitations ........... 12-34 Conclusions ............................................................................................................................. 12-39

Introduction .............................................................................................................................. 13-1 Distribution Qualiﬁcation ....................................................................................................... 13-1 Seismic Qualiﬁcation ............................................................................................................... 13-6 Clarence W. de Silva ............................................................. Introduction .............................................................................................................................. Vibration Excitations on Humans .......................................................................................... Human Response to Vibration ............................................................................................... Regulation of Human Vibration .............................................................................................

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14-1 14-1 14-2 14-3 14-6

1

Vibration Instrumentation 1.1 1.2 1.3 1.4

Clarence W. de Silva The University of British Columbia

Introduction ........................................................................ Vibration Exciters ...............................................................

Shaker Selection

†

Dynamics of Electromagnetic Shakers

1-1 1-3

Control System ................................................................... 1-15 Components of a Shaker Controller Equipment

†

Signal-Generating

Performance Speciﬁcation ................................................. 1-21 Parameters for Performance Speciﬁcation † Linearity Instrument Ratings † Accuracy and Precision

†

1.5

Motion Sensors and Transducers ...................................... 1-27

1.6

Torque, Force, and Other Sensors .................................... 1-50

Potentiometer † Variable-Inductance Transducers † Mutual-Induction Proximity Sensor † Selﬁnduction Transducers † Permanent-Magnet Transducers † Alternating Current Permanent-Magnet Tachometer † Alternating Current Induction Tachometer † Eddy Current Transducers † Variable-Capacitance Transducers † Piezoelectric Transducers Strain Gage Sensors

†

Miscellaneous Sensors

Appendix 1A Virtual Instrumentation for Data Acquisition, Analysis, and Presentation ........................... 1-73

Summary Devices useful in instrumenting a mechanical vibrating system are presented in this chapter. Shakers, which generate vibration excitations, are discussed and compared. A variety of sensors, including motion sensors, proximity sensors, force/torque sensors, and other miscellaneous sensors, are considered. Performance speciﬁcation in the time domain and the frequency domain is addressed. Rating parameters of instruments are given.

1.1

Introduction

Measurement and associated experimental techniques play a signiﬁcant role in the practice of vibration. Academic exposure to vibration instrumentation usually arises in laboratories, in the context of learning, training, and research. In vibration practice, perhaps the most important task of instrumentation is the measurement or sensing of vibration. Vibration sensing is useful in the following applications: 1. Design and development of a product 2. Testing (screening) of a ﬁnished product for quality assurance 3. Qualiﬁcation of a good-quality product to determine its suitability for a speciﬁc application

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1-2

Vibration Monitoring, Testing, and Instrumentation

4. Mechanical aging of a product prior to carrying out a test program 5. Exploratory testing of a product to determine its dynamic characteristics such as resonances, mode shapes, and even a complete dynamic model 6. Vibration monitoring for performance evaluation 7. Control and suppression of vibration Figure 1.1 indicates a procedure typical of experimental vibration, highlighting the essential instrumentation. Vibrations are generated in a device, the test object, in response to some excitation. In some experimental procedures, primarily in vibration testing (see Figure 1.1), the excitation signal has to be generated in a signal generator in accordance with some requirement (speciﬁcation), and applied to the object through an exciter after ampliﬁcation and conditioning. In some other situations, primarily in performance monitoring and vibration control, the excitations are generated as an integral part of the operating environment of the vibrating object and may originate either within the object (e.g., engine excitations in an automobile) or in the environment with which the object interacts during operation (e.g., road disturbances on an automobile). Sensors are needed to measure vibrations in the test object. In particular, a control sensor is used to check whether the speciﬁed excitation is applied to the object, and one or more response sensors may be used to measure the resulting vibrations at key locations of the object. The sensor signals have to be properly conditioned, for example by ﬁltering and ampliﬁcation, and modiﬁed, for example through modulation, demodulation, and analog-to-digital conversion, prior to recording, analyzing, and display. The purpose of the controller is to guarantee that the excitation is correctly applied to the test object. If the signal from the control sensor deviates from the required excitation, the controller modiﬁes the signal to the exciter so as to reduce this deviation. Furthermore, the controller will stabilize or limit (compress) the vibrations in the object. It follows that instruments in experimental vibration may be generally classiﬁed into the following categories: 1. 2. 3. 4. 5. 6. 7.

Signal-generating devices Vibration exciters Sensors and transducers Signal conditioning/modifying devices Signal analysis devices Control devices Vibration recording and display devices Response Sensor Mounting Fixtures Power Amplifier

Filter/ Amplifier

Test Object

Exciter

Control Sensor

Filter/ Amplifier

Digital Analog/ Signal Digital Recorder, Interface Analyzer, Display

Swivel Base Signal Generator and Exciter Controller Reference (Required) Signal (Specification)

FIGURE 1.1

Typical instrumentation in experimental vibration.

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Vibration Instrumentation

1-3

Note that one instrument may perform the tasks of more than one category listed here. Also, more than one instrument may be needed to carry out tasks in a single category. In the following sections we will provide some examples of the types of vibration instrumentation, giving characteristics, operating principles, and important practical considerations. Also, we will describe several experiments which can be found in a typical vibration laboratory. An experimental vibration system generally consists of four main subsystems: 1. 2. 3. 4.

Test object Excitation system Control system Signal acquisition and modiﬁcation system

Signal Modification System

Test Object

Control System

Vibration Exciter (Shaker) System

FIGURE 1.2 Interactions between major subsystems of an experimental vibration system.

These are schematically shown in Figure 1.2. Note that various components shown in Figure 1.1 may be incorporated into one of these subsystems. In particular, component matching hardware and object mounting ﬁxtures may be considered interfacing devices that are introduced through the interaction between the main subsystems, as shown in Figure 1.2. Some important issues of vibration testing and instrumentation are summarized in Box 1.1.

1.2

Vibration Exciters

Vibration experimentation may require an external exciter to generate the necessary vibration. This is the case in controlled experiments such as product testing where a speciﬁed level of vibration is applied to the test object and the resulting response is monitored. A variety of vibration exciters are available, with different capabilities and principles of operation. Three basic types of vibration exciters (shakers) are widely used: hydraulic shakers, inertial shakers, and electromagnetic shakers. The operation-capability ranges of typical exciters in these three categories are summarized in Table 1.1. Stroke, or maximum displacement, is the largest displacement the exciter is capable of imparting onto a test object whose weight is assumed to be within its design load limit. Maximum velocity and acceleration are similarly deﬁned. Maximum force is the largest force that could be applied by the shaker to a test object of acceptable weight (one within the design load). The values given in Table 1.1 should be interpreted with caution. Maximum displacement is achieved only at very low frequencies. The achievement of maximum velocity corresponds to intermediate frequencies in the operating frequency range of the shaker. Maximum acceleration and force ratings are usually achieved at high frequencies. It is not feasible, for example, to operate a vibration exciter at its maximum displacement and its maximum acceleration simultaneously. Consider a loaded exciter that is executing harmonic motion. Its displacement is given by x ¼ s sin vt

ð1:1Þ

in which s is the displacement amplitude (or stroke). Corresponding velocity and acceleration are x_ ¼ sv cos vt

ð1:2Þ

x€ ¼ 2sv2 sin vt

ð1:3Þ

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1-4

Vibration Monitoring, Testing, and Instrumentation

Box 1.1 VIBRATION INSTRUMENTATION Vibration Testing Applications for Products: *

*

*

Design and Development Production Screening and Quality Assessment Utilization and Qualiﬁcation for Special Applications

Testing Instrumentation: *

*

*

*

*

Exciter (excites the test object) Controller (controls the exciter for accurate excitation) Sensors and Transducers (measure excitations and responses and provide excitation error signals to controller) Signal Conditioning (converts signals to appropriate form) Recording and Display (perform processing, storage, and documentation)

Exciters: *

*

Shakers 1. Electrodynamic (high bandwidth, moderate power, complex and multifrequency excitations) 2. Hydraulic (moderate to high bandwidth, high power, complex and multifrequency excitations) 3. Inertial (low bandwidth, low power, single-frequency harmonic excitations) Transient/Initial Condition 1. Hammers (impulsive, bump tests) 2. Cable Release (step excitations) 3. Drop (impulsive)

Signal Conditioning: *

*

*

*

Filters Ampliﬁers Ampliﬁers Modulators/Demodulators ADC/DAC

Sensors: *

*

Motion (displacement, velocity, acceleration) Force (strain, torque)

If the velocity amplitude is denoted by v and the acceleration amplitude by a, it follows from Equation 1.2 and Equation 1.3 that v ¼ vs ð1:4Þ and a ¼ vv

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ð1:5Þ

Vibration Instrumentation

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TABLE 1.1

Typical Operation-Capability Ranges for Various Shaker Types

Shaker Type

Typical Operational Capabilities Frequency

Maximum Displacement (Stroke)

Maximum Velocity

Maximum Acceleration

Intermediate (50 in/sec; 125 cm/sec) Intermediate (50 in/sec; 125 cm/sec) Intermediate (50 in/sec; 125 cm/sec)

Intermediate (20 g)

High (100,000 lbf; 450,000 N)

Average ﬂexibility (simple to complex and random)

Intermediate (20 g)

Intermediate (1,000 lbf; 4,500 N)

Sinusoidal only

High (100 g)

Low to intermediate (450 lbf; 2,000 N)

High ﬂexibility and accuracy (simple to complex and random)

Hydraulic (electrohydraulic)

Low (0.1 –500 Hz)

High (20 in; 50 cm)

Inertial (counter-rotating mass)

Intermediate (2– 50 Hz)

Low (1 in; 2.5 cm)

Electromagnetic (electrodynamic)

High (2– 10,000 Hz)

Low (1 in; 2.5 cm)

Maximum Force

Excitation Waveform

1-5

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1-6

Vibration Monitoring, Testing, and Instrumentation

Peak Velocity (cm/s)

Peak Velocity (cm/s)

An idealized performance curve of a shaker Max. has a constant displacement–amplitude region, a Velocity Ac M constant velocity –amplitude region, and a conit ce ax m i 100 ler . stant acceleration – amplitude region for low, L e ati k o on intermediate, and high frequencies, respectively, r St in the operating frequency range. Such an ideal Full No 10 performance curve is shown in Figure 1.3(a) on a Load Load frequency –velocity plane. Logarithmic axes are used. In practice, typical shaker performance 1 curves would be fairly smooth yet nonlinear, 0.1 1 10 100 curves, similar to those shown in Figure 1.3(b). (a) Frequency (Hz) As the mass increases, the performance curve compresses. Note that the acceleration limit of a shaker depends on the mass of the test object (load). Full load corresponds to the heaviest object that could be tested. The “no load” condition 100 corresponds to a shaker without a test object. To standardize the performance curves, they are usually deﬁned at the rated load of the shaker. A No 10 Full Load performance curve in the frequency – velocity Load plane may be converted to a curve in the 1 frequency –acceleration plane simply by increasing 0.1 1 10 100 the slope of the curve by a unit magnitude (i.e., (b) Frequency (Hz) 20 db/decade). Several general observations can be made from Equation 1.4 and Equation 1.5. In the constant- FIGURE 1.3 Performance curve of a vibration exciter peak displacement region of the performance in the frequency– velocity plane (log): (a) ideal; (b) curve, the peak velocity increases proportionally typical. with the excitation frequency, and the peak acceleration increases with the square of the excitation frequency. In the constant-peak velocity region, the peak displacement varies inversely with the excitation frequency, and the peak acceleration increases proportionately. In the constant-peak acceleration region, the peak displacement varies inversely with the square of the excitation frequency, and the peak velocity varies inversely with the excitation frequency. This further explains why rated stroke, maximum velocity, and maximum acceleration values are not simultaneously realized.

1.2.1

Shaker Selection

Vibration testing is accomplished by applying a speciﬁed excitation to the test package, using a shaker apparatus, and monitoring the response of the test object. Test excitation may be represented by its response spectrum. The test requires that the response spectrum of the actual excitation, known as the test response spectrum (TRS), envelops the response spectrum speciﬁed for the particular test, known as the required response spectrum (RRS). A major step in the planning of any vibration testing program is the selection of a proper shaker (exciter) system for a given test package. The three speciﬁcations that are of primary importance in selecting a shaker are the force rating, the power rating, and the stroke (maximum displacement) rating. Force and power ratings are particularly useful in moderate to high frequency excitations and the stroke rating is the determining factor for low frequency excitations. In this section, a procedure is given to determine conservative estimates for these parameters in a speciﬁed test for a given test package. Frequency domain considerations are used here.

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Vibration Instrumentation

1.2.1.1

1-7

Force Rating

In the frequency domain, the (complex) force at the exciter (shaker) head is given by F ¼ mHðvÞas ðvÞ

ð1:6Þ

in which v is the excitation frequency variable, m is the total mass of the test package including mounting ﬁxture and attachments, as ðvÞ is the Fourier spectrum of the support-location (exciter head) acceleration, and H(v) is frequency response function that takes into account the ﬂexibility and damping effects (dynamics) of the test package apart from its inertia. In the simpliﬁed case where the test package can be represented by a simple oscillator of natural frequency vn and damping ratio by zt ; this function becomes HðvÞ ¼ {1 þ 2jzt v=vn }={1 2 ðv=vn Þ2 þ 2jzt v=vn }

ð1:7Þ pﬃﬃﬃﬃ in which j ¼ 21: This approximation is adequate for most practical purposes. The static weight of the test object is not included in Equation 1.6. Most heavy-duty shakers, which are typically hydraulic, have static load support systems such as pneumatic cushion arrangements that can exactly balance the dead load. The exciter provides only the dynamic force. In cases where shaker directly supports the gravity load, in the vertical test conﬁguration Equation 1.6 should be modiﬁed by adding a term to represent this weight. A common practice in vibration test applications is to specify the excitation signal by its response spectrum. This is simply the peak response of a simple oscillator expressed as a function of its natural frequency when its support location is excited by the speciﬁed signal. Clearly, the damping of the simple oscillator is an added parameter in a response spectrum speciﬁcation. Typical damping ratios ðzr Þ used in response spectra speciﬁcations are less than 0.1 (or 10%). It follows that an approximate relationship between the Fourier spectrum of the support acceleration and its response spectrum is as ¼ 2jzr ar ðvÞ

ð1:8Þ

The magnitude lar ðvÞl is the response spectrum. Equation 1.8 substituted into Equation 1.6 gives F ¼ mHðvÞ2jzr ar ðvÞ

ð1:9Þ

In view of Equation 1.7, for test packages having low damping the peak value of H(v) is approximately 1=ð2jzt Þ; this should be used in computing the force rating if the test package has a resonance within the frequency range of testing. On the other hand, if the test package is assumed to be rigid, then HðvÞ ø 1: A conservative estimate for the force rating is Fmax ¼ mðzr =zt Þlar ðvÞlmax

ð1:10Þ

It should be noted that lar ðvÞlmax is the peak value of the speciﬁed (required) response spectrum (RRS) for acceleration. 1.2.1.2

Power Rating

The exciter head does not develop its maximum force when driven at maximum velocity. Output power is determined by using p ¼ Re½Fvs ðvÞ

ð1:11Þ

in which vs ðvÞ is the Fourier spectrum of the exciter velocity, and Re [ ] denotes the real part of a complex function. Note that as ¼ jvvs : Substituting Equation 1.6 and Equation 1.8 into Equation 1.11 gives p ¼ ð4mz2r =vÞRe½jHðvÞa2r ðvÞ

ð1:12Þ

It follows that a conservative estimate for the power rating is pmax ¼ 2mðz2r =zt Þ½lar ðvÞl2 =v

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max

ð1:13Þ

1-8

Vibration Monitoring, Testing, and Instrumentation

Representative segments of typical acceleration RRS curves have slope n, as given by a ¼ k1 vn It should be clear from Equation 1.13 that the maximum output power is given by

ð1:14Þ

pmax ¼ k2 v2n21

ð1:15Þ

This is an increasing function for n . 1=2 and a decreasing function for n , 1=2: It follows that the power rating corresponds to the highest point of contact between the acceleration RRS curve and a line of slope equal to 1/2. A similar relationship may be derived if velocity RRS curves (having slopes n 2 1) are used. 1.2.1.3

Stroke Rating

From Equation 1.8, it should be clear that the Fourier spectrum, xs, of the exciter displacement time history can be expressed as xs ¼ 2zr ar ðvÞ=jv2

ð1:16Þ

An estimate for stroke rating is xmax ¼ 2zr ½lar ðvÞl=v2

ð1:17Þ

max

This is of the form xmax ¼ kvn22

ð1:18Þ

It follows that the stroke rating corresponds to the highest point of contact between the acceleration RRS curve and a line of slope equal to two.

me ace

n

sta

n Co

F,P

nst

ant

Di

spl

10

y

cit

lo

e tV

Co

Acceleration (g)

A test package of overall mass 100 kg is to be subjected to dynamic excitation represented by the acceleration RRS (at 5% damping) as shown in Figure 1.4. The estimated damping of the test package is 7%. The test package is known to have a resonance within the frequency range of the speciﬁed test. Determine the exciter speciﬁcations for the test.

nt

Example 1.1

1.0 S 0.1

Solution 0.1 1.0 10 100 Frequency (Hz) From the development presented in the previous section, it is clear that the point F (or P) in Figure 1.4 corresponds to the force and output FIGURE 1.4 Test excitation speciﬁed by an accelerapower ratings, and the point S corresponds to tion RRS (5% damping). the stroke rating. The co-ordinates of these critical points are F; P ¼ ð4:2 Hz; 4:0 gÞ; and S ¼ ð0:8 Hz; 0:75 gÞ: Equation 1.10 gives the force rating as Fmax ¼ 100 £ ð0:05=0:07Þ £ 4:0 £ 9:81 N ¼ 2803 N Equation 1.13 gives the power rating as pmax ¼ 2 £ 100 £ ð0:052 =0:07Þ £ ½ð4:0 £ 9:81Þ2 =4:2 £ 2p watts ¼ 417 W Equation 1.17 gives the stroke rating as xmax ¼ 2 £ 0:05 £ ½ð0:75 £ 9:8Þ=ð0:8 £ 2pÞ2 m ¼ 3 cm

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Vibration Instrumentation

1.2.1.4

1-9

Hydraulic Shakers

A typical hydraulic shaker consists of a piston-cylinder arrangement (also called a ram), a servo-valve, a ﬂuid pump, and a driving electric motor. Hydraulic ﬂuid (oil) is pressurized (typical operating pressure: 4000 psi) and pumped into the cylinder through a servo-valve by means of a pump that is driven by an electric motor (typical power, 150 hp). The ﬂow (typical rate: 100 gal/min) that enters the cylinder is controlled (modulated) by the servo-valve, which, in effect, controls the resulting piston (ram) motion. A typical servo-valve consists of a two-stage spool valve, which provides a pressure difference and a controlled (modulated) ﬂow to the piston, which sets it in motion. The servo-valve itself is moved by means of a linear torque motor, which is driven by the excitationinput signal (electrical). A primary function of the servo-valve is to provide a stabilizing feedback to the ram. In this respect, the servo-valve complements the main control system of the test setup. The ram is coupled to the shaker table by means of a link with some ﬂexibility. The cylinder frame is mounted on the support foundation with swivel joints. This allows for some angular and lateral misalignment, which might be caused primarily by test-object dynamics as the table moves. Two-degree-of-freedom (Two-DoF) testing requires two independent sets of actuators, and three-DoF testing requires three independent actuator sets. Each independent actuator set can consist of several actuators operated in parallel, using the same pump and the same excitation-input signal to the torque motors. If the test table is directly supported on the vertical actuators, they must withstand the total dead weight (i.e., the weight of the test table, the test object, the mounting ﬁxtures, and the instrumentation). This is requirement is usually prevented by providing a pressurized air cushion in the gap between the test table and the foundation walls. Air should be pressurized so as to balance the total dead weight exactly (typical required gage pressure: 3 psi). Figure 1.5(a) shows the basic components of a typical hydraulic shaker. The corresponding operational block diagram is shown in Figure 1.5(b). It is desirable to locate the actuators in a pit in the test laboratory so that the test tabletop is ﬂushed with the test laboratory ﬂoor under no-load conditions. This minimizes the effort required to place the test object on the test table. Otherwise, the test object has to be lifted onto the test table with a forklift. Also, installation of an aircushion to support the system dead weight is difﬁcult under these circumstances of elevated mounting. Hydraulic actuators are most suitable for heavy load testing and are widely used in industrial and civil engineering applications. They can be operated at very low frequencies (almost direct current [DC]), as well as at intermediate frequencies (see Table 1.1). Large displacements (stroke) are possible at low frequencies. Hydraulic shakers have the advantage of providing high ﬂexibility of operation during the test; their capabilities include variable-force and constant-force testing and wide-band random-input testing. The velocity and acceleration capabilities of hydraulic shakers are moderate. Although any general excitationinput motion (for example, sine wave, sine beat, wide-band random) can be used in hydraulic shakers, faithful reproduction of these signals is virtually impossible at high frequencies because of distortion and higher-order harmonics introduced by the high noise levels that are common in hydraulic systems. This is only a minor drawback in heavy-duty, intermediate-frequency applications. Dynamic interactions are reduced through feedback control. 1.2.1.5

Inertial Shakers

In inertial shakers, or “mechanical exciters,” the force that causes the shaker-table motion is generated by inertia forces (accelerating masses). Counter-rotating-mass inertial shakers are typical in this category. To understand their principle of operation, consider two equal masses rotating in opposite directions at the same angular speed v and in the same circle of radius r (see Figure 1.6). This produces a resultant force equal to 2mv2 r cos vt in a ﬁxed direction (the direction of symmetry of the two rotating arms). Consequently, a sinusoidal force with a frequency of v and an amplitude proportional to v2 is generated. This reaction force is applied to the shaker table. Figure 1.7 shows a sketch of a typical counter-rotating-mass inertial shaker. It consists of two identical rods rotating at the same speed in opposite directions. Each rod has a series of slots in which to

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1-10

FIGURE 1.5

Vibration Monitoring, Testing, and Instrumentation

A typical hydraulic shaker arrangement: (a) schematic diagram; (b) operational block diagram.

place weights. In this manner, the magnitude of the eccentric mass can be varied to achieve various force capabilities. The rods are driven by a variable-speed electric motor through a gear mechanism that usually provides several speed ratios. A speed ratio is selected depending on the required test-frequency range. The whole system is symmetrically supported on a carriage that is directly connected to the test table. The test object is mounted on the test table. The preferred mounting conﬁguration is horizontal so that the excitation force is applied to the test object in a horizontal direction. In this conﬁguration, there are no variable gravity moments (weight £ distance to center of gravity) acting on the drive mechanism. Figure 1.7 shows the vertical conﬁguration. In dynamic testing of large structures, the carriage can be mounted directly on the structure at a location where the excitation force should be applied. By incorporating two pairs of counter-rotating masses, it is possible to generate test moments as well as test forces.

2mw 2r cos wt

m

m w

wt

wt

w

FIGURE 1.6 Principle of operation of a counterrotating-mass inertial shaker.

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Vibration Instrumentation

FIGURE 1.7

1-11

Sketch of a counter-rotating-mass inertial shaker.

Reaction-type shakers driven by inertia are widely used for the prototype testing of civil engineering structures. Their ﬁrst application dates back to 1935. Inertial shakers are capable of producing intermediate excitation forces. The force generated is limited by the strength of the carriage frame. The frequency range of operation and the maximum velocity and acceleration capabilities are also intermediate for inertial shakers whereas the maximum displacement capability is typically low. A major limitation of inertial shakers is that their excitation force is exclusively sinusoidal and that the force amplitude is directly proportional to the square of the excitation frequency. As a result, complex and random excitation testing, constant-force testing (for example, transmissibility tests and constant-force sine-sweep tests), and ﬂexibility to vary the force amplitude or the displacement amplitude during a test are not generally feasible with this type of shakers. Excitation frequency and amplitude can be varied during testing, however, by incorporating a variable-speed drive for the motor. The sinusoidal excitation generated by inertial shakers is virtually undistorted, which gives them an advantage over the other types of shakers when used in sine-dwell and sine-sweep tests. Small portable shakers with low-force capability are available for use in on-site testing. 1.2.1.6

Electromagnetic Shakers

In electromagnetic shakers or “electrodynamic exciters,” the motion is generated using the principle of operation of an electric motor. Speciﬁcally, the excitation force is produced when a variable excitation signal (electrical) is passed through a moving coil placed in a magnetic ﬁeld. The components of a commercial electromagnetic shaker are shown in Figure 1.8. A steady magnetic ﬁeld is generated by a stationary electromagnet that consists of ﬁeld coils wound on a ferromagnetic base that is rigidly attached to a protective shell structure. The shaker head has a coil wound around it. When the excitation electrical signal is passed through this drive coil, the shaker head, which is supported on ﬂexure mounts, will be set in motion. The shaker head consists of the test table on which the test object is mounted. Shakers with interchangeable heads are available. The choice of appropriate shaker head is based on the geometry and mounting features of the test object. The shaker head can be turned to different angles by means of a swivel joint. In this manner, different directions of excitation (in biaxial and triaxial testing) can be obtained.

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1-12

Vibration Monitoring, Testing, and Instrumentation

FIGURE 1.8 Denmark.

1.2.2

Schematic sectional view of a typical electromagnetic shaker, manufactured by Bruel and Kjaer,

Dynamics of Electromagnetic Shakers

Consider a single axis electromagnetic shaker (Figure 1.8) with a test object having a single natural frequency of importance within the test frequency range. The dynamic interactions between the shaker and the test object give rise to two signiﬁcant natural frequencies (and correspondingly, two signiﬁcant resonances). These appear as peaks in the frequency response curve of the test setup. Furthermore, the natural frequency (resonance) of the test package alone causes a “trough” or depression (antiresonance) in the frequency response curve of the overall test setup. To explain this characteristic, consider the dynamic model shown in Figure 1.9. The following mechanical parameters are deﬁned for Figure 1.9(a): m, k, and b are the mass, stiffness, and equivalent viscous damping constant, respectively, of the test package, and me, ke, and be are the corresponding parameters of the exciter (shaker). Also, in the equivalent electrical circuit of the shaker head, as shown in Figure 1.9(b), the following electrical parameters are deﬁned: Re and Le are the resistance and (leakage) inductance and kb is the back electromotive force (back emf) of the linear motor. Assuming that the gravitational forces are supported

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Vibration Instrumentation

1-13

FIGURE 1.9 Dynamic models of an electromagnetic shaker and a ﬂexible test package: (a) mechanical model; (b) electrical model.

by the static deﬂections of the ﬂexible elements, and that the displacements are measured from the static equilibrium position, we have the following system equations: Test object: m€y ¼ 2kðy 2 ye Þ 2 bð_y 2 y_ e Þ

ð1:19Þ

Shaker head: me y€ e ¼ fe þ kðy 2 ye Þ þ bð_y 2 y_ e Þ 2 ke y 2 be y_ e

ð1:20Þ

Electrical: Le

die þ Re ie þ kb y_ e ¼ vðtÞ dt

ð1:21Þ

The electromagnetic force fe generated in the shaker head is a result of the interaction of the magnetic ﬁeld generated by the current ie with coil of the moving shaker head and the constant magnetic ﬁeld (stator) in which the head coil is located. Here, we have fe ¼ kb ie

ð1:22Þ

Note that v(t) is the voltage signal that is applied by the ampliﬁer to the shaker coil, ye is the displacement of the shaker head, and y is the displacement response of the test package. It is assumed that kb has consistent electrical and mechanical units (V/m/sec and N/A). Usually, the electrical time constant of the shaker is quite small compared with the primarily mechanical time constants of the shaker and the test package. In such cases, the Le die =dt term in Equation 1.21 may be neglected. Consequently, the equations from Equation 1.19 through Equation 1.22 may be expressed in the Laplace (frequency) domain, with the Laplace variable s taking the place of the derivative d=dt; as ðms2 þ bs þ kÞy ¼ ðbs þ kÞye

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ð1:23Þ

1-14

Vibration Monitoring, Testing, and Instrumentation

½me s2 þ ðb þ be Þs þ ðk þ ke Þ ye ¼ ðbs þ kÞy þ

kb k2 s v 2 b ye Re Re

ð1:24Þ

It follows that the transfer function of the shaker head motion with respect to the excitation voltage is given by ye k DðsÞ ¼ b v Re Dd ðsÞ

ð1:25Þ

where DðsÞ is the characteristic function of the primary dynamics of the test object DðsÞ ¼ ms2 þ bs þ k

ð1:26Þ

and Dd ðsÞ is the characteristic function of the primary dynamic interactions between the shaker and the test object Dc ðsÞ ¼ mme s4 þ ½mðbe þ b þ bo Þ þ me b s3 þ ½mðke þ kÞ þ me k þ bðbe þ bo Þ s2 þ ½bke þ ðbe þ bo Þk s þ kke

ð1:27Þ

where bo ¼

k2b Re

ð1:28Þ

It is clear that under low damping conditions Dd ðsÞ will produce two resonances as it is fourth order in s, and similarly DðsÞ will produce one antiresonance (trough) corresponding to the resonance of the test object. Note that in the frequency domain, s ¼ jv; and hence the frequency response function given by Equation 1.25, is in fact ye k DðjvÞ ¼ b v Rb Dd ðjvÞ

ð1:29Þ

Shaker Displacement Magnitude

The magnitude of this frequency response function for a typical test system is sketched in Figure 1.10. Note that this curve is for the “open-loop” case where there is no feedback from the shaker controller. In practice, the shaker controller will be able to compensate for the resonances and anti10.0 Resonance resonances to some degree, depending on its effectiveness. Resonance The main advantages of electromagnetic shakers are their high frequency range of operation, their high degree of operating ﬂexi1.0 bility, and the high level of accuracy of the generated shaker motion. Faithful reproduction of complex excitations is possible because of the advanced electronics and control systems used in Antiresonance this type of shakers. Electromagnetic shakers are not suitable for heavy-duty applications (large test objects), however. High test-input accelerations 1000 1 10 100 are possible at high frequencies when electromagExcitation Frequency (Hz) netic shakers are used, but their displacement and velocity capabilities are limited to low or FIGURE 1.10 Frequency response curve of a typical intermediate values (see Table 1.1). electromagnetic shaker with a test object.

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Vibration Instrumentation

1.2.2.1

1-15

Transient Exciters

Other varieties of exciters are commonly used in transient-type vibration testing. In these tests, either an impulsive force or an initial excitation is applied to the test object and the resulting response is monitored. The excitations and the responses are “transient” in this case. Hammer test, drop tests, and pluck tests fall into this category. For example, a hammer test may be conducted by hitting the object with an instrumented hammer and then measuring the response of the object. The hammer has a force sensor at its tip, as sketched in Figure 1.11. A piezoelectric or strain-gage type force sensor may be used. More sophisticated hammers have impedance heads in place of force sensors. An impedance head measures force and acceleration simultaneously. The results of a hammer test will depend on many factors; for example, the dynamics of the hammer body, how ﬁrmly the hammer is held during the impact, how quickly the impact is applied, and whether there are multiple impacts.

1.3

FIGURE 1.11 An instrumented hammer used in bump tests or hammer tests.

Control System

The two primary functions of the shaker control system in vibration testing are (1) to guarantee that the speciﬁed excitation is applied to the test object and (2) to ensure that the dynamic stability (motion constraints) of the test setup is preserved. An operational block diagram illustrating these control functions is given in Figure 1.12. The reference input to the control system represents the desired excitation force that should be applied to the test object. In the absence of any control, however, the force reaching the test object will be distorted, primarily because of: (1) dynamic interactions and nonlinearities of the shaker, the test table, the mounting ﬁxtures, the auxiliary instruments, and the test object itself; (2) noise and errors in the signal generator, ampliﬁers, ﬁlters, and other equipment; and (3) external loads and disturbances acting on the test object and other components (for example, external restraints, aerodynamic forces, friction). To compensate for these distorting factors, response measurements (displacements, velocities, acceleration, and so on) are made at various locations in the test setup and are used to control the system dynamics. In particular, the responses of the shaker, the test table, and the test object are measured. These responses are used to compare the actual excitation felt by Excitation Input (Reference)

Controller and Amplifier

Drive Signal

Exciter (Shaker) (Ram)

Shaker Response

Test Table

Test Table Response

Test Object

Feedback Paths

FIGURE 1.12

Operational block diagram illustrating a general shaker control system.

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Test Object Response

1-16

Vibration Monitoring, Testing, and Instrumentation

the test object at the shaker interface, with the desired (speciﬁed) input. The drive signal to the shaker is modiﬁed, depending on the error that is present. Two types of control are commonly employed in shaker apparatus: simple manual control and complex automatic control. Manual control normally consists of simple, open-loop, trial-and-error methods of manual adjustments (or calibration) of the control equipment to obtain a desired dynamic response. The actual response is usually monitored (on an oscilloscope or frequency analyzer screen, for example) during manual-control operations. The pretest adjustments in manual control can be very time-consuming; as a result, the test object might be subjected to overtesting, which could produce cumulative damage, is undesirable, and could defeat the test purpose. Furthermore, the calibration procedure for the experimental setup must be repeated for each new test object. The disadvantages of manual control suggest that automatic control is desirable in complex test schemes in which high accuracy of testing is desired. The ﬁrst step of automatic control involves automatic measurement of the system response, using control sensors and transducers. The measurement is then fed back into the control system, which instantaneously determines the best drive signal to actuate the shaker in order to get the desired excitation. This may be done by either analog or digital methods. Primitive control systems require an accurate mathematical description of the test object. This dependency of the control system on the knowledge of test-object dynamics is clearly undesirable. Performance of a good control system should not be considerably affected by the dynamic interactions and nonlinearities of the test object or by the nature of the excitation. Proper selection of feedback signals and control-system components can reduce such effects and will make the system robust. In the response-spectrum method of vibration testing, it is customary to use displacement control at low frequencies, velocity control at intermediate frequency, and acceleration control at high frequencies. This necessitates feedback of displacement, velocity, and acceleration responses. Generally, however, the most important feedback is the velocity feedback. In sine-sweep tests, the shaker velocity must change steadily over the frequency band of interest. In particular, the velocity control must be precise near the resonances of the test object. Velocity (speed) feedback has a stabilizing effect on the dynamics, which is desirable. This effect is particularly useful in ensuring stability in motion when testing is done near resonances of lightly damped test objects. On the contrary, displacement (position) feedback can have a destabilizing effect on some systems, particularly when high feedback gains are used. The controller usually consists of various instruments, equipment, and computation hardware and software. Often, the functions of the data-acquisition and processing system overlap with those of the controller to some extent. As an example, consider the digital-controller of vibration testing apparatus. First, the responses are measured through sensors (and transducers), ﬁltered, and ampliﬁed (conditioned). These data channels may be passed through a multiplexer, whose purpose is to select one data channel at a time for processing. Most modern data acquisition hardware does not need a separate multiplexer to handle multiple signals. The analog data are converted into digital data using analog-to-digital converters (ADCs). The resulting sampled data are stored on a disk or as a block data in the computer memory. The reference input signal (typically, a signal recorded on an FM tape) is also sampled (if it is not already in the digital form), using an ADC, and fed into the computer. Digital processing is done on the reference signal and the response data, with the objective of computing the command signal to drive the shaker. The digital command signal is converted into an analog signal, using a digital-to-analog converter (DAC), and ampliﬁed (conditioned) before it is used to drive the exciter. The nature of the control components depends to a large extent on the nature and objectives of the particular test to be conducted. Some of the basic components in a shaker controller are described in the following subsections.

1.3.1 1.3.1.1

Components of a Shaker Controller Compressor

A compressor circuit is incorporated in automatic excitation control devices to control the excitationinput level automatically. The level of control depends on the feedback signal from a control sensor and

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Vibration Instrumentation

1-17

the speciﬁed (reference) excitation signal. Usually, the compressor circuit is included in the excitationsignal generator (for example, in a sine generator). The control by this means may be done on the basis of a single-frequency component (e.g., the fundamental frequency). 1.3.1.2

Equalizer (Spectrum Shaper)

Random-signal equalizers are used to shape the spectrum of a random signal in a desired manner. In essence, and equalizer consists of a bank of narrow-band ﬁlters (for example, 80 ﬁlters) in parallel over the operating frequency range. By passing the signal through each ﬁlter, the spectral density (or the mean square value) of the signal in that narrow frequency band (for example, each one-third-octave band) is determined. This is compared with the desired spectral level, and automatic adjustment is made in that ﬁlter in case there is an error. In some systems, response-spectrum analysis is made in place of power spectral density analysis. In that case, the equalizer consists of a bank of simple oscillators, whose resonant frequencies are distributed over the operating frequency range of the equalizer. The feedback signal is passed through each oscillator, and the peak value of its output is determined. This value is compared with the desired response spectrum value at that frequency. If there is an error, automatic gain adjustment is made in the appropriate excitation signal components. Random-noise equalizers are used in conjunction with random signal generators. They receive feedback signals from the control sensors. In some digital control systems, there are algorithms (software) that are used to iteratively converge the spectrum of the excitation signal felt by the test object into the desired spectrum. 1.3.1.3

Tracking Filter

Many vibration tests are based on single-frequency excitations. In such cases, the control functions should be performed on the basis of the amplitudes of the fundamental-frequency component of the signal. A tracking ﬁlter is simply a frequency-tuned band-pass ﬁlter. It automatically tunes the center frequency of its very narrow-band-pass ﬁlter to the frequency of a carrier signal. Then, when a noisy signal is passed through the tuned ﬁlter, the output of the ﬁlter will be the required fundamental frequency component in the signal. Tracking ﬁlters also are useful in obtaining amplitude –frequency plots using an X – Y plotter. In such cases, the frequency value comes from the signal generator (sweep oscillator), which produces the carrier signal to the tracking ﬁlter. The tracking ﬁlter then determines the corresponding amplitude of a signal that is fed into it. Most tracking ﬁlters have dual channels so that two signals can be handled (tracked) simultaneously. 1.3.1.4

Excitation Controller (Amplitude Servo-Monitor)

An excitation controller is typically an integral part of the signal generator. It can be set so that automatic sweep between two frequency limits can be performed at a selected sweep rate (linear or logarithmic). More advanced excitation controllers have the capability of an automatic switch-over between constantdisplacement, constant-velocity and constant-acceleration excitation-input control at speciﬁed frequencies over the sweep frequency interval. Consequently, integrator circuits should be present within the excitation controller unit to determine velocities and displacements from acceleration signals. Sometimes, integration is performed by a separate unit called a vibration meter. This unit also offers the operator the capability of selecting the desired level of each signal (acceleration, velocity, or displacement). There is an automatic cut-off level for large displacement values that could result from noise in acceleration signals. A compressor is also a subcomponent of the excitation controller. The complete unit is sometimes known as an amplitude servo-monitor.

1.3.2

Signal-Generating Equipment

Shakers are force-generating devices that are operated using drive (excitation) signals generated from a source. The excitation-signal source is known as the signal generator. Three major types of signal generators are used in vibration testing applications: (1) oscillators, (2) random-signal generators, and (3) storage devices. In some units, oscillators and random-signal generators are combined. We shall

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discuss these two generators separately, however, because of their difference in function. It also should be noted that almost any digital signal (deterministic or random) can be generated by a digital computer using a suitable computer program; the signal eventually can be passed through a DAC to obtain the corresponding analog signal. These ‘digital’ signal generators along with analog sources such as magnetic tape players (FM) are classiﬁed into the category of storage devices. The dynamic range of equipment is the ratio of the maximum and minimum output levels (expressed in decibels) at which it is capable of operating without signiﬁcant error. This is an important speciﬁcation for many types of equipment, particularly for signal-generating devices. The output level of the signal generator should be set to a value within its dynamic range. 1.3.2.1

Oscillators

Oscillators are essentially single-frequency generators. Typically, sine signals are generated, but other waveforms (such as rectangular and triangular pulses) are also available in most oscillators. Normally, an oscillator has two modes of operation: (1) sweeping up and down between two frequency limits and (2) dwelling at a speciﬁed frequency. In the sweep operation, the sweep rate should be speciﬁed. This can be done either on a linear scale (Hz/min) or on a logarithmic scale (octaves/min). In the dwell operation, the frequency points (or intervals) should be speciﬁed. In either case, a desired signal level can be chosen using the gain-control knob. An oscillator that is operated exclusively in the sweep mode is called a sweep oscillator. The early generation of oscillators employed variable inductor-capacitor types of electronic circuits to generate signals oscillating at a desired frequency. The oscillator is tuned to the required frequency by varying the capacitance or inductance parameters. A DC voltage is applied to energize the capacitor and to obtain the desired oscillating voltage signal, which subsequently is ampliﬁed and conditioned. Modern oscillators use operational ampliﬁer circuits along with resistor, capacitor, and semiconductor (SC) elements. Also common are crystal (quartz) parallel-resonance oscillators, used to generate voltage signals accurately at a ﬁxed frequency. The circuit is activated using a DC-voltage source. Other frequencies of interest are obtained by passing this high-frequency signal through a frequency converter. The signal is then conditioned (ampliﬁed and ﬁltered). Required shaping (for example, rectangular pulse) is obtained using a shape circuit. Finally, the required signal level is obtained by passing the resulting signal through a variable-gain ampliﬁer. A block diagram of an oscillator, illustrating various stages in the generation of a periodic signal, is given in Figure 1.13. A typical oscillator offers a choice of several (typically six) linear and logarithmic frequency ranges and a sizable level of control capability (for example, 80 dB). Upper and lower frequency limits in a sweep can be preset on the front panel to any of the available frequency ranges. Sweep-rate settings are continuously

DC Voltage

Oscillator

Frequency Specification

Frequency Converter

Signal Specification

Level Specification

Shaper

Variable-Gain Output Amplifier

Filter/ Amplifier

Fixed-Frequency Signal Frequency Counter

FIGURE 1.13

Block diagram of an oscillator-type signal generator.

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variable (typically, 0 to 10 octaves/min in the logarithmic range, and 0 to 60 kHz/min in the linear range), but one value must be selected for a given test or part of a test. Most oscillators have a repetitive-sweep capability, which allows the execution of more than one sweep continuously, for example, for mechanical aging and in product-qualiﬁcation single-frequency tests. Some oscillators also have the capability of varying the signal level (amplitude) during each test cycle (sweep or dwell). This is known as level programming. Also, automatic switching between acceleration, velocity, and displacement excitations at speciﬁed frequency points in each test cycle can be implemented with some oscillators. A frequency counter, which is capable of recording the fundamental frequency of the output signal, is usually an integral component of the oscillator. 1.3.2.2

Random Signal Generators

In modern random-signal generators, SC devices (e.g., zener diodes) are used to generate a random signal that has a required (e.g., Gaussian) distribution. This is accomplished by applying a suitable DC voltage to a SC circuit. The resulting signal is then ampliﬁed and passed through a bank of conditioning ﬁlters, which effectively acts as a spectrum shaper. In this manner, the bandwidth of the signal can be adjusted in a desired manner. Extremely wideband signals (white noise), for example, can be generated for random-excitation vibration testing in this manner. The block diagram in Figure 1.14 shows the essential steps in a random-signal generation process. A typical random-signal generator has several (typically eight) bandwidth selections over a wide frequency range (for example, 1 Hz to 100 kHz). A level-control capability (typically 80 dB) is also available. 1.3.2.3

Tape Players

Vibration testing for product qualiﬁcation may be performed using a tape player as the signal source. A tape player is essentially a signal reproducer. The test-input signal that has a certain speciﬁed response spectrum is obtained by playing a magnetic tape and mixing the contents in the several tracks of the tape in a desirable ratio. Typically, each track contains a sine-beat signal, with a particular beat frequency, amplitude, and number of cycles per beat, or a random-signal component with a desired spectral characteristic). In frequency modulation (FM) tapes, the signal amplitude is proportional to the frequency of a carrier signal. The carrier signal is recorded on the tape. When played back, the actual signal is reproduced, based on detecting the frequency content of the carrier signal in different time points. The FM method is usually favorable, particularly for low-frequency testing (below 100 Hz). Performance of a tape player is determined by several factors, including tape type and quality, signal reproduction and recording circuitry, characteristics of the magnetic heads, and the tape-transport mechanism. Some important speciﬁcations for tape players are (1) the number of tracks per tape (for example, 14 or 28); (2) the available tape speeds (for example, 3.75, 7.5, 15, or 30 in./sec); (3) reproduction ﬁlter-ampliﬁer capabilities (for example, 0.5% third-harmonic distortion in a 1 kHz signal recorded at 15 in./sec tape speed, peak-to-peak output voltage of 5 V at 100 V load, signal-to-noise ratio of 45 dB, output impedance of 50 V); and (4) the available control options and their capabilities (for example, stop, play, reverse, fast-forward, record, speed selection, channel selection). Tape player speciﬁcations for vibration testing are governed by an appropriate regulatory agency, according to a DC Voltage

Zener Diode Noise Source

Amplifier

Gaussian Random Noise

FIGURE 1.14

Band Width Specification

Level Specification

Conditioning Filters

Variable-Gain Output Amplifier

Block diagram of a random signal generator.

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speciﬁed standard (e.g., the Communication and Telemetry Standard of the Intermediate Range Instrumentation Group (IRIG Standard 106-66). A common practice in vibration testing is to generate the test-input signal by repetitively playing a closed tape loop. In this manner, the input signal becomes periodic but has the desired frequency content. Frequency modulation players can be ﬁtted with special loop adaptors for playing tape loops. In spectral (Fourier) analysis of such signals, the analyzing-ﬁlter bandwidth should be several times more than the repetition frequency (tape speed/loop length). Extraneous noise is caused by discontinuities at the tape joint. This can be suppressed by using suitable ﬁlters or gating circuits. A technique that can be employed to generate low-frequency signals with high accuracy is to record the signal ﬁrst at a very low tape speed and then play it back at a high tape speed (for example, r times higher). This has the effect of multiplying all frequency components in the signal by the speed ratio (r). Consequently, the ﬁlter circuits in the tape player will allow some low-frequency components in the signal that would normally be cut off and will cut off some high-frequency components that would normally be allowed. Hence, this process is a way of emphasizing the low-frequency components in a signal. 1.3.2.4

Data Processing

A controller generally has some data processing functions, as well. A data-acquisition and processing system usually consists of response sensors (and transducers), signal conditioners, an input–output (I/O) board including a multiplexer, ADCs, etc., and a digital computer, with associated I/O devices. The functions of a digital data-acquisition and processing system may be quite general, as listed below: 1. Measuring, conditioning, sampling, and storing the response signals and operational data of test object (using input commands, as necessary) 2. Digital processing of the measured data according to the test objectives (and using input commands, as necessary) 3. Generating drive signals for the control system 4. Generating and recording test results (responses) in a required format The capacity and the capabilities of a data-acquisition and processing systems are determined by such factors as: 1. 2. 3. 4. 5. 6. 7.

The number of response data channels that can be handled simultaneously The data-sampling rate (samples per second) for each data channel Computer memory size Computer processing speed External storage capability (hard disks, ﬂoppy disks, and so forth) The nature of the input and output devices Software features

Commercial data-acquisition and processing systems with a wide range of processing capabilities are available for use in vibration testing. Some of the standard processing capabilities are the following: 1. 2. 3. 4. 5.

Response-spectrum analysis FFT analysis (spectral densities, correlations, coherence, Fourier spectra, and so on) Frequency-response function, transmissibility, and mechanical-impedance analysis Natural-frequency and mode-shape analysis System-parameter identiﬁcation (for example, damping parameters)

Most processing is done in real time, which means that the signals are analyzed as they are being measured. The advantage of this is that outputs and command signals are available simultaneously as the monitoring is done, so that any changes can be detected as they occur (for example, degradation in the

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test object or deviations in the excitation signal from the desired form) and automatic feedback control can be effected. For real-time processing to be feasible, the data-acquisition rate (sampling rate) and the processing speed of the computer should be sufﬁciently fast. In real-time frequency analysis, the entire frequency range is analyzed at a given instant, as opposed to analyzing narrow bands separately. Results are presented as Fourier spectra, power spectral densities, cross-spectral densities, coherence functions, correlation functions, and response-spectra curves. Averaging of frequency plots can be done over small frequency bands (for example, one-third-octave analysis), or the running average of each instantaneous plot can be determined.

1.4

Performance Speciﬁcation

Proper selection and integration of sensors and transducers are crucial in “instrumenting” a vibrating system. The response variable that is being measured (for example, acceleration) is termed the measurand. A measuring device passes through two stages in making a measurement. First, the measurand is sensed. Then, the measured signal is transduced (converted) into a form that is particularly suitable for signal conditioning, processing, or recording. Often, the output from the transducer stage is an electrical signal. It is common practice to identify the combined sensor–transducer unit as either a sensor or a transducer. The measuring device itself might contain some of the signal-conditioning circuitry and recording (or display) devices or meters. These are components of an overall measuring system. For our purposes, we shall consider these components separately. In most applications, the following four variables are particularly useful in determining the response and structural integrity of a vibrating system (in each case the usual measuring devices are indicated in parentheses): 1. 2. 3. 4.

Displacement (potentiometer or LVDT) Velocity (tachometer) Acceleration (accelerometer) Stress and strain (strain gage)

It is somewhat common practice to measure acceleration ﬁrst and then determine velocity and displacement by direct integration. Any noise and DC components in the measurement, however, could give rise to erroneous results in such cases. Consequently, it is good practice to measure displacement, velocity, and acceleration by using separate sensors, particularly when the measurements are employed in feedback control of the vibratory system. It is not recommended to differentiate a displacement (or velocity) signal to obtain velocity (or acceleration), because this process would amplify any noise present in the measured signal. Consider, for example, a sinusoidal signal give by A sin vt: Since d=dtðA sin vtÞ ¼ Av cos vt; it follows that any high-frequency noise would be ampliﬁed by a factor proportional to its frequency. Also, any discontinuities in noise components would produce large deviations in the results. Using the same argument, it may be concluded that the acceleration measurements are desirable for high-frequency signals and the displacement measurements are desirable for low-frequency signals. It follows that the selection of a particular measurement transducer should depend on the frequency content of the useful portion of the measured signal. Transducers are divided into two broad categories: active transducers and passive transducers. Passive transducers do not require an external electric source for activation. Some examples are electromagnetic, piezoelectric, and photovoltaic transducers. Active transducers, however, do not possess selfcontained energy sources and thus need external activation. A good example is a resistive transducer, such as a potentiometer. In selecting a particular transducer (measuring device) for a speciﬁc vibration application, special attention should be give to its ratings, which usually are provided by the manufacturer, and the required performance speciﬁcations as provided by the customer (or developed by the system designer).

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Parameters for Performance Speciﬁcation

A perfect measuring device can be deﬁned as one that possesses the following characteristics: 1. Output instantly reaches the measured value (fast response). 2. Transducer output is sufﬁciently large (high gain, low output impedance, high sensitivity). 3. Output remains at the measured value (without drifting or being affected by environmental effects and other undesirable disturbances and noise) unless the measurand itself changes (stability and robustness). 4. The output signal level of the transducer varies in proportion to the signal level of the measurand (static linearity). 5. Connection of a measuring device does not distort the measurand itself (loading effects are absent and impedances are matched). 6. Power consumption is small (high input impedance). All these properties are based on dynamic characteristics and therefore can be explained in terms of dynamic behavior of the measuring device. In particular, items 1 to 4 can be speciﬁed in terms of the device (response), either in the time domain or in the frequency domain. Items 2, 5, and 6 can be speciﬁed using the impedance characteristics of a device. First, we shall discuss response characteristics that are important in performance speciﬁcation of a sensor/transducer unit. 1.4.1.1

Time-Domain Speciﬁcations

Several parameters that are useful for the time-domain performance speciﬁcation of a device are as follows: 1. Rise time ðTr Þ: This is the time taken to pass the steady-state value of the response for the ﬁrst time. In overdamped systems, the response is nonoscillatory; consequently, there is no overshoot. So that the deﬁnition is valid for all systems, rise time is often deﬁned as the time taken to pass 90% of the steady-state value for the ﬁrst time. Rise time is often measured from 10% of the steady-state value in order to leave out irregularities occurring at start-up and time lags that might be present in a system. Rise time represents the speed of response of a device: a small rise time indicates a fast response. 2. Delay time (Td): This is usually deﬁned as the time taken to reach 50% of the steady-state value for the ﬁrst time. This parameter is also a measure of the speed of response. 3. Peak time (Tp): This is the time at the ﬁrst peak. This parameter also represents the speed of response of the device. 4. Settling time (Ts): This is the time taken for the device response to settle down within a certain percentage (e.g., ^2%) of the steady-state value. This parameter is related to the degree of damping present in the device as well as the degree of stability. 5. Percentage overshoot (PO): This is deﬁned as PO ¼ 100ðMp 2 1Þ%

ð1:30Þ

using the normalized-to-unity step response curve, where Mp is the peak value. Percentage overshoot is a measure of damping or relative stability in the device. 6. Steady-state error: This is the deviation of the actual steady-state value from the desired value. Steady-state error may be expressed as a percentage with respect to the (desired) steady-state value. In a measuring device, steady-state error manifests itself as an offset. This is a systematic (deterministic) error that normally can be corrected by recalibration. In servo-controlled devices, steady-state error can be reduced by increasing the loop gain or by introducing a lag compensation. Steady-state error can be completely eliminated using the integral control (reset) action.

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For the best performance of a measuring device, we wish to have the values of all the foregoing parameters as small as possible. In actual practice, however, it might be difﬁcult to meet all speciﬁcations, particularly under conﬂicting requirements. For instance, Tr can be decreased by increasing the dominant natural frequency vn of the device. This, however, increases the PO and sometimes the Ts. On the other hand, the PO and Ts can be decreased by increasing device damping, but this has the undesirable effect of increasing Tr. 1.4.1.2

Frequency-Domain Speciﬁcations

Since any time signal can be decomposed into sinusoidal components through Fourier transformation, it is clear that the response of a system to an arbitrary input excitation also can be determined using transfer-function (frequency response-function) information for that system. For this reason, one could argue that it is redundant to use both time-domain speciﬁcations and frequency-domain speciﬁcations, as they carry the same information. Often, however, both speciﬁcations are used simultaneously, because this can provide a better understanding of the system performance. Frequency-domain parameters are more suitable in representing some characteristics of a system under some types of excitation. Consider a device with the frequency-response function (transfer function) Gð jvÞ: Some useful parameters for performance speciﬁcation of the device in the frequency domain are: 1. Useful frequency range (operating interval): This is given by the ﬂat region of the frequency response magnitude, lGð jvÞl; of the device. 2. Bandwidth (speed of response): This may be represented by the primary natural frequency (or resonant frequency) of the device. 3. Static gain (steady-state performance): Since static conditions correspond to zero frequencies; this is given by Gð0Þ: 4. Resonant frequency (speed and critical frequency region) vr: This corresponds to the lowest frequency at which lGð jvÞl peaks. 5. Magnitude at resonance (stability): This is given by lGðjvr Þl: 6. Input impedance (loading, efﬁciency, interconnectability): This represents the dynamic resistance as felt at the input terminals of the device. This parameter will be discussed in more detail under component interconnection and matching. 7. Output impedance (loading, efﬁciency, interconnectability): This represents the dynamic resistance as felt at the output terminals of the device. 8. Gain margin (stability): This is the amount by which the device gain could be increased before the system becomes unstable. 9. Phase margin (stability): This is the amount by which the device phase lead could be decreased (i.e., phase lag increased) before the system becomes unstable.

1.4.2

Linearity

A device is considered linear if it can be modeled by linear differential equations, with time t as the independent variable. Nonlinear devices are often analyzed using linear techniques by considering small excursions about an operating point. This linearization is accomplished by introducing incremental variables for the excitations (inputs) and responses (outputs). If one increment can cover the entire operating range of a device with sufﬁcient accuracy, it is an indication that the device is linear. If the input/output relations are nonlinear algebraic equations, that represents a static nonlinearity. Such a situation can be handled simply by using nonlinear calibration curves, which linearize the device without introducing nonlinearity errors. If, on the other hand, the input/output relations are nonlinear differential equations, analysis usually becomes quite complex. This situation represents a dynamic nonlinearity. Transfer-function representation is a “linear” model of an instrument. Hence, it implicitly assumes linearity. According to industrial terminology, a linear measuring instrument provides a measured value that varies linearly with the value of the measurand. This is consistent with the deﬁnition of static linearity.

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All physical devices are nonlinear to some degree. This stems from any deviation from the ideal behavior, due to causes such as saturation, deviation from Hooke’s Law in elastic elements, Coulomb friction, creep at joints, aerodynamic damping, backlash in gears and other loose components, and component wearout. Nonlinearities in devices are often manifested as some peculiar characteristics. In particular, the following properties are important in detecting nonlinear behavior in dynamic systems: 1. Saturation: The response does not increase when the excitation is increased beyond some level. This may result from such causes as magnetic saturation, which is common in transformer devices such as differential transformers, plasticity in mechanical components, or nonlinear deformation in springs. 2. Hysteresis: In this case, the input/output curve changes depending on the direction of motion, resulting in a hysteresis loop. This is common in: loose components such as gears, which have backlash; in components with nonlinear damping, such as Coulomb friction; and in magnetic devices with ferromagnetic media and various dissipative mechanisms (e.g., eddy current dissipation). 3. The jump phenomenon: Some nonlinear devices exhibit an instability known as the jump phenomenon (or fold catastrophe). Here, the frequency response (transfer) function curve suddenly jumps in magnitude at a particular frequency, while the excitation frequency is increased or decreased. A device with this nonlinearity will exhibit a characteristic “tilt” of its resonant peak either to the left (softening nonlinearity) or to the right (hardening nonlinearity). Furthermore, the transfer function itself may change with the level of input excitation in the case of nonlinear devices. 4. Limit cycles: A limit cycle is a closed trajectory in the state space that corresponds to sustained oscillations without decay or growth. The amplitude of these oscillations is independent of the location at which the response began. In the case of a stable limit cycle, the response will return to the limit cycle irrespective of the location near the limit cycle from which the response was initiated. In the case of an unstable limit cycle, the response will steadily move away from the location with the slightest disturbance. 5. Frequency creation: At steady state, nonlinear devices can create frequencies that are not present in the excitation signals. These frequencies might be harmonics (integer multiples of the excitation frequency), subharmonics (integer fractions of the excitation frequency), or nonharmonics (usually rational fractions of the excitation frequency). Several methods are available to reduce or eliminate nonlinear behavior in vibrating systems. They include calibration (in the static case), use of linearizing elements, such as resistors and ampliﬁers to neutralize the nonlinear effects, and the use of nonlinear feedback. It is also good practice to take the following precautions: 1. 2. 3. 4. 5.

1.4.3

Avoid operating the device over a wide range of signal levels. Avoid operation over a wide frequency band. Use devices that do not generate large mechanical motions. Minimize Coulomb friction. Avoid loose joints and gear coupling (i.e., use direct-drive mechanisms).

Instrument Ratings

Instrument manufacturers do not usually provide complete dynamic information for their products. In most cases, it is unrealistic to expect complete dynamic models (in the time or the frequency domain) and associated parameter values for complex instruments. Performance characteristics provided by manufacturers and vendors are primarily static parameters. Known as instrument ratings, these are available as parameter values, tables, charts, calibration curves, and empirical equations. Dynamic characteristics such as transfer functions (e.g., transmissibility curves expressed with respect to excitation

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frequency) might also be provided for more sophisticated instruments, but the available dynamic information is never complete. Furthermore, the deﬁnitions of rating parameters used by manufacturers and vendors of instruments are in some cases not the same as analytical deﬁnitions used in textbooks. This is particularly true in relation to the term linearity. Nevertheless, instrument ratings provided by manufacturers and vendors are very useful in the selection, installation, operation, and maintenance of instruments. Some of these performance parameters are indicated below. 1.4.3.1

Rating Parameters

Typical rating parameters supplied by instrument manufacturers are: 1. 2. 3. 4. 5. 6. 7. 8.

Sensitivity Dynamic range Resolution Linearity Zero drift and full-scale drift Useful frequency range Bandwidth Input and output impedances

The conventional deﬁnitions given by instrument manufacturers and vendors are summarized below. Sensitivity of a transducer is measured by the magnitude (peak, root-mean-square [RMS] value, etc.) of the output signals corresponding to a unit input of the measurand. This may be expressed as the ratio of (incremental output)/(incremental input) or, analytically, as the corresponding partial derivative. In the case of vectorial or tensorial signals (e.g., displacement, velocity, acceleration, strain, force), the direction of sensitivity should be speciﬁed. Cross-sensitivity is the sensitivity along directions that are orthogonal to the direction of primary sensitivity; it is expressed as a percentage of the direct sensitivity. High sensitivity and low crosssensitivity are desirable for measuring instruments. Sensitivity to parameter changes, disturbances, and noise has to be small in any device, however; this is an indication of its robustness. Often, sensitivity and robustness are conﬂicting requirements. Dynamic range of an instrument is determined by the allowed lower and upper limits of its input or output (response) so as to maintain a required level of measurement accuracy. This range is usually expressed as a ratio, in decibels. In many situations, the lower limit of the dynamic range is equal to the resolution of the device. Hence, the dynamic range is usually expressed as the ratio (range of operation)/(resolution), in decibels. Resolution is the smallest change in a signal that can be detected and accurately indicated by a transducer, a display unit, or other instrument. It is usually expressed as a percentage of the maximum range of the instrument or as the inverse of the dynamic range ratio, as deﬁned above. It follows that dynamic range and resolution are very closely related. Linearity is determined by the calibration curve of an instrument. The curve of output amplitude (a peak or rms value) vs. input amplitude under static conditions within the dynamic range of an instrument is known as the static calibration curve. Its closeness to a straight line measures the degree of linearity. Manufacturers provide this information either as the maximum deviation of the calibration curve from the least squares straight-line ﬁt of the calibration curve or from some other reference straight line. If the least squares ﬁt is used as the reference straight line, the maximum deviation is called independent linearity (or more correctly, the independent nonlinearity, because the larger the deviation, the greater the nonlinearity). Nonlinearity may be expressed as a percentage of either the actual reading at an operating point or the full-scale reading. Zero drift is deﬁned as the drift from the null reading of the instrument when the measurand is maintained steady for a long period. Note that in this case, the measurand is kept at zero or any other level that corresponds to null reading of the instrument. Similarly, full-scale drift is deﬁned with respect to the full-scale reading (the measurand is maintained at the full-scale value). Usual causes of drift include

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instrument instability (e.g., instability in ampliﬁers), ambient changes (e.g., changes in temperature, pressure, humidity, and vibration level), changes in power supply (e.g., changes in reference DC voltage or alternating current [AC] line voltage), and parameter changes in an instrument (due to aging, wearout, nonlinearities, etc.). Drift due to parameter changes that are caused by instrument nonlinearities is known as parametric drift, sensitivity drift, or scale-factor drift. For example, a change in spring stiffness or electrical resistance due to changes in ambient temperature results in a parametric drift. Note that the parametric drift depends on the measurand level. Zero drift, however, is assumed to be the same at any measurand level if the other conditions are kept constant. For example, a change in reading caused by thermal expansion of the readout mechanism due to changes in the ambient temperature is considered a zero drift. In electronic devices, drift can be reduced by using AC circuitry rather than direct current (DC) circuitry. For example, AC-coupled ampliﬁers have fewer drift problems than DC ampliﬁers. Intermittent checking for the instrument response level for zero input is a popular way to calibrate for zero drift. In digital devices, this can be done automatically and intermittently, between sample points, when the input signal can be bypassed without affecting the system operation. Useful frequency range corresponds to the interval of both ﬂat gain and zero phase in the frequency response characteristics of an instrument. The maximum frequency in this band is typically less than half (say, one ﬁfth of) the dominant resonant frequency of the instrument. This is a measure of instrument bandwidth. Bandwidth of an instrument determines the maximum speed or frequency at which the instrument is capable of operating. High bandwidth implies faster speed of response. Bandwidth is determined by the dominant natural frequency, vn; or the dominant resonant frequency, vr; of the transducer. (Note: For low damping, vr is approximately equal to vn.) It is inversely proportional to the rise time and the dominant time constant. Half-power bandwidth is also a useful parameter. Instrument bandwidth must be several times greater than the maximum frequency of interest in the measured signal. The bandwidth of a measuring device is important, particularly when measuring transient signals. Note that the bandwidth is directly related to the useful frequency range.

1.4.4

Accuracy and Precision

The instrument ratings mentioned above affect the overall accuracy of an instrument. Accuracy can be assigned either to a particular reading or to an instrument. Note that instrument accuracy depends not only on the physical hardware of the instrument but also on the operating conditions (e.g., design conditions that are the normal, steady operating conditions or extreme transient conditions, such as emergency start-up and shutdown). Measurement accuracy determines the closeness of the measured value to the true value. Instrument accuracy is related to the worst accuracy obtainable within the dynamic range of the instrument in a speciﬁc operating environment. Measurement error is deﬁned as Error ¼ ðmeasured valueÞ 2 ðtrue valueÞ

ð1:31Þ

Correction, which is the negative of error, is deﬁned as Correction ¼ ðtrue valueÞ 2 ðmeasured valueÞ

ð1:32Þ

Each of these can also be expressed as a percentage of the true value. The accuracy of an instrument may be determined by measuring a parameter whose true value is known, and is near the extremes of the dynamic range of the instrument, under certain operating conditions. For this purpose, standard parameters or signals that can be generated at very high levels of accuracy would be needed. The National Institute for Standards and Testing (NIST) is usually responsible for the generation of these standards. Nevertheless, accuracy and error values cannot be determined to 100% exactness in typical applications, because the true value is not known. In a given situation, we can only make estimates for accuracy, by using ratings provided by the instrument manufacturer or by analyzing data from previous measurements and models.

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Causes of error include instrument instability, external noise (disturbances), poor calibration, inaccurate information (e.g., poor analytical models, inaccurate control), parameter changes (e.g., due to environmental changes, aging, and wearout), unknown nonlinearities, and improper use of the instrument. Errors can be classiﬁed as deterministic (or systematic) and random (or stochastic). Deterministic errors are those caused by well-deﬁned factors, including nonlinearities and offsets in readings. These usually can be removed by applying proper calibration and analytical practices. Error ratings and calibration charts are used to remove systematic errors from instrument readings. Random errors are caused by uncertain factors entering into the instrument response. These include device noise, line noise, and the effects of unknown random variations in the operating environment. A statistical analysis using sufﬁciently large amounts of data is necessary to estimate random errors. The results are usually expressed as a mean error, which is the systematic part of random error, and a standard deviation or conﬁdence interval for instrument response. Precision is not synonymous with accuracy. Reproducibility (or repeatability) of an instrument reading determines the precision of an instrument. Two or more identical instruments that have the same high offset error might be able to generate responses at high precision, even though these readings are clearly inaccurate. For example, consider a timing device (clock) that very accurately indicates time increments (say, up to the nearest microsecond). If the reference time (starting time) is set incorrectly, the time readings will be in error, even though the clock has a very high precision. Instrument error may be represented by a random variable that has a mean value me and a standard deviation se. If the standard deviation is zero, the variable is considered deterministic. In that case, the error is said to be deterministic or repeatable. Otherwise, the error is said to be random. The precision of an instrument is determined by the standard deviation of error in the instrument response. Readings of an instrument may have a large mean value of error (e.g., large offset), but if the standard deviation is small, the instrument has a high precision. Hence, a quantitative deﬁnition for precision is Precision ¼ ðmeasurement rangeÞ=se

ð1:33Þ

Lack of precision originates from random causes and poor construction practices. It cannot be compensated for by recalibration, just as the precision of a clock cannot be improved by resetting the time. On the other hand, accuracy can be improved by recalibration. Repeatable (deterministic) accuracy is inversely proportional to the magnitude of the mean error me. In selecting instruments for a particular application, in addition to matching instrument ratings with speciﬁcations, several additional features should be considered. These include geometric limitations (size, shape, etc.); environmental conditions (e.g., chemical reactions including corrosion, extreme temperatures, light, dirt accumulation, electromagnetic ﬁelds, radioactive environments, shock and vibration); power requirements; operational simplicity; availability; the past record and reputation of the manufacturer and of the particular instrument; and cost-related economic aspects (initial cost, maintenance cost, cost of supplementary components such as signal-conditioning and processing devices, design life and associated frequency of replacement, and cost of disposal and replacement). Often, these considerations become the ultimate deciding factors in the selection process.

1.5

Motion Sensors and Transducers

Motion sensing is considered the most important measurement in vibration applications. Other variables, such as force, torque, stress, strain, and material properties, are also important, either directly or indirectly, in the study of vibration. This section will describe some useful measuring devices of motion in the ﬁeld of mechanical vibration.

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1.5.1

Potentiometer

The potentiometer, or pot, is a displacement transducer. This active transducer consists of a uniform coil of wire or a ﬁlm of high-resistive material, such as carbon, platinum, or conductive plastic, whose resistance is proportional to its length. A ﬁxed voltage, vre,f is applied across the coil (or ﬁlm) using an external, constant DC voltage supply. The transducer output signal, vo, is the DC voltage between the movable contact (wiper arm) sliding on the coil and one terminal of the coil, as shown schematically in Figure 1.15(a). Slider displacement x is proportional to the output voltage vo ¼ kx

ð1:34Þ

This relationship assumes that the output terminals are open-circuit, which means that inﬁniteimpedance load (or resistance in the present DC case) is present at the output terminal, so that the output current is zero. In actual practice, however, the load (the circuitry into which the pot signal is fed, e.g., conditioning or processing circuitry) has a ﬁnite impedance. Consequently, the output current (the current through the load) is nonzero, as shown in Figure 1.15(b). The output voltage thus drops to vo ; even if the reference voltage, vref, is assumed to remain constant under load variations (i.e., the voltage source has zero output impedance). This consequence is known as the loading effect of the transducer. Under these conditions, the linear relationship given by Equation 1.34 is no longer valid. This causes an error in the displacement reading. Loading can affect the transducer reading in two ways: by changing the reference voltage (i.e., loading the voltage source) or by loading the transducer. To reduce these effects, a voltage source that is not seriously affected by load variations (e.g., a regulated or stabilized power supply that has low output impedance) and data acquisition circuitry (including signal-conditioning circuitry) that has high input impedance should be used. The resistance of a potentiometer should be chosen with care. On the one hand, an element with high resistance is preferred because this results in reduced power dissipation for a given voltage, which has the added beneﬁt of reduced thermal effects. On the other hand, increased resistance increases the output impedance of the potentiometer and results in loading nonlinearity error unless the load resistance is also increased proportionately. Low-resistance pots have resistances less than 10 V. High-resistance pots can have resistances on the order of 100 kV. Conductive plastics can provide high resistances, typically about 100 V/mm, and are increasingly used in potentiometers. Reduced friction (low mechanical loading), reduced wear, reduced weight, and increased resolution are advantages of using conductive plastics in potentiometers. 1.5.1.1

Potentiometer Resolution

The force required to move the slider arm comes from the motion source, and the resulting energy is dissipated through friction. This energy conversion, unlike pure mechanical-to-electrical conversions, Resistive Element

vref (Supply)

Wiper Arm

+ vo (Measurement) i _ No Current

(a)

∼ Z vo

i

(b) FIGURE 1.15

+

vref

x (Measurand)

_ Nonzero Current

(a) Schematic diagram of a potentiometer; (b) potentiometer loading.

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Load Impedance

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1-29

involves relatively high forces, and the energy is wasted rather than converted into the output signal of the transducer. Furthermore, the electrical energy from the reference source is also dissipated through the resistor coil (or ﬁlm), resulting in an undesirable temperature rise. These are two obvious disadvantages of this resistively coupled transducer. Another disadvantage is the ﬁnite resolution in coil-type pots. Coils, instead of straight wire, are used to increase the resistance per unit travel of the slider arm in the coil-type pot. However, the slider contact jumps from one turn to the next in this case. Accordingly, the resolution of a coil-type potentiometer is determined by the number of turns in the coil. For a coil that has N turns, the resolution, r, expressed as a percentage of the output range, is given by r¼

100 % N

ð1:35Þ

Resolutions better (smaller) than 0.1% (i.e., 1000 turns) are available with coil potentiometers. Inﬁnitesimal (incorrectly termed inﬁnite) resolutions are now possible with high-quality resistive ﬁlm potentiometers that use conductive plastics, for example. In this case, the resolution is limited by other factors, such as mechanical limitations and signal-to-noise ratio. Nevertheless, resolutions on the order of 0.01 mm are possible with good rectilinear potentiometers. Some limitations and disadvantages of potentiometers as displacement measuring devices are as follows: 1. The force needed to move the slider (against friction and arm inertia) is provided by the vibration source. This mechanical loading distorts the measured signal itself. 2. High-frequency (or highly transient) measurements are not feasible because of such factors as slider bounce, friction and inertia resistance, and induced voltages in the wiper arm and primary coil. 3. Variations in the supply voltage cause error. 4. Electrical loading error can be signiﬁcant when the load resistance is low. 5. Resolution is limited by the number of turns in the coil and by the coil uniformity. This will limit small-displacement measurements such as ﬁne vibrations. 6. Wearout and heating up (with associated oxidation) in the coil (ﬁlm) and slider contact cause accelerated degradation. There are several advantages associated with potentiometer devices, however, including the following: 1. They are relatively less costly. 2. Potentiometers provide high-voltage (low-impedance) output signals, requiring no ampliﬁcation in most applications. Transducer impedance can be varied simply by changing the coil resistance and supply voltage. 1.5.1.2

Optical Potentiometer

The optical potentiometer, shown schematically in Figure 1.16(a), is a displacement sensor. A layer of photoresistive material is sandwiched between a layer of regular resistive material and a layer of conductive material. The layer of resistive material has a total resistance of Rc, and it is uniform (i.e., it has a constant resistance per unit length). The photoresistive layer is practically an electrical insulator when no light is projected on it. The displacement of the moving object whose displacement is being measured causes a moving light beam to be projected on a rectangular area of the photoresistive layer. This light-projected area attains a resistance of Rp, which links the resistive layer, which is above the photoresistive layer, and the conductive layer, which is below it. The supply voltage to the potentiometer is vref, and the length of the resistive layer is L. The light spot is projected at a distance x from one end of the resistive element, as shown in Figure 1.16(a). An equivalent circuit for the optical potentiometer is shown in Figure 1.16(b). Here, it is assumed that a load of resistance RL is present at the output of the potentiometer, the voltage across which is vo. Current through the load is vo/RL. Hence, the voltage drop across ð1 2 aÞRc þ RL ; which is also the voltage across Rp, is given by ½ð1 2 aÞRc þ RL vo =RL : Note that a ¼ x=L is the fractional position of

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Resistive Layer (Rc)

Light Spot (Rp)

Photoresistive Layer

+ Output vo

Supply vref

_ Measurand x (a)

Conductive Layer

L aRc

(1-a)Rc vo

vref

Rp

RL

(b) FIGURE 1.16

(a) An optical potentiometer; (b) equivalent circuit ða ¼ x=LÞ:

the light spot. The current balance at the junction of the three resistors in Figure 1.16(b) is vref 2 ½ð1 2 aÞRc þ RL vo =RL v ½ð1 2 aÞRc þ RL vo =RL ¼ o þ aRc Rp RL which can be written as vo vref

(

Rc x Rc þ1þ RL L Rp

x Rc 12 þ1 L RL

) ¼1

ð1:36Þ

When the load resistance RL is quite large in comparison to the element resistance Rc , we have Rc =RL . 0: Hence, Equation 1.36 becomes vo 1 # ¼ " vref x Rc þ1 L Rp

ð1:37Þ

This relationship is still nonlinear in vo =vref vs. x=L: The nonlinearity decreases, however, with decreasing Rc =Rp :

1.5.2

Variable-Inductance Transducers

Motion transducers that employ the principle of electromagnetic induction are termed variableinductance transducers. When the ﬂux linkage (deﬁned as magnetic ﬂux density times the number of turns in the conductor) through an electrical conductor changes, a voltage is induced in the conductor. This, in turn, generates a magnetic ﬁeld that opposes the primary ﬁeld. Hence, a mechanical force is necessary to sustain the change of ﬂux linkage. If the change in ﬂux linkage is brought about by a relative motion, the mechanical energy is directly converted (induced) into electrical energy. This is the basis of electromagnetic induction, and it is the principle of operation of electrical generators and variableinductance transducers. Note that, in these devices, the change of ﬂux linkage is caused by a mechanical motion, and mechanical-to-electrical energy transfer takes place under near-ideal conditions.

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The induced voltage or change in inductance may be used as a measure of the motion. Variableinductance transducers are generally electromechanical devices coupled by a magnetic ﬁeld. There are many different types of variable-inductance transducers. Three primary types are: 1. Mutual-induction transducers 2. Selﬁnduction transducers 3. Permanent-magnet transducers Variable-inductance transducers that use a nonmagnetized ferromagnetic medium to alter the reluctance (magnetic resistance) of the ﬂux path are known as variable-reluctance transducers. Some of the mutualinduction transducers and most of the selﬁnduction transducers are of this type. Permanent-magnet transducers do not fall into category of variable-reluctance transducers. 1.5.2.1

Mutual-Induction Transducers

The basic arrangement of a mutual-induction transducer is constituted of two coils, the primary windings and the secondary windings. One of the coils, the primary windings, carries an AC excitation that induces a steady AC voltage in the other coil, the secondary windings. The level (amplitude, RMS value, etc.) of the induced voltage depends on the ﬂux linkage between the coils. In mutual-induction transducers, a change in the ﬂux linkage is effected by one of the two common techniques. One technique is to move an object made of ferromagnetic material within the ﬂux path. This changes the reluctance of the ﬂux path, with an associated change of the ﬂux linkage in the secondary coil. This is the operating principle of the linear-variable differential transformer (LVDT), the rotatory-variable differential transformer (RVDT), and the mutual-induction proximity probe. All of these are variable-reluctance transducers. The other common way to change the ﬂux linkage is to move one coil with respect to the other. This is the operating principle of the resolver, the synchro-transformer, and some types of AC tachometer. These are not variable-reluctance transducers, however. The motion can be measured by using the secondary signal in several ways. For example, the AC signal in the secondary windings may be demodulated by rejecting the carrier frequency (primary-winding excitation frequency) and directly measuring the resulting signal, which represents the motion. This method is particularly suitable for measuring transient motions. Alternatively, the amplitude or the rms value of the secondary (induced) voltage may be measured. Another method is to measure the change of inductance in the secondary circuit directly by using a device such as an inductance bridge circuit. 1.5.2.2

Linear-Variable Differential Transformer

The LVDT is a displacement (vibration) measuring device, which can overcome most of the shortcomings of the potentiometer. It is considered a passive transducer because the measured displacement provides energy for “changing” the induced voltage, even though an external power supply is used to energize the primary coil, which in turn induces a steady carrier voltage in the secondary coil. The LVDT is a variable-reluctance transducer of the mutual induction type. In its simplest form, the LVDT consists of an insulating, nonmagnetic cylinder that has a primary coil in the midsegment and a secondary coil symmetrically wound in the two end segments, as depicted schematically in Figure 1.17(a). The primary coil is energized by an AC supply of voltage vref. This will generate, by mutual induction, an AC of the same frequency in the secondary winding. A core made of ferromagnetic material is inserted coaxially into the cylindrical form without actually touching it, as shown. As the core moves, the reluctance of the ﬂux path changes. Hence, the degree of ﬂux linkage depends on the axial position of the core. The two secondary coils are connected in series opposition so that the potentials induced in these two coil segments oppose each other, therefore, the net induced voltage is zero when the core is centered between the two secondary winding segments. This is known as the null position. When the core is displaced from this position, a nonzero induced voltage will be generated. At steady state, the amplitude vo of this induced voltage is proportional, in the linear (operating) region, to the core displacement x. Consequently, vo may be used as a measure of the displacement. Note that, because of its opposed secondary windings, the LVDT

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Vibration Monitoring, Testing, and Instrumentation

vo (Measurement)

Primary Coil

Insulating Form

Ferromagnetic Core

Housing

(a)

Core Displacement x (Measurand)

Secondary Coil Segment

vref

Secondary Coil Segment

Voltage level

vo

Displacement x

(b) FIGURE 1.17

Linear Range (a) Schematic diagram of an LVDT; (b) a typical operating curve.

provides the direction as well as the magnitude of displacement. If the output signal is not demodulated, the direction is determined by the phase angle between the primary (reference) voltage and the secondary (output) voltage, including the carrier signal. For an LVDT to measure transient motions accurately, the frequency of the reference voltage (the carrier frequency) has to be about ten times larger than the largest signiﬁcant frequency component in the measured motion. For quasi-dynamic displacements and slow transients on the order of a few hertz, a standard AC supply (at 60 Hz line frequency) is adequate. The performance, particularly the sensitivity and accuracy, is known to improve with the excitation frequency, however. Since the amplitude of the output signal is proportional to the amplitude of the primary signal, the reference voltage should be regulated to obtain accurate results. In particular, the power source should have a low output impedance. The output signal from a differential transformer is normally not in phase with the reference voltage. Inductance in the primary windings and the leakage inductance in the secondary windings are mainly responsible for this phase shift. Since demodulation involves extraction of the modulating signal by rejecting the carrier frequency component from the secondary signal, it is important to understand the size of this phase shift. An error known as null voltage is present in some differential transformers. This manifests itself as a nonzero reading at the null position (i.e., at zero displacement). This is usually 908 out of phase from the main output signal and, hence, is known as quadrature error. Nonuniformities in the windings (unequal impedances in the two segments of the secondary windings) are a major reason for this error. The null voltage may also result from harmonic noise components in the primary signal and nonlinearities in the device. Null voltage is usually negligible (typically about 0.1% of the full scale). This error can be eliminated from the measurements by employing appropriate signal-conditioning and calibration practices.

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1.5.2.3

1-33

Signal Conditioning

Signal conditioning associated with differential transformers includes ﬁltering and ampliﬁcation. Filtering is needed to improve the signal-to-noise ratio of the output signal. Ampliﬁcation is necessary to increase the signal strength for data acquisition and processing. Since the reference frequency (carrier frequency) is embedded in the output signal, it is also necessary to interpret the output signal properly, particularly for transient motions. Two methods are commonly used to interpret the amplitudemodulated output signal from a differential transformer: (1) rectiﬁcation; (2) demodulation. In the ﬁrst method, rectiﬁcation, the AC output from the differential transformer is rectiﬁed to obtain a DC signal. This signal is ampliﬁed and then low-pass ﬁltered to eliminate any high-frequency noise components. The amplitude of the resulting signal provides the transducer reading. In this method, phase shift in the LVDT output must be checked separately to determine the direction of motion. In the second method, demodulation, the carrier frequency component is rejected from the output signal by comparing it with a phase-shifted and amplitude-adjusted version of the primary (reference) signal. Note that phase shifting is necessary because the output signal is not in phase with the reference signal. The modulating signal which is extracted in this manner is subsequently ampliﬁed and ﬁltered. As a result of advances in miniature integrated circuit (LSI and VLSI) technology, differential transformers with builtin microelectronics for signal conditioning are commonly available today. DC differential transformers have built-in oscillator circuits to generate the carrier signal powered by a DC supply. The supply voltage is usually on the order of 25 V, and the output voltage is about 5 V. Let us illustrate the demodulation approach to signal conditioning for an LVDT, using an example.

Example 1.2 Figure 1.18 shows a schematic diagram of a simpliﬁed signal conditioning system for an LVDT. The system variables and parameters are as indicated in Figure 1.18. In particular: u(t) ¼ displacement of the LVDT core (to be measured) wc ¼ frequency of the carrier voltage vo ¼ output signal of the system (measurement) The resistances R1, R2, and R, and the capacitance C are as marked. In addition, you may introduce a transformer parameter r for the LVDT, as required. 1. Explain the functions of the various components of the system shown in Figure 1.18. 2. Write equations for the ampliﬁer and ﬁlter circuits and, using them, give expressions for the voltage signals v1, v2, v3, and vo marked in Figure 1.18. Note that the excitation in the primary coil is vp sin vc t: R

x(t)

C Carrier Signal vp sin wct

v1

+ −

v2

v3

Amplifier FIGURE 1.18

−

Output vo

R2 R1

LVDT

R1

+ R1

Multiplier

Signal-conditioning system for an LVDT.

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Low-pass Filter

1-34

Vibration Monitoring, Testing, and Instrumentation

3. Suppose that the carrier frequency is vc ¼ 500 rad=s and the ﬁlter resistance is R ¼ 100 k V: If no more than 5% of the carrier component passes through the ﬁlter, estimate the required value of the ﬁlter capacitance, C. Also, what is the useful frequency range (measurement bandwidth) of the system in rad/sec, with these parameter values?

Solution 1. The LVDT has a primary coil that is excited by an AC voltage of vp sin vc t: The ferromagnetic core is attached to the moving object whose displacement x(t) is to be measured. The two secondary coils are connected in series opposition so that the LVDT output is zero at the null position, and that the direction of motion can be detected as well. The ampliﬁer is a noninverting type. It ampliﬁes the output of the LVDT which is an AC (carrier) signal of frequency vc ; which is modulated by the core displacement x(t). The multiplier circuit determines the product of the primary (carrier) signal and the secondary (LVDT output) signal. This is an important step in demodulating the LVDT output. The product signal from the multiplier has a high-frequency (2vc ) carrier component, added to the modulating component ðxðtÞÞ: The low-pass ﬁlter removes this unnecessary high-frequency component to obtain the demodulated signal which is proportional to the core displacement x(t). 2. Noninverting Ampliﬁer: Note that the potentials at the positive and negative terminals of the operational ampliﬁer (opamp) are nearly equal. Also, currents through these leads are nearly zero. (These are the two common assumptions used for an opamp.) Then, the current balance at node A gives v2 2 v1 v ¼ 1 R2 R1 or

v2 ¼

R1 þ R2 R1

v1

Then, v 2 ¼ k v1

with k¼

ðiÞ

R1 þ R2 ¼ amplifier gain R1

Loss-Pass Filter: Since the þ lead of the opamp has approximately a zero potential (ground), the voltage at point B is also approximately zero. The current balance for node B gives v3 v þ o þ C_vo ¼ 0 R1 R Hence,

t

dvo R þ vo ¼ 2 v3 R1 dt

ðiiÞ

where t ¼ RC ¼ ﬁlter time constant. The transfer function of the ﬁlter is vo ko ¼2 v3 ð1 þ tsÞ

ðiiiÞ

with the ﬁlter gain ko ¼ R/R1. In the frequency domain vo ko ¼2 v3 ð1 þ tjvÞ Finally, neglecting the phase shift in the LVDT, we have v1 ¼ vp r uðtÞ sin vc t v2 ¼ vp rk uðtÞ sin vc t

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ð1:38Þ

Vibration Instrumentation

1-35

v3 ¼ vp2 rk uðtÞ sin2 vc t

or

v3 ¼

vp2 rk uðtÞ½1 2 cos 2 vc t 2

ðivÞ

Owing to the low-pass ﬁlter, with an appropriate cut-off frequency, the carrier signal will be ﬁltered out. Then, vo ¼

vp2 rko uðtÞ 2

ð1:39Þ

pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 3. Filter magnitude ¼ ko = 1 þ t 2 v 2 : For no more than 5% of the carrier ð2vc Þ component to pass through, we must have ko 5 pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ # ko 2 2 100 1 þ t ð2vc Þ

ðvÞ

or, 400 # 1 þ 4t2 v2c ; or, t2 v2c $ 399=4; or tvc $ 10 (approximately). Pick tvc ¼ 10: With R ¼ 100 kV and vc ¼ 500 rad=s, we have, C £ 100 £ 103 £ 500 ¼ 10. Hence, C ¼ 0.2 m F. According to the carrier frequency (500 rad/sec), we should be able to measure displacements u(t) up to about 50 rad/sec. However, the ﬂat region of the ﬁlter extends to approximately vt ¼ 0:1; which, with the present value of t ¼ 0.02 sec, gives a bandwidth of only 5 rad/sec. Advantages of the LVDT include the following: 1. It is essentially a noncontacting device with no frictional resistance. Its near-ideal electromechanical energy conversion and lightweight core will result in very small resistive forces. Hysteresis (both magnetic hysteresis and mechanical backlash) is negligible. 2. It has low output impedance, typically on the order of 100 V. (signal ampliﬁcation is usually not needed). 3. Directional measurements (positive/negative) are obtained. 4. It is available in small sizes (e.g., 1 cm long with maximum travel of 2 mm). 5. It has a simple and robust construction (inexpensive and durable). 6. Fine resolutions are possible (theoretically, inﬁnitesimal resolution; practically, much ﬁner resolution than that of a coil potentiometer). The RVDT operates using the same principle as the LVDT, except that in an RVDT, a rotating ferromagnetic core is used. The RVDT is used for measuring angular displacements. The rotating core is shaped such that a reasonably wide linear operating region is obtained. Advantages of the RVDT are essentially the same as those cited for the LVDT. The linear range is typically ^ 408 with a nonlinearity error less than 1%. In variable-inductance devices, the induced voltage is generated through the rate of change of the magnetic ﬂux linkage. Therefore, displacement readings are distorted by velocity, and similarly, velocity readings are affected by acceleration. For the same displacement value, the transducer reading will depend on the velocity at that displacement. This error is known to increase with the ratio: (cyclic velocity of the core)/(carrier frequency). Hence, these rate errors can be reduced by increasing the carrier frequency. The reason for this is as follows. At high frequencies, the induced voltage due to the transformer effect (frequencies of the primary signal) is greater than the induced voltage due to the rate (velocity) effect of the moving member. Hence, the error will be small. To estimate a lower limit for the carrier frequency in order to reduce rate effects,

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we may proceed as follows: 1. For LVDT: max speed of operation=stroke of LVDT ¼ vo The excitation frequency of the primary coil should be chosen as at least 5vo : 2. For RVDT: for vo use the maximum angular frequency of operation (of the rotor).

1.5.3

Mutual-Induction Proximity Sensor

This displacement transducer operates on the mutual-induction principle. A simpliﬁed schematic diagram of such a device is shown in Figure 1.19(a). The insulated “E core” carries the primary windings in its middle limb. The two end limbs carry secondary windings that are connected in series. Unlike the LVDT and the RVDT, the two voltages induced in the secondary winding segments are additive in this case. The region of the moving surface (target object) that faces the coils has to be made of ferromagnetic material so that as it moves, the magnetic reluctance and the ﬂux linkage will change. This, in turn, changes the induced voltage in the secondary windings, and this change is a measure of the displacement. Note that, unlike the LVDT, which has an “axial” displacement conﬁguration, the proximity probe has a “transverse” displacement conﬁguration. Hence, it is particularly suitable for measuring transverse displacements or proximities of moving objects (e.g., transverse vibrations of a beam or whirling of a rotating shaft). We can see from the operating curve shown in Figure 1.19(b) that the displacementvoltage relation of a proximity probe is nonlinear. Hence, these proximity sensors should be used only for measuring small displacements, such as linear vibrations (e.g., a linear range of 5.0 mm or 0.2 in.), unless accurate nonlinear calibration curves are available. Since the proximity sensor is a noncontacting device, mechanical loading is small and the product life is long. Because a ferromagnetic object is used to alter the reluctance of the ﬂux path, the mutual-induction proximity sensor is a variable-reluctance device. The operating frequency limit is about one tenth the excitation frequency of the primary coil (carrier frequency). As for an LVDT, demodulation of the induced voltage (secondary) would be required to obtain direct (DC) output readings.

vo (Measurement)

~ Primary Coil

Secondary Coil

Secondary Coil

vref

Ferromagnetic Target Object x (Measurand)

(a)

Output Voltage

vo

Proximity

(b) FIGURE 1.19

x

(a) Schematic diagram of the mutual-induction proximity sensor; (b) operating curve.

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1.5.4

1-37

Selﬁnduction Transducers

These transducers are based on the principle of AC selﬁnduction. Unlike mutual-induction transduSupply vref cers, only a single coil is employed. This coil is Inductance activated by an AC supply voltage, vref. The current ~ Measuring produces a magnetic ﬂux, which is linked with the Circuit coil. The level of ﬂux linkage (or selﬁnductance) can be varied by moving a ferromagnetic object within the magnetic ﬁeld. This changes the reluctance of the ﬂux path and the inductance in the coil. This change is a measure of the displacement of the ferromagnetic object. Ferromagnetic The change in inductance is measured using an Target Object inductance measuring circuit (e.g., an inductance x bridge). Note that selﬁnduction transducers are (Measurand) usually variable-reluctance devices. A typical selﬁnduction transducer is a self- FIGURE 1.20 Schematic diagram of a selﬁnduction induction proximity sensor. A schematic diagram proximity sensor. of this device is shown in Figure 1.20. This device can be used as a displacement or vibration sensor for transverse displacements. For instance, the distance between the sensor tip and ferromagnetic surface of a moving object, such as a beam or shaft, can be measured. Applications are essentially the same as those for mutual-induction proximity sensors. Highspeed displacement (vibration) measurements can result in velocity error (rate error) when variableinductance displacement sensors, including selﬁnduction transducers, are used. This effect may be reduced by increasing the carrier frequency, as in other AC-powered variable-inductance sensors.

1.5.5

Permanent-Magnet Transducers

In discussing this third type of variable-inductance transducer, we will ﬁrst consider the permanentmagnet DC velocity sensors (DC tachometers). A distinctive feature of permanent-magnet transducers is that they have a permanent magnet to generate a uniform and steady magnetic ﬁeld. A relative motion between the magnetic ﬁeld and an electrical conductor induces a voltage that is proportional to the speed at which the conductor crosses the magnetic ﬁeld. In some designs, a unidirectional magnetic ﬁeld generated by a DC supply (i.e., an electromagnet) is used in place of a permanent magnet. Nevertheless, this is generally termed a permanent-magnet transducer. The principle of electromagnetic induction between a permanent magnet and a conducting coil is used in speed measurement by permanent-magnet transducers. Depending on the conﬁguration, either rectilinear speeds or angular speeds can be measured. Schematic diagrams of the two conﬁgurations are shown in Figure 1.21. Note that these are passive transducers, because the energy for the output signal vo is derived from the motion (measured signal) itself. The entire device is usually enclosed in a steel casing to isolate it from ambient magnetic ﬁelds. In the rectilinear velocity transducer, as shown in Figure 1.21(a), the conductor coil is wrapped on a core and placed centrally between two magnetic poles, which produce a cross-magnetic ﬁeld. The core is attached to the moving object whose velocity must be measured. The velocity v is proportional to the induced voltage, vo. An alternative design, consisting of a moving-magnet and ﬁxed-coil arrangement, may be used as well, thus eliminating the need for any sliding contacts (slip rings and brushes) for the output leads, and thereby reducing mechanical loading error, wearout, and related problems. The tachogenerator (or tachometer) is a very common permanent-magnet device. The principle of operation of a DC tachogenerator is shown in Figure 1.21(a). The rotor is directly connected to the rotating object. The output signal that is induced in the rotating coil is picked up as DC voltage, vo, using a suitable

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Vibration Monitoring, Testing, and Instrumentation

Permanent Magnet

Output vo (Measurement)

Moving Coil

Velocity v (Measurand)

(a) Commutator

Permanent Magnet

Speed wc

wc

S

N

h

Rotating Coil vo

(b) FIGURE 1.21

Rotating Coil

2r

Permanent-magnet transducers: (a) rectilinear velocity transducer; (b) DC tachometer-generator.

commutator device, which typically consists of a pair of low-resistance carbon brushes, that is stationary but makes contact with the rotating coil through split slip rings so as to maintain the positive direction of induced voltage throughout each revolution. The induced voltage is given by vo ¼ ð2nhr bÞvc

ð1:40Þ

for a coil of height h and width 2r that has n turns, moving at an angular speed vc in a uniform magnetic ﬁeld of ﬂux density b: This proportionality between vo and vc is used to measure the angular speed vc : When tachometers are used to measure transient velocities, some error will result from the rate (acceleration) effect. This error generally increases with the maximum signiﬁcant frequency that must be retained in the transient velocity signal. Output distortion can also result because of reactive (inductive and capacitive) loading of the tachometer. Both types of error can be reduced by increasing the load impedance. For an illustration, consider the equivalent circuit of a tachometer with an impedance load, R as shown in Figure 1.22. The induced voltage kvc is represented by a voltage source. Note that the constant k depends on the coil geometry, the + Ll number of turns, and the magnetic ﬂux density Load (see Equation 1.40). Coil resistance is denoted by vo ZL + Induced R, and leakage inductance is denoted by L‘ : The Voltage − load impedance is ZL. From straightforward circuit kw c − analysis in the frequency domain, the output voltage at the load is given by vo ¼

ZL kvc R þ jvL‘ þ ZL

ð1:41Þ

FIGURE 1.22 Equivalent circuit for a tachometer with an impedance load.

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1-39

It can be seen that because of the leakage inductance, the output signal attenuates more at higher frequencies v of the velocity transient. In addition, loading error is present. If ZL is much larger than the coil impedance, however, the ideal proportionality, as given by vo ¼ kvc

ð1:42Þ

is achieved. Some tachometers operate in a different manner. For example, digital tachometers generate voltage pulses at a frequency proportional to the angular speed. These are considered to be digital transducers.

1.5.6

Alternating Current Permanent-Magnet Tachometer

This device has a permanent magnet rotor and two AC separate sets of stator windings as schematically Output Carrier shown in Figure 1.23(a). One set of windings is ~ vo Source energized using an AC reference voltage. Induced v ref Permanentvoltage in the other set of windings is the Primary Magnet Secondary tachometer output. When the rotor is stationary Rotor (a) Stator Stator or moving in a quasi-static manner, the output voltage is a constant-amplitude signal much like AC Output the reference voltage. As the rotor moves at a ﬁnite Carrier ~ vo speed, an additional induced voltage that is Source proportional to the rotor speed is generated in v ref Shorted Secondary Primary the secondary windings. This is due to the rate of Rotor Coil (b) Stator Stator change of ﬂux linkage from the magnet in the secondary coil. The net output is an amplitudemodulated signal whose amplitude is proportional FIGURE 1.23 (a) AC permanent-magnet tachometer; to the rotor speed. For transient velocities, it will (b) AC induction tachometer. be necessary to demodulate this signal in order to extract the transient velocity signal (i.e., the modulating signal) from the modulated output. The direction of velocity is determined from the phase angle of the modulated signal with respect to the carrier signal. Note that in an LVDT, the amplitude of the AC magnetic ﬂux is altered by the position of the ferromagnetic core. But in an AC permanent-magnet tachometer, the DC magnetic ﬂux generated by the magnetic rotor is linked with the stator windings, and the associated induced voltage is caused by the speed of rotation of the rotor. For low-frequency applications (5 Hz or less), a standard AC supply (60 Hz) may be used to power an AC tachometer. For moderate-frequency applications, a 400 Hz supply is widely used. Typical sensitivity of an AC permanent-magnet tachometer is on the order of 50 to 100 mV/rad/sec.

1.5.7

Alternating Current Induction Tachometer

These tachometers are similar in construction to the two-phase induction motors. The stator arrangement is identical to that of the AC permanent-magnet tachometer. The rotor, however, has windings that are shorted and not energized by an external source, as shown in Figure 1.23(b). One set of stator windings is energized with an AC supply. This induces a voltage in the rotor windings, and it has two components. One component is due to the direct transformer action of the supply AC. The other component is induced by the speed of rotation of the rotor and its magnitude is proportional to the speed of rotation. The nonenergized stator windings provide the output of the tachometer. Voltage induced in the output stator windings is due to both the primary stator windings and the rotor windings. As a result, the tachometer output has a carrier AC component and a modulating component that is proportional to the speed of rotation. Demodulation would be needed to extract the output component that is proportional to the angular speed of the rotor.

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Vibration Monitoring, Testing, and Instrumentation

The main advantage of AC tachometers over their DC counterparts is the absence of slip-ring and brush devices. In particular, the signal from a DC tachometer usually has a voltage ripple, known as commutator ripple, which is generated as the split segments of the slip ring pass over the brushes. The frequency of the commutator ripple depends on the speed of operation; consequently, ﬁltering out its effects using a notch ﬁlter is difﬁcult (a tunable notch ﬁlter is necessary). Also, there are problems with frictional loading and contact bounce in DC tachometers, and these problems are absent in AC tachometers. It is known, however, that the output of an AC tachometer is somewhat nonlinear at high speeds due to the saturation effect. Furthermore, for measuring transient speeds, a sufﬁciently high carrier frequency and signal demodulation would be necessary. Another disadvantage of AC tachometers is that their output signal level depends on the supply voltage; hence, a stabilized voltage source that has a very small output impedance is necessary for accurate measurements.

1.5.8

Eddy Current Transducers

If a conducting (i.e., low-resistivity) medium is subjected to a ﬂuctuating magnetic ﬁeld, eddy currents are generated in the medium. The strength of eddy currents increases with the strength of the magnetic ﬁeld and the frequency of the magnetic ﬂux. This principle is used in eddy current proximity sensors. Eddy current sensors may be used as either dimensional gagging devices or high-frequency vibration sensors. A schematic diagram of an eddy current proximity sensor is shown in Figure 1.24(a). Unlike variableinductance proximity sensors, the target object of the eddy current sensor does not have to be made of a ferromagnetic material. A conducting target object is needed; however, a thin ﬁlm conducting material, such as household aluminum foil glued onto a nonconducting target object, is adequate. The probe head has two identical coils, which form two arms of an impedance bridge. The coil closer to the probe face is the active coil. The other coil is the compensating coil. It compensates for ambient changes, particularly thermal effects. The other two arms of the bridge consist of purely resistive elements (see Figure 1.24(b)). The bridge is excited by a radio-frequency voltage supply. The frequency can range from 1 to 100 MHz. This signal is generated from a radio-frequency converter (an oscillator) that is typically powered by a 20 V DC supply. In the absence of the target object, the output of the impedance bridge is zero, which corresponds to the balanced condition. When the target object is moved close to the sensor, eddy currents are generated in the conducting medium because of the radio-frequency magnetic ﬂux from the active coil. The magnetic ﬁeld of the eddy currents opposes the primary ﬁeld that generates these currents. Hence, the inductance of the active coil increases, creating an imbalance in the bridge. The resulting output from the bridge is an amplitude-modulated signal containing the radio-frequency carrier. This signal is demodulated by removing the carrier. The resulting signal (the modulating signal) measures the transient displacement (vibration) of the target object. Low-pass ﬁltering is used to remove the high-frequency leftover noise in the output signal once the carrier is removed. For large displacements, the output is not linearly related to the displacement. Furthermore, the sensitivity of the eddy current probe depends nonlinearly on the nature of the conducting medium, particularly its resistivity. For example, for low resistivities, sensitivity increases with resistivity; for high resistivities, sensitivity decreases with resistivity. A calibrating unit is usually available with commercial eddy current sensors to accommodate various target objects and nonlinearities. The gage factor is usually expressed in volts/millimeter. Note that eddy current probes can also be used to measure resistivity and surface hardness (which affects resistivity) in metals. The facial area of the conducting medium on the target object has to be slightly larger than the frontal area of the eddy current probe head. If the target object has a curved surface, its radius of curvature has to be at least four times the diameter of the probe. These are not serious restrictions, because the typical diameter of the probe head is about 2 mm. Eddy current sensors are medium-impedance devices; 1000 V output impedance is typical. Sensitivity is on the order of 5 V/mm. Since the carrier frequency is very high, eddy current devices are suitable for highly transient vibration measurements — for example,

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Vibration Instrumentation

1-41

Compensating Coil

Coaxial Cable Output vo

Calibrating Unit

Impedance Bridge Demodulator Low-Pass Filter

(Measurand) x Target Object Conducting Surface

RF Signal (100 MHz)

Active Coil

Radio Frequency Converter (Oscillator)

20 V DC Supply

(a)

C Compensating Coil

R2

R1 L

Bridge Output (to Demodulator)

RF Generator

~ L + ∆L

Active Coil C

R1

R2

(b) FIGURE 1.24

Eddy current proximity sensor: (a) schematic diagram; (b) impedance bridge.

bandwidths up to 100 kHz. Another advantage of an eddy current sensor is that it is a noncontacting device; there is no mechanical loading on the moving (target) object.

1.5.9

Variable-Capacitance Transducers

Capacitive or reactive transducers are commonly used to measure small transverse displacement such as vibrations, large rotations, and ﬂuid level oscillations. They may also be employed to measure angular velocities. In addition to analog capacitive sensors, digital (pulse-generating) capacitive tachometers are also available. Capacitance C of a two-plate capacitor is given by C¼

kA x

ð1:43Þ

where A is the common (overlapping) area of the two plates, x is the gap width between the two plates, and k is the dielectric constant which depends on dielectric properties of the medium between the two plates. A change in any one of these three parameters may be used in the sensing process.

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Vibration Monitoring, Testing, and Instrumentation

Schematic diagrams for measuring devices that use this feature are shown in Figure 1.25. In Figure 1.25(a), angular displacement of one of the plates causes a change in A. In Figure 1.25(b), a transverse displacement of one of the plates changes x. Finally, in Figure 1.25(c), a change in k is produced as the ﬂuid level between the capacitor plates changes. Liquid oscillations may be sensed in this manner. In all three cases, the associated change in capacitance is measured directly or indirectly, and is used to estimate the measurand. A popular method is to use a capacitance bridge circuit to measure the change in capacitance, in a manner similar to that by which an inductance bridge is used to measure changes in inductance. Other methods include measuring a change in such quantities as charge (using a charge ampliﬁer), voltage (using a high input-impedance device in parallel), and current (using a very low impedance device in series), as these changes will be results of the change in capacitance in a suitable circuit. An alternative method is to make the capacitor appart ﬃﬃﬃﬃ of an inductance–capacitance (L 2 C) oscillator circuit; the natural frequency of the oscillator ð1= LC Þ measures the capacitance. (Incidentally, this method may also be used to measure inductance.) Capacitance Bridge

DC Output vo Fixed Plate

Rotation

A

q Rotating Plate

(a)

Capacitance Bridge

Fixed Plate

Position x (b)

vo

Moving Plate (e.g., Diaphragm) Capacitance Bridge

vo

Fixed Plate

(c)

Level h

k

Liquid

FIGURE 1.25 Schematic diagrams of capacitive sensors: (a) capacitive rotation sensor; (b) capacitive displacement sensor; (c) capacitive liquid oscillation sensor.

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Vibration Instrumentation

1.5.9.1

1-43

Capacitive Displacement Sensors C = K/x

Consider the arrangement shown in Figure 1.25(a). Since the common area A is proportional to the angle of rotation u, Equation 1.43 may be written as C ¼ Ku

Cref

ð1:44Þ

where K is a sensor constant. This is a linear relationship between C and u. The capacitance may be measured by any convenient method. The sensor is linearly calibrated to give the angle of rotation. For the arrangement shown in Figure 1.25(b), the sensor relationship is

Supply Voltage vref

+

A −

+ Output vo −

+ Opamp

−

FIGURE 1.26 Inverting ampliﬁer circuit used to linearize the capacitive transverse displacement sensor.

K ð1:45Þ x The constant K has a different meaning here. Note that Equation 1.45 is a nonlinear relationship. A simple way to linearize this transverse displacement sensor is to use an inverting ampliﬁer, as shown in Figure 1.26. Note that Cref is a ﬁxed, reference capacitance. Since the gain of the operational ampliﬁer is very high, the voltage at the negative lead (point A) is zero for most practical purposes (because the positive lead is grounded). Furthermore, since the input impedance of the opamp is also very high, the current through the input leads is negligible. These are the two common assumptions used in opamp analysis. Accordingly, the charge balance equation for node point A is C¼

vref Cref þ vo C ¼ 0 Now, in view of Equation 1.45, we obtain the following linear relationship for the output voltage, vo, in terms of the displacement, x: v C ð1:46Þ vo ¼ 2 ref ref x K Hence, measurement of vo gives the displacement through linear calibration. The sensitivity of the device can be increased by increasing vref and Cref. The reference voltage could be DC as well as AC. With an AC reference voltage, the output voltage is a modulated signal that has to be demodulated to measure transient displacements. 1.5.9.2

Capacitive Angular Velocity Sensor

The schematic diagram for an angular velocity sensor that uses a rotating-plate capacitor is shown in Figure 1.27. Since the current sensor has negligible resistance, the voltage across the capacitor is almost equal to the supply voltage vref, which is constant. It follows that the current in the circuit is given by d dC i¼ ðCvref Þ ¼ vref dt dt

i Supply + Voltage − vref

+

Current Sensor

v~ – vref

C = Kq −

FIGURE 1.27 ocity sensor.

which, in view of Equation 1.44, may be expressed as du i ¼ Kvref dt

Rotating-plate capacitive angular vel-

ð1:47Þ

This is a linear relationship for the angular velocity in terms of the measured current i. Care must be exercised to guarantee that the current-measuring device does not interfere with the basic circuit.

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Vibration Monitoring, Testing, and Instrumentation

An advantage of capacitance transducers is that, because they are noncontacting devices, mechanical loading effects are negligible. There is some loading due to inertial and frictional resistance in the moving plate. This can be eliminated by using the moving object itself to function as the moving plate. Variations in the dielectric properties due to humidity, temperature, pressure, and impurities introduce errors. A capacitance bridge circuit can compensate for these effects. Extraneous capacitances, such as cable capacitance, can produce erroneous readings in capacitive sensors. This problem can be reduced by using a charge ampliﬁer to condition the sensor signal. Another drawback of capacitance displacement sensors is low sensitivity. For a transverse displacement transducer, the sensitivity is typically less than one picofarad (pF) per millimeter (1 pF ¼ 10212 F). This problem is not serious, because high supply voltages and ampliﬁer circuitry can be used to increase the sensor sensitivity. 1.5.9.3

Capacitance Bridge Circuit

Sensors that are based on the change in capacitance (reactance) will require some means of measuring that change. Furthermore, changes in capacitance that are not caused by a change in the measurand, for example, changes in humidity, temperature, and so on, will cause errors and should be compensated for. Both these goals are accomplished using a capacitance bridge circuit. An example is shown in Figure 1.28. In this circuit: Z2 ¼ reactance (i.e., capacitive impedance) of the capacitive sensor (of capacitance C2) ¼

Compensator Sensor Z1 Z2 AC Excitation v ref

Bridge Output vo

v − + Z3

Z4

Bridge Completion

FIGURE 1.28

A bridge circuit for capacitive sensors.

1 jvC2

Z1 ¼ reactance of the compensating capacitor C1 ¼

1 jvC1

Z4, Z3 ¼ bridge completing impedances (typically reactances) vref ¼ excitation AC voltage ¼ va sin vt vo ¼ bridge output ¼ vb sin(vt 2 f ) f ¼ phase lag of the output with respect to the excitation. Using the two assumptions for the opamp (potentials at the negative and positive leads are equal and the current through these leads is zero), we can write the current balance equations vref 2 v v 2v þ o ¼0 ðiÞ Z1 Z2 vref 2 v 02v þ ¼0 Z3 Z4

ðiiÞ

where v is the common voltage at the opamp leads. Next, eliminate v in Equation i and Equation ii to obtain vo ¼

ðZ4 =Z3 2 Z2 =Z1 Þ vref 1 þ Z4 =Z3

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ð1:48Þ

Vibration Instrumentation

It is noted that, when

1-45

Z2 Z ¼ 4 Z1 Z3

ð1:49Þ

the bridge output vo ¼ 0, and the bridge is said to be balanced. Since all capacitors in the bridge are similarly affected by ambient changes, a balanced bridge will maintain that condition even under ambient changes, unless the sensor reactance Z2 is changed due to the measurand itself. It follows that the ambient effects are compensated for (at least up to the ﬁrst order) by a bridge circuit. From Equation 1.48 it is clear that the bridge output due to a sensor change of dZ; starting from a balanced state, is given by vref dZ ð1:50Þ dvo ¼ 2 Z1 ð1 þ Z4 =Z3 Þ The amplitude and phase angle of dvo with respect to vref will determine dZ; assuming that Z1 and Z4/Z3 are known.

1.5.10

Piezoelectric Transducers

Some substances, such as barium titanate and Equivalent single-crystal quartz, can generate an electrical Capacitance charge and an associated potential difference when C they are subjected to mechanical stress or strain. This piezoelectric effect is used in piezoelectric transducers. Direct application of the piezoelectric Charge effect is found in pressure and strain measuring Source q devices, and many indirect applications also exist. They include piezoelectric accelerometers and velocity sensors, and piezoelectric torque sensors and force sensors. It is also interesting to note that FIGURE 1.29 Equivalent circuit representation of a piezoelectric materials deform when subjected to a piezoelectric sensor. potential difference (or charge). Some delicate test equipments (e.g., that used for vibration testing) use piezoelectric actuating elements (reverse piezoelectric action) to create ﬁne motions. Also, piezoelectric valves (e.g., ﬂapper valves), which are directly actuated using voltage signals, are used in pneumatic and hydraulic control applications and in ink-jet printers. Miniature stepper motors based on the reverse piezoelectric action are available. Consider a piezoelectric crystal in the form of a disc with two electrodes plated on the two opposite faces. It is essentially a charge source. Furthermore, since the crystal is a dielectric medium, this device has a capacitor, which may be modeled by a capacitance, C, as in Equation 1.43. Accordingly, a piezoelectric sensor may be represented as a charge source with a capacitive impedance in series (Figure 1.29), in an equivalent circuit. The impedance from the capacitor is given by Z¼

1 jvC

ð1:51Þ

As is clear from Equation 1.51, the output impedance of piezoelectric sensors is very high, particularly at low frequencies. For example, a quartz crystal may present an impedance of several megohms at 100 Hz, increasing hyperbolically with decreasing frequencies. This is one reason why piezoelectric sensors have a limitation on the useful lower frequency. The other reason is charge leakage. 1.5.10.1

Sensitivity

The sensitivity of a piezoelectric crystal may be represented either by its charge sensitivity or by its voltage sensitivity. Charge sensitivity is deﬁned as Sq ¼

›q ›F

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ð1:52Þ

1-46

Vibration Monitoring, Testing, and Instrumentation

where q denotes the generated charge and F denotes the applied force. For a crystal with surface area A, Equation 1.52 may be expressed as 1 ›q ð1:53Þ Sq ¼ A ›p where p is the stress (normal or shear) or pressure applied to the crystal surface. Voltage sensitivity Sv is given by the change in voltage due to a unit increment in pressure (or stress) per unit thickness of the crystal. Thus, in the limit, we have 1 ›v d ›p

ð1:54Þ

dq ¼ Cdv

ð1:55Þ

Sv ¼ where d denotes the crystal thickness. Now, since

by using Equation 1.43 for a capacitor element, the following relationship between charge sensitivity and voltage sensitivity is obtained: Sq ¼ kSv

ð1:56Þ

Note that k is the dielectric constant of the crystal capacitor, as deﬁned by Equation 1.43.

Example 1.3 A barium titanate crystal has a charge sensitivity of 150.0 picocoulombs per newton (pC/N). (Note: 1 pC ¼ 1 £ 10212 coulombs; coulombs ¼ farads £ volts.) The dielectric constant for the crystal is 1.25 £ 1028 farads per meter (F/m). What is the voltage sensitivity of the crystal?

Solution The voltage sensitivity of the crystal is given by Sv ¼ or

150:0 pC=N 150:0 £ 10212 C=N ¼ 28 1:25 £ 10 F=m 1:25 £ 1028 F=m

Sv ¼ 12:0 £ 1023 V·m=N ¼ 12:0 mV·m=N

The sensitivity of a piezoelectric element depends on the direction of loading. This is because the sensitivity depends on the crystal axis. Sensitivities of several piezoelectric materials along their most sensitive crystal axis are listed in Table 1.2. 1.5.10.2

Piezoelectric Accelerometer

Next we will discuss a piezoelectric motion transducer or vibration sensor, the piezoelectric accelerometer, in more detail. A piezoelectric velocity transducer is simply a piezoelectric accelerometer with a built-in integrating ampliﬁer in the form of a miniature integrated circuit. Accelerometers are acceleration-measuring devices. It is known from Newton’s Second Law that a force ( f ) is necessary to accelerate a mass (or inertia element), and its magnitude is given by the TABLE 1.2

Sensitivities of Several Piezoelectric Materials

Material Lead zirconate titanate (PZT) Barium titanate Quartz Rochelle salt

Charge Sensitivity Sq (pC/N) 110 140 2.5 275

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Voltage Sensitivity Sv (mV m/N) 10 6 50 90

Vibration Instrumentation

1-47

Accelaerometer Signal (dB)

product of mass (M) and acceleration (a). This product (Ma) is commonly termed inertia force. The rationale for this terminology is that if a force of magnitude Ma were applied to the accelerating mass in the direction opposing the acceleration, then the system could be analyzed using static equilibrium considerations. This is known as d’Alembert’s principle. The force that causes acceleration is itself a measure of the acceleration (mass is kept constant). Accordingly, mass can serve as a frontend element to convert acceleration into a force. This is the principle of operation of common accelerometers. There are many different types of accelerometers, ranging from strain gage devices to those that use electromagnetic induction. For example, force that causes acceleration can be converted into a proportional displacement using a spring element, and this displacement may be measured using a convenient displacement sensor. Examples of instruments of which operate this way are differential-transformer accelerometers, potentiometer accelerometers, and variablecapacitance accelerometers. Alternatively, the strain at a suitable location of a member that was deﬂected due to inertia force may be determined using a strain gage. This method is used in strain gage accelerometers. Vibrating-wire accelerometers use the accelerating force to create tension in a wire. The force is measured by detecting the natural frequency of vibration of the wire (which is proportional to the square root of tension). In servo force-balance (or null-balance) accelerometers, the inertia element is restrained from accelerating by detecting its motion and feeding back a force (or torque) to exactly cancel out the accelerating force (torque). This feedback force is determined, for instance, by knowing the motor current, and it is a measure of the acceleration. The advantages of piezoelectric accelerometers (also known as crystal accelerometers) over other types of accelerometers are their light weight and high-frequency response (up to about 1 MHz). However, piezoelectric transducers are inherently high-output-impedance devices that generate small voltages (on the order of 1 mV). For this reason, special impedance-transforming ampliﬁers (e.g., charge ampliﬁers) have to be employed to condition the output signal and to reduce loading error. A schematic diagram for a compressionSpring type piezoelectric accelerometer is shown in Direction of Sensitivity Figure 1.30. The crystal and the inertia mass are (Input) Inertia Mass restrained by a spring of very high stiffness. Piezoelectric Consequently, the fundamental natural freOutput Element v o quency or resonant frequency of the device becomes high (typically 20 kHz). This gives a Electrodes reasonably wide, useful range (typically up to 5 kHz). The lower limit of the useful range FIGURE 1.30 A compression-type piezoelectric (typically 1 Hz) is set by factors such as the accelerometer. limitations of the signal-conditioning systems, the mounting methods, the charge leakage in the Resonace piezoelectric element, the time constant of the charge-generating dynamics, and the signal-tonoise ratio. A typical frequency response curve for a piezoelectric accelerometer is shown in Figure 1.31. In compression-type crystal accelerometers, the inertia force is sensed as a compressive normal Useful Range stress in the piezoelectric element. There are also piezoelectric accelerometers that sense the inertia 1 5,000 20,000 force as a shear strain or tensile strain. For an Frequency (Hz) accelerometer, acceleration is the signal that is being measured (the measurand). Hence, accelerometer sensitivity is commonly expressed in FIGURE 1.31 A typical frequency-response curve for a terms of electrical charge per unit acceleration or piezoelectric accelerometer.

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voltage per unit acceleration (compare this with Equation 1.53 and Equation 1.54). Acceleration is measured in units of acceleration due to gravity (g), and charge is measured in picocoulombs (pC), which are units of 10212 coulombs (C). Typical accelerometer sensitivities are 10 pC/g and 5 mV/g. Sensitivity depends on the piezoelectric properties and on the mass of the inertia element. If a large mass is used, the reaction inertia force on the crystal will be large for a given acceleration, thus generating a relatively large output signal. A large accelerometer mass results in several disadvantages, however. In particular: 1. The accelerometer mass distorts the measured motion variable (mechanical loading effect). 2. A heavier accelerometer has a lower resonant frequency and, hence, a lower useful frequency range (Figure 1.31). For a given accelerometer size, improved sensitivity can be obtained by using the shear-strain conﬁguration. In this conﬁguration, several shear layers can be used (e.g., in a delta arrangement) within the accelerometer housing, thereby increasing the effective shear area and, hence, the sensitivity in proportion to the shear area. Another factor that should be considered in selecting an accelerometer is its cross-sensitivity or transverse sensitivity. Cross-sensitivity primarily results from manufacturing irregularities in the piezoelectric element, such as material unevenness and incorrect orientation of the sensing element. Cross-sensitivity should be less than the maximum error (a percentage) that is allowed for the device (typically 1%). The technique employed to mount the accelerometer to an object can signiﬁcantly affect the useful frequency range of the accelerometer. Some common mounting techniques are: 1. 2. 3. 4. 5.

Screw-in base Glue, cement, or wax Magnetic base Spring-base mount Hand-held probe

Drilling holes in the object can be avoided by using the second to ﬁfth methods, but the useful range can decrease signiﬁcantly when spring-base mounts or hand-held probes are used (giving a typical upper limit of 500 Hz). The ﬁrst two methods usually maintain the full useful range, whereas the magnetic attachment method reduces the upper frequency limit to some extent (typically 1.5 kHz). Piezoelectric signals cannot be read using low-impedance devices. The two primary reasons for this are: 1. High output impedance in the sensor results in small output signal levels and large loading errors. 2. The charge can quickly leak out through the load. 1.5.10.3

Charge Ampliﬁer

The charge ampliﬁer, which has a very high input impedance and a very low output impedance, is the commonly used signal-conditioning device for piezoelectric sensors. Clearly, the impedance at the charge ampliﬁer output is much smaller than the output impedance of the piezoelectric sensor. These impedance characteristics of a change ampliﬁer virtually eliminate loading error. Also, by using a charge ampliﬁer circuit with a large time constant, charge leakage speed can be decreased. For example, consider a piezoelectric sensor and charge ampliﬁer combination, as represented by the circuit in Figure 1.32. Let us examine how the charge leakage rate is slowed down by using this arrangement. Sensor capacitance, feedback capacitance of the charge ampliﬁer, and feedback resistance of the charge ampliﬁer are denoted by C, Cf ; and Rf ; respectively. The capacitance of the cable that connects the sensor to the charge ampliﬁer is denoted by Cc. For an opamp of gain, K, the voltage at its negative input is 2vo/K, where vo is the voltage at the ampliﬁer output. Note that the positive input of the opamp is grounded (zero potential). Current balance

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at point A gives q_ þ Cc

Rf

v_ o v_ v þ vo =K þ Cf v_ o þ o þ o ¼0 K K Rf

Cf

ð1:57Þ A

Since gain, K; is very large (typically 105 to 109) compared to unity, this differential equation may be approximated to dv dq Rf Cf o þ vo ¼ 2Rf dt dt

C q

ð1:58Þ

− vo /K

+

K

Output vo

Cc

Piezoelectric Cable Sensor

+

Charge Amplifier

−

Alternatively, instead of using Equation 1.57 it is possible to directly obtain Equation 1.58 from the two common assumptions (equal inverting and FIGURE 1.32 A piezoelectric sensor and charge noninverting lead potentials and zero lead cur- ampliﬁer combination. rents) for an opamp. In accordance with these assumptions the potential at the negative (inverting) lead is zero, as the positive lead is grounded. Also, as a result, the voltage across Cc is zero. Hence, the current balance at point A gives q_ þ

vo þ Cf v_ o ¼ 0 Rf

This is identical to Equation 1.58. The corresponding transfer function is vo ðsÞ Rf s ¼2 qðsÞ ½Rf Cf s þ 1

ð1:59Þ

where s is the Laplace variable. Now, in the frequency domain ðs ¼ jvÞ; we have vo ðjvÞ Rf jv ¼2 qðjvÞ ½Rf Cf jv þ 1

ð1:60Þ

Note that the output is zero at zero frequency ðv ¼ 0Þ: Hence, a piezoelectric sensor cannot be used for measuring constant (DC) signals. At very high frequencies, on the other hand, the transfer function approaches the constant value 21/Cf, which is the calibration constant for the device. From Equation 1.58 or Equation 1.59, which represent a ﬁrst-order system, it is clear that the time constant tc of the sensor-ampliﬁer unit is

t c ¼ Rf C f

ð1:61Þ

Suppose that the charge ampliﬁer is properly calibrated (by the factor 2 1/Cf ) so that the frequency transfer function (Equation 1.60) can be written as GðjvÞ ¼

jt c v ½jtc v þ 1

ð1:62Þ

Magnitude M of this transfer function is given by

tc v M ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 t cv 2 þ 1

ð1:63Þ

As v ! 1; note that M ! 1: Hence, at inﬁnite frequency, there is no error. Measurement accuracy depends on the closeness of M to 1. Suppose that we want the accuracy to be better than a speciﬁed value Mo. Accordingly, we must have

tc v pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ . M0 2 2 tc v þ 1

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ð1:64Þ

1-50

Vibration Monitoring, Testing, and Instrumentation

or M0 ﬃ tc v . qﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 M02

ð1:65Þ

If the required lower frequency limit is vmin, the time constant requirement is

tc .

M0 qﬃﬃﬃﬃﬃﬃﬃﬃﬃ ﬃ vmin 1 2 M02

ð1:66Þ

or Rf Cf .

M0 ﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃ vmin 1 2 M02

ð1:67Þ

It follows that a speciﬁed lower limit on the frequency of operation, for a speciﬁed level of accuracy, may be achieved by increasing the charge-ampliﬁer time constant (i.e., by increasing Rf, Cf, or both). For instance, an accuracy better than 99% is obtained if

tc v pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ . 0:99; or tc v . 7:0 2 t cv 2 þ 1 The minimum frequency of a transient signal that can tolerate this level of accuracy is vmin ¼ 7:0=tc : Now vmin can be set by adjusting the time constant.

1.6

Torque, Force, and Other Sensors

The forced vibrations in a mechanical system depend on the forces and torques (excitations) applied to the system. Also, the performance of the system may be speciﬁed in terms of forces and torques that are generated, as for machine-tool operations such as grinding, cutting, forging, extrusion, and rolling. Performance monitoring and evaluation, failure detection and diagnosis, and vibration testing may depend considerably on the accurate measurement of associated forces and torques. In mechanical applications such as parts assembly, slight errors in motion can generate large forces and torques. These observations highlight the importance of measuring forces and torques. The strain gage is a sensor that is commonly used in this context. There are numerous other types of sensors and transducers that are useful in the context of mechanical vibration. In this section, we will outline several of these sensors.

1.6.1

Strain Gage Sensors

Many types of force and torque sensors, as well as motion sensors such as accelerometers, are based on strain gage measurements. Hence, strain gages are very useful in vibration instrumentation. Although strain gages measure strain, the measurements can be directly related to stress and force. Note, however, that strain gages may be used in a somewhat indirect manner, using auxiliary front-end elements, to measure other types of variables, including displacement and acceleration. 1.6.1.1

Equations for Strain Gage Measurements

The change of electrical resistance in material when it is mechanically deformed is the property used in resistance-type strain gages. The resistance R of a conductor that has length ‘ and area of cross section A, is given by R¼r

‘

A

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ð1:68Þ

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where r denotes the resistivity of the material. Taking the logarithm of Equation 1.68, we attain log R ¼ log r þ logð‘=AÞ: Now, taking the differential, we obtain dR dr dð‘=AÞ ¼ þ R ‘=A r

ð1:69Þ

The ﬁrst term on the right-hand side of Equation 1.69 depends on the change in resistivity, and the second term represents deformation. It follows that the change in resistance comes from the change in shape as well as from the change in the resistivity of the material. For linear deformations, the two terms on the right-hand side of Equation 1.69 are linear functions of strain, 1; the proportionality constant of the second term, in particular, depends on Poisson’s ratio of the material. Hence, the following relationship can be written for a strain gage element: dR ¼ Ss 1 R

ð1:70Þ

The constant Ss is known as the sensitivity or gage factor of the strain gage element. The numerical value of this constant ranges from 2 to 6 for most metallic strain gage elements and from 40 to 200 for SC strain gages. These two types of strain gage will be discussed later. The change in resistance of a strain gage element, which determines the associated strain (Equation 1.70), is measured using a suitable electrical circuit. Resistance strain gages are based on resistance change due to strain, or the piezoresistive property of materials. Early strain gages were ﬁne metal ﬁlaments. Modern strain gages are manufactured primarily as metallic foil (for example, using the copper–nickel alloy known as constantan) or SC elements (e.g., silicon with trace impurity boron). They are manufactured by ﬁrst forming a thin ﬁlm (foil) of metal or a single crystal of SC material and then cutting it into a suitable grid pattern, either mechanically or by using photoetching (chemical) techniques. This process is much more economical and is more precise than making strain gages with metal ﬁlaments. The strain gage element is formed on a backing ﬁlm of electrically insulated material (e.g., plastic). This element is cemented onto the member whose strain is to be measured. Alternatively, a thin ﬁlm of insulating ceramic substrate is melted onto the measurement surface, on which the strain gage is mounted directly. The direction of sensitivity is the major direction of elongation of the strain gage element (Figure 1.33(a)). To measure strains in more than one direction, multiple strain gages (e.g., various rosette conﬁgurations) are available as single units. These units have more than one direction of sensitivity. Principal strains in a given plane (the surface of the object on which the strain gage is mounted) can be determined by using these multiple strain gage units. Typical foil-type strain gages produce a relatively large output signal. A large accelerometer mass results in several disadvantages, however. In particular: 1. The accelerometer mass distorts the measured motion variable (mechanical loading effect). 2. A heavier accelerometer has a lower resonant frequency and, hence, a lower useful frequency range (Figure 1.31). A direct way to obtain strain gage measurement is to apply a constant DC voltage across a seriesconnected strain gage element and a suitable resistor, and to measure the output voltage vo across the strain gage under open-circuit conditions using a voltmeter with high input impedance. It is known as a potentiometer circuit or ballast circuit (see Figure 1.34(a)). This arrangement has several weaknesses. Any ambient temperature variation will directly introduce some error because of associated change in the strain gage resistance and the resistance of the connecting circuitry. Also, measurement accuracy will be affected by possible variations in the supply voltage vref. Furthermore, the electrical loading error will be signiﬁcant unless the load impedance is very high. Perhaps, the most serious disadvantage of this circuit is that the change in signal due to strain is usually a very small percentage of the total signal level in the circuit output.

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 1.33

(a) Strain gage nomenclature; (b) typical foil-type strain gages; (c) a SC strain gage.

A more favorable circuit for use in strain gage measurements is the Wheatstone bridge, shown in Figure 1.34(b). One or more of the four resistors R1, R2, R3, and R4 in the circuit may represent strain gages. To obtain the output relationship for the Wheatstone bridge circuit, assume that the load impedance RL is very high. Hence, the load current, i; is negligibly small. Then, the potentials at nodes A and B are vA ¼

R1 R3 v and vB ¼ v ðR1 þ R2 Þ ref ðR3 þ R4 Þ ref

and the output voltage vo ¼ vA 2 vB is given by vo ¼

R1 R3 2 v ðR1 þ R2 Þ ðR3 þ R4 Þ ref

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ð1:71Þ

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By using straightforward algebra, we obtain

+

vo ¼

ðR1 R4 2 R2 R3 Þ v ðR1 þ R2 ÞðR3 þ R4 Þ ref

ð1:72Þ

When this output voltage is zero, the bridge is said to be “balanced.” It follows from Equation 1.72 that, for a balanced bridge R1 R ¼ 3 R2 R4

ð1:73Þ

DC Supply

+ vref

Output Signal vo −

−

Resistor Rc

(a) A

Note that Equation 1.73 is valid for any value of RL, not just for large RL, because when the bridge is balanced, current i will be zero, even for small RL. 1.6.1.2

Strain Gage R

R1

Bridge Sensitivity

+ i

R2

Strain gage measurements are calibrated with Load Output respect to a balanced bridge. When the strain RL vo gages in the bridge deform, the balance is upset. If one of the arms of the bridge has a variable R4 resistor, it can be changed to restore the balance. R3 − The amount of this change corresponds to the amount by which the resistance of the strain gages B changed, thereby measuring the applied strain. This is known as the null-balance method of − + strain measurement. This method is inherently vref (b) slow because of the time required to balance the bridge each time a reading is taken. Hence, the null-balance method is generally not suitable for FIGURE 1.34 (a) A potentiometer circuit (ballast dynamic (time-varying) measurements. This circuit) for strain gage measurements; (b) a Wheatstone approach to strain measurement can be sped up bridge circuit for strain gage measurements. by using servo balancing, whereby the output error signal is fed back into an actuator that automatically adjusts the variable resistance so as to restore the balance. A more common method, which is particularly suitable for making dynamic readings from a strain gage bridge, is to measure the output voltage resulting from the imbalance caused by the deformation of active strain gages in the bridge. To determine the calibration constant of a strain gage bridge, the sensitivity of the bridge output to changes in the four resistors in the bridge should be known. For small changes in resistance, this may be determined using the differential relation (or, equivalently, the ﬁrstorder approximation for the Taylor series expansion): dvo ¼

4 X ›vo dR › Ri i i¼1

ð1:74Þ

The partial derivatives are obtained directly from Equation 1.71. Speciﬁcally,

›vo R2 ¼ v ›R1 ðR1 þ R2 Þ2 ref

ð1:75Þ

R1 ›vo v ¼ ðR1 þ R2 Þ2 ref ›R2

ð1:76Þ

R4 ›vo v ¼ ðR1 þ R4 Þ2 ref ›R3

ð1:77Þ

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Vibration Monitoring, Testing, and Instrumentation

›vo R3 ¼ v › R4 ðR3 þ R4 Þ2 ref

ð1:78Þ

The required relationship is obtained by substituting the equations from Equation 1.75 to Equation 1.78 into Equation 1.74; thus dvo ðR dR 2 R1 dR2 Þ ðR dR 2 R3 dR4 Þ ¼ 2 1 2 4 3 vref ðR1 þ R2 Þ2 ðR3 þ R4 Þ2

ð1:79Þ

This result is subject to Equation 1.73, because changes are measured from the balanced condition. Note that, from Equation 1.79, if all four resistors are identical (in value and material), resistance changes due to ambient effects cancel out among the ﬁrst-order terms ðdR1 ; dR2 ; dR3 ; dR4 Þ; producing no net effect on the output voltage from the bridge. Closer examination of Equation 1.79 will reveal that only the adjacent pairs of resistors (e.g., R1 with R2 and R3 with R4) have to be identical in order to achieve this environmental compensation. Even this requirement can be relaxed. Compensation is achieved if R1 and R2 have the same temperature coefﬁcient and if R3 and R4 have the same temperature coefﬁcient. 1.6.1.3

The Bridge Constant

Numerous activating combinations of strain gages are possible in a bridge circuit. For example, there might be tension in R1 and compression in R2, as in the case of two strain gages mounted symmetrically at 458 about the axis of a shaft in torsion. In this manner, the overall sensitivity of a strain gage bridge can be increased. It is clear from Equation 1.79 that, if all four resistors in the bridge are active, the best sensitivity is obtained if all four differential terms have the same sign, for example, when R1 and R4 are in tension and R2 and R3 are in compression. If more than one strain gage is active, the bridge output may be expressed as dvo dR ¼k vref 4R

Axial Gage

ð1:80Þ 1

where bridge output in the general case k¼ bridge output if only one strain gage is active This constant is known as the bridge constant. The larger the bridge constant is, the better the sensitivity of the bridge.

Cross Section of Sensing Member (a)

4

2

1 +

Example 1.4 A strain gage load cell (force sensor) consists of four identical strain gages, which form a Wheatstone bridge and are mounted on a rod that has a square cross section. One opposite pair of strain gages is mounted axially and the other pair is mounted in the transverse direction, as shown in Figure 1.35(a). To maximize the bridge sensitivity, the strain gages are connected to the bridge as shown in Figure 1.35(b). Determine the bridge constant k in terms of Poisson’s ratio y of the rod material.

Transverse Gage

2

3

vo − 3

(b)

4

−

vref

+

FIGURE 1.35 A strain-gage force sensor: (a) mounting conﬁguration; (b) bridge circuit.

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Solution Suppose that dR1 ¼ dR: Then, for the given conﬁguration, we have dR2 ¼ 2y dR dR3 ¼ 2y dR dR4 ¼ dR Note that, from the deﬁnition of Poisson’s ratio, transverse strain ¼ (2 y ) £ longitudinal strain. Now, it follows from Equation 1.79 that dvo dR ¼ 2ð1 þ y Þ ð1:81Þ vref 4R according to which the bridge constant is given by k ¼ 2ð1 þ y Þ 1.6.1.4

The Calibration Constant

The calibration constant, C; of a strain gage bridge relates the strain that is measured to the output of the bridge. Speciﬁcally, dvo ¼ C1 vref

ð1:82Þ

Now, in view of Equation 1.70 and Equation 1.80, the calibration constant may be expressed as C¼

k S 4 s

ð1:83Þ

where k is the bridge constant and Ss is the sensitivity or gage factor of the strain gage. Ideally, the calibration constant should remain constant over the measurement range of the bridge (i.e., independent of strain 1 and time t) and should be stable with respect to ambient conditions. In particular, there should not be any creep, nonlinearities such as hysteresis, or thermal effects.

W

(a)

(b)

Example 1.5 A schematic diagram of a strain gage accelerometer is shown in Figure 1.36(a). A point mass of weight W is used as the acceleration sensing element, and a light cantilever with a rectangular cross section, which is mounted inside the accelerometer casing, converts the inertia force of the mass into a strain. The maximum bending strain at the root of the cantilever is measured using four identical active SC strain gages. Two of the strain gages (A and B) are mounted axially on the top surface of the cantilever, and the remaining two (C and D) are mounted on the bottom surface, as shown in Figure 1.36(b). In order to maximize the sensitivity of the accelerometer, indicate the

A

C + dvo −

D

(c)

B

−

vref

+

FIGURE 1.36 A strain gage accelerometer: (a) schematic diagram; (b) strain gage mounting conﬁguration; (c) bridge connections.

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Vibration Monitoring, Testing, and Instrumentation

manner in which the four strain gages (A, B, C, and D) should be connected to a Wheatstone bridge circuit. What is the bridge constant of the resulting circuit? Obtain an expression relating the applied acceleration a (in units of g, which denotes acceleration due to gravity) to the bridge output dvo (measured using a bridge balanced at zero acceleration) in terms of the following parameters: W ¼ weight of the seismic mass at the free end of the cantilever element E ¼ Young’s modulus of the cantilever ‘ ¼ length of the cantilever b ¼ cross-sectional width of the cantilever h ¼ cross-sectional height of the cantilever Ss ¼ sensitivity (gage factor) of each strain gage vref ¼ supply voltage to the bridge If W ¼ 0.02 lb, E ¼ 10 £ 106 lbf/in.2, ‘ ¼ 1 in., b ¼ 0.1 in., h ¼ 0.05 in., Ss ¼ 200, and vref ¼ 20 V, determine the sensitivity of the accelerometer in mV/g. If the yield strength of the cantilever element is 10 £ l03 lbf/in.2, what is the maximum acceleration that could be measured using the accelerometer? Is the cross-sensitivity (i.e., the sensitivity in the two directions orthogonal to the direction of sensitivity shown in Figure 1.36(a)) small given your arrangement of the strain gage bridge? Explain. Note: For a cantilever subjected to force F at the free end, the maximum stress at the root is given by

s¼

6F ‘ bh2

ð1:84Þ

with the present notation.

Solution The bridge sensitivity is maximized by connecting the strain gages A, B, C, and D to the bridge as shown in Figure 1.36(c). This follows from Equation 1.79, noting that the contributions from all four strain gages are positive when dR1 and dR4 are positive and dR2 and dR3 are negative. The bridge constant for the resulting arrangement is k ¼ 4. Hence, from Equation 1.80, we have dvo dR ¼ vref R or, from Equation 1.82 and Equation 1.83 dvo ¼ Ss 1 vref Also, 1¼

s 6F ‘ ¼ E Ebh2

F¼

W x€ ¼ Wa g

where F denotes the inertia force

Note that x€ is the acceleration in the direction of sensitivity and x€ =g ¼ a is the acceleration in units of g. Thus, 1¼

6W ‘ a Ebh2

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ð1:85Þ

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or dvo ¼

6W ‘ Sv a Ebh2 s ref

ð1:86Þ

Now, with the given values dvo 6 £ 0:02 £ 1 £ 200 £ 20 ¼ V=g ¼ 0:192 V=g ¼ 192 mV=g a 10 £ 106 £ 0:1 £ ð0:05Þ2 1 1 dvo 0:192 ¼ ¼ strain=g a Ss vref a 200 £ 20 yield strain ¼

yield strength 10 £ 103 ¼ ¼ 1 £ 1023 strain E 10 £ 106

Hence, number of gs to yielding ¼

1 £ 1023 g ¼ 20:8g 48 £ 1026

Cross-sensitivity comes from accelerations in the two directions (y and z) orthogonal to the direction of sensitivity (x). In the lateral (y) direction, the inertia force causes lateral bending. This will produce equal tensile (or compressive) strains in B and D, and equal compressive (or tensile) strains in A and C. According to the bridge circuit, we see that these contributions cancel each other. In the axial (z) direction, the inertia force causes equal tensile (or compressive) stresses in all four strain gages. These also will cancel out, as is clear from the following relationship for the bridge: dvo ðR dR 2 RA dRC Þ ðR dR 2 RD dRB Þ ¼ C A 2 B D vref ðRA þ RC Þ2 ðRD þ RB Þ2

ð1:87Þ

with RA ¼ RB ¼ RC ¼ RD ¼ R which gives dvo ðdRA 2 dRC 2 dRD þ dRB Þ ¼ vref 4R

ð1:88Þ

It follows that this arrangement is good with respect to cross-sensitivity problems. 1.6.1.5

Data Acquisition

As noted earlier, the two common methods of measuring strains using a Wheatstone bridge circuit are (1) the null-balance method and (2) the imbalance output method. One possible scheme for using the ﬁrst method is shown in Figure 1.37(a). In this particular arrangement, two bridge circuits are used. The active bridge contains the active strain gages, dummy gages, and bridge-completion resistors. The reference bridge has four resistors, one of which is micro-adjustable, either manually or automatically. The output from the each of the two bridges is fed into a difference ampliﬁer, which provides an ampliﬁed difference of the two signals. This error signal is indicated on a null detector, such as a galvanometer. Initially, both bridges are balanced. When the measurement system is in use, the active gages are subjected to the strain that is being measured. This upsets the balance, giving a net output that is indicated on the null detector. In manual operation of the null-balance mechanism, the resistance knob in the reference bridge is adjusted carefully until the galvanometer indicates a null reading. The knob can be calibrated to indicate the measured strain directly. In servo operation, which is much faster than the manual method, the error signal is fed into an actuator that automatically adjusts the variable resistor in the reference bridge until the null balance is achieved. Actuator movement measures the strain. For measuring dynamic strains in vibrating systems, either the servo null-balance method or the imbalance output method should be employed. A schematic diagram for the imbalance output method is

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Vibration Monitoring, Testing, and Instrumentation

Strain

Active Bridge

Difference Amplifier

Power Supply

Null Detector (Galvanometer)

Reference Bridge

Null Adjustment (Manual or Automatic) (a) Dynamic Strain

(b)

FIGURE 1.37

Bridge Circuit

Error Feedback (Manual or Servo) Amplifier/ Filter

Calibration

Strain Measurement

Power Supply

Strain gage bridge measurement: (a) null-balance method; (b) imbalance output method.

shown in Figure 1.37(b). In this method, the output from the active bridge is directly measured as a voltage signal and calibrated to provide the measured strain. An AC bridge may be used, where the bridge is powered by an AC voltage. The supply frequency should be about ten times the maximum frequency of interest in the dynamic strain signal (bandwidth). A supply frequency on the order of 1 kHz is typical. This signal is generated by an oscillator and is fed into the bridge. The transient component of the output from the bridge is very small (typically less than 1 mV and sometimes a few microvolts). This signal must be ampliﬁed, demodulated (especially if the signals are transient), and ﬁltered to provide the strain reading. The calibration constant of the bridge should be known in order to convert the output voltage to strain. Strain gage bridges powered by DC voltages are very common. They have the advantages of portability and simplicity with regard to necessary circuitry. The advantages of AC bridges include improved stability (reduced drift), improved accuracy, and reduced power consumption. 1.6.1.6

Accuracy Considerations

Foil gages are available with resistances as low as 50 V and as high as several kilohms. The power consumption of the bridge decreases with increased resistance. This has the added advantage of decreased heat generation. Bridges with a high range of measurement (e.g., a maximum strain of 0.01 m/m) are available. The accuracy depends on the linearity of the bridge, environmental (particularly temperature) effects, and mounting techniques. For example, a calibration error occurs in the case of zero shift, due to the strains produced when the cement that is used to mount the strain gage dries. Creep will introduce errors during static and low-frequency measurements. Flexibility and hysteresis of the bonding cement will bring about errors during high-frequency strain measurements. Resolutions on the order of 1 mm/m (i.e., one microstrain) are common. The cross-sensitivity should be small (say, less than 1% of the direct sensitivity). Manufacturers usually provide the values of the cross-sensitivity factors for their strain gages. This factor, when multiplied by the cross strain present in a given application, gives the error in the strain reading due to cross-sensitivity.

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Often, measurements of strains in moving members are needed, for example, in real-time monitoring and failure detection in machine tools. If the motion is small or the device has a limited stroke, strain gages mounted on the moving member can be connected to the signal-conditioning circuitry and the power source using coiled ﬂexible cables. For large motions, particularly in rotating shafts, some form of commutating arrangement must be used. Slip rings and brushes are commonly used for this purpose. When AC bridges are used, a mutual-induction device (rotary transformer) can be used, with one coil located on the moving member and the other coil stationary. To accommodate and compensate for errors caused by commutation (e.g., losses and glitches in the output signal), it is desirable to place all four arms of the bridge, rather than just the active arms, on the moving member. 1.6.1.7

Semiconductor Strain Gages

In some low-strain applications (e.g., dynamic Conductor torque measurement), the sensitivity of foil gages Ribbons is not adequate to produce an acceptable strain gage signal. SC strain gages are particularly Single Crystal of useful in such situations. The strain element of a Semiconductor SC strain gage is made of a single crystal of piezoresistive material such as silicon, doped with a trace impurity such as boron. A typical construcGold Leads tion is shown in Figure 1.38. The sensitivity (gage factor) of a SC strain gage is about two orders of magnitude higher than that of a metallic Phenolic Glass foil gage (typically, 40 to 200). The resistivity is Backing Plate also higher, providing reduced power consumption and heat generation. Another advantage of SC FIGURE 1.38 Details of a semiconductor strain gage. strain gages is that they deform elastically until fracture. In particular, mechanical hysteresis is negligible. Furthermore, they are smaller and lighter, providing less cross-sensitivity, reduced distribution error (i.e., improved spatial resolution), and negligible error due to mechanical loading. The maximum strain that is measurable using a SC strain gage is typically 0.003 m/m (i.e., 3000 m1). Strain gage resistance can be several hundred ohms (typically, 120 V or 350 V). There are several disadvantages associated with SC strain gages, however, which can be interpreted as advantages of foil gages. Undesirable characteristics of SC gages include the following: 1. The strain–resistance relationship is more nonlinear. 2. They are brittle and difﬁcult to mount on curved surfaces. 3. The maximum strain that can be measured is an order of magnitude smaller (typically, less than 0.01 m/m). 4. They are more costly. 5. They have a much higher temperature sensitivity. The ﬁrst disadvantage is illustrated in Figure 1.39. There are two types of SC strain gages: the P-type and the N-type. In P-type strain gages, the direction of sensitivity is along the ð1; 1; 1Þ crystal axis, and the element produces a “positive” (P) change in resistance in response to a positive strain. In N-type strain gages, the direction of sensitivity is along the ð1; 0; 0Þ crystal axis, and the element responds with a “negative” (N) change in resistance to a positive strain. In both types, the response is nonlinear and can be approximated by the quadratic relationship dR ¼ S1 1 þ S2 1 2 R

ð1:89Þ

The parameter S1 represents the linear sensitivity, which is positive for P-type gages and negative for N-type gages. Its magnitude is usually somewhat larger for P-type gages, thereby providing

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better sensitivity. The parameter S2 represents the degree of nonlinearity, which is usually positive for both types of gage. Its magnitude, however, is typically a little smaller for P-type gages. It follows that P-type gages are less nonlinear and have higher strain sensitivities. The nonlinear relationship given by Equation 1.89 or the nonlinear characteristic curve (Figure 1.39) should be used when measuring moderate to large strains with SC strain gages. Otherwise, the nonlinearity error will be excessive. 1.6.1.8

Force and Torque Sensors

Torque and force sensing is useful in vibration applications, including the following:

Resistance Change δR R

0.4 0.3 0.2

me = 1 Microstrain = Strain of 1×10−6

0.1 −3

−2

−1 −0.1

1

2 Strain

3

×103 me

−0.2 −0.3

(a)

Resistance Change

1. In vibration control of machinery where a 0.4 small motion error can cause large damagδR 0.3 ing forces or performance degradation. R 0.2 2. In high-speed vibration control when 0.1 motion feedback alone is not fast enough (here, force feedback and feedforward force −3 −2 −1 1 2 3 3 control can be used to improve the accuracy Strain ×10 me −0.1 and bandwidth). −0.2 3. In vibration testing, monitoring, an diag−0.3 nostic applications, where torque and force (b) sensing can detect, predict, and identify abnormal operation, malfunction, com- FIGURE 1.39 Nonlinear behavior of a semiconductor ponent failure, or excessive wear (e.g., in (silicon/boron) strain gage: (a) a P-type gage; (b) an monitoring machine tools such as milling N-type gage. machines and drills). 4. In experimental modal analysis where both excitation forces and response motioning may be needed to experimentally determine the system model. In most applications, torque (or force) is sensed by detecting either an effect or the cause of torque (or force). There are also methods for measuring torque (or force) directly. Common methods of torque sensing include the following: 1. Measuring the strain in a sensing member between the drive element and the driven load, using a strain gage bridge. 2. Measuring the displacement in a sensing member (as in the ﬁrst method), either directly, using a displacement sensor, or indirectly, by measuring a variable, such as magnetic inductance or capacitance, that varies with displacement. 3. Measuring the reaction in the support structure or housing (by measuring a force) and the associated lever arm length. 4. In electric motors, measuring the ﬁeld or armature current that produces motor torque; in hydraulic or pneumatic actuators, measuring the actuator pressure. 5. Measuring the torque directly, for example, using piezoelectric sensors. 6. Employing the servo method to balance the unknown torque with a feedback torque generated by an active device (say, a servomotor) whose torque characteristics are known precisely. 7. Measuring the angular acceleration in a known inertia element when the unknown torque is applied.

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Note that force sensing may be accomplished by essentially the same techniques. Some types of force sensor (e.g., the strain gage force sensor) have been introduced before. Now, we will limit our discussion primarily to torque sensing. The extension of torque-sensing techniques to the task of force sensing is somewhat straightforward. 1.6.1.9

Strain Gage Torque Sensors

The most straightforward method of torque sensing is to connect a torsion member between the drive unit and the load-in series, and to measure the torque in the torsion member. If a circular shaft (solid or hollow) is used as the torsion member, the torque–strain relationship is relatively simple. A complete development of the relationship is found in standard textbooks on elasticity, solid mechanics, or strength of materials. With reference to Figure 1.40, it can be shown that the torque, T, may be expressed in terms of the direct strain, 1; on the shaft surface along a principal stress direction (i.e., at 458 to the shaft axis) as 2GJ 1 r

T¼

ð1:90Þ

where G ¼ shear modulus of the shaft material, J ¼ polar moment of area of the shaft, and r ¼ shaft radius (outer). This is the basis of torque sensing using strain measurements. Using the general bridge Equation 1.82 along with Equation 1.83 in Equation 1.90, we can obtain torque, T, from bridge output, dvo: T¼

8GJ dvo kSs r vref

ð1:91Þ

where Ss is the gage factor (or sensitivity) of the strain gages. The bridge constant, k, depends on the number of active strain gages used. Strain gages are assumed to be mounted along a principal direction. Circular Shaft (Solid)

rmax r

(a)

Torque T

t

T t

x 45° (b)

tmax

T

y

C −s

T

r

s Shear Stress t B

(c)

s

t

0

A B C D

= Stress Along Principal Direction x = Circumferential Stress = Stress Along Principal Direction y = Axial (Longitudinal) Stress

A s Tensile Stress

−t D

FIGURE 1.40 (a) Linear distribution of shear stress in a circular shaft under pure torsion; (b) pure shear state of stress and principal directions x and y; (c) Mohr’s circle.

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2 1

2

1

2

1 R1

2

2

1

1

T Bridge Constant (k): Axial Loads Compensated: Bending Loads Compensated:

FIGURE 1.41

2 Yes Yes (a)

3

4

2

1

3

4

T 2 Yes Yes (b)

T 4 Yes Yes (c)

R2 + dvo − R4

R3

vref Strain Gage Bridge

Strain gage conﬁgurations for a circular shaft torque sensor.

Three possible conﬁgurations are shown in Figure 1.41. In conﬁgurations (a) and (b), only two strain gages are used, and the bridge constant, k, is equal to 2. Note that both axial loads and bending are compensated with the given conﬁgurations because resistance in both gages will be changed by the same amount (the same sign and same magnitude) that cancels out, up to ﬁrst order, for the bridge circuit connection shown in Figure 1.41. Conﬁguration (c) has two pairs of gages, mounted on the two opposite surfaces of the shaft. The bridge constant is doubled in this conﬁguration, and here again, the sensor clearly selfcompensates for axial and bending loads up to ﬁrst order ½OðdRÞ : For a circular-shaft torque sensor that uses SC strain gages, design criteria for obtaining a suitable value for the polar moment of area (J) are listed in Table 1.3. Note that f is a safety factor. Although the manner in which strain gages are conﬁgured on a torque sensor can be exploited to compensate for cross-sensitivity effects arising from factors such as tensile and bending loads, it is advisable to use a torque-sensing element that inherently possesses low sensitivity to these factors that cause error in a torque measurement. A tubular torsion element is convenient for analytical purposes because of the simplicity of the associated expressions for design parameters. Unfortunately, such an TABLE 1.3

Design Criteria for a Strain Gage Torque-Sensing Element Criterion

Speciﬁcation

Strain capacity of strain gage element

1max

Strain gage nonlinearity

Np ¼

Governing Formula for the Polar Moment of Area (J) $

fr Tmax 2G 1max

Max strain error £ 100% Strain range

$

25frS2 Tmax GS1 Np

Sensor sensitivity (output voltage)

vo ¼ Ka dvo where Ka ¼ transducer gain

#

Ka kSs rvref Tmax 8G vo

Sensor stiffness (system bandwidth and gain)

K¼

$

L K G

Torque Twist angle

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element is not very rigid to bending and tensile Strain loading. Alternative shapes and structural arrangeGage ments have to be considered if inherent rigidity (insensitivity) to cross-loads is needed. FurtherA more, a tubular element has the same strain at all locations on the element surface. This does not provide a choice with respect to mounting locations of strain gages in order to maximize (a) the torque sensor sensitivity. Another disadvantage Connected to of the basic tubular element is that the surface is Drive Member curved; therefore, much care is needed in mountA ing fragile SC gages, which could be easily damaged by even slight bending. Hence, a sensor Connected to element that has ﬂat surfaces to mount the strain Driven Member gages would be desirable. A torque-sensing element that has the aforementioned desirable A characteristics (i.e., inherent insensitivity to crossloading, nonuniform strain distribution on the surface, and the availability of ﬂat surfaces to A = Torque Sensing (b) Elements mount strain gages) is shown in Figure 1.42. Note that two sensing elements are connected radially between the drive unit and the driven member. FIGURE 1.42 Use of a bending element in torque The sensing elements undergo bending while sensing: (a) sensing element; (b) element conﬁguration. transmitting a torque between the driver and the driven member. Bending strains are measured at locations of high sensitivity and are taken to be proportional to the transmitted torque. Analytical determination of the calibration constant is not easy for such complex sensing elements, but experimental determination is straightforward. Note that the strain gage torque sensor measures the direction as well as the magnitude of the torque transmitted through it. 1.6.1.10

Deﬂection Torque Sensors

Instead of measuring strain in the sensor element, the actual deﬂection (twisting or bending) can be measured and used to determine torque, through a suitable calibration constant. For a circular-shaft (solid or hollow) torsion element, the governing relationship is given by T¼

GJ u L

ð1:92Þ

The calibration constant GJ/L must be small in order to achieve high sensitivity. This means that the element stiffness should be low. This will limit the bandwidth (which measures speed of response) and gain (which determines steady-state error) of the overall system. The twist angle, u, is very small (e.g., a fraction of a degree) in systems with high bandwidth. This requires very accurate measurement of u in order to determine the torque T. A type of displacement sensor that could be used is described as follows. Two ferromagnetic gear wheels are splined at two axial locations of the torsion element. Two stationary proximity probes of the magnetic induction type (selﬁnduction or mutual induction) are placed radially, facing the gear teeth, at the two locations. As the shaft rotates, the gear teeth change the ﬂux linkage of the proximity sensor coils. The resulting output signals of the two probes are pulse sequences, shaped somewhat like sine waves. The phase shift of one signal with respect to the other determines the relative angular deﬂection of one gear wheel with respect to the other, assuming that the two probes are synchronized under no-torque conditions. Both the magnitude and the direction of the transmitted torque are determined using this method. A 3608 phase shift corresponds to a relative deﬂection by an integer multiple of the gear pitch. It follows that deﬂections less than half the pitch can be measured

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without ambiguity. Assuming that the output signals of the two probes are sine waves (narrow-band ﬁltering can be used to achieve this), the phase shift will be proportional to the angle of twist, u. 1.6.1.11

Variable-Reluctance Torque Sensor

A torque sensor that is based on the sensor element deformation and that does not require a contacting commutator is a variable-reluctance device that operates like a differential transformer (RVDT or LVDT). The torque-sensing element is a ferromagnetic tube that has two sets of slits, typically oriented along the two principal stress directions of the tube (458) under torsion. When a torque is applied to the torsion element, one set of gaps closes and the other set opens as a result of the principal stresses normal to the slit axes. Primary and secondary coils are placed around the slitted tube, and they remain stationary. One segment of the secondary coil is placed around one set of slits, and the second segment is placed around the other, perpendicular, set. The primary coil is excited by an AC supply, and the induced voltage, vo, in the secondary coil is measured. As the tube deforms, it changes the magnetic reluctance in the ﬂux linkage path, thus changing the induced voltage. The two segments of the secondary coil should be connected so that the induced voltages are absolutely additive (algebraically subtractive), because one voltage increases and the other decreases, to obtain the best sensitivity. The output signal should be demodulated, by removing the carrier frequency component, to measure transient torques effectively. Note that the direction of torque is given by the sign of the demodulated signal. 1.6.1.12

Reaction Torque Sensors

The foregoing methods of torque sensing use a sensing element that is connected between the drive member and the driven member. A major drawback of such an arrangement is that the sensing element modiﬁes the original system in an undesirable manner, particularly by decreasing the system stiffness and adding inertia. Not only will the overall bandwidth of the system decrease, but the original torque will also be changed (mechanical loading) because of the inclusion of an auxiliary sensing element. Furthermore, under dynamic conditions, the sensing element will be in motion, thereby making the torque measurement more difﬁcult. The reaction method of torque sensing eliminates these problems to a large degree. This method can be used to measure torque in a rotating machine. The supporting structure (or housing) of the rotating machine (e.g., a motor, pump, compressor, turbine, or generator) is cradled by releasing its ﬁxtures, and the effort necessary to keep the structure from moving is measured. A schematic representation of the method is shown in Figure 1.43(a). Ideally, a lever arm is mounted on the cradled housing, and the force required to ﬁx the housing is measured using a force sensor (load cell). The reaction torque on the housing is given by TR ¼ FR ·L

ð1:93Þ

where FR ¼ reaction force measured using load cell L ¼ lever arm length Alternatively, strain gages or other types of force sensors could be mounted directly at the ﬁxture locations (e.g., at the mounting bolts) of the housing to measure the reaction forces without cradling the housing. Then, the reaction torque is determined with a knowledge of the distance of the ﬁxture locations from the shaft axis. The reaction-torque method of torque sensing is widely used in dynamometers (reaction dynamometers) that determine the transmitted power in rotating machinery through torque and shaft speed measurements. A drawback of reaction-type torque sensors can be explained using Figure 1.43(b). A motor with rotor inertia, J, which rotates at angular acceleration, u€; is shown. By Newton’s Third Law (action equals reaction), the electromagnetic torque generated at the rotor of the motor, Tm, and the frictional torques, Tf1 and Tf2, will be reacted back onto the stator and housing. By applying

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Motor Housing (Stator)

1-65

L

Frictionless Bearing

F Lever Arm

Force Sensor (Load Cell)

FR

(a) Motor Torque Tm

Reaction Torque TR Frictional Torque Tf1 .. q

Tm Tf1

Rotor J

Stator Housing Frictional Torque Tf 2 Tf 2

Load Torque TL

To Load

Bearings (b) FIGURE 1.43 (a) Schematic representation of a reaction torque sensor setup (reaction dynamometer); (b) various torque components.

Newton’s Second Law to the motor rotor and the housing combination, we obtain TL ¼ TR 2 J u€

ð1:94Þ

Note that TL is the variable which must be measured. Under accelerating or decelerating conditions, the reaction torque, Tr, is not equal to the actual torque, TL, that is transmitted. One method of compensating for this error is to measure the shaft acceleration, compute the inertia torque, and adjust the measured reaction torque using this inertia torque. Note that the frictional torque in the bearings does not enter the ﬁnal Equation 1.94. This is an advantage of this method.

1.6.2

Miscellaneous Sensors

Motion and force/torque sensors of the types described thus far are widely used in vibration instrumentation. Several other types of sensors are also useful. A few of them are indicated now. 1.6.2.1

Stroboscope

Consider an object that executes periodic motions such as vibrations or rotations in a fairly dark environment. Suppose that a light is ﬂashed at the object at the same frequency as the moving object. Since the object completes a full cycle of motion during the time period between two adjacent ﬂashes,

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the object will appear to be stationary. This is the principle of operation of a stroboscope. The main components of a stroboscope are a high-intensity strobe lamp and circuitry to vary the frequency of the electrical pulse signal that energizes the lamp. The ﬂashing frequency may be varied either manually using a knob or according to the frequency of an external periodic signal (trigger signal) that is applied to the stroboscope. It is clear that by synchronizing the stroboscope with a moving (vibrating, rotating) object so that the object appears stationary, and then noting the ﬂashing (strobe) frequency, the frequency of vibration or speed of rotation of the object can be measured. In this sense, the stroboscope is a noncontacting vibration frequency sensor or a tachometer (rotating speed sensor). Note that the object appears stationary for any integer multiple of the synchronous ﬂashing frequency. Hence, once the strobe is synchronized with the moving object, it is good practice to check whether the strobe also synchronizes at an integer fraction of that ﬂashing frequency. (Typically, trying 1/2, 1/3, 1/5, and 1/7 the original synchronous frequency is adequate.) The lowest synchronous frequency thus obtained is the correct speed (frequency) of the object. Since the frequency of visual persistence of a human is about 15 Hz, the stationary appearance will not be possible using a stroboscope below this frequency. Hence, the lowfrequency limit for a stroboscope is about 15 Hz. In addition to serving as a sensor for vibration frequency and rotating speed, the stroboscope has many other applications. For example, by maintaining the strobe (ﬂashing) frequency close (but not equal) to the object frequency, the object will appear to move very slowly. In this manner, visual inspection of objects that execute periodic motions at high speed is possible. Also, stroboscopes are widely used in dynamic balancing of rotating machinery. In this case, it is important to measure the phase angle of the resultant imbalance force with respect to a coordinate axis (direction) that is ﬁxed to the rotor. Suppose that a radial line is marked on the rotor. If we synchronize a stroboscope with the rotor such that the marked line appears not only stationary but also oriented in a ﬁxed direction (e.g., horizontal or vertical), we in effect make the strobe signal in phase with the rotation of the rotor. Then by comparing the imbalance force signal of the rotor (obtained, for example, by an accelerometer or a force sensor at the bearings of the rotor) with the synchronized strobe signal (with a ﬁxed reference), by means of an oscilloscope or a phase meter, it is possible to determine the orientation of the imbalance force with respect to a ﬁxed body reference of the rotating machine. 1.6.2.2

Fiber Optic Sensors and Lasers

The characteristic component in a ﬁber optic sensor is a bundle of glass ﬁbers (typically a few hundred) that can carry light. Each optical ﬁber may have a diameter on the order of 0.01 mm. There are two basic types of ﬁber optic sensors. In one type, the “indirect” or the extrinsic type, the optical ﬁber acts only as the medium in which the sensed light is transmitted. In this type, the sensing element itself does not consist of optical ﬁbers. In the second type, the “direct” or the intrinsic type, the optical ﬁber bundle itself acts as the sensing element. When the conditions of the sensed medium change, the light-propagation properties of the optical ﬁbers change, providing a measurement of the change in the conditions. Examples of the ﬁrst (extrinsic) type of sensor include ﬁber optic position sensors and tactile (distributed touch) sensors. The second (intrinsic) type of sensor is found, for example, in ﬁber optic gyroscopes, ﬁber optic hydrophones, and some types of micro-displacement or force sensors. A schematic representation of a ﬁber optic position sensor (or proximity sensor or displacement sensor) is shown in Figure 1.44(a). The optical ﬁber bundle is divided into two groups: transmitting ﬁbers and receiving ﬁbers. Light from the light source is transmitted along the ﬁrst bundle of ﬁbers to the target object whose position is being measured. Light reﬂected onto the receiving ﬁbers by the surface of the target object is carried to a photodetector. The intensity of the light received by the photodetector will depend on the position, x, of the target object. In particular, if x ¼ 0, the transmitting bundle will be completely blocked off and the light intensity at the receiver will be zero. As x is increased, the received light intensity will increase, because more light will be

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FIGURE 1.44

1-67

(a) A ﬁber-optic proximity sensor; (b) nonlinear characteristic curve.

reﬂected onto the receiving bundle tip. This will reach a peak at some value of x. When x is increased beyond that value, more light will be reﬂected outside the receiving bundle; hence, the intensity of the received light will decrease. Hence, in general, the proximity– intensity curve for an optical proximity sensor will be nonlinear and will have the shape shown in Figure 1.44(b). Using this (calibration) curve, we can determine the position (x) once the intensity of the light received at the photosensor is known. The light source could be a laser (or light ampliﬁcation by stimulated emission of radiation; structured light), infrared light-source, or some other type, such as a lightemitting diode (LED). The light sensor (photodetector) could be some light-sensitive discrete SC element such as a photodiode or a photo ﬁeld effect transistor (photo FET). Very ﬁne resolutions, better than 1 £ 1026 cm, can be obtained using a ﬁber optic position sensor. An optical encoder is a digital (or pulse-generating) motion transducer. Here, a light beam is intercepted by a moving disk that has a pattern of transparent windows. The light that passes through, as detected by a photosensor, provides the transducer output. These sensors may also be considered in the extrinsic category. The advantages of ﬁber optics include insensitivity to electrical and magnetic noise (due to optical coupling), safe operation in explosive, high-temperature, hazardous environments and high sensitivity. Furthermore, mechanical loading and wear problems do not exist because ﬁber optic position sensors are noncontacting devices with stationary sensor heads. The disadvantages include direct sensitivity to variations in the intensity of the light source and dependence on ambient conditions (ambient light, dirt, moisture, smoke, etc.). As an example of an intrinsic application of ﬁber optics in sensing, consider a straight optical ﬁber element that is supported at each end. In this conﬁguration almost 100% of the light at the source end will transmit through the optical ﬁber and will reach the detector (receiver) end. Then, suppose that a slight load is applied to the optical ﬁber segment at its mid span. The ﬁber will deﬂect slightly due to the

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load, and as a result the amount of light received at the detector can signiﬁcantly drop. For example, a deﬂection of just 50 mm can result in a drop in intensity at the detector by a factor of 25. Such an arrangement may be used in deﬂection, force, and tactile sensing. Another intrinsic application is the ﬁber optic gyroscope, as described below. 1.6.2.3

Fiber-Optic Gyroscope

This is an angular speed sensor that uses ﬁber optics. Contrary to the implication of its name, however, it is not a gyroscope in the conventional sense. Two loops of optical ﬁbers wrapped around a cylinder are used in this sensor. One loop carries a monochromatic light (or laser) beam in the clockwise direction, and the other loop carries a beam from the same light (or laser) source in the counterclockwise direction. Since the laser beam traveling in the direction of rotation of the cylinder has a higher frequency than that of the other beam, the difference in frequencies of the two laser beams received at a common location will measure the angular speed of the cylinder. This may be accomplished through interferometry, as the light and dark patterns of the detected light will measure the frequency difference. Note that the length of the optical ﬁber in each loop can exceed 100 m. Angular displacements can be measured with the same sensor simply by counting the number of cycles and clocking the fractions of cycles. Acceleration can be determined by digitally determining the rate of change of speed. 1.6.2.4

Laser Doppler Interferometer

The laser produces electromagnetic radiation in the ultraviolet, visible, or infrared bands of the spectrum. A laser can provide a single-frequency (monochromatic) light source. Furthermore, the electromagnetic radiation in a laser is coherent in the sense that all waves generated have constant phase angles. The laser uses oscillations of atoms or molecules of various elements. The helium –neon (HeNe) laser and the SC laser are commonly used in industrial applications. As noted earlier, the laser is useful in ﬁber optics, but it can also be used directly in sensing and gaging applications. The laser Doppler interferometer is one such sensor. It is useful in the accurate measurement of small displacements, for example, in strain measurements. To explain the operation of this device, we should explain two phenomena: the Doppler effect and light wave interference. Consider a wave source (e.g., a light source or sound source) that is moving with respect to a receiver (observer). If the source moves toward the receiver, the frequency of the received wave appears to have increased; if the source moves away from the receiver, the frequency of the received wave appears to have decreased. The change in frequency is proportional to the velocity of the source relative to the receiver. This phenomenon is known as the Doppler effect. Now consider a monochromatic (single-frequency) light wave of frequency, f (say, 5 £ 1014 Hz), emitted by a laser source. If this ray is reﬂected by a target object and received by a light detector, the frequency of the received wave is f2 ¼ f þ Df

ð1:95Þ

The frequency increase Df will be proportional to the velocity, v, of the target object, which is assumed to be positive when moving toward the light source. Hence, Df ¼ cv

ð1:96Þ

Now by comparing the frequency, f2, of the reﬂected wave, with the frequency f1 ¼ f

ð1:97Þ

of the original wave, we can determine Df and, hence, the velocity, v, of the target object. The change in frequency Df due to the Doppler effect can be determined by observing the fringe pattern due to light wave interference. To understand this, consider the two waves v1 ¼ a sin 2pf1 t

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and

Reflector

v2 ¼ a sin 2pf2 t

ð1:99Þ

Speed v

If we add these two waves, the resulting wave is

Target Object

v ¼ v1 þ v2 ¼ aðsin 2pf1 t þ sin 2pf2 tÞ Laser

which can be expressed as v ¼ 2a sin pðf2 þ f1 Þt cos pðf2 2 f1 Þt

ð1:100Þ

Beam Splitter

It follows that the combined signal will beat at the beat frequency Df =2: When f2 is very close to f1 (i.e., when Df is small compared with f), these beats will appear as dark and light lines (fringes) in the resulting light wave. This is known as wave interference. Note that Df can be determined by two methods: 1. By measuring the spacing of the fringes 2. By counting the beats in a given time interval or by timing successive beats using a high-frequency clock signal

Photosensor Signal Processor Speed, Position Readings FIGURE 1.45

A laser Doppler interferometer for

The velocity of the target object is determined in measuring velocity and displacement. this manner. Displacement can be obtained simply by digital integration or by accumulating the count. A schematic diagram for the laser Doppler interferometer is shown in Figure 1.45. Industrial interferometers usually employ a HeNe laser that has waves of two frequencies close together. In that case, the arrangement shown in Figure 1.45 has to be modiﬁed to take into account the two frequency components. Note that there are laser interferometers that directly measure displacement rather than speed. They are based on measuring phase difference between the direct and the returning laser, not the Doppler effect (frequency difference). In this case, integration is not needed to obtain displacement from a measured velocity. 1.6.2.5

Ultrasonic Sensors

Audible sound waves have frequencies in the range of 20 Hz to 20 kHz. Ultrasound waves are pressure waves, just like sound waves, but their frequencies are higher than the audible frequencies. Ultrasonic sensors are used in many applications, including displacement and vibration sensing, medical imaging, ranging for cameras with autofocusing capability, level sensing, machine monitoring, and speed sensing. For example, in medical applications, ultrasound probes of frequencies 40 kHz, 75 kHz, 7.5 MHz and 10 MHz are commonly used. Ultrasound can be generated according to several principles. For example, high-frequency (gigahertz) oscillations in piezoelectric crystals subjected to electrical potentials are used to generate very high-frequency ultrasound. Another method is to use the magnetostrictive property of ferromagnetic material. Ferromagnetic materials deform when subjected to magnetic ﬁelds. Respondent oscillations generated by this principle can produce ultrasonic waves. Another method of generating ultrasound is to apply a high-frequency voltage to a metal-ﬁlm capacitor. A microphone can serve as an ultrasound detector (receiver). Analogous to the case of ﬁber-optic sensing, there are two common ways of employing ultrasound in a sensor. In one approach, the intrinsic method, the ultrasound signal undergoes change as it passes through an object, due to acoustic impedance and the absorption characteristics of the object. The resulting signal (image) may be interpreted to determine properties of the object, such as texture, ﬁrmness, and deformation. This approach is utilized, for example, in machine monitoring and object

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ﬁrmness sensing. In the other approach, the Transmitter/ Ultrasound extrinsic method, the time for an ultrasound Target Receiver Generator burst to travel from its source to some object Object and then back to a receiver is measured. This Signal approach is used in distance, position, and Processor vibration measurement and in dimensional gauging. In distance (vibration, proximity, displaceDistance Reading ment) measurement using ultrasound, a burst of ultrasound is projected at the target object, and FIGURE 1.46 An ultrasonic position sensor. the time taken for the echo to be received is clocked. A signal processor computes the position of the target object, possibly compensating for environmental conditions. This conﬁguration is shown in Figure 1.46. Alternatively, the velocity of the target object can be measured, using the Doppler effect, by measuring (clocking) the change in frequency between the transmitted wave and the received wave. The “beat” phenomenon may be employed here. Position measurements with ﬁne resolution (e.g., a fraction of a millimeter) can be achieved using the ultrasonic method. Since the speed of ultrasonic wave propagation depends on the temperature of the medium (typically air), errors will enter into the ultrasonic readings unless the sensor is adjusted to compensate for temperature variations. 1.6.2.6

Gyroscopic Sensors

Consider a rigid body spinning about an axis at angular speed, v: If the moment of inertia of the body about that axis is J, the angular momentum H about the same axis is given by H ¼ Jv ð1:101Þ Newton’s Second Law (torque ¼ rate of change of angular momentum) tells us that to rotate (precess) the spinning axis slightly, a torque has to be applied, because precession causes a change in the spinning angular momentum vector (the magnitude remains constant but the direction changes), as shown in Figure 1.47(a). This is the principle of operation of a gyroscope. Gyroscopic sensors are commonly used in control systems for stabilizing vehicle systems.

θ

Frictionless Bearings Spinning Disk

ω

H1 = Jw H2 = Jw ∆q = Angle of Procession ∆H

Gimbal

Torque Motor

H2

Spin Axis

∆q (a)

H1

(b)

Gimbal Axis

FIGURE 1.47 (a) Illustration of the gyroscopic torque needed to change the direction of an angular momentum vector; (b) a simple single-axis gyroscope for sensing angular displacements.

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TABLE 1.4

Rating Parameters of Several Sensors and Transducers

Transducer

Measurand

Potentiometer LVDT Resolver

Displacement Displacement Angular displacement

Tachometer Eddy current proximity sensor Piezoelectric accelerometer Semiconductor strain gage Loadcell Laser Optical encoder

Measurand Frequency Max/Min

Output Impedance

Typical Resolution

Accuracy

Sensitivity

Low Moderate Low

0.1 mm 0.001 mm or less 2 min

0.1% 0.3% 0.2%

200 mV/mm 50 mV/mm 10 mV/deg

Velocity

10 Hz/DC 2500 Hz/DC 500 Hz/DC (limited by excitation frequency) 700 Hz/DC

Moderate (50 V)

0.2 mm/sec

0.5%

Displacement Acceleration (and velocity, etc.) Strain (displacement, acceleration, etc.) Force (10– 1000 N) Displacement/shape Motion

100 kHz/DC 25 kHz/1 Hz 1 kHz/DC (limited by fatigue) 500 Hz/DC 1 kHz/DC 100 kHz/DC

Moderate High 200

0.001 mm 0.05% full scale 1 mm/sec2 1 to 10 msec(1 msec ¼ 1026 unity strain) 0.01 N 1.0 mm 10 bit

0.5% 1% 1%

5 mV/mm/sec; 75 mV/rad/sec 5 V/mm 0.5 mV/m/sec2 1 V/1, 2000 msec max 1 mV/N 1 V/mm 104/rev

Moderate 100 V 500 V

0.05% 0.5% ^1/2 bit

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Consider the gyroscope shown in Figure 1.47(b). The disk is spun about frictionless bearings using a torque motor. Since the gimbal (the framework on which the disk is supported) is free to turn about the frictionless bearings on the vertical axis, it will remain ﬁxed with respect to an inertial frame, even if the bearing housing (the main structure in which the gyroscope is located) rotates. Hence, the relative angle between the gimbal and the bearing housing (angle u in the ﬁgure) can be measured, and this gives the angle of rotation of the main structure. In this manner, angular displacements in systems such as aircraft, space vehicles, ships, and land vehicles can be measured and stabilized with respect to an inertial frame. Note that bearing friction introduces an error that must be compensated for, perhaps by recalibration before a reading is taken. The rate gyro, which has the same arrangement as shown in Figure 1.47(b), except with a slight modiﬁcation, can be used to measure angular speeds. In this case, the gimbal is not free but is restrained by a torsional spring. A viscous damper is provided to suppress any oscillations. By analyzing this gyro as a mechanical tachometer, we will note that the relative angle of rotation, u; gives the angular speed of the structure about the gimbal axis. Several areas can be identiﬁed where new developments and innovations are being made in sensor technology: 1. Microminiature sensors: IC-based, with built-in signal processing. 2. Intelligent sensors: built-in reasoning or information preprocessing to provide high-level knowledge. 3. Integrated and distributed sensors: sensors are integral with the components and agents of the overall multiagent system that communicate with each other. 4. Hierarchical sensory architectures: low level sensory information is preprocessed to match higher level requirements. These four areas of activity are also representative of future trends in sensor technology development. To summarize, rating parameters of a selected set of sensors/transducers are listed in Table 1.4.

References Broch, J.T. 1980. Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum, Denmark. Buzdugan, G., Mihaiescu, E., and Rades, M. 1986. Vibration Measurement, Martinus Nijhoff Publishers, Dordrecht, The Netherlands. de Silva, C.W., Selection of Shaker speciﬁcations in seismic qualiﬁcation tests, J. Sound Vib., 91, 2, 21–26, 1983. de Silva, C.W., Shaker test-ﬁxture design, Meas. Control, 17, 6, 152 –155, 1983. de Silva, C.W., Sensory information acquisition for monitoring and control of intelligent mechatronic systems, Int. J. Inf. Acquisition, 1, 1, 89 –99, 2004. de Silva, C.W. 1989. Control Sensors and Actuators, Prentice Hall, Englewood Cliffs, NJ. de Silva, C.W. 2005. MECHATRONICS—An Integrated Approach, Taylor & Francis, CRC Press, Boca Raton, FL. de Silva, C.W., Price, T.E., and Kanade, T., A torque sensor for direct-drive manipulators, J. Eng. Ind., Trans. ASME, 109, 2, 122 –127, 1987. de Silva, C.W. 2006. VIBRATION—Fundamentals and Practice, 2nd ed., Taylor & Francis, CRC Press, Boca Raton, FL. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, England. McConnell, K.G. 1995. Vibration Testing, Wiley, New York. Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark.

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Appendix 1A Virtual Instrumentation for Data Acquisition, Analysis, and Presentation Michael Sedlak, and Chris “Flip” DeFilippo National Instruments

Source: Lab VIEW Sound and Vibration User Manual, April 2004 Edition. With permission. Summary The proper acquisition, analysis, and presentation of shock and vibration data demand careful conﬁguration and execution of the measurement system. This Appendix provides a review of the fundamental considerations, which have to be made when acquiring, analyzing, and presenting shock as vibration data. Additionally, the Appendix indicates some tips and techniques for programming such tests using modern software packages. Speciﬁcally, this Appendix refers to the graphical programming environment LabVIEW and the Sound and Vibration Toolkit, and supplements the material presented in Chapter 1 to Chapter 4.

List of Abbreviations Symbol Quantity

Symbol Quantity

ADC AI ANSI AO DAC DAQ DFT DOF DSA DUT DZT EU FFT FRF IEC

IEPE

1A.1

Analog-to-Digital Converter (A/D) Analog Input American National Standards Institute Analog Output Digital-to-Analog Converter (D/A) Data Acquisition Discrete Fourier Transform Degree of Freedom Dynamic Signal Acquisition Device Under Test Discrete Zak Transform Engineering Unit Fast Fourier Transform Frequency Response Function International Electrotechnical Commission

IMD JTFA NI PXI RMS RPM SDOF SRS STFT SVL SVT THD VI

Integrated Electronic Piezoelectric Excitation Intermodulation Distortion Joint Time Frequency Analysis National Instruments PCI eXtensions For Instrumentation Root Mean Square Revolutions per Minute Single Degree of Freedom Shock Response Spectrum Short Time Fourier Transform Sound and Vibration Library Sound and Vibration Toolkit Total Harmonic Distortion LabVIEW Virtual Instrument

Dynamic Signals

This Appendix introduces how to properly obtain data to analyze with the LabVIEW Sound and Vibration Toolkit, as well as issues that can affect the quality of the data. One can simulate data with the generation Virtual Instruments (VIs) located on the Generation palette as well as with other VIs. The Appendix particularly supplements the material presented in Chapter 1 to Chapter 4, on vibration instrumentation, testing, data acquisition, and analysis.

1A.1.1

Acquiring and Simulating Dynamic Signals

This section discusses obtaining data and some key issues when acquiring or simulating dynamic signals to ensure valid measurement results. The three techniques that allow one to obtain data

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are as follows: *

*

*

Acquisition of data with a data acquisition (DAQ) device or system Reading of data from a ﬁle Simulation of data with a generation VI (LabVIEW Virtual Instrument) or other source

It is important that you keep certain considerations in mind when you obtain your data. Measurement and analysis software such as the LabVIEW Sound and Vibration Toolkit does not compensate for inaccurate data. Therefore, the test equipment and test procedure should be calibrated to ensure accurate results. Generally, the test equipment should have speciﬁcations at least ten times better than those of the device under test (DUT). Use a veriﬁable and repeatable test procedure to get accurate results. Whether one is obtaining the data from a DAQ system, reading the data from a ﬁle, or simulating the data, aliasing and time continuity are common issues, which should be considered in the measurement analysis. 1A.1.1.1

Aliasing

When a dynamic signal is discretely sampled, aliasing is the phenomenon in which frequency components greater than the Nyquist frequency are erroneously shifted to lower frequencies. The Nyquist frequency is calculated with the following formula: fNyquist ¼ sample rate =2 When acquiring data with an NI Dynamic Signal Acquisition (DSA) device, aliasing protection is automatic in any acquisition. The sharp antialiasing ﬁlters on DSA devices track the sample rate and ﬁlter out (attenuate) all frequencies above the Nyquist frequency. When performing frequency measurements with an NI E Series DAQ device, you must take steps to eliminate aliasing. These antialiasing steps can include the following actions: *

*

*

Increasing the sample rate Applying an external low-pass ﬁlter Using an inherently band-limited DUT

FIGURE 1A.1

Simulated data aliasing.

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Simulated data also can exhibit aliasing. The signals often are generated according to a time-domain expression and, therefore, have high-frequency components that are aliased in the discretely sampled data. Figure 1A.1 shows an example of this aliasing for a simulated square wave. The only way to protect data from aliasing is to apply appropriate aliasing protection before the data are generated or acquired. Aliasing occurs when the data are generated or sampled, and it is not possible to remove aliased components from the data without detailed knowledge of the original signal. In general, it is not possible to distinguish between true frequency components and aliased frequency components. Therefore, accurate frequency measurements require adequate alias protection. 1A.1.1.2

Time Continuity

When you acquire data in a continuous acquisition, you can use the t0 parameter in the waveform datatype to ensure there are no gaps between successive blocks of waveforms returned by sequential calls to the DAQmx Read VI or AI Read VI. When signals are generated with one of the waveform generation VIs, in the Generation palette or the Waveform Generation palette, the t0 of the current waveform is one sample period later than the timestamp of the last sample in the previous waveform. Continuity is enforced in this way until the generation is reset. The waveform data type is integral for testing time continuity in the Sound and Vibration Toolkit. If you read data from a ﬁle or simulate a signal using one of the VIs in the Signal Generation palette, wire a t0 that meets the continuous timestamp condition to the waveform data type connected to the measurement analysis VIs. This action prevents unexpected resets of the measurement analysis due to detected discontinuities in the input signal.

1A.2

Measurement Conﬁguration Considerations

This section describes how the analog input (AI), analog output (AO), timing, and triggering conﬁguration affect your measurements.

1A.2.1

Input Signal Considerations

One must consider the following conﬁguration before acquiring dynamic signals and performing shock and vibration measurements. 1A.2.1.1

Input Pseudodifferential and Differential Conﬁguration

DSA devices such as the National Instruments PXI-4461 supports two terminal conﬁgurations for AI, differential and pseudodifferential. The term pseudodifferential refers to the fact that there is a 50 W resistance between the outer BNC shell and chassis ground. One can conﬁgure the NI PXI-4461 input channels on a per-channel basis. Therefore, you can have one channel conﬁgured for differential mode and the other channel conﬁgured for pseudodifferential mode. Conﬁgure the channels based on how the signal source or DUT is referenced. Refer to Table 1A.1 to determine how to conﬁgure the channel based on the source reference. If the signal source is ﬂoating, use the pseudodifferential channel conﬁguration. A ﬂoating signal source does not connect to the building ground system. Instead, the signal source has an isolated groundreference point. Some examples of ﬂoating signal sources are outputs of transformers without grounded center taps, battery-powered devices, nongrounded accelerometers, and most instrumentation TABLE 1A.1

Input Channel Conﬁguration

Source Reference

Channel Conﬁguration

Floating, ground referenced Ground referenced

Pseudodifferential Differential

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microphones. An instrument or device that has an isolated output is considered to be a ﬂoating signal source. It is important to provide a ground reference for a ﬂoating signal. If no ground-reference point is provided — for example, in selecting differential mode with a ﬂoating microphone — the microphone outputs can drift outside the NI PXI-4461 common-mode range. If the signal source is ground-referenced, use either the differential or pseudodifferential channel conﬁgurations. A ground-referenced signal source connects in some way to the building system ground. Therefore, it is already connected to a ground-reference point with respect to the NI PXI-4461, assuming the PXI or CompactPCI chassis and controller are plugged into the same power system. Nonisolated outputs of instruments and devices that plug into the building power system fall into this category. Provide only one ground-reference point for each channel by properly selecting differential or pseudodifferential conﬁguration. If you provide two ground-reference points — for example, if you select pseudodifferential mode with a grounded accelerometer — the difference in ground potential results in currents in the ground system that can cause measurement errors. The 50 W resistor on the signal ground is usually sufﬁcient to reduce this current to negligible levels, but results can vary depending on the system setup. The NI PXI-4461 is automatically conﬁgured for differential mode when powered on or when power is removed from the device. This conﬁguration protects the 50 W resistor on the signal ground. 1A.2.1.2

Gain

DSA devices such as the NI PXI-4461 often offer variable gain settings for each AI channel. Each gain setting corresponds to a particular AI range, and each range is centered on 0 V. The gain settings are speciﬁed in decibels (dB), where the 0 dB reference is the default input range of ^10 V. Positive gain values amplify the signal before the analog-to-digital converter (ADC) digitizes it. This signal ampliﬁcation reduces the range of the measurement. However, amplifying the signal before digitization allows better resolution by strengthening weak signal components before they reach the ADC. Conversely, negative gains attenuate the signal before they reach the ADC. This attenuation increases the effective measurement range though it sacriﬁces some resolution for weak signal components. In general, select the voltage range that provides the greatest dynamic range and the least distortion. For example, consider an accelerometer with a 100 mV/g sensitivity rating with an absolute maximum output voltage of 5 Vpk. In this case, the ^10 Vpk is appropriate, corresponding to 0 dB gain. However, the ^ 3.16 Vpk setting maximizes the dynamic range if one knows the stimulus is limited, for example, to 20 g or 2 Vpk. 1A.2.1.3

Input Coupling

One can conﬁgure each AI channel for either alternating current (AC) or direct current (DC) coupling. If you select DC coupling, any DC offset present in the source signal is passed to the ADC. The DC-coupling conﬁguration is usually best if the signal source has only small amounts of offset voltage or if the DC content of the acquired signal is important. If the source has a signiﬁcant amount of unwanted offset, select AC coupling to take full advantage of the input dynamic range. 1A.2.1.4

Integrated Electronic Piezoelectric Excitation

If you attach an Integrated Electronic Piezoelectric Excitation (IEPE) accelerometer or microphone to an AI channel that requires excitation from your DSA device, you must enable the IEPE excitation circuitry for that channel to generate the required current. One can independently conﬁgure IEPE signal conditioning on a per-channel basis. It is common to set the excitation from 0 to 20 mA with 20 mA resolution. A DC voltage offset is generated equal to the product of the excitation current and sensor impedance when IEPE signal conditioning is enabled. To remove the unwanted offset, enable AC coupling. Using DC coupling with IEPE excitation enabled is appropriate only if the offset does not exceed the voltage range of the channel.

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Nyquist Frequency and Bandwidth

Further discussion of DSA measurement conﬁguration requires a brief introduction of two concepts: *

*

Nyquist frequency Nyquist bandwidth

Any sampling system, such as an ADC, is limited in the bandwidth of the signals it can represent. Speciﬁcally, a sampling rate of fs can only represent signals with a maximum frequency of fs =2: This maximum frequency is known as the Nyquist frequency. The bandwidth from 0 Hz to the Nyquist frequency is the Nyquist bandwidth. 1A.2.1.6

Analog-to-Digital Conversion

ADC is discussed in Chapter 2. DSA devices commonly use a conversion method known as delta –sigma modulation. If the data rate is 51.2 kS/sec, each ADC actually samples its input signal at 6.5536 MS/sec, 128 times the data rate, and produces one-bit samples that are applied to the digital ﬁlter. This ﬁlter then expands the data to 24 bits, rejects signal components greater than the Nyquist frequency of 25.6 kHz, and digitally resamples the data at 51.2 kS/sec. The one-bit, 6.5536 MS/sec data stream from the ADC contains all of the information necessary to produce 24-bit samples at 51.2 kS/sec. The delta–sigma ADC achieves this conversion from high speed to high resolution by adding a large amount of random noise to the signal so that the resulting quantization noise, although large, is restricted to frequencies above the Nyquist frequency, 25.6 kHz in this case. This noise is not correlated with the input signal and is almost completely rejected by the digital ﬁlter. The resulting output of the ﬁlter is a band-limited signal with a large dynamic range. One of the advantages of a delta –sigma ADC is that it uses a one-bit digital-to-analog converter (DAC) as an internal reference. As a result, the delta– sigma ADC is free from the kind of differential nonlinearity (DNL) and associated noise that is inherent in most high-resolution ADCs. 1A.2.1.7

Antialias Filters

A digitizer may sample signals containing frequency components above the Nyquist limit. The process by which the digitizer modulates out-of-band components, returning them to the Nyquist bandwidth, is known as aliasing. The greatest danger of aliasing is that there is no straightforward way to know whether it has happened by looking at the ADC output. If an input signal contains several frequency components or harmonics, some of these components maybe represented correctly while others are aliased. Low-pass ﬁltering to eliminate components above the Nyquist frequency, either before or during the digitization process, can guarantee that the digitized data set is free of aliased components. The NI PXI4461 employs both digital and analog low-pass ﬁlters to achieve this protection. In addition to the ADC built-in digital ﬁltering, DSA devices may also feature a ﬁxed-frequency analog ﬁlter. The analog ﬁlter removes high-frequency components in the analog signal path before they reach the ADC. This ﬁltering addresses the possibility of high-frequency aliasing from the narrow-bands that are not covered by the digital ﬁlter. 1A.2.1.8

Input Filter Delay

The input ﬁlter delay is the time required for digital data to propagate through the ADC digital ﬁlter, For example, a signal experiences a delay equal to 6.3 msec at 10 kS/sec. This delay is an important factor for stimulus –response measurements, control applications, or any application where loop time is critical. In this case, it is often advantageous to maximize the sample rate and minimize the time required for 63 sample clock cycles to elapse. The input ﬁlter delay also makes an external digital trigger appear to occur 63 sample clocks later than expected. Alternatively, the acquired buffer appears to begin 63 samples earlier than expected. This delay occurs because external digital triggering is a predigitization event.

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Overload Detection

It is desirable to ensure that the DSA device includes overload detection in both the analog domain (predigitization) and digital domain (postdigitization). An analog overrange can occur independently from a digital overrange, and vice versa. For example, an IEPE accelerometer might have a resonant frequency that, when stimulated, can produce an overrange in the analog signal. However, because the delta –sigma technology of the ADC uses very sharp antialiasing ﬁlters, the overrange is not passed into the digitized signal. Conversely, a sharp transient on the analog side might not overrange, but the step response of the delta–sigma antialiasing ﬁlters might result in clipping in the digital data. Modern DSA devices allow you to programmatically poll the digital and analog overload detection circuitry on a per-channel basis to monitor for an overload condition. If an overload is detected, consider any data acquired at that time corrupt.

1A.2.2

Output Signal Considerations

This section describes the theory of operation of the output components of DSA devices such as the NI PXI-4461. 1A.2.2.1

Output Pseudodifferential and Differential Conﬁguration

The output channel terminal conﬁguration options are very similar to those for the input channels. The NI PXI-4461 output channels are conﬁgurable on a per-channel basis. As with the input channels, you should conﬁgure the output channel based on how the DUT is referenced. Refer to Table 1A.2 to determine how to conﬁgure the output channel based on the DUT reference. If the DUT inputs are ﬂoating, use the pseudodifferential channel conﬁguration. The term pseudodifferential refers to the fact that there is a 50 W resistance between the outer BNC shell and chassis ground. A ﬂoating DUT does not connect in any way to the building ground system. Instead, the DUT has an isolated ground-reference point. Transformer inputs without center ground taps, battery-powered devices, or any instruments that have an isolated input are all examples of ﬂoating DUTs. One should provide a ground-reference for a ﬂoating DUT input. If no ground-reference point is provided — for example, in selecting differential mode with a ﬂoating shaker table input ampliﬁer — the outputs can ﬂoat outside the common-mode range of the ampliﬁer input. If the DUT input is ground referenced, use the differential channel conﬁguration. A single-ended DUT connects in some way to the building system ground. Therefore, it is already connected to a groundreference point with respect to the NI PXI-4461, assuming the PXI or CompactPCI chassis and controller are plugged into the same power system. Nonisolated inputs of instruments that plug into the building power system fall into this category. You should provide only one ground-reference point for each channel by properly selecting the differential or pseudodifferential conﬁguration. If you provide two ground-reference points — for example, by selecting the pseudodifferential output mode for a single-ended ampliﬁer as the DUT — the difference in ground potential results in currents in the ground system that can cause errors in the output signal. The 50 W resistor on the signal ground is usually sufﬁcient to reduce this current to negligible levels, but results can vary depending on the system setup. The NI PXI-4461 is automatically conﬁgured for the differential mode when powered on or when power is removed from the device. Using the differential mode by default protects the 50 W resistor on the signal ground. TABLE 1A.2

Output Channel Conﬁguration

DUT Reference

Output Channel Conﬁguration

Floating Ground referenced

Pseudodifferential Differential

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TABLE 1A.3

NI PXI-4461 Gain Ranges

Gain (Referenced to ^10 Vpk) (dB) 0 220 240

1A.2.2.2

Voltage Range (Vpk) ^ 10 ^1 ^ 0.1

Attenuation

Modern DSA devices offer variable gain settings for AO. Most gain settings correspond to a particular AO range, always centered at 0 V. These gain settings can be speciﬁed in dB, where the 0 dB reference corresponds to the default output range. Table 1A.3 summarizes the three output gain options available on the NI PXI-4461. In general, select the gain that provides the greatest dynamic range and the least distortion. The ^ 1 Vpk setting maximizes the dynamic range if you know the stimulus is limited to, for example, 0.5 Vpk. You can minimize system distortion by providing sufﬁcient headroom between the stimulus setting (0.5 Vpk) and the range setting (^ 1 Vpk). In some cases in which distortion performance is critical, you can reduce the overall dynamic range to improve the distortion characteristics by selecting the ^10 Vpk setting. 1A.2.2.3

Digital-to-Analog Conversion

Digital-to-analog conversion (DAC) is discussed in Chapter 2. The delta–sigma DACs on the NI PXI4461 function in a way analogous to delta –sigma ADCs. The digital data ﬁrst passes through a digital interpolation ﬁlter, then the resampling ﬁlter of the DAC, and ﬁnally goes to the delta –sigma modulator. In the ADC, the delta–sigma modulator is an analog circuit that converts high-resolution analog signals to high-rate, 1-bit digital data, whereas in the DAC the delta–sigma modulator is a digital circuit that converts high-resolution digital data to high-rate, 1-bit digital data. As in the ADC, the modulator frequency shapes the quantization noise so that almost all of its energy is above the Nyquist frequency. The digital 1-bit data is then sent directly to a 1-bit DAC. This DAC can have only one of two analog values, and therefore is inherently perfectly linear. 1A.2.2.4

Anti-imaging and Interpolation Filters

A sampled signal repeats itself throughout the frequency spectrum. These repetitions begin above one half the sample rate, fs ; and, theoretically, continue up through the spectrum to inﬁnity. Images remain in the sample data because the data actually represent only the frequency components below one half fs (the baseband). 1A.2.2.5

Output Filter Delay

Output ﬁlter delay, or the time required for digital data to propagate through the DAC and interpolation digital ﬁlters, varies depending on the sample rate. This delay is an important factor for stimulus – response measurements, control applications, and every application where loop time is critical.

1A.3

Scaling and Calibration

This Appendix discusses using the SVL Scale Voltage to EU VI located on the Scaling palette to scale a signal to engineering units (EU) and using the Calibration VIs located on the Calibration palette.

1A.3.1

Scaling to Engineering Units

This section discusses scaling data to the appropriate EU so one can perform measurement analysis. Typically, scaling a signal to the appropriate EU occurs before any analysis is performed. Use the SVL Scale Voltage to EU VI to scale the signal to the appropriate EU. All measurement VIs in the Sound and

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Vibration Toolkit expect input signals and return results with the appropriate units, such as time-domain signals in the correct EU, frequency spectra in decibels with the proper reference, phase information in degrees or radians, and so on. To handle units properly, the high-level VIs need the signal to be scaled to the appropriate EU. If you use any method outside of the Sound and Vibration Toolkit to apply scaling to a waveform, do not use the SVL Scale Voltage to EU VI. NI provides several tools and methods to apply scaling to a waveform. These include, but are not limited to, NI-DAQmx tasks or global channels created with Measurement and Automation Explorer (MAX), the DAQ Assistant, or the DAQmx Create Virtual Channel VI.

1A.3.2

Performing System Calibration

One typically performs system calibration with a dedicated calibrator, such as a pistonphone for microphones or a handheld shaker for accelerometers. If you are calibrating a microphone, consider using the SVL Calibrate Microphone VI. If you are calibrating an accelerometer, consider using the SVL Calibrate Accelerometer VI. These VIs are very similar to the general-purpose SVL Calibrate Sensor VI, but they offer the advantage of having default values commonly found for pistonphones or hand-held shakers. All of the Calibration VIs use the characteristics of the calibrator, such as reference calibration value and frequency, to perform the calibration. 1A.3.2.1

Propagation Delay Calibration

The Sound and Vibration Toolkit provides VIs for calibrating the propagation delay of the measurement system. National Instruments DSA devices like the NI PXI-4461 and NI PCI-4451 can acquire and generate signals on the same device. The input and output channels have analog and digital circuitry, such as antialiasing and anti-imaging ﬁlters, that introduce a certain delay to the signal. The propagation delay is the number of samples ranging from the time a sample is ﬁrst written to the output channel, to the time when that sample is digitized on the input channel, assuming there is no delay from the output channel to the input channel. This delay varies by DSA device. There are two ways to determine the propagation delay of the DSA device. You can refer to the documentation for the DSA device to ﬁnd the propagation delay speciﬁcations, also referred to as group delay. You also can measure the propagation delay in samples with the SVL Measure Propagation Delay VIs. The SVL Measure Propagation Delay VIs allow you to measure the delay introduced in the input and output circuitry for a speciﬁc device at the desired sample rate. Connect the DSA device output channel directly to the input channel, as displayed in Figure 1A.2, to measure the device propagation delay. Note: Do not put a DUT in the signal path when measuring the propagation delay for a DAQ device. For an E or S Series DAQ device from NI, you should expect to measure a one-sample propagation delay due to the time required for the signal to traverse the signal path between the DAC on the analog output channel and the ADC on the analog input channel. Figure 1A.3 shows the time domain data for the propagation delay measurement of an FIGURE 1A.2 1A.2 Measuring the device propagation NI PCI-6052E. delay.

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FIGURE 1A.3

FIGURE 1A.4

Propagation delay measurement of an NI PCI-6052E.

NI PXI-4461 propagation delay with a 204.8 kHz sample rate.

For DSA devices, or any other device which has onboard ﬁltering on either the input, output, or both channels, you should expect to measure a propagation delay consistent with the sum of the delays speciﬁed for the onboard ﬁlters on the input and output channels. Figure 1A.4 shows the delay of a smooth pulse generated and acquired by an NI PXI-4461 with a 204.8 kHz sample rate. Not all DSA devices have a constant propagation delay across the entire range of supported sample rates. For example, the NI PXI-4461 propagation FIGURE 1A.5 NI PXI-4461 propagation delay vs. sample rate. delay is dependent on the output update rate. Figure 1A.5 shows the total propagation delay vs. sample rate relationship for the NI PXI-4461 from output to input as a function of the sample rate. As illustrated by Figure 1A.3, Figure 1A.4, and Figure 1A.5, the propagation delay can vary signiﬁcantly with different sample rates and devices. To ensure measurement accuracy in your I/O applications, determine and account for the propagation delay of the DAQ device at the same sample rate used in your application. It is important to remove the effects of the delay due to the data acquisition system for two reasons. First, there is always a delay between the generated output signal and the acquired input on the device even when the output and input channels are hardware synchronized. Second, the anti-imaging and antialiasing ﬁlters of the device introduce additional delays. You must account for this delay to perform accurate dynamic measurements. Use the device propagation delay [samples ] input on the examples found in the LabVIEW program directory under “\examples\Sound” and “Vibration\Audio Measurements\” to remove the delay due to the DAQ device. The anti-imaging and antialiasing ﬁlters have a low-pass ﬁlter effect on the data. This effect results in a transient response at sharp transitions in the data. These transitions are common at the start and stop of a

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generation, at a change in frequency (swept sine), and when the amplitude changes (amplitude sweep). The swept-sine analysis and audio measurements examples in the Sound and Vibration Toolkit account for this transient behavior in the device response to achieve the highest degree of accuracy. The propagation delay of the DUT is also an important speciﬁcation in some applications. For example, the propagation delay for the DUT is a required input when performing audio measurements and when measuring the frequency response using swept sine. If the DUT and the FIGURE 1A.6 Measuring the DUT propagation delay. propagation medium can successfully pass the pulse signal used by the SVL Measure Propagation Delay VIs without excessive attenuation, then this measurement also applies when measuring the propagation delay of the DUT and the propagation medium. Figure 1A.6 shows the wiring diagram for this conﬁguration. The DUT propagation delay is the delay of the entire system minus the device delay. Remember to measure the device delay without the DUT connected. The propagation delay for an analog DUT is a constant time delay rather than a delay of samples. Use the following equation to convert the measured delay in samples to the equivalent delay in seconds: delay½sec ¼ delay½samples = sample rate½Hz

1A.4

Limit Testing Analysis

This Appendix discusses using the polymorphic SVT Limit Testing VI located on the Limit Testing palette. You can use Limit Testing to perform analysis on any type of measured result produced by the Sound and Vibration Toolkit, including the following measurements: *

*

*

*

*

*

Waveform Spectrum Peak Octave Swept sine Scalar

1A.4.1

Limit Testing Overview

You can use the SVT Limit Testing VI to analyze almost any measured result produced by the Sound and Vibration Toolkit. Refer to Table 1A.4 for examples of data-types supported by the SVT Limit Testing VI and VIs that generate supported data-types.

1A.4.2

Using the SVT Limit Testing VI

Limit testing allows one to specify an envelope around the data to deﬁne a pass range. You can enter a scalar to the upper limit, lower limit, or both to specify a constant ceiling and ﬂoor for the data to perform tests such as range detection. You can enter an upper limit mask, lower limit mask, or both to the SVT Limit Testing VI to deﬁne a pass range that varies in shape and level based on acceptable results at any given point in the measurement. You also can create a discontinuous mask which allows you to perform limit testing on only a part of the results while ignoring the rest.

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Compatible Data Types for SVT Limit Testing VI

Data-type

Output VIs

Waveform measurement

AI read, DAQmx read, waveform generation, weighting, integration, vibration level, sound level

Frequency spectrum measurement

Baseband FFT, baseband subset FFT, zoom FFT, extended measurements

XY data

ANSI and IEC octave, swept sine

Peak measurement

Distortion, single-tone, extended measurements

Scalar measurement

Calibration, vibration level, sound level, ANSI and IEC octave, distortion, single-tone, extended measurements

You must enter at least one limit, or the SVT Limit Testing VI returns an error. You can visually display the input signal, failures, upper limit, and lower limit by creating an indicator from the output values terminal. The upper limit and lower limit inputs to the SVT Limit Testing VI must be compatible with the input signal. Table 1A.5 lists the criteria that must be met for each input signal type that is compatible with the SVT Limit Testing VI. In Table 1A.5, the following abbreviations apply: *

*

*

*

*

*

*

dt is the time spacing, in seconds, between elements. df is the frequency spacing, in hertz, between elements. N is the number of elements in the array. f ðiÞ is the ith frequency element. S is the signal. U is the upper mask limit. L is the lower mask limit.

Limit testing covers a broad range of data testing from range detection to discontinuous mask testing of a swept-sine frequency response spectrum. Figure 1A.7, Figure 1A.9, Figure 1A.11, and TABLE 1A.5

Criteria for Upper and Lower Limits Input Signal Type

Criteria on Input Limit Masks

Waveform data type ðt0; dt; ½signal Þ Frequency spectrum ðf 0; df ; ½spectrum Þ Octave spectrum, swept-sine spectrum, XY data ([X ], [Y ]) Identiﬁed peaks, harmonic components, multitone phases ([frequency, amplitude])

dt . 0; dtS ¼ dtU ¼ dtL , NS ¼ NU ¼ NL f 0S ¼ f 0U ¼ f 0L , dfS ¼ dfU ¼ dfL , NS ¼ NU ¼ NL [X ]S ¼ [X ]U ¼ [X ]L, NS ¼ NU ¼ NL f(i)S ¼ f(i)U ¼ f(i)L, NS ¼ NU ¼ NL

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FIGURE 1A.7

Range detection performed in engineering units.

Figure 1A.13 illustrate some, but not all, of the different ways one can use the SVT Limit Testing VI in your application. Figure 1A.7 illustrates a range-detection test. Scaled waveform data and upper and lower limits are input to the SVT Limit Testing VI. The VI checks that the data fall within the envelope speciﬁed by the upper and lower limits. Figure 1A.8 shows the output results for the range detection test. Figure 1A.9 shows a pass/fail test on the measured THD. This test only checks the upper limit of the measurement, therefore, only the upper limit is wired to the VI. The upper limit should have the same units as the input measure- FIGURE 1A.8 Range detection test on a time-domain ment. In this case, both the THD and the upper signal. limit are expressed as percentages. Figure 1A.10 shows the THD test output results. Figure 1A.11 shows a continuous mask test on a power spectrum. Formula nodes deﬁne both the upper and lower limits in this VI, making this a more complex test than the one in Figure 1A.9. Figure 1A.12 shows the output graph for the power spectrum continuous mask test. Figure 1A.13 shows a discontinuous mask test on a swept-sine frequency response. A discontinuous mask test can track and test the results at different magnitudes and ranges, as well as stop testing at deﬁned intervals. For example, one might use the envelope deﬁned by the upper and lower limit masks in this example for a DUT such as a notch ﬁlter. Figure 1A.14 shows the output graph for the discontinuous mask test.

1A.5

Integration

This Appendix discusses the integration process, including basic theory and implementation in the time and frequency domains.

FIGURE 1A.9

Test scalar measurement.

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The Sound and Vibration Toolkit contains the following integration VIs: *

*

SVT Integration VI located on the Integration palette for time-domain integration SVT Integration (frequency) VI located on the Frequency Analysis .. Extended Measurements palette for frequencydomain integration

1A.5.1

Introduction to Integration

The conversion between acceleration, velocity, and displacement is based on one of the fundamental laws in Newtonian physics, represented by the following equations: x_ ¼

d ðxÞ dt x€ ¼

FIGURE 1A.10

Limit testing on THD measurements.

d d2 ð_xÞ ¼ 2 ðxÞ dt dt

Velocity is the ﬁrst derivative of displacement with respect to time. Acceleration is the ﬁrst derivative of velocity and the second derivative of displacement with respect to time. Therefore, given acceleration, perform a single integration with respect to time to compute the velocity or perform a double integration with respect to time to compute the displacement. When representing the acceleration of a point by a simple sinusoid, the velocity and the displacement of the point are well known and represented by the following equations: a ¼ A sinðvtÞ A A p v ¼ 2 cosðvtÞ ¼ sin vt 2 2 v v A A d ¼ 2 2 sinðvtÞ ¼ 2 sinðvt 2 pÞ v v

ð1A:1Þ ð1A:2Þ

Note: The initial condition is arbitrarily set to zero in Equation 1A.1 and Equation 1A.2. The amplitude of the velocity is inversely proportional to the frequency of vibration. The amplitude of the displacement is inversely proportional to the square of the frequency of vibration. Furthermore, the phase of the velocity lags the acceleration by 908. The phase of the displacement lags the acceleration by 1808. Figure 1A.15 illustrates the relationship between acceleration, velocity, and displacement. The integration of a sinusoid is known in closed form. Integration of an arbitrary waveform typically requires a numerical approach. You can use several numerical integration schemes to evaluate an integral in the time domain.

FIGURE 1A.11

Continuous mask test on power spectrum.

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In the frequency domain, you can deﬁne any arbitrary band-limited waveform as a sum of sinusoids. Because the amplitude and phase relationships are known for sinusoids, you can carry out the integration in the frequency domain.

1A.5.2

Implementing Integration

If you need to perform measurements on velocity or displacement data when you have only acquired acceleration or velocity data, respectively, integrate the measured signal to yield the desired data. You can perform integration either in the time domain as a form of signal conditioning or in the frequency domain as a stage of analysis. When performed in the frequency domain, integration is one of the extended measurements for frequency analysis. 1A.5.2.1

FIGURE 1A.12 spectrum.

Continuous mask test on a power

Challenges when Integrating Vibration Data

Converting acceleration data to velocity or displacement data presents a pair of unique challenges. First, measured signals typically contain some unwanted DC components. The second challenge is the fact that many transducers, especially vibration transducers, have lower-frequency limits. A transducer cannot accurately measure frequency components below the lower-frequency limit of the transducer. 1A.5.2.1.1 DC Component Even though a DC component in the measured signal might be valid, the presence of a DC component indicates that the DUT has a net acceleration along the axis of the transducer. For a typical vibration measurement, the DUT is mounted or suspended in the test setup. The net acceleration of the DUT is zero. Therefore, any DC component in the measured acceleration is an artifact and should be ignored. 1A.5.2.1.2 Transducers Most acceleration and velocity transducers are not designed to accurately measure frequency components close to DC (see Chapter 1). Closeness to DC is relative and depends on the speciﬁc transducer. A typical accelerometer can accurately measure components down to about 10 Hz. A typical velocity probe can accurately measure components down to 2 to 3 Hz. Inaccurately measured low-frequency vibrations can dominate the response when the signal is integrated because integration attenuates low-frequency components less than high-frequency components.

FIGURE 1A.13

Discontinuous mask test on swept-sine frequency response.

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1A.5.2.1.3 Implementing Integration Using the Sound and Vibration Toolkit Both the SVT Integration VI and the SVT Integration (frequency) VI address the challenges of converting acceleration data to velocity or displacement data.

1A.5.3

Time-Domain Integration

This section presents examples of and discussion about time-domain integration. 1A.5.3.1 Single-Shot Acquisition and Integration The following example shows how one can use integration to convert acceleration data into displacement data in a single-shot acquisition and integration. In this example, the acquired waveform is sampled at 51.2 kHz and is double integrated. Figure 1A.16 shows the block diagram for the VI. Because the integration is implemented with ﬁlters, there is a transient response associated with integration while the ﬁlters settle. You should take care to avoid the transient region when making further measurements. Figure 1A.17 shows the results of a single-shot acquisition and integration of a 38 Hz sine wave. You can see the transient response in the ﬁrst 200 msec of the integrated signal. 1A.5.3.2

FIGURE 1A.14 Discontinuous mask test on a sweptsine frequency response.

FIGURE 1A.15

Integration of a 0.5 Hz sine wave.

Continuous Acquisition and Integration

The more common case for time-domain integration occurs with continuous acquisition. Figure 1A.18 shows the block diagram for a VI designed for continuous acquisition and integration. In this example, the high-pass cut-off frequency used for the integration is 10 Hz. Additionally, the integration is explicitly reset in the ﬁrst iteration of the VI and performed continuously thereafter. In this example, this additional wiring is optional because the SVT Integration VI automatically resets the ﬁrst time it is called and runs continuously thereafter. If you use the block diagram in Figure 1A.18 in a larger application that requires starting and stopping the data acquisition process more than once, NI suggests setting the reset ﬁlter control to “TRUE” for the ﬁrst iteration of the while loop. Setting the reset ﬁlter control to TRUE causes the ﬁlter to reset every time the data acquisition process starts. Set the reset ﬁlter control to “FALSE” for subsequent iterations of the while loop.

FIGURE 1A.16

Block diagram for single-shot acquisition and integration.

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Figure 1A.19 shows the results of the continuous acquisition and integration of the same 38 Hz sinusoid used in the single-shot acquisition and integration example. As in single-shot acquisition and integration, continuous acquisition and integration has an initial transient response. Take care to avoid making additional measurements until the response of the ﬁlters settles. Once the ﬁlters settle, you can use the integrated signals for additional analysis. Figure 1A.20 shows the frequency response for time-domain single integration. Figure 1A.21 shows the frequency response for time-domain double integration. In Figure 1A.20, one can see the characteristic 20 dB per decade roll-off of the magnitude response of the single integration. In Figure 1A.21, one can see the characteristic 40 dB per decade roll-off of the magnitude response of the double integration. Upper and lower frequency limits exist for which you can obtain a speciﬁed degree of accuracy in the magnitude response. For example, sampling at a rate of 51.2 kHz, the magnitude response of the integrator is accurate to within 1 dB from 1.17 to 9.2 kHz for single integration and from 1.14 to 6.6 kHz for double integration. The accuracy ranges change with the sampling frequency and the high-pass cut-off frequency. The attenuation of the single integration ﬁlter at 9.2 kHz is 2 95 dB. The attenuation of the double integration ﬁlter at 6.6 kHz is 2 185 dB. Accuracy at high frequencies usually is not an issue.

FIGURE 1A.17 Transient response in single-shot acquisition and integration.

FIGURE 1A.18 integration.

Continuous

acquisition

and

1A.5.4 Frequency-Domain Integration You can use the following strategies to obtain the spectrum of an integrated signal: *

*

FIGURE 1A.19

Settled response of continuous acqui-

Perform the integration in the time domain sition and integration. before computing the spectrum. Compute the spectrum before performing the integration in the frequency domain.

The following example demonstrates the implementation of the strategies used to obtain the spectrum of an integrated signal. Figure 1A.22 shows the block diagram for the example VI. The high-pass cutoff frequency parameter of the SVT Integration VI is wired with a constant of 10 Hz. The SVT Integration (frequency) VI does not have a high-pass cutoff frequency parameter. Instead, the SVT Integration (frequency) VI sets the DC component of the integrated signal to zero if the spectrum scale is linear or to negative inﬁnity (2 Inf) if the spectrum scale is in decibels. Figure 1A.23 shows the results of integrating in the time and frequency domains.

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The power spectrum is computed after the timedomain integration ﬁlters settle. The frequencydomain integration scales the spectrum at each frequency line. No settling time is necessary for the frequency-domain integration because integration ﬁlters are not involved in the frequency-domain integration. Perform frequency-domain integration in the following situations to maximize performance: *

*

When the integrated signal is not needed in the time domain When spectral measurements are made

1A.6 Vibration-Level Measurements This Appendix brieﬂy discusses the analysis concepts associated with performing vibrationlevel measurements and how one can use the Vibration Level VIs located on the Vibration Level palette to perform vibration-level measurements.

FIGURE 1A.20 integration.

Frequency response for single

FIGURE 1A.21 integration.

Frequency response for double

1A.6.1 Measuring the Root Mean Square Level A basic requirement of vibration measurements is measuring the level of the signal returned by an accelerometer. The level of the accelerometer signal generally is expressed in root-mean-square (RMS) acceleration ðgrms Þ: 1A.6.1.1

Single-Shot Buffered Acquisition

The block diagram in Figure 1A.24 illustrates a VI designed to perform a single-shot acquisition and compute the RMS levels. The sampling frequency is 10 kS/sec. A buffer containing 1 sec of data is returned by the read VI.

FIGURE 1A.22

Integration in the time domain and in the frequency domain.

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FIGURE 1A.23

FIGURE 1A.24

FIGURE 1A.25

1A.6.1.2

Power spectra of the integrated signal.

Single-shot buffered acquisition and RMS level VI.

Continuous data acquisition and RMS level VI.

Continuous Signal Acquisition

One can use the block diagram in Figure 1A.24 with a while loop to continuously acquire signals from an accelerometer and display the vibration level in a chart. The block diagram in Figure 1A.25 illustrates how to measure the RMS value once every 100 msec and display the results in a strip chart. In this example, the RMS value is computed based on the last 100 msec of acquired data.

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FIGURE 1A.26

Running RMS VI.

Note: Set the restart averaging control on the SVT RMS Level VI to TRUE. Otherwise, the SVT RMS Level VI accumulates intermediate results to compute the RMS vibration level over the entire data acquisition instead of just over the last block of data.

1A.6.2

Performing a Running RMS Level Measurement

The SVT Running RMS Level VI returns the RMS value computed over the last N seconds, which is the integration time. The block diagram in Figure 1A.26 illustrates an application using the SVT Running RMS Level VI. The sampling frequency is 10 kS/sec. The read VI reads 1000 samples at a time.

1A.6.3

Computing the Peak Level

Use the SVT Peak Level VI to compute the peak level of a signal. In peak-hold averaging, the largest measured level value of all previous values is computed and returned until a new value exceeds the current maximum. The new value becomes the new maximum value and is the value returned until a new value exceeds it.

1A.6.4

Computing the Crest Factor

The crest factor is the ratio of the peak value over the RMS value of a given signal and indicates the shape of the waveform. The crest factor is deﬁned by the following equation: where

Fc ¼

Vpk Vrms

Fc is the crest factor. Vpk is the peak value of the signal. Vrms is the RMS value of the signal. The block diagram in Figure 1A.27 illustrates an application using the SVT Crest Factor VI. Along with the crest factors, the SVT Crest Factor VI also returns the peak and RMS levels.

FIGURE 1A.27

Crest factor VI.

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Vibration Monitoring, Testing, and Instrumentation

Frequency Analysis

This Appendix discusses methods used by the Frequency Analysis VIs located on the Frequency Analysis palette for windowing, averaging, and performing frequency-domain measurements. The Frequency Analysis VIs offer various frequency measurements based on the discrete Fourier transform (DFT). For simplicity, the remainder of this document uses the term FFT to denote both the FFT and the DFT.

1A.7.1

FFT Fundamentals

The FFTresolves the time waveform into its sinusoidal components. The FFT takes a block of time-domain data and returns the frequency spectrum of that data. The FFT is a digital implementation of the Fourier transform. Thus, the FFT does not yield a continuous spectrum. Instead, the FFT returns a discrete spectrum where the frequency content of the waveform is resolved into a ﬁnite number of frequency lines, or bins. 1A.7.1.1

Number of Samples

The computed spectrum is completely determined by the sampled time waveform input to the FFT. If an arbitrary signal is sampled at a rate equal to fs over an acquisition time, T; N samples are acquired. Compute T with the following equation: T¼

N fs

where T is the acquisition time. N is the number of samples acquired. fs is the sampling frequency. Compute N with the following equation: N ¼ Tfs where N is the number of samples acquired. T is the acquisition time. fs is the sampling frequency. 1A.7.1.2

Frequency Resolution

Because of the properties of the FFT, the spectrum computed from the sampled signal has a frequency resolution, df : Calculate the frequency resolution with the following equation: df ¼

1 f ¼ s T N

where df is the frequency resolution. T is the acquisition time. fs is the sampling frequency. N is the number of samples. Note: The frequency resolution is determined solely by the acquisition time. The frequency resolution improves as the acquisition time increases.

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Maximum Resolvable Frequency

The sampling rate of the time waveform determines the maximum resolvable frequency. According to the Shannon Sampling Theorem, the maximum resolvable frequency must be half the sampling frequency. To calculate the maximum resolvable frequency, use the following equation: fmax ¼ fNyquist ¼

fs 2

where fmax is the maximum resolvable frequency fNyquist is the Nyquist frequency fs is the sampling frequency 1A.7.1.4

Minimum Resolvable Frequency

The minimum resolvable frequency is 0 (DC). The term baseband analysis is often used to describe analysis from 0 to fNyquist : 1A.7.1.5

Number of Spectral Lines

The number of lines in the spectrum is half the number of samples, N; in the waveform. Directly specify the number of lines in the spectrum for the Zoom FFT VIs, in the Sound and Vibration Toolkit. Specify the number of data samples to control the number of spectral lines for the Baseband FFT and the Baseband Subset VIs. 1A.7.1.6 Relationship between Time-Domain and Frequency-Domain Speciﬁcations and Parameters Table 1A.6 summarizes the relationship of time-domain speciﬁcations to frequency-domain parameters. Use the information in Table 1A.7 if you prefer to specify the spectrum parameters and determine the required data-acquisition parameters from these speciﬁcations. TABLE 1A.6

Time-Domain Speciﬁcations to Frequency-Domain Parameters

Time Domain

fs N T

Frequency Domain fNyquist

fmax

Number of Lines

fs 2

fs ·Eb

Eb ·N

df 1 f ¼ s T N

fs is the sampling frequency, Eb is the effective bandwidth, N is the number of samples acquired, and T is the acquisition time.

TABLE 1A.7

Frequency-Domain Speciﬁcations to Time-Domain Parameters

Frequency Domain

fmax Number of lines in the spectrum df

Time Domain fs

N

T

fmax Eb

Number of lines Eb

1 Number of lines ¼ df fmax

fmax is the maximum resolvable frequency, Eb is the effective bandwidth, and df is the frequency resolution.

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In Table 1A.6 and Table 1A.7, Eb is the ratio of the alias-free bandwidth to the sampling frequency. In traditional instruments, Eb is typically 1/256. However, the alias-free bandwidth depends on the hardware used to acquire the dynamic signal. Eb can have a maximum value of 0.5. This maximum value corresponds to a perfect antialiasing ﬁlter. For example, if 1024 samples are input to the FFT algorithm, the computed spectrum has 512 nonDC spectral lines. The computed spectrum has a total of 513 lines including the DC component. Acquire these same 1024 samples on an instrument with a standard 1/256 effective bandwidth, then use the equation in Table 1A.6 to ﬁnd the expected number of alias-free lines in the computed spectrum. Complete the necessary calculations with the following equations: Number of lines ¼ Eb N Number of lines ¼

1 £ 1024 ¼ 400 lines 2:56

Eb is entirely a hardware property. However, mathematically, you can use the FFT to compute the frequency spectrum up to the Nyquist frequency. Remember to account for the presence or absence of an antialiasing ﬁlter when performing the frequency analysis. The Frequency Analysis VIs compute every spectral line, alias-free or not. Use the frequency range to limit the analysis to the alias-free region of the spectrum with FFT subset and zoom FFT measurements. Use the SVT Get Spectrum Subset VI to limit the analysis to the alias-free region of the spectrum with baseband FFT measurements. Note: Table 1A.6 shows that the sampling frequency and the block size acquired during each cycle of a continuous acquisition completely determine the frequency-domain parameters in baseband FFT analysis. However, many stand-alone instruments are operated by specifying the frequency range of interest and the number of lines in the FFT. Table 1A.7 shows how a stand-alone instrument uses the frequency range of interest and the number of lines in the FFT to determine an appropriate sampling frequency and block size.

1A.7.2

Increasing Frequency Resolution

Increasing the frequency resolution helps one to distinguish two individual tones that are close together. For example, if you analyze a signal that contains two tones at 1000 and 1100 Hz, use a sampling frequency of 10,000 Hz. Acquire data for 10 msec and the frequency resolution is 100 Hz. Figure 1A.28 shows the results of this analysis. In this particular case, you cannot distinguish one tone from the other. Increase the acquisition time to 1 sec to achieve a frequency resolution of 1 Hz. Figure 1A.29 shows the results obtained with a 1 sec acquisition FIGURE 1A.28 Power spectrum obtained with an acquisition time of 10 msec. time. You can distinguish the individual tones with the increased acquisition time. The following strategies achieve a ﬁner frequency resolution: *

*

Decrease the sampling frequency, fs : Decreasing fs usually is not practical because decreasing fs reduces the frequency range. Increase the number of samples, N: Increasing N yields an increased number of lines over the original frequency range.

Implement the decreased fs strategy with zoom FFT analysis. Use baseband FFT and FFT-subset analyses to implement the increased N strategy. Baseband FFT analysis and FFT-subset analysis both achieve the same frequency resolution. However, FFT subset analysis only computes a narrow subset of the spectrum.

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Refer to Section 1A.7.2.1 and Section 1A.7.2.5 for examples that demonstrate the importance of frequency resolution in frequency analysis. The examples illustrate how to achieve a ﬁner frequency resolution with the frequency analysis tools in the Sound and Vibration Toolkit. 1A.7.2.1

Zoom FFT Analysis

In some applications, it is necessary to obtain the spectral information with a very ﬁne frequency resolution over a limited portion of the baseband span. In other words, you must zoom in on a FIGURE 1A.29 Power spectrum obtained with an spectral region to observe the details of that acquisition time of 1 sec. spectral region. Use the zoom FFT to obtain spectral information over a limited portion of the baseband span and with greater resolution. Just as in baseband analysis, the acquisition time determines the frequency resolution of the computed spectrum. The number of samples used in the transform determines the number of lines computed in the spectrum. Zoom FFT analysis achieves a ﬁner frequency resolution than the baseband FFT. The Zoom FFT VI acquires multiple blocks of data and downsamples to simulate a lower sampling frequency. The block size is decoupled from the achievable frequency resolution because the Zoom FFT VI accumulates the decimated data until you acquire the required number of points. Because the transform operates on a decimated set of data, you only need to compute a relatively small spectrum. The data are accumulated, so do not think of the acquisition time as the time required to acquire one block of samples. Instead, the acquisition time is the time required to accumulate the required set of decimated samples. The Zoom FFT VIs complete the following steps to process the sampled data: 1. Modulate the acquired data to center the analysis band at 0 frequency. 2. Filter the modulated data in the time domain to isolate the analysis band and prevent aliasing when the data are resampled at a lower sampling frequency. 3. Decimate the ﬁltered data to reduce the effective sampling frequency. 4. Accumulate the decimated data until sufﬁcient samples are available to compute the spectrum. 5. Use the Discrete Zak Transform (DZT) to efﬁciently compute the desired spectral lines. 6. Demodulate, or shift, the computed spectrum. 1A.7.2.2

Frequency Resolution of the Zoom FFT VIs

Use the Zoom FFT VIs to compute the spectrum of a signal over a narrow frequency range with an arbitrarily ﬁne frequency resolution. To get an approximation of the frequency resolution seen with the Zoom FFT VIs, use the following formula: frequency resolution <

stop frequency 2 start freqeuncy number of lines

Note: The exact frequency resolution is returned as df in the spectrum computed by the Zoom FFT VIs. 1A.7.2.3

Zoom Measurement

The following example demonstrates a zoom measurement of the power spectrum. Acquire a sine wave at 1390 Hz with a National Instruments DSA device and the VI displayed in Figure 1A.30. Acquire the signal at 51.2 kHz. The VI reads the data in blocks of 2048 samples. Compute the frequency resolution of this measurement using baseband analysis with the following equation: 51; 200 Hz ¼ 25 Hz 2048

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FIGURE 1A.30

Performing a zoom power spectrum measurement.

Use the SVT Zoom Power Spectrum VI located on the Zoom FFT palette to analyze a narrowband with a much ﬁner frequency resolution. Figure 1A.31 shows the result of limiting the measurement to the frequency band between 1 and 2 kHz and computing 400 lines. Derive the frequency resolution of the computed spectrum with the following equation: 2000 2 1000 Hz ¼ 2:5 Hz 400 lines

1A.7.2.4

FIGURE 1A.31 results.

Zoom power spectrum measurement

Zoom Settings

Figure 1A.32 shows the zoom settings control used to acquire the zoom measurement results displayed in Figure 1A.31. Use this control to specify the frequency range, window, number of lines, and percent overlap used in the zoom analysis. 1A.7.2.5

Subset Analysis

FIGURE 1A.32

Zoom settings control.

The Baseband Subset VIs located on the Baseband Subset palette allow one to compute a subset of the baseband FFT measurement. Subset analysis uses the DZT to compute a subset of the baseband FFT. The frequency resolution for spectral measurements computed with the Baseband Subset VIs equals the frequency resolution for measurements made with the Baseband FFT VIs. The acquisition time determines the frequency resolution. The only way to achieve a ﬁner frequency resolution is to increase the length of the time record. In the case of baseband or subset analysis, a longer time record implies a larger block size. The Baseband Subset VIs algorithm computes only the desired spectral lines. The only programming difference between the Baseband Subset VIs and the Baseband FFT VIs is the additional parameter frequency range. The frequency range parameter speciﬁes which spectral lines the Baseband Subset VI computes. The computed spectral lines are always inclusive of the start frequency and the stop frequency. Note: Setting the start frequency to 0 Hz and the stop frequency to fmax yields the same spectrum as the corresponding Baseband FFT VI. If you set the stop frequency to 2 1, the baseband subset VIs return the Nyquist frequency as the highest frequency in the computed spectrum.

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The following consideration can help you decide when to use the Baseband Subset VIs instead of the Baseband FFT VIs: *

*

The required block size yields an acceptable frequency resolution. The analysis of a narrow subset of the baseband span requires better processing performance than the Baseband FFT VI can provide.

1A.8

Transient Analysis

This Appendix discusses performing transient analysis with the Transient Analysis VIs located on the Transient Analysis palette.

1A.8.1

Transient Analysis with the Sound and Vibration Toolkit

Transient analysis is the analysis of nonstationary signals. The Transient Analysis VIs offer two different techniques for extracting information about transient signals. Use the short-time Fourier transform (STFT) for signals in which the frequency content changes relatively slowly with time. Use the shockresponse spectrum (SRS) for shock signals. You can use the STFT VIs to extract frequency information as a function of time directly from the signal of interest. Additionally, in the case of a rotating machine where a tachometer signal is simultaneously acquired with the signal of interest, the STFT VIs can extract frequency information as a function of the rotational speed. The results generated by the STFT are typically displayed on a waterfall display or on a colormap. The STFT VIs return the information needed to properly scale the axes of the displays. You can pass the information directly to a Waterfall Display VI. Use property nodes for the colormap display. You can use the SVT SRS VI to evaluate the severity of a shock signal. The results generated by the SRS are typically displayed on an X –Y graph. Note: Other LabVIEW toolkits are available that provide additional transient analysis capabilities. The Order Analysis Toolkit is designed for rotating machinery analysis and monitoring. The Signal Processing Toolkit has tools, such as wavelets and joint timefrequency analysis (JTFA), for the analysis of fast transients.

1A.8.2 Time

Performing an STFT vs.

The STFT available in the Sound and Vibration Toolkit can compute multiple Fourier transforms on the time-domain signal with or without overlapping. For example, analyze a waveform containing 10 sec of data acquired at 51.2 kS/sec. The signal is a chirp signal with the following attributes: *

*

FIGURE 1A.33

Chirp signal.

Start frequency ¼ 10 Hz End frequency ¼ 10,000 Hz

Figure 1A.33 shows the signal corresponding to the ﬁrst 200 msec of the waveform. Figure 1A.34 shows the result of applying a baseband FFT on the entire waveform. Note: No window is applied on the signal.

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FIGURE 1A.34

Baseband FFT on a chirp signal.

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The spectrum is ﬂat from 10 Hz to 10 kHz. Only noise is measured at frequencies above 10 kHz. Unfortunately, this measurement does not provide any information about how the frequency content of the signal changes with time. However, the STFT can reveal useful information about the time dependence of the frequency content. Instead of computing a single FFT on the whole data set, you can divide the data set into smaller FIGURE 1A.35 Time segment control. blocks and compute FFTs on these smaller data blocks. For example, divide the signal into 100 msec blocks and perform an FFT on each of the blocks with the SVT STFT vs. Time VI. Subdivide the time-domain signal by conﬁguring the time segment control displayed in Figure 1A.35. Leave from [s] and to [s] each equal to 2 1.00 to ensure that the entire signal is used in the STFT computation. In this particular example, the 21.00 setting in both from [s] and to [s] is equivalent to setting from [s] to 0 and to [s] to 10. Create a 100 msec time increment by setting time increment to 100.00 and time increment units (%) to msec. The 100 msec time increment causes the SVT STFT vs. Time VI to compute one FFT every 100 msec. Setting time increment is independent from selecting the FFT block size. 1A.8.2.1

Selecting the FFT Block Size

In addition to the time segment, one can adjust the FFT block size. For example, analyze a chirp signal having the following attributes: *

*

Start frequency ¼ 10 Hz End frequency ¼ 10,000 Hz

The measurement is performed using the following settings: *

*

*

*

Acquisition time ¼ 10 sec Sampling frequency ¼ 51.2 kS/sec FFT block size ¼ 1024 samples or 512 lines (400 alias-free lines) Time increment ¼ 100 msec.

Based on the sampling frequency of 51,200 Hz, a 1024 sample FFT requires a 20 msec block of data, leading to a frequency resolution of 50 Hz. Because the time increment is 100 msec and a 1024 sample FFT only requires a 20 msec block, only one block out of ﬁve is used for computation. Figure 1A.36 shows the result obtained with a 1024 sample FFT.

FIGURE 1A.36

STFT using a 1024 sample block size.

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FIGURE 1A.37

1-99

STFT using a 4096 sample block size.

If you select an FFT Block size of 4096 samples, or 1600 alias-free lines, the resolution improves, as illustrated in Figure 1A.37. However, the increased resolution comes with the expense of extra processing. 1A.8.2.2

Overlapping

Overlapping is a method that uses a percentage of the previous data block to compute the FFT of the current data block. When combined with windowing, overlapping maximizes the use of the entire data set. If no overlapping is used, the part of the signal close to the window edges becomes greatly attenuated. The attenuation of the signal near the window edges could result in the loss of information in the region near the window edges. Note: Set the desired overlapping rate by FIGURE 1A.38 50% overlap. specifying % in the time increment units (%) in the time increment control. Refer to Figure 1A.35 for the location of this control. No overlapping, or 0%, corresponds to a time increment of 100%. An overlapping of 75% corresponds to a time increment of 25%. An overlapping of 50% corresponds to a time increment of 50%, and so forth. The advantage of using the time increment control is that one can specify values greater than 100%. For example, a time increment of 200% corresponds to computing an FFT on every other block of data. Figure 1A.38 and Figure 1A.39 illustrate the overlapping process. Figure 1A.38 shows a 50% overlap. Figure 1A.39 shows the resulting subdivisions when one uses a 50% overlap and a Hamming window. 1A.8.2.3

Using the SVT STFT vs. Time VI

The following example illustrates how to use the SVT STFT vs. Time VI. Figure 1A.40 shows the block diagram. The example in Figure 1A.40 acquires 10 sec of data at a sample rate of 51.2 kHz. After scaling, the signal is sent to the SVT STFT vs. Time VI. The result is displayed on the intensity graph in Figure 1A.41. Note: Use the X scale and Y scale offset and multiplier properties to properly scale the axes of the intensity graph. In this example, the X scale range is 0 to 10 sec. The Y scale range is 0 to 25,600 Hz. The Nyquist frequency is 25,600 Hz. You can adjust the Z scale so that only the relevant part of the signal is displayed. In other words, you can hide noise in the displayed signal by increasing the minimum limit of the Z-axis. Refer to the LabVIEW Help for information about the offset and multiplier properties for graph controls.

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FIGURE 1A.39

Subdivisions of the time-domain waveform.

FIGURE 1A.40

Use of the SVT STFT vs. Time VI.

FIGURE 1A.41

1A.8.3

STFT vs. time graph.

Performing an STFT vs. Rotational Speed

Analyzing the frequency content as a function of the rotational speed is helpful when dealing with measurements on rotating machinery. Use the SVT STFT vs. RPM (analog) VI to analyze the frequency content as a function of the rotational speed.

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1A.8.3.1

1-101

Converting the Pulse Train to Rotational Speed

Use the SVT Convert to RPM (analog) VI to convert a pulse train acquired by a tachometer or encoder to the rotational speed expressed in rotations per minute (RPM). Note: For simplicity, the remainder of this Appendix uses the term tachometer to denote both a tachometer and an encoder. In this example, an accelerometer is mounted at the test location for an engine run-up. A tachometer is used to measure the speed of the shaft and returns one pulse per revolution as a transistor–transistor logic (TTL) signal. Use the tach info control to specify the characteristics of the pulses generated by the tachometer. Figure 1A.42 shows the settings for the tachometer info control. Figure 1A.43 shows a simulated tachometer signal. You can use the SVT Convert to RPM (analog) VI to measure the rotational speed in RPM as a function of time. Figure 1A.44 shows the result obtained with the SVT Convert to RPM (analog) VI and a simulated tachometer signal. 1A.8.3.2

FIGURE 1A.42

Tachometer info control.

FIGURE 1A.43

Tachometer signal.

STFT vs. RPM

You also can display the STFT of an input signal as a function of the rotational speed based on the tachometer signal. Two input signals are needed, the signal of interest and the signal from the tachometer. Again, an engine run-up serves as a good example of computing an STFT as a function of the rotational speed. During an engine run-up, the sound pressure close to the engine is measured with a microphone. Figure 1A.45 shows the signal acquired by the microphone. The signal from the tachometer is also acquired. The measured tachometer signal is converted to RPM with the SVT Convert to RPM (analog) VI. Figure 1A.46 shows the rotational speed as a function of time, as computed by the SVT Convert to RPM (analog) VI. Using the SVT STFT vs. RPM (analog) VI allows you to measure the frequency content of the signal as a function of the rotational speed of the engine. Figure 1A.47 displays the results obtained with the SVT STFT vs. RPM (analog) VI on an intensity graph.

FIGURE 1A.44 (analog) VI.

Result from SVT convert to RPM

FIGURE 1A.45 engine run-up.

Microphone signal obtained during

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1A.8.4

Measuring a Shock Response Spectrum

Obtain the SRS by applying the acquired shock pulse to a series of single degree of freedom (SDOF) systems. Plot the system maximum response as resonance frequency of the system. An SDOF mechanical system consists of the following components: *

*

*

Mass, whose value is represented with the variable m Spring, whose stiffness is represented with the variable k Damper, whose damping coefﬁcient is represented with the variable c

FIGURE 1A.46 Rotational speed as a function of time during engine run-up.

The resonance frequency fN, and the critical damping factor, z; characterize an SDOF system, where sﬃﬃﬃﬃ 1 k fN ¼ 2p m c z ¼ pﬃﬃﬃﬃ 2 km For light damping, where z is less than or equal to 0.05, the peak value of the frequency response occurs in the immediate vicinity of fN and is given by the following equation, where Q is the resonant gain: Q¼

FIGURE 1A.47 Intensity graph of sound pressure level for an engine run-up.

1 2z

Figure 1A.48 illustrates the response of an single-DoF system to a half-sine pulse with a 10g acceleration amplitude and 10 msec duration. The top graph shows the time-domain acceleration. The middle graph is the single-DoF system response with a 50 Hz resonance frequency. The bottom graph is the single-DoF system response with a 150 Hz resonance frequency. In both cases, z is 0.05. Use the signals shown in Figure 1A.48 to construct the SRS. For example, the maximax, the absolute maximum response of the calculated shock response signal over the entire signal duration, uses the absolute maximum system FIGURE 1A.48 Single-DoF system response to a halfresponse as a function of the system natural sine shock. frequency. Figure 1A.49 illustrates the maximax SRS for the same half-sine pulse. Note: Each computed SRS is speciﬁc to the pulse used to perform the measurement. You can use other types of shock spectra depending on the application. These spectra include the initial shock response from the system response over the pulse duration or from the residual shock

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spectrum from the system response after the pulse. You can use the positive maximum, the negative maximum, or the absolute maximum response signal value. The Sound and Vibration Toolkit uses the Smallwood algorithm to compute the SRS. The SVT Shock Response Spectrum VI also offers the ability to preprocess the time-domain signal to improve SRS results. You can remove any DC component or apply a low-pass ﬁlter with a selectable cut-off frequency. The SVT Shock Response Spectrum VI can compute the SRS from the absolute acceleration response or from the relative displacement response. Use the model control on the SVT Shock Response Spectrum VI to select the appropriate response. Figure 1A.50 shows how to use the SVT Shock FIGURE 1A.49 Half-sine pulse SRS (Maximax). Response Spectrum VI. The example in Figure 1A.50 acquires 1000 samples of data from an accelerometer during a shock. The shock signal triggers the acquisition. The program stores 100 samples before the trigger to properly capture the entire shock signal. Figure 1A.51 displays the acquired time-domain signal and the computed SRS.

1A.9

Waterfall Display

This Appendix discusses using the Waterfall Display VIs located on the Waterfall Display palette.

1A.9.1

Using the Display VIs

Waterfall display is a visualization technique that permits the visual representation of various analyses of nonstationary signals, such as machine run-up, coast down, transients, and others. Use the Waterfall Display VIs to display FFT spectra from frequency analysis or transient analysis and octave spectra from octave analysis in waterfall graphs. Refer to Front Panel Displays in the LabVIEW Help for more information about displaying octave results. Speciﬁc Waterfall Display VIs display the results of frequency analysis and octave analysis in a waterfall graph. The waterfall display opens in an external window called the Waterfall window.

FIGURE 1A.50

Using the SVT shock response spectrum VI.

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Complete the following basic steps to display results on a waterfall graph: 1. Initialize the display. 2. Send data to the display. 3. Close the waterfall display. Figure 1A.52 shows the Waterfall window. 1A.9.1.1

Initializing the Display

Use the SVL Initialize Waterfall Display VI to create a reference to a waterfall display. If you are displaying octave spectra, use the SVT Initialize Waterfall Display for Octave VI. Both of the initializing VIs also enable one to deﬁne graph properties, including the window title, the bounds of the external window, and the colors used in the waterfall display. 1A.9.1.2

Sending Data to the Display

FIGURE 1A.51

Acquired SRS (Maximax).

The Waterfall window does not open until it receives data sent to it. Use the SVL Send Data to Waterfall VI to send data to a waterfall display. Use the SVT Send Data to Waterfall for Octave VI to send octave data to a waterfall display. The SVL Send Data to Waterfall VI is polymorphic and accepts an array of spectra, such as that returned by a power spectrum, a twodimensional (2D) array, or a STFT. 1A.9.1.3 Waterfall Display for Frequency Analysis The following example shows how to accumulate FIGURE 1A.52 Waterfall display for frequency 20 spectra and display them in a waterfall graph. analysis. Figure 1A.53 shows the block diagram for the VI. Twenty data blocks of 1024 samples are acquired. The power spectrum is computed on each block. The autoindexing capability of the For Loop is used to build an array of 20 spectra. The array or spectra is sent to the waterfall display. Refer to the LabVIEW Help for information about autoindexing. 1A.9.1.4

Waterfall Display for Transient Analysis

This example illustrates how to use the waterfall display in conjunction with the Transient Analysis VIs. Figure 1A.54 shows the block diagram for the example VI.

FIGURE 1A.53

Waterfall display for frequency analysis VI block diagram.

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FIGURE 1A.54

STFT VI block diagram.

The data are scaled and sent to the SVT STFT vs. Time VI. The SVT STFT vs. Time VI returns a 2D array. You can use the results in the 2D array in an intensity graph or connect the 2D array directly to the SVL Send Data to Waterfall VI. Figure 1A.54 shows the 2D array connected directly to the Waterfall VI. The while loop keeps the waterfall display open until the Stop control is set to TRUE. Note: Connect f 0 and delta f and y 0 and delta y on the SVL Send Data to Waterfall VI to ensure the graph shows the proper scales. Figure 1A.55 shows the result obtained with the STFT VI illustrated in Figure 1A.54. 1A.9.1.5

FIGURE 1A.55

STFT waterfall display.

Waterfall Display for Octave Spectra

To display octave spectra in a waterfall display, use the SVT Initialize Waterfall Display for Octave and SVT Send Data to Waterfall for Octave VIs. Figure 1A.56 shows the block diagram for a VI displaying octave spectra in a waterfall display. Figure 1A.57 shows the waterfall display created by the VI in Figure 1A.56.

1A.10

Swept-Sine Measurements

This Appendix discusses using the swept-sine VIs located on the Swept Sine palette (see Chapter 3). The swept-sine measurements include dynamic measurements for stimulus level, response level, frequency response (gain and phase), THD, and individual harmonic distortion.

1A.10.1

Swept-Sine Overview

Swept sine is a technique for characterizing the frequency response of the DUT. Two techniques are commonly used in swept-sine measurements. The ﬁrst technique slowly sweeps through a range of

FIGURE 1A.56

Block diagram for VI displaying octave spectra.

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frequencies in a manner similar to a chirp. Figure 1A.58 shows an example of the excitation signal for this form of swept-sine measurement. The second technique steps through a range of frequencies. Figure 1A.59 shows an example of the excitation signal for this form of swept-sine measurement. The swept sine implemented in the Sound and Vibration Toolkit generates an excitation signal that steps through a range of test frequencies, similar to the signal in Figure 1A.59. Both techniques can yield similar results. However, they require very different measurement analysis FIGURE 1A.57 Waterfall display for octave spectra processes. analysis. Swept-sine frequency-response measurements compare a response signal to the stimulus tone in order to compute the FRF of the DUT. The magnitude of the FRF is equivalent to gain and represents the ratio of the output level to the input level for each test frequency. The phase of the FRF is equivalent to the phase lag introduced by the DUT for each test frequency. Swept-sine measurements require a signal source. The stimulus signal is always a single tone that excites the DUT at the test frequency. Since the stimulus is a single tone, swept-sine analysis can measure the harmonic distortion while simultaneously measuring the linear response.

1A.10.2

Choosing Swept-Sine vs. FFT Measurements

The frequency response of the DUT is a useful tool. The Sound and Vibration Toolkit provides two distinct techniques to measure the frequency response. The swept-sine technique performs single –tone

FIGURE 1A.58

Sweeping swept sine example.

FIGURE 1A.59

Stepping swept sine example.

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Swept-Sine Frequency Response

FFT-Based Frequency Response

Single-tone excitation Can measure harmonics Arbitrary test frequencies Longer test time for many test frequencies Better dynamic range possibility

Broadband excitation Cannot measure harmonics Linearly spaced frequency resolution

measurements at each test frequency. The FFTbased technique measures the response over the entire acquisition bandwidth. Table 1A.8 lists the basic differences between swept-sine and FFTbased techniques for measuring frequency response. Swept-sine measurements offer superior dynamic range over FFT-based measurements because you can optimize the signal level and input ranges at each test frequency. FFT-based FIGURE 1A.60 Swept-sine and FFT measurements. techniques must specify a signal level and input ranges appropriate for the maximum broadband response. Figure 1A.60 shows the simulated frequency response function for a four-DoF system. The peak at 17.6 Hz has a magnitude roughly 1000 times larger than the peak at 5.8 Hz. To use an FFT-based technique, use broadband excitation to excite the entire frequency range of interest, to measure the frequency response. This situation forces one to set the input range so that the overall response does not overload the DUT or the acquisition device. Therefore, when you measure the response at 5.8 Hz, you lose 60 dB of measurement dynamic range. The swept-sine technique allows you to tailor the excitation amplitude to the speciﬁc test frequency, preserving the full measurement dynamic range. FFT-based measurements are limited to a linearly spaced frequency resolution determined by the sample rate and the block size. Refer to Appendix 1A.7 for more information on FFT-based measurements. When the response changes rapidly, this frequency resolution may not yield enough information about the dynamic response. Also, a linear resolution may yield an excessive amount of information in frequency regions where the dynamic response is relatively constant. Swept-sine analysis has the ability to test arbitrary frequency resolutions that are linear, logarithmic, or adapted to the dynamic response of the DUT. When the frequency resolution is adapted to the DUT dynamic response, you can test more frequencies in regions where the dynamic response is of interest to your application and fewer where it is not. The main beneﬁt of swept-sine analysis is the ability to measure harmonic distortion simultaneously with linear response. FFT-based analysis offers a speed advantage for broadband measurements with many test frequencies.

1A.10.3

Taking a Swept-Sine Measurement

Use the SVT Initialize Swept Sine VI to create a new swept-sine task for the designated device, source channel settings, and acquisition channel settings. Swept sine in the Sound and Vibration Toolkit only supports measurements on a single device with output and input capabilities. Use conﬁgure swept sine VIs in the conﬁgure swept sine palette to conﬁgure the scaling, test frequencies, averaging, delays, and other measurement settings. These conﬁguration VIs allow control over basic and advanced measurement parameters. The order in which you place the conﬁguration VIs is important, as it allows you to customize a swept-sine measurement. For example, you can easily generate

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FIGURE 1A.61

Customizing a swept-sine measurement.

100 logarithmically spaced test frequencies in the audio range, then apply inverse A-weighted scaling to the excitation level by adding code similar to that in Figure 1A.61 into your swept-sine application. You can use the swept-sine conﬁguration VIs to customize your swept-sine application. For example, to speed up a swept-sine measurement, reduce the settling or integration time speciﬁed by the SVT Set Swept Sine Averaging VI. You also can conﬁgure the device IEPE with the SVT Set Swept Sine Coupling and IEPE Excitation (DAQmx) VI. You also can reduce the block duration input to SVT Set Swept Sine Block Duration VI. Note: The minimum block duration is limited by the capabilities of the computer processing the measurement. A very small block duration can result in a loss of continuous processing, causing the swept-sine measurement to stop and return an error. Use the SVT Start Swept Sine VI to begin the generation and acquisition. The VI ﬁlls the device output buffer with zeros before writing the ﬁrst test frequency excitation. The SVT Swept Sine Engine VI continually acquires data and processes it to remove samples acquired during delays, transitions, and settling periods. The SVT Swept Sine Engine VI performs measurement analysis on samples acquired during integration periods. The SVT Swept Sine Engine VI updates the excitation to excite the DUT at the next test frequency after it integrates sufﬁcient data at the current test frequency. Note: The transition to the next excitation tone, both frequency and amplitude, always occurs at a zero crossing to minimize transients introduced to the DUT. Use the Read Swept Sine Measurements VIs in the Read Swept Sine Measurements palette to read the raw measurements, scale the measurements, and perform additional conversions to display and report the data in the desired format. Use the SVT Close Swept Sine VI to stop the generation and acquisition, and clear the sweptsine task.

1A.10.4

Swept-Sine Measurement Example

This example of a swept-sine application measures the frequency response and harmonic distortion of a notch ﬁlter. In this example, a NI PXI-4461 generates the excitation signal and acquires the stimulus and response signals. Figure 1A.62 illustrates the connection scheme used in this example to measure the dynamic response of the DUT using a swept-sine measurement. The acquired stimulus signal on the analog input channel 0, the AI0, is the generated excitation signal from the analog output channel 0, AO0. The NI PXI-4461 converts the desired stimulus signal from digital data to an analog signal and outputs that signal on AO0. The excitation signal is

FIGURE 1A.62 diagram.

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Swept-sine measurement connection

Vibration Instrumentation

FIGURE 1A.63 TABLE 1A.9

2 3 4

5

6 7

8

9 10 11 12

Block diagram of SVXMPL_swept sine FRF DAQmx VI.

Swept Sine Measurement Steps

Step Number 1

1-109

Description Initialize a swept-sine measurement by specifying the hardware device and channel settings Specify the scaling that will be applied to the acquired stimulus and response data. Conﬁgure the source by specifying the test frequencies, amplitude, and whether or not the sweep automatically restarts after completion Set the settling and integration parameters to allow sufﬁcient time for the DUT to settle before the measurement is performed at the new test frequency and that there is sufﬁcient integration time to achieve the desired level of accuracy Set the block duration input terminal for the measurement to be small enough to give a reasonable test time and large enough so that it does not put the test computer at risk of being unable to continuously generate and read the signals. The smaller the block size, the faster the swept sine can transition from one test frequency to the next Explicitly set the sample rate for the measurement. The rate is automatically selected if this VI is not used. The same rate is used for input and output channels Specify the propagation time terminal input speciﬁc to the DAQ device being used for the measurement. You can measure the device propagation time using the SVL Measure Propagation Delay VI. Refer to Appendix 3, Scaling and Calibration, for more information Conﬁgure the harmonic distortion measurement by specifying the maximum harmonic to use in the computation of the THD. Only those harmonics speciﬁed in the harmonics to visualize array return individual harmonic components Start the swept sine to perform the hardware conﬁguration and start the output and input tasks. Channel synchronization is performed internally in this VI Generate the excitation and acquire the stimulus and response data at each test frequency Convert the raw data to the speciﬁed format in order to display and report measurement results Stop the swept-sine measurement and clear the output and input tasks to release the device

Optional or Required Required Optional Required Required

Optional

Optional Optional

Required if performing distortion measurements Required Required Required Required

connected to both the stimulus input channel AI0 and the input terminal of the DUT. The response signal is connected from the output terminal of the DUT to the response input channel AI1. The DUT for this example is a notch ﬁlter centered at 1 kHz. Figure 1A.63 shows the block diagram of the example SVXMPL_swept sine FRF DAQmx VI, which ships with the Sound and Vibration Toolkit.

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1-110

Vibration Monitoring, Testing, and Instrumentation

FIGURE 1A.64

FIGURE 1A.65

Time-domain results.

Magnitude and phase response of a 1 kHz notch ﬁlter.

FIGURE 1A.66

THD vs. frequency.

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Vibration Instrumentation

1-111

FIGURE 1A.67

THD vs. frequency results.

Table 1A.9 documents the actions performed by the VIs in Figure 1A.63. Some steps are required and must be done for the VI to function correctly. The optional steps allow you to customize your measurement. The while loop in Figure 1A.63 controls the synchronized generation and acquisition. Display controls and measurement indicators are updated inside the while loop. This loop allows for the monitoring of intermediate results. Many of the steps in Table 1A.9 are conﬁguration steps. Through the Sound and Vibration Toolkit swept-sine conﬁguration VIs, you can specify numerous conﬁguration parameters to achieve ﬁne control of the swept-sine measurement parameters. For many applications two or three conﬁguration VIs are sufﬁcient. It is important to allow for the propagation delay of the DAQ or DSA device. This delay is speciﬁc to the device used to perform the measurement. To determine the device propagation delay, refer to the device documentation or measure the delay with the SVL Measure Propagation Delay VI. Figures 1A.64 to 1A.67 display measurement results obtained with the SVXMPL_swept sine FRF DAQmx VI example program. Figure 1A.64 shows the time-domain stimulus and response signals for the 138.49 Hz test frequency. From the time-domain data, you can see that the notch ﬁlter has attenuated the signal and introduced a phase shift. Figure 1A.65 shows the magnitude and phase responses of the notch ﬁlter at all the test frequencies in the magnitude and phase spectra in the Bode plot. In addition to measuring the frequency response, this example simultaneously measures the harmonic distortion at each test frequency. Figure 1A.66 shows the graph of THD vs. frequency. You expect to see a peak in the THD at the notch frequency. The peak occurs because the fundamental frequency is attenuated at the notch frequency. However, the graph indicates that this measurement has failed to accurately identify the power in the harmonic distortion components. For the example in Figure 1A.66, the number of integration cycles is two. More integration cycles must be speciﬁed to perform accurate harmonic distortion measurements. If you change the number of integration cycles to ten and rerun the example, you obtain the THD vs. frequency results displayed in Figure 1A.67. Now, with a sufﬁcient number of integration cycles speciﬁed, you can see the characteristic peak in the THD at the center frequency of the notch ﬁlter.

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1-112

Vibration Monitoring, Testing, and Instrumentation

Bibliography Crocker and Malcolm, J. 1998. Handbook of Acoustics, Wiley, New York. Design Response of Weighting Networks for Acoustical Measurements, ANSI S1.42-2001, American National Standards Institute, Washington, 1986. Electroacoustics — Sound Level Meters, International Standard IEC 61672, 1st ed., 2002–2005, International Electrotechnical Commission, Geneva, Switzerland, 2002. Hassall, J.R. and Zaveri, K. 1988. Acoustic Noise Measurements, Bruel & Kjær, Nærum, Denmark. Measurement of Audio-Frequency Noise Voltage Level in Sound Broadcasting, ITU-R Recommendation 468-4, 1986. Octave-Band and Fractional Octave-Band Filters, International Standard IEC 1260, 1st ed., 1995-07. International Electrotechnical Commission, Geneva, Switzerland, 1995. Preferred Frequencies for Measurements, International Standard IEC 266, 1st ed., 1975-07-15, International Electrotechnical Commission, Geneva, Switzerland, 1975. Psophometer for Use on Telephone-Type Circuits, ITU-T Recommendation O.41, Revised, 1993–1996. Telecommunication Standardization Sector of the International Telecommunication Union, 1995. Randall, R.B. 1987. Frequency Analysis, BrY¨el & Kjær, Nærum, Denmark. Smallwood, D., An improved recursive formula for calculating shock response spectra, Shock Vib. Bull., 51, Pt 2, 211 –217, 1981, May 1981. Speciﬁcation for Octave-Band and Fractional Octave-Band Analog and Digital Filters, ANSI S1.11-1986, Acoustical Society of America, New York, 1986b. Speciﬁcation for Sound Level Meters, ANSI S1.4-1983, American National Standards Institute, Washington, 1983.

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2

Signal Conditioning and Modiﬁcation 2.1 2.2

2.3 2.4 2.5

2.6 2.7

2.8

Clarence W. de Silva The University of British Columbia

2.9

Introduction .......................................................................... 2-2 Ampliﬁers ............................................................................... 2-2

Operational Ampliﬁer † Use of Feedback in Opamp † Voltage, Current, and Power Ampliﬁers † Instrumentation Ampliﬁers † Ampliﬁer Performance Ratings † Component Interconnection

Analog Filters ......................................................................... 2-15

Passive Filters and Active Filters † Low-Pass Filters † High-Pass Filters † Band-Pass Filters † Band-Reject Filters

Modulators and Demodulators ........................................... 2-29

Amplitude Modulation † Application of Amplitude Modulation † Demodulation

Analog – Digital Conversion ................................................. 2-37

Digital-to-Analog Conversion † Analog-to-Digital Conversion † Analog-to-Digital Converter Performance Characteristics † Sample-and-Hold Circuitry † Digital Filters

Bridge Circuits ....................................................................... 2-43

Wheatstone Bridge † Constant-Current Bridge Ampliﬁers † Impedance Bridges

†

Bridge

Linearizing Devices ............................................................... 2-49

Linearization by Software † Linearization by Hardware Logic † Analog Linearizing Circuitry † Offsetting Circuitry † Proportional-Output Circuitry

Miscellaneous Signal Modiﬁcation Circuitry ..................... 2-56

Phase Shifters † Voltage-to-Frequency Converter † Frequency-to-Voltage Converter † Voltage-to-Current Converters † Peak-Hold Circuits

Signal Analyzers and Display Devices ................................. 2-62

Signal Analyzers

†

Oscilloscopes

Summary This chapter concerns the conditioning of signals in a vibrating system and the conversion of signals in one form to another as needed. Ampliﬁcation, ﬁltering, modulation, demodulation, analog/digital conversion, voltage/ frequency conversion, voltage/current conversion, linearization, bridge circuits, and signal analysis and display devices are presented. Hardware, software, and techniques are considered. Issues of impedance and loading associated with the interconnection of components are addressed.

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2-2

2.1

Vibration Monitoring, Testing, and Instrumentation

Introduction

Signal modiﬁcation is an important function in many applications of vibration. The tasks of signal modiﬁcation may include: signal conditioning (e.g., ampliﬁcation, and analog and digital ﬁltering); signal conversion (e.g., analog-to-digital conversion, digital-to-analog conversion, voltage-to-frequency conversion, and frequency-to-voltage conversion); modulation (e.g., amplitude modulation, frequency modulation, phase modulation, pulse-width modulation, pulse-frequency modulation, and pulse-code modulation); and demodulation (the reverse process of modulation). In addition, many other types of useful signal modiﬁcation operations can be identiﬁed. For example, sample and hold circuits are used in digital data acquisition systems. Devices such as analog and digital multiplexers and comparators are needed in many applications of data acquisition and processing. Phase shifting, curve shaping, offsetting, and linearization can also be classiﬁed as signal modiﬁcation. This chapter describes signal conditioning and modiﬁcation operations that are useful in vibration applications. Signal modiﬁcation plays a crucial role in component interfacing. When two devices are interfaced, it is essential to ensure that a signal leaving one device and entering the other will do so at proper signal levels (voltage, current, power), in the proper form (analog, digital), and without distortion (loading and impedance considerations). A signal should be properly modiﬁed for transmission by ampliﬁcation, modulation, digitizing, and so on, so that the signal/noise ratio of the transmitted signal is sufﬁciently large at the receiver. The signiﬁcance of signal modiﬁcation is clear from these observations.

2.2

Ampliﬁers

The level of an electrical signal can be represented by variables such as voltage, current, and power. Analogous across variables, through variables, and power variables can be deﬁned for other types of signals (e.g., mechanical) as well. Signal levels at various interface locations of components in a vibratory system have to be properly adjusted for proper performance of these components and of the overall system. For example, input to an actuator should possess adequate power to drive the actuator. A signal should maintain its signal level above some threshold during transmission so that errors due to signal weakening will not be excessive. Signals applied to digital devices must remain within the speciﬁed, logic levels. Many types of sensors produce weak signals that have to be upgraded before they can be fed into a monitoring system, data processor, controller, or data logger. Signal ampliﬁcation concerns the proper adjustment of a signal level for performing a speciﬁc task. Ampliﬁers are used to accomplish signal ampliﬁcation. An ampliﬁer is an active device that needs an external power source to operate. Even though active circuits, ampliﬁers in particular, can be developed in the monolithic form using an original integrated-circuit (IC) layout so as to accomplish a particular ampliﬁcation task, it is convenient to study their performance using the operational ampliﬁer (opamp) as the basic element. Of course, operational ampliﬁers are widely used not only for modeling and analyzing other types of ampliﬁer but also as basic elements in building other kinds of ampliﬁer. For these reasons, our discussion on ampliﬁers will revolve around the operational ampliﬁer.

2.2.1

Operational Ampliﬁer

The origin of the operational ampliﬁer dates to the 1940s when the vacuum tube operational ampliﬁer was introduced. The operational ampliﬁer, or opamp, got its name due to the fact that originally it was used almost exclusively to perform mathematical operations; for example, it was used in analog computers. Subsequently, in the 1950s, the transistorized opamp was developed. It used discrete elements such as bipolar junction transistors and resistors. The opamp was still too large in size, consumed too much power, and was too expensive for widespread use in general applications. This situation changed in the late 1960s when IC opamp was developed in the monolithic form as a single IC chip. Today, the IC opamp, which consists of a large number of circuit elements on a substrate, typically of a single silicon crystal (the monolithic form), is a valuable component in almost any signal modiﬁcation device.

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Signal Conditioning and Modiﬁcation

2-3

An opamp could be manufactured in the discrete-element form using perhaps ten bipolar junction transistors and as many discrete resistors; alternatively (and preferably), it may be manufactured in the modern monolithic form as an IC chip that may be equivalent to over 100 discrete elements. In any form, the device has an input impedance, Zi ; an output impedance, Zo ; and a gain, K: Hence, a schematic model for an opamp can be given as in Figure 2.1(a). The conventional symbol of an opamp is shown in Figure 2.1(b). Typically, there are about six terminals (lead connections) to an opamp. For example, there may be two input leads (a positive lead with voltage vip and a negative load with voltage vin ), an output lead (voltage vo ), two bipolar power supply leads ðþvs and 2vs Þ; and a ground lead. Note from Figure 2.1(a) that, under open-loop (no feedback) conditions

vs (Power Supply) +

vip Inputs

vi

vin

Zo

Zi

Output vo= K vi

+ Kv i −

−

(a) vin (b)

vip

− +

vo

FIGURE 2.1 Operational ampliﬁer: (a) a schematic model; (b) conventional symbol.

vo ¼ Kvi

ð2:1Þ

in which the input voltage, vi ; is the differential input voltage deﬁned as the algebraic difference between the voltages at the positive and negative lead; thus vi ¼ vip 2 vin

ð2:2Þ

The open loop voltage gain K is very high (105 to 109) for a typical opamp. Furthermore, the input impedance, Zi ; could be as high as 1 MV and the output impedance is low, of the order of 10 V. Since vo is typically 1 to 10 V, from Equation 2.1 it follows that vi ø 0 since K is very large. Hence, from Equation 2.2, we have vip ø vin : In other words, the voltages at the two input leads are nearly equal. Now, if we apply a large voltage differential vi (say, 1 V) at the input then, according to Equation 2.1, the output voltage should be extremely high. This never happens in practice, however, since the device saturates quickly beyond moderate output voltages (of the order of 15 V). From Equation 2.1 and Equation 2.2, it is clear that if the negative input lead is grounded (i.e., vin ¼ 0), then vo ¼ Kvip

ð2:3Þ

and, if the positive input lead is grounded (i.e., vip ¼ 0) vo ¼ 2Kvin

ð2:4Þ

Accordingly, vip is termed noninverting input and vin is termed inverting input.

Example 2.1 Consider an opamp having an open-loop gain of 1 £ 105. If the saturation voltage is 15 V, determine the output voltage in the following cases: 1. 2. 3. 4. 5. 6.

5 mV at the positive lead and 2 mV at the negative lead. 2 5 mV at the positive lead and 2 mV at the negative lead. 5 mV at the positive lead and 22 mV at the negative lead. 2 5 mV at the positive lead and 2 2 mV at the negative lead. 1 V at the positive lead and negative lead grounded. 1 V at the negative lead and positive lead grounded.

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2-4

Vibration Monitoring, Testing, and Instrumentation TABLE 2.1 vip 5 mV 25 mV 5 mV 25 mV 1V 0

Solution to Example 2.1 vin

vi

vo

2 mV 2 mV 22 mV 22 mV 0 1V

3 mV 27 mV 7 mV 23 mV 1V 21 V

0.3 V 20.7 V 0.7 V 20.3 V 15 V 215 V

Solution This problem can be solved using Equation 2.1 and Equation 2.2. The results are given in Table 2.1. Note that, in the last two cases, the output will saturate and Equation 2.1 will no longer hold. Field effect transistors (FET), for example, metal oxide semiconductor ﬁeld effect transistors (MOSFET), could be used in the IC form of an opamp. The MOSFET type has advantages over many other types; for example, such opamps have higher input impedance and more stable output (almost equal to the power supply voltage) at saturation. This makes the MOSFET opamps preferable over bipolar junction transistor opamps in many applications. In analyzing operational ampliﬁer circuits under unsaturated conditions, we use the following two characteristics of an opamp: 1. Voltages of the two input leads should be (almost) equal. 2. Currents through each of the two input leads should be (almost) zero. As explained earlier, the ﬁrst property is credited to high open-loop gain and the second property to high input impedance in an operational ampliﬁer. We shall repeatedly use these two properties to obtain input –output equations for ampliﬁer systems.

2.2.2

Use of Feedback in Opamp

The operation ampliﬁer is a very versatile device, primarily due to its very high input impedance, low output impedance, and very high gain. However, it cannot be used without modiﬁcation as an ampliﬁer because it is not very stable, as shown in Figure 2.1. Two factors that contribute to this problem are: 1. Frequency response 2. Drift Stated in another way, opamp gain, K; does not remain constant; it can vary with the frequency of the input signal (i.e., frequency-response function is not ﬂat in the operating range); also, it can vary with time (i.e., drift). The frequency-response problem arises due to circuit dynamics of an operational ampliﬁer. This problem is usually not severe unless the device is operated at very high frequencies. The drift problem arises due to the sensitivity of gain, K; to environmental factors such as temperature, light, humidity, and vibration, and as a result of variation of K due to aging. Drift in an opamp can be signiﬁcant and steps should be taken to remove that problem. It is virtually impossible to avoid drift in gain and frequency-response error in an operational ampliﬁer. However, an ingenious way has been found to remove the effect of these two problems at the ampliﬁer output. Since gain K is very large, by using feedback we can virtually eliminate its effect at the ampliﬁer output. This closed loop form of an opamp is preferred in almost every application. In particular, the voltage follower and charge ampliﬁer are devices that use the properties of high Zi ; low Zo ; and high K of an opamp, along with feedback through a precision resistor, to eliminate errors due to nonconstant K: In summary, the operational ampliﬁer is not very useful in its open-loop form, particularly because gain, K; is not steady. However, since K is very large, the problem can be removed by using feedback. It is this closed-loop form that is commonly used in the practical applications of an opamp.

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Signal Conditioning and Modiﬁcation

2-5

In addition to the nonsteady nature of gain, there are other sources of error that contribute to the less than ideal performance of an operational ampliﬁer circuit. Noteworthy are: 1. offset current present at input leads due to bias currents that are needed to operate the solid-state circuitry 2. offset voltage that might be present at the output even when the input leads are open 3. unequal gains corresponding to the two input leads (i.e., the inverting gain not equal to the noninverting gain) Such problems can produce nonlinear behavior in opamp circuits, and they can be reduced by proper circuit design and through the use of compensating circuit elements.

2.2.3

Voltage, Current, and Power Ampliﬁers

Any type of ampliﬁer can be constructed from scratch in the monolithic form as an IC chip, or in the discrete form as a circuit containing several discrete elements such as discrete bipolar junction transistors or discrete FETs, discrete diodes, and discrete resistors. However, almost all types of ampliﬁers can also be built using operational ampliﬁer as the basic element. Since we are already familiar with opamps and since opamps are extensively used in general ampliﬁer circuitry, we prefer to use the latter approach, which uses discrete opamps for the modeling of general ampliﬁers. If an electronic ampliﬁer performs a voltage ampliﬁcation function, it is termed a voltage ampliﬁer. These ampliﬁers are so common that, the term “ampliﬁer” is often used to denote a voltage ampliﬁer. A voltage ampliﬁer can be modeled as in which

vo ¼ K v vi

ð2:5Þ

vo ¼ output voltage vi ¼ input voltage Kv ¼ voltage gain Voltage ampliﬁers are used to achieve voltage compatibility (or level shifting) in circuits. Current ampliﬁers are used to achieve current compatibility in electronic circuits. A current ampliﬁer may be modeled by io ¼ Ki ii ð2:6Þ in which io ¼ output current ii ¼ input current Ki ¼ current gain Note that voltage follower has Kv ¼ 1 and, hence, it may be considered to be a current ampliﬁer. Also, it provides impedance compatibility and acts as a buffer between a low-current (high-impedance) output device (the device that provides the signal) and a high-current (low-impedance) input device (the device that receives the signal) that are interconnected. Hence, the name buffer ampliﬁer or impedance transformer is sometimes used for a current ampliﬁer with unity voltage gain. If the objective of signal ampliﬁcation is to upgrade the associated power level, then a power ampliﬁer should be used for that purpose. A simple model for a power ampliﬁer is po ¼ Kp Pi in which po ¼ output power pi ¼ input power Kp ¼ power gain

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ð2:7Þ

2-6

Vibration Monitoring, Testing, and Instrumentation

It is easy to see from Equation 2.5 to Equation 2.7 that Kp ¼ Kv Ki

ð2:8Þ

Note that all three types of ampliﬁcation could be achieved simultaneously from the same ampliﬁer. Furthermore, a current ampliﬁer with unity voltage gain (for example, a voltage follower) is a power ampliﬁer as well. Usually, voltage ampliﬁers and current ampliﬁers are used in the ﬁrst stages of a signal path (e.g., sensing, data acquisition, and signal generation) where signal levels and power levels are relatively low. Power ampliﬁers are typically used in the ﬁnal stages (e.g., actuation, recording, and display) where high signal levels and power levels are usually required. Figure 2.2(a) shows an opamp-based voltage ampliﬁer. Note the feedback resistor, Rf ; that serves the purposes of stabilizing the opamp and providing an accurate voltage gain. The negative lead is grounded through an accurately known resistor, R: To determine the voltage gain, recall that the voltages at the two input leads of an opamp should be virtually equal. The input voltage, vi , is applied to the positive lead of Input vi

Output vo

+ A −

R

Rf

(a)

R

B ii A Input

Rf

RL Load

−

ii

io (Output)

+

(b) Cf Feedback Capacitor −vo /k −

− Sensor Charge q (c) FIGURE 2.2

K Zi +

vo k +

Ce +

−

Zo vo

Voltage Drop Across Zo = 0

+ Output vo −

(a) A voltage ampliﬁer; (b) a current ampliﬁer; (c) a charge ampliﬁer.

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Signal Conditioning and Modiﬁcation

2-7

the opamp. Then the voltage at point A should also be equal to vi. Next, recall that the current through the input lead of an opamp is virtually zero. Hence, by writing the current balance equation for the node point A, we have vo 2 vi v ¼ i Rf R This gives the ampliﬁer equation vo ¼ 1 þ

Rf v R i

ð2:9aÞ

Hence, the voltage gain is given by Kv ¼ 1 þ

Rf R

ð2:9bÞ

Note the Kv depends on R and Rf and not on the opamp gain. Hence, the voltage gain can be accurately determined by selecting the two resistors, R and Rf ; precisely. Also note that the output voltage has the same sign as the input voltage. Hence, this is a noninverting ampliﬁer. If the voltages are of the opposite sign, we will have an inverting ampliﬁer. A current ampliﬁer is shown in Figure 2.2(b). The input current, ii ; is applied to the negative lead of the opamp as shown and the positive lead is grounded. There is a feedback resistor Rf connected to the negative lead through the load RL : The resistor Rf provides a path for the input current since the opamp takes in virtually zero current. There is a second resistor R through which the output is grounded. This resistor is needed for current ampliﬁcation. To analyze the ampliﬁer, note that the voltage at point A (i.e., at the negative lead) should be zero because the positive lead of the opamp is grounded (zero voltage). Furthermore, the entire input current, ii ; passes through resistor, Rf ; as shown. Hence, the voltage at point B is Rf ii : Consequently, current through resistor R is Rf ii =R; which is positive in the direction shown. It follows that the output current, io ; is given by io ¼ ii þ

Rf i R i

io ¼ 1 þ

Rf i R i

or ð2:10aÞ

The current gain of the ampliﬁer is Ki ¼ 1 þ

Rf R

ð2:10bÞ

This gain can be accurately set using the high-precision resistors, R and Rf . 2.2.3.1

Charge Ampliﬁers

The principle of capacitance feedback is utilized in charge ampliﬁers. These ampliﬁers are commonly used for conditioning the output signals from piezoelectric transducers. A schematic diagram for the charge ampliﬁer is shown in Figure 2.2(c). The feedback capacitance is denoted by Cf and the connecting cable capacitance by Cc : The charge ampliﬁer views the sensor as a charge source (q), even though there is an associated voltage. Using the fact that charge ¼ voltage £ capacitance, a charge balance equation can be written: qþ

vo v C c þ vo þ o C f ¼ 0 K K

ð2:11Þ

From this, we obtain vo ¼ 2

K q ðK þ 1ÞCf þ Cc

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ð2:12aÞ

2-8

Vibration Monitoring, Testing, and Instrumentation

If the feedback capacitance is large in comparison with the cable capacitance, the latter can be neglected. This is desirable in practice. In any event, for large values of gain, K; we have the approximate relationship vo ¼ 2

q Cf

ð2:12bÞ

Note that the output voltage is proportional to the charge generated at the sensor and depends only on the feedback parameter, Cf : This parameter can be appropriately chosen in order to obtain the required output impedance characteristics. Actual charge ampliﬁers also have a feedback resistor, Rf , in parallel with the feedback capacitor, Cf : Then, the relationship corresponding to Equation 2.12a becomes a ﬁrstorder ordinary differential equation, which in turn determines the time constant of the charge ampliﬁer. This time constant should be high. If it is low, the charge generated by the piezoelectric sensor will leak out quickly, giving erroneous results at low frequencies.

2.2.4

Instrumentation Ampliﬁers

An instrumentation ampliﬁer is typically a special-purpose voltage ampliﬁer dedicated to a particular instrumentation application. Examples include ampliﬁers used for producing the output from a bridge circuit (bridge ampliﬁer) and ampliﬁers used with various sensors and transducers. An important characteristic of an instrumentation ampliﬁer is the adjustable gain capability. The gain value can be adjusted manually in most instrumentation ampliﬁers. In more sophisticated instrumentation ampliﬁers, gain is programmable and can be set by means of digital logic. Instrumentation ampliﬁers are normally used with low-voltage signals. 2.2.4.1

Differential Ampliﬁer

Usually, an instrumentation ampliﬁer is also a differential ampliﬁer (sometimes termed difference ampliﬁer). Note that in a differential ampliﬁer both input leads are used for signal input, whereas in a single-ended ampliﬁer one of the leads is grounded and only one lead is used for signal input. Groundloop noise can be a serious problem in single-ended ampliﬁers. Ground-loop noise can be effectively eliminated by using a differential ampliﬁer, because noise loops are formed with both inputs of the ampliﬁer using a differential ampliﬁer allows that these noise signals are subtracted at the ampliﬁer output. Since the noise level is almost the same for both inputs, it is canceled out. Note that any other noise (e.g., 60 Hz line noise) that might enter both inputs with the same intensity will also be canceled out in the output of a differential ampliﬁer. A basic differential ampliﬁer that uses a single opamp is shown in Figure 2.3(a). The input–output equation for this ampliﬁer can be obtained in the usual manner. For instance, since current through the opamp is negligible, current balance at point B gives vi2 2 vB v ¼ B R Rf

ðiÞ

in which vB is the voltage at B. Similarly, current balance at point A gives vo 2 vA v 2 vi1 ¼ A Rf R

ðiiÞ

vA ¼ vB

ðiiiÞ

Now, we use the property

for an operational ampliﬁer to eliminate vA and vB from Equation i and Equation ii. This gives vi2 ðv R=Rf þ vi1 Þ ¼ o ð1 þ R=Rf Þ ð1 þ R=Rf Þ

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Signal Conditioning and Modiﬁcation

2-9

Rf

Inputs

R

vi1 vi2

R

A − Output vo

B + Rf

(a) +

vi1

A

− 1

R3

R3

2 R1

− vi2

FIGURE 2.3

−

R2

Inputs

(b)

R4

R1

+

+

Output vo

R4+δ R4

B

(a) A basic differential ampliﬁer; (b) a basic instrumentation ampliﬁer.

or vo ¼

Rf ðv 2 vi1 Þ R i2

ð2:13Þ

Two things are clear from Equation 2.13. First, the ampliﬁer output is proportional to the difference between, and not the absolute value of, the two inputs vi1 and vi2 : Second, voltage gain of the ampliﬁer is Rf =R: This is known as the differential gain. Note that the differential gain can be accurately set by using high-precision resistors R and Rf : The basic differential ampliﬁer, shown in Figure 2.3(a) and discussed above, is an important component of an instrumentation ampliﬁer. In addition, an instrumentation ampliﬁer should possess the adjustable gain capability. Furthermore, it is desirable to have a very high input impedance and very low output impedance at each input lead. An instrumentation ampliﬁer that possesses these basic requirements is shown in Figure 2.3(b). The ampliﬁer gain can be adjusted using the precisely variable resistor, R2 : Impedance requirements are provided by two voltage-follower-type ampliﬁers, one for each input, as shown. The variable resistance, dR4 ; is necessary to compensate for errors due to unequal common-mode gain. Let us ﬁrst consider this aspect and then obtain an equation for the instrumentation ampliﬁer. 2.2.4.2

Common Mode

The voltage that is “common” to both input leads of a differential ampliﬁer is known as the commonmode voltage. This is equal to the smaller of the two input voltages. If the two inputs are equal, then the common-mode voltage is obviously equal to each one of the two inputs. When vi1 ¼ vi2 ; ideally, the output voltage vo should be zero. In other words, ideally, common-mode signals are rejected by a

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Vibration Monitoring, Testing, and Instrumentation

differential ampliﬁer. However, since the operational ampliﬁers are not ideal and since they usually do not have exactly identical gains with respect to the two input leads, the output voltage vo will not be zero when the two inputs are identical. This common-mode error can be compensated for by providing a variable resistor with ﬁne resolution at one of the two input leads of the differential ampliﬁer. As shown in Figure 2.3(b), to compensate for the common-mode error (i.e., to achieve a satisfactory level of common-mode rejection), ﬁrst the two inputs are made equal and then dR4 is varied carefully until the output voltage level is sufﬁciently small (minimum). Usually, the dR4 that is required to achieve this compensation is small compared with the nominal feedback resistance R4 : Since ideally dR4 ¼ 0; we shall neglect dR4 in the derivation of the instrumentation ampliﬁer equation. Now, note from the basic characteristics of an opamp with no saturation (voltages at the two input leads have to be almost identical) that, in Figure 2.3(b), the voltage at point 2 should be vi2 and the voltage at point 1 should be vi1 : Furthermore, current through each input lead of an opamp is negligible. Hence, current through the circuit path B ! 2 ! 1 ! A has to be the same. This gives the current continuity equations vB 2 vi2 v 2 vi1 v 2 vA ¼ i2 ¼ i1 R1 R2 R1 in which VA and VB are the voltages at points A and B, respectively. Hence, we obtain the two equations vB ¼ vi2 þ

R1 ðv 2 vi1 Þ R2 i2

vA ¼ vi1 2

R1 ðv 2 vi1 Þ R2 i2

Now, by subtracting the second equation from the ﬁrst, we have the equation for the ﬁrst stage of the ampliﬁer; thus 2R1 vB 2 vA ¼ 1 þ ðvi2 2 vi1 Þ ðiÞ R2 From the previous result (see Equation 2.13) for a differential ampliﬁer, we have (with dR4 ¼ 0) R vo ¼ 4 ðvB 2 vA Þ ðiiÞ R3 Note that only the resistor R2 is varied to adjust the gain (differential gain) of the ampliﬁer. In Figure 2.3(b), the two input opamps (the voltage-follower opamps) do not have to be exactly identical as long as the resistors R1 and R2 are chosen so that they are accurate. This is so because the opamp parameters such as open-loop gain and input impedance do not enter the ampliﬁer equations provided that their values are sufﬁciently high, as noted earlier.

2.2.5

Ampliﬁer Performance Ratings

Main factors that affect the performance of an ampliﬁer are: 1. Stability 2. Speed of response (bandwidth, slew rate) 3. Unmodeled signals We have already discussed the signiﬁcance of some of these factors. The level of stability of an ampliﬁer, in the conventional sense, is governed by the dynamics of the ampliﬁer circuitry and may be represented by a time constant. However, a more important consideration for an ampliﬁer is the “parameter variation” due to aging, temperature, and other environmental factors. Parameter variation is also classiﬁed as a stability issue in the context of devices such as ampliﬁers, because it pertains to the steadiness of the response when the input is maintained steady. Of particular importance is temperature drift. This may be speciﬁed as drift in the output signal per unit change in temperature (e.g., mV/8C).

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Signal Conditioning and Modiﬁcation

2-11

The speed of response of an ampliﬁer dictates the ability of the ampliﬁer to faithfully respond to transient inputs. Conventional time-domain parameters such as rise time may be used to represent this. Alternatively, in the frequency domain, speed of response may be represented by a bandwidth parameter. For example, the frequency range over which the frequency-response function is considered constant (ﬂat) may be taken as a measure of bandwidth. Since there is some nonlinearity in any ampliﬁer, bandwidth can depend on the signal level itself. Speciﬁcally, small-signal bandwidth refers to the bandwidth that is determined using small input signal amplitudes. Another measure of the speed of response is the slew rate. Slew rate is deﬁned as the largest possible rate of change in the ampliﬁer output for a particular frequency of operation. Since, for a given input amplitude, the output amplitude depends on the ampliﬁer gain, slew rate is usually deﬁned for unity gain. Ideally, for a linear device, the frequency-response function (transfer function) does not depend on the output amplitude (i.e., the product of the DC gain and the input amplitude). However, for a device that has a limited slew rate, the bandwidth, or the maximum operating frequency at which output distortions may be neglected, will depend on the output amplitude. The larger the output amplitude, the smaller the bandwidth for a given slew rate limit. We have noted that stability problems and frequency-response errors are prevalent in the open-loop form of an operational ampliﬁer. These problems can be eliminated using feedback because the effect of the open-loop transfer function on the closed loop transfer function is negligible if the open-loop gain is very large, which is the case for an operational ampliﬁer. Unmodeled signals can be a major source of ampliﬁer error. Unmodeled signals include: 1. 2. 3. 4.

Bias currents Offset signals Common-mode output voltage Internal noise

In analyzing operational ampliﬁers, we assume that the current through the input leads is zero. This is not strictly true because bias currents for the transistors within the ampliﬁer circuit have to ﬂow through these leads. As a result, the output signal of the ampliﬁer will deviate slightly from the ideal value. Another assumption that we make in analyzing opamps is that the voltage is equal at the two input leads. However, in practice, offset currents and voltages are present at the input leads, due to minute discrepancies inherent to the internal circuits within an opamp. 2.2.5.1

Common-Mode Rejection Ratio

Common-mode error in a differential ampliﬁer was discussed earlier. We noted that ideally the commonmode input voltage (the voltage common to both input leads) should have no effect on the output voltage of a differential ampliﬁer. However, since a practical ampliﬁer has imbalances in the internal circuitry (for example, gain with respect to one input lead is not equal to the gain with respect to the other input lead and, furthermore, bias signals are needed for operation of the internal circuitry), there will be an error voltage at the output that depends on the common-mode input. The common-mode rejection ratio (CMRR) of a differential ampliﬁer is deﬁned as CMRR ¼ in which

Kvcm vocm

ð2:14Þ

K ¼ gain of the differential ampliﬁer (i.e., differential gain) vcm ¼ common-mode voltage (i.e., voltage common to both input leads) vocm ¼ common-mode output voltage (i.e., output voltage due to common-mode input voltage) Note that, ideally, vocm ¼ 0 and CMRR should be inﬁnity. It follows that the larger the CMRR, the better the differential ampliﬁer performance. The three types of unmodeled signals mentioned above can be considered as noise. In addition, there are other types of noise signals that degrade the performance of an ampliﬁer. For example, ground-loop

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Vibration Monitoring, Testing, and Instrumentation

noise can enter the output signal. Furthermore, stray capacitances and other types of unmodeled circuit effects can generate internal noise. Usually in ampliﬁer analysis, unmodeled signals (including noise) can be represented by a noise voltage source at one of the input leads. Effects of unmodeled signals can be reduced by using suitably connected compensating circuitry, including variable resistors that can be adjusted to eliminate the effect of unmodeled signals at the ampliﬁer output (e.g., see dR4 in Figure 2.3(b)). Some useful information about operational ampliﬁers is summarized in Box 2.1.

Box 2.1 OPERATIONAL AMPLIFIERS Ideal Opamp Properties: *

*

*

*

*

Inﬁnite open-loop differential gain Inﬁnite input impedance Zero output impedance Inﬁnite bandwidth Zero output for zero differential input

Ideal Analysis Assumptions: *

*

Voltages at the two input leads are equal. Current through either input lead is zero.

Deﬁnitions: *

*

Open-loop gain ¼

Output voltage with no feedback Voltage difference at input leads

Voltage between an input lead and ground with other input lead Current through that lead grounded and the output in open circuit Input impedance ¼

Voltage between output lead and ground in open circuit Current through that lead with normal input conditions

*

Output impedance ¼

*

Bandwidth ¼ frequency range in which the frequency response is ﬂat (gain is constant)

*

Input bias current ¼ average (DC) current through one input lead

*

Input offset current ¼ difference in the two input bias currents

*

Differential input voltage ¼ voltage at one input lead with the other grounded when the output voltage is zero Output voltage when input leads are at the same voltage Common input voltage

*

Common-mode gain ¼

*

Common-mode rejection ratio ðCMRRÞ ¼

*

Slew rate ¼ speed at which steady output is reached for a step input

Open loop differential gain Common-mode gain

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Signal Conditioning and Modiﬁcation

2.2.6

2-13

Component Interconnection

When two or more components are interconnected, the behavior of the individual components in the overall system can deviate signiﬁcantly from their behavior of each component when they operate independently. The matching of components in a multicomponent system should be done carefully in order to improve system performance and accuracy, particularly with respect to their impedance characteristics. This is particularly true in vibration instrumentation. 2.2.6.1

Impedance Characteristics

When components such as measuring instruments, digital processing boards, process (plant) hardware, and signal-conditioning equipment are interconnected, it is necessary to match impedances properly at each interface in order to realize the devices’ rated performance level. One adverse effect of improper impedance matching is the loading effect. For example, in a measuring system, the measuring instrument can distort the signal that is being measured. The resulting error can far exceed other types of measurement error. Loading errors will result from connecting a measuring device with low input impedance to a signal source. Impedance can be interpreted either in the traditional electrical sense or in the mechanical sense, depending on the signal that is being measured. For example, a heavy accelerometer can introduce an additional dynamic load that will modify the actual acceleration at the monitoring location. Similarly, a voltmeter can modify the currents (and voltages) in a circuit. In mechanical and electrical systems, loading errors can appear as phase distortions as well. Digital hardware also can produce loading errors. For example, an ADC board can load the ampliﬁer output from a strain-gage bridge circuit, thereby signiﬁcantly affecting digitized data. Another adverse effect of improper impedance consideration is inadequate output signal levels, which can make signal processing and transmission very difﬁcult. Many types of transducers (e.g., piezoelectric accelerometers, impedance heads, and microphones) have high output impedances in the order of a thousand megohms. These devices generate low output signals, and they require conditioning to step up the signal level. Impedance-matching ampliﬁers, which have high input impedances (megohms) and low output impedances (a few ohms), are used for this purpose (e.g., charge ampliﬁers are used in conjunction with piezoelectric sensors). A device with a high input impedance has the further advantage that it usually consumes less power (v2 =R is low) for a given input voltage. The fact that a low input impedance device extracts a high level of power from the preceding output device may transpire to be the reason for a loading error. 2.2.6.2

Cascade Connection of Devices

Consider a standard two-port electrical device. The output impedance, Zo ; of such a device is deﬁned as the ratio of the open-circuit (i.e., no-load) voltage at the output port to the short-circuit current at the output port. Open-circuit voltage at the output is the output voltage present when there is no current ﬂowing at the output port. This is the case if the output port is not connected to a load (impedance). As soon as a load is connected at the output of the device, a current will ﬂow through it and the output voltage will drop to a value less than that of the open-circuit voltage. To measure open-circuit voltage, the rated input voltage is applied at the input port and maintained constant, and the output voltage is measured using a voltmeter that has a very high (input) impedance. To measure short-circuit current, a very low-impedance ammeter is connected at the output port. The input impedance, Zi ; is deﬁned as the ratio of the rated input voltage to the corresponding current through the input terminals while the output terminals are maintained as an open circuit. Note that these deﬁnitions are associated with electrical devices. A generalization is possible that includes both electrical and mechanical devices; one must interpret voltage and velocity as across variables, and current and force as through variables. Then, mechanical mobility can be used in place of electrical impedance in the associated analysis.

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2-14

Vibration Monitoring, Testing, and Instrumentation

Example 2.2 Input impedance, Zi , and output impedance, Zo ; can be represented schematically as in Figure 2.4(a). Note that vo is the open-circuit output voltage. When a load is connected at the output port, the voltage across the load will be different from vo : This is caused by the presence of a current through Zo : In the frequency domain, vi and vo are represented by their respective Fourier spectra. The corresponding transfer relation can be expressed in terms of the complex frequency-response (transfer) function G (jv) under open-circuit (no-load) conditions: vo ¼ Gvi

ð2:15Þ

Now, consider two devices connected in cascade, as shown in Figure 2.4(b). It can be easily veriﬁed that the following relations apply:

+ Input

vi

−

(a)

vo = Gvi G Zo + Zi vo −

+ vi (b)

−

Zi1

G2 G1 + + +Z vo1 o1 Zi2 vi2 vo Zo2 − −

FIGURE 2.4 (a) Schematic representation of input impedance and output impedance; (b) the inﬂuence of cascade connection of devices on the overall impedance characteristics.

vo1 ¼ G1 vi vi2 ¼

Output

Zi2 v Zo1 þ Zi2 o1

vo ¼ G2 vi2

ðiÞ ðiiÞ ðiiiÞ

These relations can be combined to give the overall input/output relation vo ¼

Zi2 GGv Zo1 þ Zi2 2 1 i

ð2:16aÞ

We see from Equation 2.16a that the overall frequency-transfer function differs from the ideally expected product ðG2 G1 Þ by the factor Zi2 1 ¼ Zo1 þ Zi2 Zo1 =Zi2 þ 1

ð2:16bÞ

Note that cascading has “distorted” the frequency-response characteristics of the two devices. If Zo1 =Zi2 p 1; this deviation becomes insigniﬁcant. From this observation, it can be concluded that, when frequency-response characteristics (i.e., dynamic characteristics) are important in a cascaded device, cascading should be done such that the output impedance of the ﬁrst device is much smaller than the input impedance of the second device. 2.2.6.3

AC-Coupled Ampliﬁers

The DC component of a signal can be blocked off by connecting that signal through a capacitor. (Note that the impedance of a capacitor is 1=ðjvCÞ and, hence, at zero frequency there will be an inﬁnite impedance.) If the input lead of a device has a series capacitor, we say that the input is AC coupled and, if the output lead has a series capacitor, then the output is AC coupled. Typically, an AC-coupled ampliﬁer has a series capacitor both at the input lead and the output lead. Hence, its frequency-response function will have a high-pass characteristic; in particular, the DC components will be ﬁltered out. Errors due to bias currents and offset signals are negligible for an AC-coupled ampliﬁer. Furthermore, in an AC-coupled ampliﬁer, stability problems are not very serious.

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Signal Conditioning and Modiﬁcation

2.3

Analog Filters

Unwanted signals can seriously degrade the performance of a vibration monitoring and analysis system. External disturbances, error components in excitations, and noise generated internally within system components and instrumentation are such spurious signals. A ﬁlter is a device that allows only the desirable part of a signal to pass through, rejecting the unwanted part. In typical applications of acquisition and processing of a vibration signal, the ﬁltering task requires allowing certain frequency components through and ﬁltering out certain other frequency components in the signal. In this context, we can identify four broad categories of ﬁlters: 1. 2. 3. 4.

2-15

Low-pass ﬁlters High-pass ﬁlters Band-pass ﬁlters Band-reject (or notch) ﬁlters

Magnitude G( f ) 1 (a)

fc = Cutoff Frequency

0

fc

Frequency f

G( f ) 1 (b)

0

fc

f

G( f ) 1 (c)

0 G( f ) 1

fc1

fc2

f

The ideal frequency-response characteristic of each of these four types of ﬁlter is shown in Figure 0 2.5. Note that only the magnitude of the fc1 f fc2 (d) frequency-response function is shown. It is understood, however, that the phase distortion of the input signal also should be FIGURE 2.5 Ideal ﬁlter characteristics: (a) low-pass ﬁlter; (b) high-pass ﬁlter; (c) band-pass ﬁlter; (d) bandsmall, within the pass band (the allowed frequency reject (notch) ﬁlter. range). Practical ﬁlters are less than ideal. Their frequency-response functions do not exhibit sharp cutoffs as in Figure 2.5 and, furthermore, some phase distortion will be unavoidable. Input Channel 1 Output Channel 1 A special type of band-pass ﬁlter that is widely used in acquisition and monitoring of Tracking Filter vibration signals (e.g., in vibration testing) is the Input Channel 2 Output Channel 2 tracking ﬁlter. This is simply a band-pass ﬁlter with a narrow pass band that is frequency tunable. The center frequency (the mid-value) of the pass band is variable, usually by coupling it Carrier Input to the frequency of a carrier signal. In this (Tracking Frequency) manner, signals whose frequency varies with some basic variable in the system (e.g., rotor speed, frequency of a harmonic excitation signal, FIGURE 2.6 Schematic representation of a twofrequency of a sweep oscillator) can be accu- channel tracking ﬁlter. rately tracked in the presence of noise. The inputs to a tracking ﬁlter are the signal that is being tracked and the variable tracking frequency (carrier input). A typical tracking ﬁlter that can simultaneously track two signals is schematically shown in Figure 2.6. Filtering can be achieved using digital ﬁlters as well as analog ﬁlters. Before digital signal processing became efﬁcient and economical, analog ﬁlters were exclusively used for signal ﬁltering and they are still widely used. In an analog ﬁlter, the signal is passed through an analog circuit.

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Vibration Monitoring, Testing, and Instrumentation

The dynamics of the circuit will be such that the desired signal components will be passed through and the unwanted signal components will be rejected. Earlier versions of analog ﬁlters employed discrete circuit elements such as discrete transistors, capacitors, resistors, and even discrete inductors. Since inductors have several shortcomings, including susceptibility to electromagnetic noise, unknown resistance effects, and large size. These days, they are rarely used in ﬁlter circuits. Furthermore, owing to well-known advantages of IC devices, analog ﬁlters in the form of monolithic IC chips are today extensively used in modem applications and are preferred over discrete-element ﬁlters. Digital ﬁlters that employ digital signal processing to achieve ﬁltering are also widely used nowadays.

2.3.1

Passive Filters and Active Filters

Passive analog ﬁlters employ analog circuits containing only passive elements, such as resistors and capacitors (and sometimes inductors). An external power supply is not needed in a passive ﬁlter. Active analog ﬁlters employ active elements and components, such as transistors and operational ampliﬁers in addition to passive elements. Since external power is needed for the operation of the active elements and components, an active ﬁlter is characterized by the need of an external power supply. Active ﬁlters are widely available in a monolithic IC form and are usually preferred over passive ﬁlters. Advantages of active ﬁlters include the following: 1. Loading effects are negligible because active ﬁlters can provide a very high input impedance and very low output impedance. 2. They can be used with low-level signals because signal ampliﬁcation and ﬁltering can be provided by the same active circuit. 3. They are widely available in a low cost and compact IC form. 4. They can be easily integrated with digital devices. 5. They are less susceptible to noise from electromagnetic interference than passive ﬁlters. Commonly mentioned disadvantages of active ﬁlters are the following: 1. They need an external power supply. 2. They are susceptible to “saturation”-type nonlinearity at high signal levels. 3. They can introduce many types of internal noise and unmodeled signal errors (offset, bias signals, etc.). Note that advantages and disadvantages of passive ﬁlters can be directly inferred from the disadvantages and advantages of active ﬁlters as given above. 2.3.1.1

Number of Poles

Analog ﬁlters are dynamic systems and they can be represented by transfer functions, assuming linear dynamics. The number of poles of a ﬁlter is the number of poles in the associated transfer function. This is also equal to the order of the characteristic polynomial of the ﬁlter transfer function (i.e., order of the ﬁlter). Note that poles (or eigenvalues) are the roots of the characteristic equation. In our discussion, we will show simpliﬁed versions of ﬁlters, typically consisting of a single ﬁlter stage. The performance of such a basic ﬁlter can be improved at the expense of circuit complexity (and an increased pole count). Only simple discrete-element circuits are shown for passive ﬁlters. Simple operational-ampliﬁer circuits are given for active ﬁlters. Even here, much more complex devices are commercially available, but our purpose is to illustrate underlying principles rather than to provide descriptions and data sheets for commercial ﬁlters.

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Signal Conditioning and Modiﬁcation

2.3.2

2-17

Low-Pass Filters

The purpose of a low-pass ﬁlter is to allow through all signal components below a certain (cutoff) frequency and block all signal components above that cutoff. Analog low-pass ﬁlters are widely used as antialiasing ﬁlters in digital signal processing. An error known as aliasing will enter the digitally processed results of a signal if the original signal has frequency components above half the sampling frequency. (Half the sampling frequency is called the Nyquist frequency.) Hence, aliasing distortion can be eliminated if, prior to sampling and digital processing, the signal is ﬁltered using a low-pass ﬁlter with its cutoff set at Nyquist frequency. This is one of numerous applications of analog low-pass ﬁlters. Another typical application would be to eliminate high-frequency noise in a measured vibration response. A single-pole, passive low-pass ﬁlter circuit is shown in Figure 2.7(a). An active ﬁlter corresponding to the same low-pass ﬁlter is shown in Figure 2.7(b). It can be shown that the two circuits have identical transfer functions. Hence, it might seem that the opamp in Figure 2.7(b) is redundant. This is not true, however. If two passive ﬁlter stages, each similar to Figure 2.7(a), are connected together, the overall transfer function is not equal to the product of the transfer functions of the individual stages. The reason for this apparent ambiguity is the circuit loading that arises due to the fact that the input impedance of the second stage is not sufﬁciently larger than the output impedance of the ﬁrst stage. However, if two active ﬁlter stages, similar to those in Figure 2.7(b), are connected together, such loading errors will be negligible because the opamp with feedback (i.e., a voltage follower) introduces a very high input impedance and very low output impedance, while maintaining the voltage gain at unity. To obtain the ﬁlter equation for the scenario depicted in Figure 2.7(a), note that, since the output is open circuit (zero load current), the current through capacitor C is equal to the current through resistor R: Hence, C

dvo v 2 vo ¼ i dt R

t

dvo þ vo ¼ vi dt

or ð2:17Þ

where the ﬁlter time constant is

t ¼ RC

ð2:18Þ

From Equation 2.17, it follows that the ﬁlter transfer function is vo 1 ¼ GðsÞ ¼ vi ðts þ 1Þ

ð2:19Þ

From this transfer function, it is clear that an analog low-pass ﬁlter is essentially a lag circuit (i.e., it provides a phase lag). It can be shown that the active ﬁlter stage in Figure 2.7(b) has the same input/output equation. First, since current through an opamp lead is almost zero, we have from the previous analysis of the passive circuit stage vA 1 ¼ vi ðts þ 1Þ

ðiÞ

in which vA is the voltage at the node point A. Now, since the opamp with feedback resistor is in fact a voltage follower, we have vo ¼1 vA

ðiiÞ

Next, by combining Equation i and Equation ii, we obtain Equation 2.19, as required. As mentioned earlier, a main advantage of the active ﬁlter version is that the resulting loading error is negligible.

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2-18

Vibration Monitoring, Testing, and Instrumentation

Input vi

(a)

R

+

+ Output v − o

C − Rf −

R Input vi

A

C

Output vo

+

(b) Magnitude (Log)

Slope = −20 dB/decade

0 dB −3 dB

wc, w b

(c)

Frequency (Log) w

C1 R1 Input vi

R2

B

A

+ Output vo

− C2

Rf

(d) FIGURE 2.7 A single-pole low-pass ﬁlter: (a) a passive ﬁlter stage; (b) an active ﬁlter stage; (c) the frequencyresponse characteristic; (d) a two-pole, low-pass Butterworth ﬁlter.

The frequency-response function corresponding to Equation 2.19 is obtained by setting s ¼ jv; thus GðjvÞ ¼

1 ðtjv þ 1Þ

ð2:20Þ

This gives the response of the ﬁlter when a sinusoidal signal of frequency, v; is applied. The magnitude lGðjvÞl of the frequency-transfer function gives the signal ampliﬁcation and phase angle /GðjvÞ gives the phase lead of the output signal with respect to the input. The magnitude curve (Bode magnitude curve) is shown in Figure 2.7(c). Note from Equation 2.20 that, for small frequencies (i.e., v p 1=t), the magnitude is approximately unity. Hence, 1=t can be considered the

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cutoff frequency vc :

vc ¼

1 t

ð2:21Þ

Example 2.3 Show that the cutoff frequency given by Equation 2.21 is also the half-power bandwidth for the low-pass ﬁlter. Show that, for frequencies much larger than this, the ﬁlter transfer function on the Bode magnitude plane (i.e., log magnitude vs. log frequency) can be approximated by a straight line with slope 220 dB/ decade. This slope is known as the roll-off rate. Solution The frequency corresponding to half power (or 1/2 magnitude) is given by 1 1 ¼ pﬃﬃ lt jv þ 1l 2 or 1 1 ¼ 2 t 2v 2 þ 1 or

t 2 v2 þ 1 ¼ 2 or

t 2 v2 ¼ 1 Hence, the half-power bandwidth is

vb ¼

1 t

ð2:22Þ

This is identical to the cutoff frequency given by Equation 2.11. Now, for v q 1=t (i.e., tv q 1) Equation 2.20 can be approximated by Gð jvÞ ¼

1 tjv

lGð jvÞl ¼

1 tv

This has the magnitude

In the log scale log10 lGð jvÞl ¼ 2log10 v 2 log10 t It follows that the log10 (magnitude) vs. log10 (frequency) curve is a straight line with slope 2 1. In other words, when frequency increases by a factor of ten (i.e., a decade), the log10 magnitude decreases by unity (i.e., by 20 dB). Hence, the roll-off rate is 2 20 dB/decade. These observations are shown in pﬃﬃ Figure 2.7(c). Note that an amplitude change by a factor of 2 (or power by a factor of 2) corresponds to 3 dB. Hence, when the DC (zero-frequency) magnitude is unity (0 dB), the half power magnitude is 2 3 dB. Cutoff frequency and the roll-off rate are the two main design speciﬁcations for a low-pass ﬁlter. Ideally, we would like a low-pass ﬁlter magnitude curve to be ﬂat until the required pass-band limit (cutoff frequency) and then roll off very rapidly. The low-pass ﬁlter shown in Figure 2.7 only

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Vibration Monitoring, Testing, and Instrumentation

approximately meets these requirements. In particular, the roll-off rate is not as large as is desirable. We would like a roll-off rate of at least 240 dB/decade and, preferably, 2 60 dB/decade in practical ﬁlters. This can be realized by using a higher order ﬁlter (i.e., a ﬁlter having many poles). The low-pass Butterworth ﬁlter is a widely used ﬁlter of this type. 2.3.2.1

Low-Pass Butterworth Filter

A low-pass Butterworth ﬁlter having two poles can provide a roll-off rate of 2 40 dB/decade, and one having three poles can provide a roll-off rate of 2 60 dB/decade. Furthermore, the steeper the slope of the roll-off, the ﬂatter is the ﬁlter magnitude curve within the pass band. A two-pole, low-pass Butterworth ﬁlter is shown in Figure 2.7(d). We could construct a two-pole ﬁlter simply by connecting two single-pole stages of the type shown in Figure 2.7(b). Then, we would require two opamps, whereas the circuit shown in Figure 2.7(d) achieves the same objective by using only one opamp (i.e., at a lower cost).

Example 2.4 Show that the opamp circuit in Figure 2.7(d) is a low-pass ﬁlter having two poles. What is the transfer function of the ﬁlter? Estimate the cutoff frequency under suitable conditions. Show that the roll-off rate is 240 dB/decade. Solution To obtain the ﬁlter equation, we write the current balance equations. Speciﬁcally, the sum of the currents through R1 and C1 passes through R2 : The same current passes through C2 because current through the opamp lead must be zero. Hence, vi 2 vA d v 2 vB dv þ C1 ðvo 2 vA Þ ¼ A ¼ C2 B R1 dt R2 dt

ðiÞ

Also, since the opamp with a feedback resistor Rf is a voltage follower (with unity gain), we have vB ¼ vo

ðiiÞ

vi 2 vA dv dv dv þ C1 o 2 C1 A ¼ C2 o R1 dt dt dt

ðiiiÞ

vA 2 vo dv ¼ C2 o R2 dt

ðivÞ

From Equation i and Equation ii, we obtain

Now, deﬁning the constants

t1 ¼ R1 C1

ð2:23Þ

t2 ¼ R2 C2

ð2:24Þ

t3 ¼ R1 C2

ð2:25Þ

and introducing the Laplace variable, s; we can eliminate vA by substituting Equation iv into Equation iii; thus

v2n vo 1 ¼ ¼ ½s2 þ 2zv2n þ v2n vi ½t1 t2 s2 þ ðt2 þ t3 Þs þ 1

ð2:26Þ

This second-order transfer function becomes oscillatory if ðt2 þ t3 Þ2 , 4t1 t2 : Ideally, wepwould like to ﬃﬃ have a zero resonant frequency, which corresponds to a damping ratio value z ¼ 1= 2: Since the undamped natural frequency is 1 vn ¼ pﬃﬃﬃﬃﬃﬃ t1 t2

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ð2:27Þ

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the damping ratio is

t þt z ¼ p2ﬃﬃﬃﬃﬃﬃﬃ3 4t 1 t 2

ð2:28Þ

and the resonant frequency is qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vr ¼ 1 2 2z 2 vn

ð2:29Þ

we have, under ideal conditions (i.e., for vr ¼ 0), ðt2 þ t3 Þ2 ¼ 2t1 t2

ð2:30Þ

The frequency-response function of the ﬁlter is (see Equation 2.26) GðjvÞ ¼

½v2n

v2n 2 v þ 2jzvn v 2

ð2:31Þ

Now, for v p vn ; the ﬁlter frequency response is ﬂat with a unity gain. For v q vn ; the ﬁlter frequency response can be approximated by Gð jvÞ ¼ 2

v2n v2

In a log (magnitude) vs. log (frequency) scale, this function is a straight line with slope equals to 2 2. Hence, when the frequency increases by a factor of ten (i.e., one decade), the log10 (magnitude) drops by 2 units (i.e., 40 dB). In other words, the roll-off rate is 2 40 dB/decade. Also, vn can be taken as the ﬁlter cutoff frequency. Hence, 1 vc ¼ pﬃﬃﬃﬃﬃﬃ t1 t2

ð2:32Þ

pﬃﬃ It can be easily veriﬁed that, when z ¼ 1= 2; the frequency is identical pﬃﬃto the half-power bandwidth (i.e., the frequency at which the transfer function magnitude becomes 1= 2). Note that, if two single-pole stages (of the type shown in Figure 2.7(b)) are cascaded, the resulting two-pole pﬃﬃ ﬁlter has an overdamped (nonoscillatory) transfer function, and it is not possible to achieve z ¼ 1= 2; as in the present case. Also, note that a three-pole, low-pass Butterworth ﬁlter can be obtained by cascading the two-pole unit shown in Figure 2.7(d) with a single-pole unit as shown in Figure 2.7(b). Higher order low-pass Butterworth ﬁlters can be obtained in a similar manner by cascading an appropriate selection of basic units.

2.3.3

High-Pass Filters

Ideally, a high-pass ﬁlter allows through it all signal components above a certain (cutoff) frequency and blocks off all signal components below that frequency. A single-pole, high-pass ﬁlter is shown in Figure 2.8. As for the low-pass ﬁlter that was discussed earlier, the passive ﬁlter stage (Figure 2.8(a)) and the active ﬁlter stage (Figure 2.8(b)) have identical transfer functions. The active ﬁlter is desirable, however, because of its many advantages, including negligible loading error due to the high input impedance and low output impedance of the opamp voltage follower that is present in this circuit. The ﬁlter equation is obtained by considering current balance in Figure 2.8(a), noting that the output is in open circuit (zero load current). Accordingly, C

d v ðv 2 vo Þ ¼ o dt 1 R

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Vibration Monitoring, Testing, and Instrumentation

C Input vi (a)

+

+ R

−

Output vo

− Rf C

Input vi

− Output vo

+ R

(b) Magnitude (Log)

Slope = –20 dB/decade

0 dB −3 dB

wc

(c)

Frequency (Log) w

FIGURE 2.8 A single-pole high-pass ﬁlter: (a) a passive ﬁlter stage; (b) an active ﬁlter stage; (c) frequency-response characteristic.

or

t

dvo dv þ vo ¼ t i dt dt

ð2:33Þ

t ¼ RC

ð2:34Þ

in which the ﬁlter time constant is

Introducing the Laplace variable, s; the ﬁlter transfer function is obtained as vo ts ¼ GðsÞ ¼ vi ðts þ 1Þ

ð2:35Þ

Note that this corresponds to a lead circuit (i.e., an overall phase lead is provided by this transfer function). The frequency-response function is GðjvÞ ¼

t jv ðtjv þ 1Þ

ð2:36Þ

Since its magnitude is zero for v p 1=t and is unity for v q 1=t; we have the cutoff frequency

vc ¼

1 t

ð2:37Þ

Signals above this cutoff frequency are allowed undistorted by an ideal high-pass ﬁlter, and signals below the cutoff are completely blocked off. The actual behavior of the basic high-pass ﬁlter discussed

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above is not perfect, as observed from the frequency-response characteristic shown in Figure 2.8(c). It can be easily veriﬁed that the half-power bandwidth of the basic high-pass ﬁlter is equal to the cutoff frequency given by Equation 2.37, as in the case of the basic low-pass ﬁlter. The roll-up slope of the single-pole high-pass ﬁlter is 20 dB/decade. Steeper slopes are desirable. Multiple-pole, high-pass Butterworth ﬁlters can be constructed to give steeper roll-up slopes and reasonably ﬂat pass-band magnitude characteristics.

2.3.4

Band-Pass Filters

An ideal band-pass ﬁlter passes all signal components within a ﬁnite frequency band and blocks off all signal components outside that band. The lower frequency limit of the pass band is called the lower cutoff frequency ðvc1 Þ; and the upper frequency limit of the band is called the upper cutoff frequency ðvc2 Þ: The most straightforward way to form a band-pass ﬁlter is to cascade a high-pass ﬁlter of cutoff frequency vc1 with a low-pass ﬁlter of cutoff frequency vc2 : Such an arrangement is shown in Figure 2.9. The passive circuit shown in Figure 2.9(a) is obtained by connecting the circuits shown in Figure 2.7(a) and Figure 2.8(a). The passive circuit shown in Figure 2.9(b) is obtained by connecting a voltage follower opamp circuit to the original passive circuit. Passive and active ﬁlters have the same transfer function, assuming that loading problems are not present in the passive ﬁlter. Since loading errors can be serious in practice, however, the active version is preferred.

Input vi (a)

+

R1

C2

A

+ R1

C1 −

Output vo

− Rf C2

R1 Input vi

− Output vo

+ C1

R2

(b) Magnitude (Log)

20 dB/decade

−20 dB/decade

0 dB

(c)

wc1

wc2

Frequency (Log) w

FIGURE 2.9 Band-pass ﬁlter: (a) a basic passive ﬁlter stage; (b) a basic active ﬁlter stage; (c) frequency-response characteristic.

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Vibration Monitoring, Testing, and Instrumentation

To obtain the ﬁlter equation, ﬁrst consider the high-pass portion of the circuit shown in Figure 2.9(a). Since the output is open circuit (zero current), we have from Equation 2.35:

t2 s vo ¼ ðt2 s þ 1Þ vA

ðiÞ

in which

t2 ¼ R2 C2

ð2:38Þ

Next, writing the current balance at node A of the circuit, we have vi 2 vA dv d ¼ C1 A þ C2 ðvA 2 vo Þ R1 dt dt

ðiiÞ

Introducing the Laplace variable, s; we obtain vi ¼ ðt1 s þ t3 s þ 1ÞvA 2 t3 svo

ðiiiÞ

in which

t1 ¼ R1 C1

ð2:39Þ

t3 ¼ R1 C2

ð2:40Þ

and

Now, on eliminating vA by substituting Equation i in Equation iii, we obtain the band-pass ﬁlter transfer function vo t2 s ¼ GðsÞ ¼ 2 vi ½t1 t2 s þ ðt1 þ t2 þ t3 Þs þ 1

ð2:41Þ

We can show that the roots of the characteristic equation

t1 t2 s2 þ ðt1 þ t2 þ t3 Þs þ 1 ¼ 0

ð2:42Þ

are real and negative. The two roots are denoted by 2vc1 and 2vc2 and they provide the two cutoff frequencies shown in Figure 2.9(c). It can be veriﬁed that, for this basic band-pass ﬁlter, the roll-up slope is þ20 dB/decade and the roll-down slope is 220 dB/decade. These slopes are not sufﬁcient in many applications. Furthermore, the ﬂatness of the frequency response within the pass band of the basic ﬁlter is not adequate either. More complex (higher order) band-pass ﬁlters with sharper cutoffs and ﬂatter pass bands are commercially available.

2.3.4.1

Resonance-Type Band-Pass Filters

There are many applications where a ﬁlter with a very narrow pass band is required. The tracking ﬁlter mentioned in the beginning of the section on analog ﬁlters is one such application. A ﬁlter circuit with a sharp resonance can serve as a narrow-band ﬁlter. Note that the cascaded RC circuit shown in Figure 2.9 does not provide an oscillatory response (the ﬁlter poles are all real) and, hence, it does not form a resonance-type ﬁlter. A slight modiﬁcation to this circuit using an additional resistor, R1 ; as shown in Figure 2.10(a), will produce the desired effect. To obtain the ﬁlter equation, note that, for the voltage follower unit vA ¼ vo

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ðiÞ

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2-25

R1 C2

R1

+ B

Input vi

C1

Output vo

− R2 Rf

(a) Magnitude M/

M 2 ∆ω

wc1 wr wc2

(b) FIGURE 2.10 characteristic.

Frequency w

A resonance-type narrow-band-pass ﬁlter: (a) an active ﬁlter stage; (b) frequency-response

Next, since current through an opamp lead is zero, for the high-pass circuit unit (see Equation 2.35), we have vA t2 s ¼ vB ðt2 s þ 1Þ

ðiiÞ

in which

t2 ¼ R2 C2 Finally, current balance at node B gives vi 2 vB dv d v 2 vo ¼ C1 B þ C2 ðvB 2 vA Þ þ B R1 dt dt R1 or, by using the Laplace variable, we obtain vi ¼ ðt1 s þ t3 s þ 2ÞvB 2 t3 svA 2 vo

ðiiiÞ

Now, by eliminating vA and vB in the equations from Equation i to Equation iii, we obtain the ﬁlter transfer function vo t2 s ¼ GðsÞ ¼ vi ½t1 t2 s2 þ ðt1 þ t2 þ t3 Þs þ 2

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ð2:43Þ

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Vibration Monitoring, Testing, and Instrumentation

It can be shown that, unlike Equation 2.41, the present characteristic equation

t1 t2 s2 þ ðt1 þ t2 þ t3 Þs þ 2 ¼ 0

ð2:44Þ

can possess complex roots.

Example 2.5 Verify that the band-pass ﬁlter shown in Figure 2.10(a) can have a frequency response with a resonant peak as shown in Figure 2.10(b). Verify that the half-power bandwidth Dv of the ﬁlter is given by 2zvr at low damping values. (Note: z ¼ damping ratio and vr ¼ resonant frequency.) Solution We may verify that the transfer function given by Equation 2.43 can have a resonant peak by showing that the characteristic equation (Equation 2.44) can have complex roots. For example, if we use parameter values C1 ¼ 2; C2 ¼ 1; R1 ¼ 1; and R2 ¼ 2; we have t1 ¼ 2; t2 ¼ 2; and t3 ¼ 1: The corresponding characteristic equation is 4s2 þ 5s þ 2 ¼ 0 It has the roots

pﬃﬃ 5 7 2 ^j 8 8

is obviously complex. To obtain an expression for the half-power bandwidth of the ﬁlter, note that the ﬁlter transfer function may be written as GðsÞ ¼

ks ðs2 þ 2zvn s þ v2n Þ

ð2:45Þ

kjv ½v2n 2 v2 þ 2jzvn v

ð2:46Þ

in which

vn ¼ undamped natural frequency z ¼ damping ratio k ¼ a gain parameter The frequency-response function is given by GðjvÞ ¼

For low damping, resonant frequency vr ø vn : The corresponding peak magnitude M is obtained by substituting v ¼ vn in Equation 2.46 and taking the transfer function magnitude; thus M¼

k 2zvn

At half-power frequencies, we have M lGðjvÞl ¼ pﬃﬃ 2 or kv k pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ¼ pﬃﬃ 2 2 2 2 2 2 2 2zvn ðvn 2 v Þ þ 4z vn v

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ð2:47Þ

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2-27

This gives ðv2n 2 v2 Þ2 ¼ 4z2 v2n v2

ð2:48Þ

the positive roots of which provide the pass band frequencies vc1 and vc2. Note that the roots are given by

v2n 2 v2 ¼ ^2zvn v Hence, the two roots, vc1 and vc, satisfy the following two equations:

v2c1 þ 2zvn vc1 2 v2n ¼ 0 v2c2 2 2zvn vc2 2 v2n ¼ 0 Accordingly, by solving these two quadratic equations and selecting the appropriate sign, we obtain qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vc1 ¼ 2zvn þ v2n þ z2 v2n ð2:49Þ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ vc2 ¼ zvn þ v2n þ z2 v2n ð2:50Þ The half-power bandwidth is Dv ¼ vc2 2 vc1 ¼ 2zvn

ð2:51Þ

Dv ¼ 2zvr

ð2:52Þ

Now, since vn ø vr ; for low z we have A notable shortcoming of a resonance-type ﬁlter is that the frequency response within the bandwidth (pass band) is not ﬂat. Hence, quite nonuniform signal attenuation takes place inside the pass band.

2.3.5

Band-Reject Filters

Band-reject ﬁlters, or notch ﬁlters, are commonly used to ﬁlter out a narrow band of noise components from a signal. For example, 60 Hz line noise in signals can be eliminated by using a notch ﬁlter with a notch frequency of 60 Hz. An active circuit that could serve as a notch ﬁlter is shown in Figure 2.11(a). This is known as the Twin T circuit because its geometric conﬁguration resembles two T-shaped circuits connected together. To obtain the ﬁlter equation, note that the voltage at point P is vo because of unity gain of the voltage follower. Now, we write the current balance at nodes A and B; thus vi 2 vB dv v 2 vo ¼ 2C B þ B R dt R C

d v d ðv 2 vA Þ ¼ A þ C ðvA 2 vo Þ dt i R=2 dt

Next, since the current through the positive lead of the opamp (voltage follower) is zero, we have the current through point P as vB 2 vo d ¼ C ðvo 2 vA Þ R dt These three equations are written in the Laplace form as vi ¼ 2ðts þ 1ÞvB 2 vo

ðiÞ

tsvi ¼ 2ðts þ 1ÞvA 2 tsvo

ðiiÞ

vB ¼ ðts þ 1Þvo 2 tsvA

ðiiiÞ

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Vibration Monitoring, Testing, and Instrumentation

Rf R

C

Input vi

B

R

− P +

C

A

Output vo

2C

R/2

(a) Magnitude

1

0 (b) FIGURE 2.11

wo = 1 = 1 RC τ

Frequency w

A notch ﬁlter: (a) an active Twin T ﬁlter circuit; (b) frequency-response characteristic.

in which

t ¼ RC

ð2:53Þ

Finally, eliminating vA and vB in Equation i to Equation iii, we obtain vo ðt2 s2 þ 1Þ ¼ GðsÞ ¼ 2 2 ðt s þ 4ts þ 1Þ vi

ð2:54Þ

The frequency-response function of the ﬁlter is GðjvÞ ¼

ð1 2 t2 v2 Þ ð1 2 t2 v2 þ 4jtvÞ

ð2:55Þ

with s ¼ jv: Note that the magnitude of this function becomes zero at frequency

vo ¼

1 t

ð2:56Þ

This is known as the notch frequency. The magnitude of the frequency-response function of the notch ﬁlter is sketched in Figure 2.11(b). It is noticed that any signal component at frequency vo will be completely eliminated by the notch ﬁlter. Sharp roll-down and roll-up are needed to allow the other (desirable) signal components through without too much attenuation. Whereas the previous three types of ﬁlters achieve their frequency-response characteristics through the poles of the ﬁlter transfer function, a notch ﬁlter achieves its frequency-response characteristic through its zeros (roots of the numerator polynomial equation). Some useful information about ﬁlters is summarized in Box 2.2.

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Box 2.2 FILTERS Active Filters (Need External Power) Advantages: *

*

*

Smaller loading errors (have high input impedance and low output impedance, and hence do not affect the input circuit conditions and output signals) Lower cost Better accuracy

Passive Filters (No External Power, Use Passive Elements) Advantages: *

*

Useable at very high frequencies (e.g., radio frequency) No need for a power supply

Filter Types *

*

*

*

Low pass: Allows frequency components up to cutoff and rejects the higher frequency components High pass: Rejects frequency components up to cutoff and allows the higher frequency components Band pass: Allows frequency components within an interval and rejects the rest Notch (or band reject): Rejects frequency components within an interval (usually narrow) and allows the rest

Deﬁnitions *

*

*

*

*

2.4

Filter order: Number of poles in the ﬁlter circuit or transfer function Antialiasing ﬁlter: Low-pass ﬁlter with cutoff at less than half the sampling rate (i.e., Nyquist frequency), for digital processing Butterworth ﬁlter: A high-order ﬁlter with a very ﬂat pass band Chebyshev ﬁlter: An optimal ﬁlter with uniform ripples in the pass band Sallen-Key ﬁlter: An active ﬁlter whose output is in phase with input

Modulators and Demodulators

Sometimes signals are deliberately modiﬁed to maintain the accuracy during signal transmission, conditioning, and processing. In signal modulation, the data signal, known as the modulating signal, is used to vary a property (such as amplitude or frequency) of a carrier signal. We say that the carrier signal is modulated by the data signal. After transmitting or conditioning the modulated signal, the data signal is usually recovered by removing the carrier signal. This is known as demodulation or discrimination. Many modulation techniques exist, and several other types of signal modiﬁcation (e.g., digitizing) could be classiﬁed as signal modulation even though they might not be commonly termed as such. Four types of modulation are illustrated in Figure 2.12. In amplitude modulation (AM), the amplitude of a periodic carrier signal is varied according to the amplitude of the data signal (modulating signal), frequency of the carrier signal (carrier frequency) being kept constant. Suppose that the transient signal

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Time t (a)

t (b)

t (c)

(d)

t

t (e) FIGURE 2.12 (a) Modulating signal (data signal); (b) amplitude-modulated (AM) signal; (c) frequencymodulated (FM) signal; (d) pulse-width-modulated (PWM) signal; (e) pulse-frequency-modulated (PFM) signal.

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shown in Figure 2.12(a) is used as the modulating signal. A high-frequency sinusoidal signal is used as the carrier signal. The resulting amplitude-modulated signal is shown in Figure 2.12(b). Amplitude modulation is used in telecommunications, radio and TV signal transmission, instrumentation, and signal conditioning. The underlying principle is useful in other applications such as fault detection and diagnosis in rotating machinery. In frequency modulation (FM), the frequency of the carrier signal is varied in proportion to the amplitude of the data signal (modulating signal), while the amplitude of the carrier signal is kept constant. If the data signal shown in Figure 2.12(a) is used to frequency modulate a sinusoidal carrier signal, then the result will appear as shown in Figure 2.12(c). Since information is carried as frequency rather than amplitude, any noise that might alter the signal amplitude will have virtually no effect on the transmitted data. Hence, FM is less susceptible to noise than AM. Furthermore, since the carrier amplitude is kept constant in FM, signal weakening and noise effects that are unavoidable in longdistance data communication will have less effect than in the case of AM, particularly if the data signal level is low in the beginning. However, more sophisticated techniques and hardware are needed for signal recovery (demodulation) in FM transmission, because FM demodulation involves frequency discrimination rather than amplitude detection. Frequency modulation is also widely used in radio transmission and in data recording and replay. In pulse-width modulation (PWM), the carrier signal is a pulse sequence. The pulse width is changed in proportion to the amplitude of the data signal, while keeping the pulse spacing constant. This is illustrated in Figure 2.12(d). Pulse-width modulated signals are extensively used in controlling electric motors and other mechanical devices such as valves (hydraulic, pneumatic) and machine tools. Note that, in a given short time interval, the average value of the pulse-width modulated signal is an estimate of the average value of the data signal in that period. Hence, PWM signals can be used directly in controlling a process without one having to demodulate it. Advantages of PWM include better energy efﬁciency (less dissipation) and better performance with nonlinear devices. For example, a device may stick at low speeds due to Coulomb friction. This can be avoided by using a PWM signal that provides the signal amplitude that is necessary to overcome friction while maintaining the required average control signal, which might be very small. In pulse-frequency modulation (PFM) as well, the carrier signal is a pulse sequence. In this method, the frequency of the pulses is changed in proportion to the data signal level, while the pulse width is kept constant. PFM has the advantage of ordinary frequency modulation. Additional advantages result due to the fact that electronic circuits (digital circuits, in particular) can handle pulses very efﬁciently. Furthermore, pulse detection is not susceptible to noise because it involves distinguishing between the presence and absence of a pulse rather than accurate determination of the pulse amplitude (or width). PFM may be used in place of PWM in most applications with better results. Another type of modulation is phase modulation (PM). In this method, the phase angle of the carrier signal is varied in proportion to the amplitude of the data signal. Conversion of discrete (sampled) data into the digital (binary) form is also considered to be modulation. In fact, this is termed pulse-code modulation (PCM). In this case, each discrete data sample is represented by a binary number containing a ﬁxed number of binary digits (bits). Since each digit in the binary number can take only two values, 0 or 1, it can be represented by the absence or presence of a voltage pulse. Hence, each data sample can be transmitted using a set of pulses. This is known as encoding. At the receiver, the pulses have to be interpreted (or decoded) in order to determine the data value. As with any other pulse technique, PCM is quite immune to noise because decoding involves detection of the presence or absence of a pulse rather than determination of the exact magnitude of the pulse signal level. Also, since pulse amplitude is constant, long-distance signal transmission (of this digital data) can be accomplished without the danger of signal weakening and associated distortion. Of course, there will be some error introduced by the digitization process itself, which is governed by the ﬁnite word size (or dynamic range) of the binary data element. This is known as quantization error and is unavoidable in signal digitization.

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In any type of signal modulation, it is essential to preserve the algebraic sign of the modulating signal (data). Different types of modulators handle this in different ways. For example, in PCM an extra sign bit is added to represent the sign of the transmitted data sample. In AM and FM, a phase-sensitive demodulator is used to extract the original (modulating) signal with the correct algebraic sign. Note that, in these two modulation techniques, a sign change in the modulating signal can be represented by a 1808 phase change in the modulated signal. This is not noticeable in Figure 2.12(b) and (c). In PWM and PFM, a sign change in the modulating signal can be represented by changing the sign of the pulses, as shown in Figure 2.12(d) and (e). In PM, a positive range of phase angles (say 0 to p) can be assigned for the positive values of the data signal and a negative range of phase angles (say 2 p to 0) can be assigned for the negative values of the signal.

2.4.1

Amplitude Modulation

Amplitude modulation can naturally enter into many physical phenomena. More important, perhaps, is the deliberate (artiﬁcial) use of AM to facilitate data transmission and signal conditioning. Let us ﬁrst examine the related mathematics. Amplitude modulation is achieved by multiplying the data signal (modulating signal), xðtÞ; by a high frequency (periodic) carrier signal, xc ðtÞ: Hence, amplitude-modulated signal, xa ðtÞ; is given by xa ðtÞ ¼ xðtÞxc ðtÞ

ð2:57Þ

Note that the carrier could be any periodic signal such as one which is harmonic (sinusoidal), square wave, or triangular. The main requirement is that the fundamental frequency of the carrier signal (carrier frequency), fc ; be signiﬁcantly larger (say, by a factor of ﬁve or ten) than the highest frequency of interest (bandwidth) of the data signal. Analysis can be simpliﬁed by assuming a sinusoidal carrier frequency; thus xc ðtÞ ¼ ac cos 2p fc t 2.4.1.1

ð2:58Þ

Modulation Theorem

Modulation theorem is also known as the frequency-shifting theorem, and it relates the fact that if a signal is multiplied by a sinusoidal signal, the Fourier spectrum of the product signal is simply the Fourier spectrum of the original signal shifted through the frequency of the sinusoidal signal. In other words, the Fourier spectrum, Xa ðf Þ; of the amplitude-modulated signal, xa ðtÞ; can be obtained from the Fourier spectrum, Xðf Þ; of the data signal, xðtÞ; simply by shifting through the carrier frequency, fc : To mathematically explain the modulation theorem, we use the deﬁnition of the Fourier integral transform to obtain ð1 Xa ð f Þ ¼ ac xðtÞ cos 2p fc t expð2j2p ftÞdt 21

However, since cos 2p fc t ¼

1 ½expð j2p fc tÞ þ expð2j2p fc tÞ 2

we have Xa ð f Þ ¼

1 ð1 1 ð1 ac xðtÞ exp½2j2pð f 2 fc Þt dt þ ac xðtÞ exp½2j2p ð f þ fc Þt dt 2 2 21 21 1 Xa ð f Þ ¼ ac ½Xð f 2 fc Þ þ Xð f þ fc Þ 2

ð2:59Þ

Equation 2.59 is the mathematical statement of the modulation theorem. It is illustrated by an example in Figure 2.13. Consider a transient signal, xðtÞ; with a (continuous) Fourier spectrum, Xðf Þ; whose magnitude, lXðf Þl; is as shown in Figure 2.13(a). If this signal is used to modulate the AM of a high-frequency sinusoidal signal, the resulting modulated signal, xa ðtÞ; and the magnitude of its

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X( f ) M

x(t)

Time t

(a)

fb Frequency f

0

−fb Xa( f ) Mac

xa(t) = x(t)ac cos 2π fc t

2 t

−fc−fb

−fc −fc+fb 0 fc−fb

fc

fc+fb

f

(b) Xa( f ) a 2

x(t) = a cos 2πfot

t

−fo

0

fo

f

(c) xa(t) = aac cos 2πfot cos 2πfct

Xa( f ) aac 4 t

−fc−fo −fc −fc+fo 0

fc−fo

fc

fc+fo f

(d) FIGURE 2.13 Illustration of the modulation theorem: (a) a transient data signal and its Fourier spectrum magnitude; (b) amplitude-modulated signal and its Fourier spectrum magnitude; (c) a sinusoidal data signal; (d) amplitude modulation by a sinusoidal signal.

Fourier spectrum are as shown in Figure 2.13(b). It should be kept in mind that the magnitude has been multiplied by ac =2: Note that the data signal is assumed to be band limited, with bandwidth fb : Of course, the theorem is not limited to band-limited signals but, for practical reasons, we need to have some upper limit on the useful frequency of the data signal. Also for practical reasons (not for the theorem itself), the carrier frequency, fc ; should be several times larger than fb so that there is a reasonably wide frequency band from 0 to ð fc 2 fb Þ, within which the magnitude of the modulated signal is virtually zero. The signiﬁcance of this should be clear when we discuss applications of amplitude modulation. Figure 2.13 shows only the magnitude of the frequency spectra. It should be remembered, however, that every Fourier spectrum has a phase angle spectrum as well. This is not shown for conciseness, but clearly the phase-angle spectrum is also similarly affected (frequency shifted) by AM. 2.4.1.2

Side Frequencies and Side Bands

The modulation theorem, as described above, assumed transient data signals with associated continuous Fourier spectra. The same ideas are applicable to periodic signals (with discrete spectra) as well. The case

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of periodic signals is merely a special case of what was discussed above. This case can be analyzed by using Fourier integral transform itself, from the beginning. If this method is chosen, however, we will have to cope with impulsive spectral lines. Alternatively, Fourier series expansion could be employed to avoid the introduction of impulsive discrete spectra into the analysis. However, as shown in Figure 2.13(c) and (d), no analysis is actually needed for the periodic signal case because a ﬁnal answer can be deduced from the transient signal results. Speciﬁcally, each frequency component, fo ; that has amplitude a/2 in the Fourier series expansion of the data signal will be shifted by ^fc to the two new frequency locations fc þ fo and 2fc þ fo with an associated amplitude aac =4: The negative frequency component 2fo should also be considered in the same way, as illustrated in Figure 2.13(d). Note that the modulated signal does not have a spectral component at carrier frequency, fc ; but rather on each side of it, at fc ^ fo : Hence, these spectral components are termed side frequencies. When a band of side frequencies is present, we have a side band. Side frequencies are very useful in fault detection and diagnosis of rotating machinery.

2.4.2

Application of Amplitude Modulation

The main hardware component of an amplitude modulator is an analog multiplier. They are commercially available in the monolithic IC form, or one can be assembled using IC opamps and other discrete circuit elements. A schematic representation of an amplitude modulator is shown in Figure 2.14. In practice, to achieve satisfactory modulation, other components such as signal preampliﬁers and ﬁlters are needed. There are many applications of AM. In some applications, modulation is performed intentionally. In others, modulation occurs naturally as a consequence of the physical process, and the resulting signal is used to meet a practical objective. Typical applications of AM include the following: 1. Conditioning of general signals (including DC, transient, and low-frequency) by exploiting the advantages of AC signal conditioning hardware 2. Improvement of the immunity of low-frequency signals to low-frequency noise 3. Transmission of general signals (DC, low-frequency, etc.) by exploiting the advantages of AC signals 4. Transmission of low-level signals under noisy conditions 5. Transmission of several signals simultaneously through the same medium (e.g., same telephone line, same transmission antenna, etc.) 6. Fault detection and diagnosis of rotating machinery The role of AM in many of these applications should be obvious if one understands the frequencyshifting property of AM. Several other types of application are also feasible due to the fact that the power of the carrier signal can be increased somewhat arbitrarily, irrespective of the power level of the data (modulating) signal. Let us discuss, one by one, the six categories of application mentioned above. AC signal conditioning devices such as AC ampliﬁers are known to be more “stable” than their DC counterparts. In particular, drift problems are not as severe and nonlinearity effects are lower in AC signal conditioning devices. Hence, instead of conditioning a DC signal using DC hardware, we can ﬁrst use the

Modulating Input (Data) Carrier Signal

FIGURE 2.14

Multiplier

Modulated Signal

Out

Representation of an amplitude modulator.

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signal to modulate a high-frequency carrier signal. Then, the resulting high-frequency modulated signal may be conditioned more effectively using AC hardware. The frequency-shifting property of AM can be exploited in making low-frequency signals immune to low-frequency noise. Note from Figure 2.13 that, by AM, low-frequency spectrum of the modulating signal can be shifted into a very high-frequency region by choosing a sufﬁciently large carrier frequency, fc : Then, any low-frequency noise (within the band 0 to fc 2 fb ) will not distort the spectrum of the modulated signal. Hence, this noise can be removed by a high-pass ﬁlter (with cutoff at fc 2 fb ) without affecting the data. Finally, the original data signal can be recovered by demodulation. Note that the frequency of a noise component can be within the bandwidth, fb ; of the data signal and, hence, if AM is not employed, noise can directly distort the data signal. Transmission of AC signals is more efﬁcient than that of DC signals. Advantages of AC transmission include lower problems with energy dissipation. Hence, a modulated signal can be transmitted over long distances more effectively than could the original data signal alone. Furthermore, transmission of lowfrequency (large wave-length) signals requires large antennas. Hence, when AM is employed (with an associated reduction in signal wave length), the size of broadcast antenna can be effectively reduced. Transmission of weak signals over long distances is not desirable because signal weakening and corruption by noise could produce disastrous results. By increasing the power of the carrier signal to a sufﬁciently high level, the strength of the modulated signal can be elevated to an adequate level for longdistance transmission. It is impossible to transmit two or more signals in the same frequency range simultaneously using a single telephone line. This problem can be resolved by using carrier signals with signiﬁcantly different carrier frequencies to modulate the amplitude of the data signals. By choosing carrier frequencies that are sufﬁciently farther apart, the spectra of the modulated signals can be made nonoverlapping, thereby making simultaneous transmission possible. Similarly, with AM, simultaneous broadcasting by several radio (AM) broadcast stations in the same broadcast area has become possible. 2.4.2.1

Fault Detection and Diagnosis

A use of the AM principle that is particularly important in the practice of mechanical vibration is in the fault detection and diagnosis of rotating machinery. In this method, modulation is not deliberately introduced, but rather results from the dynamics of the machine. Flaws and faults in a rotating machine are known to produce periodic forcing signals at frequencies higher than, and typically at an integer multiple of, the rotating speed of the machine. For example, backlash in a gear pair will generate forces at the tooth-meshing frequency (equal to the number of teeth £ gear rotating speed). Flaws in roller bearings can generate forcing signals at frequencies proportional to the rotating speed times the number of rollers in the bearing race. Similarly, blade passing in turbines and compressors and eccentricity and unbalance in rotors can produce forcing components at frequencies that are integer multiples of the rotating speed. The resulting vibration response will be an amplitude-modulated signal, where the rotating response of the machine modulates the high-frequency forcing response. This can be conﬁrmed experimentally by Fourier analysis (fast Fourier transform or FFT) of the resulting vibration signals. For a gear box, for example, it will be noticed that, instead of obtaining a spectral peak at the gear toothmeshing frequency, two side bands are produced around that frequency. Faults can be detected by monitoring the evolution of these side bands. Furthermore, since side bands are the result of modulation of a speciﬁc forcing phenomenon (e.g., gear-tooth meshing, bearing-roller hammer, turbine-blade passing, imbalance, eccentricity, misalignment, etc.), one can trace the source of a particular fault (i.e., diagnose the fault) by studying the Fourier spectrum of the measured vibrations. Amplitude modulation is an integral part of many types of sensors. In these sensors, a high-frequency carrier signal (typically the AC excitation in a primary winding) is modulated by the motion. Actual motion can be detected by demodulation of the output. Examples of sensors that generate modulated outputs are differential transformers (LVDT, RVDT), magnetic-induction proximity sensors, eddycurrent proximity sensors, AC tachometers, and strain-gage devices that use AC bridge circuits.

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Signal conditioning and transmission is facilitated by AM in these cases. However, the signal has to be demodulated at the end for most practical purposes such as analysis and recording.

2.4.3

Demodulation

Demodulation, or discrimination or detection, is the process of extracting the original data signal from a modulated signal. In general, demodulation must be phase sensitive in the sense that the algebraic sign of the data signal should be preserved and determined by the demodulation process. In full-wave demodulation, an output is generated continuously. In half-wave demodulation, no output is generated for every alternate half-period of the carrier signal. A simple and straightforward method of demodulation is by the detection of the envelope of the modulated signal. For this method to be feasible, the carrier signal must be quite powerful (i.e., the signal level has to be high) and the carrier frequency also should be very high. An alternative method of demodulation that generally provides more reliable results involves the further step of modulation performed on the already-modulated signal, followed by low-pass ﬁltering. This method will be explained by referring to Figure 2.13. Consider the amplitude-modulated signal, xa ðtÞ; shown in Figure 2.13(b). If this signal is multiplied by the sinusoidal carrier signal, 2=ac cos 2pfc t; we obtain x~ ðtÞ ¼

2 x ðtÞcos 2p fc t ac a

ð2:60Þ

Now, by applying the modulation theorem (Equation 2.59) to Equation 2.60, we obtain the Fourier spectrum of x~ ðtÞ as ~ fÞ ¼ 1 2 Xð 2 ac

1 1 a {Xð f 2 2fc Þ þ Xð f Þ} þ ac {Xð f Þ þ Xð f þ 2fc Þ} 2 c 2

or ~ f Þ ¼ Xð f Þ þ 1 Xð f 2 2fc Þ þ 1 Xð f þ 2fc Þ Xð 2 2

ð2:61Þ

The magnitude of this spectrum is shown in Figure 2.15(a). Note that we have recovered the spectrum, Xð f Þ; of the original data signal, except for the two side bands that are present at locations far removed (centered at ^2fc ) from the bandwidth of the original signal. Hence, we can easily use a low-pass ﬁlter on this signal, x~ ðtÞ; using a ﬁlter with cutoff at fb to recover the original data signal. A schematic representation of this method of amplitude demodulation is shown in Figure 2.15(b). ~ X (f) M M 2 (a)

−2 fc

Modulated Signal xa(t) Carrier 2 cos 2pfc t (b) ac

−fb

0

fb

2 fc

Multiplier Out

~ x(t)

Low-Pass Filter

Frequency f Original Signal x(t)

Cutoff = fb

FIGURE 2.15 Amplitude demodulation: (a) spectrum of the signal after the second modulation; (b) demodulation schematic diagram (modulation þ ﬁltering).

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Analog –Digital Conversion

Data-acquisition systems in machine condition monitoring, fault detection and diagnosis, and vibration testing employ digital computers for various tasks including signal processing, data analysis and reduction, parameter identiﬁcation, and decision making. Typically, the measured response (output) of a dynamic system is available in the analog form as a continuous signal (function of continuous time). Furthermore, typically, the excitation signals (inputs) for a dynamic system have to be provided in the analog form. Inputs to a digital device, say from a digital computer, and outputs from a digital device are necessarily present in the digital form. Hence, when a digital device is interfaced with an analog device, the interface hardware and associated driver software must perform several important functions. Two of the most important interface functions are digital-to-analog conversion (DAC) and analog-to-digital conversion (ADC). A digital output from a digital device has to be converted into the analog form for it to be fed into an analog device such as actuator or analog recording or display unit. Also, an analog signal has to be converted into the digital form according to an appropriate code before it is read by a digital processor or computer. Digital-to-analog converters are simpler and lower in cost than analog-to-digital converters. Furthermore, some types of analog-to-digital converters employ a digital-to-analog converter to perform their function. For these reasons, we will ﬁrst discuss DAC.

2.5.1

Digital-to-Analog Conversion

The function of a DAC is to convert a sequence of digital words stored in a data register (called a DAC register), typically in straight binary form, into an analog signal. The data in the DAC register may come from a data bus of a computer. Each binary digit (bit) of information in the register may be present as a state of a bistable (two-stage) logic device, which can generate a voltage pulse or a voltage level to represent that bit. For example, the off state of a bistable logic element, the absence of a voltage pulse, a low level of a voltage signal, or no change in a voltage level can represent binary 0. Then, the on state of a bistable device, the presence of a voltage pulse, a high level of a voltage signal, or a change in a voltage level will represent binary 1. The combination of these bits, which form the digital word in the DAC register, will correspond to some numerical value for the output signal. The purpose of DAC is to generate an output voltage (signal level) that has this numerical value and maintain the value until the next digital word is converted. Since a voltage output cannot be arbitrarily large or small, for practical reasons, some form of scaling will have to be employed in the DAC process. This scale will depend on the reference voltage, vref, used in the particular DAC circuit. A typical DAC unit is an active circuit in the IC form, which may consist of a data register (digital circuits), solid-state switching circuits, resistors, and operational ampliﬁers powered by an external power supply that can provide a reference voltage. The reference voltage will determine the maximum value of the output ( full-scale voltage). An IC chip that represents the DAC is usually one of many components mounted on a printed circuit (PC) board. This PC board may be identiﬁed by several names including input/output (I/O) board, I/O card, interface board, and data acquisition and control board. Typically, the same board will provide both DAC and ADC capabilities for many output and input channels. There are many types and forms of DAC circuits. The form will depend mainly on the manufacturer, the requirements of the user, or of the particular application. Most DACs are variations of two basic types: the weighted (or summer or adder) type and the ladder type. The latter type of DAC is more desirable, though the former can be somewhat simpler and less expensive. 2.5.1.1

DAC Error Sources

For a given digital word, the analog output voltage from a DAC is not exactly equal to what is given by the analytical formulas. The difference between the actual output and the ideal output is the error. The DAC error can be normalized with respect to the full-scale value.

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There are many causes of DAC error. Typical error sources include parametric uncertainties and variations, circuit time constants, switching errors, and variations and noise in the reference voltage. Several types of error sources and representations are discussed below. 1. Code ambiguity: In many digital codes (for example, in the straight binary code), incrementing a number by a least signiﬁcant bit (LSB) will involve more than one bit-switching. If the speed of switching from 0 to 1 is different from that for 1 to 0, and if switching pulses are not applied to the switching circuit simultaneously, the bit-switchings will not take place simultaneously. For example, in a 4-bit DAC, incrementing from decimal 2 to decimal 4 will involve changing the digital word from 0011 to 0100. This requires two bit-switchings from 1 to 0 and one bit-switching from 0 to 1. If 1 to 0 switching is faster than the 0 to 1 switching, then an intermediate value given by 0000 (decimal 0) will be generated, with a corresponding analog output. Hence, there will be a momentary code ambiguity and associated error in the DAC signal. This problem can be reduced (and eliminated in single bit increments) if a gray code is used to represent the digital data. Improved switching circuitry will also help reduce this error. 2. Settling time: The circuit hardware in a DAC unit will have some dynamics, with associated time constants and perhaps oscillations (underdamped response). Hence, the output voltage cannot instantaneously settle to its ideal value upon switching. The time required for the analog output to settle within a certain band (say ^ 2% of the ﬁnal value or ^ 1/2 resolution), following the application of the digital data, is termed settling time. Naturally, settling time should be smaller for better (faster and more accurate) performance. As a guideline, the settling time should be approximately half the data arrival time. Note that the data arrival time is the time interval between the arrival of two successive data values, and is given by the inverse of the data arrival rate. 3. Glitches: Switching of a circuit will involve sudden changes in magnetic ﬂux due to current changes. This will induce voltages that produce unwanted signal components. In a DAC circuit, these induced voltages due to rapid switching can cause signal spikes that will appear in the output. The error due to these noise signals is not signiﬁcant at low conversion rates. 4. Parametric errors: Resistor elements in a DAC might not be precise, particularly when resistors within a wide range of magnitudes are employed, as in the case in a weighted-resistor DAC. These errors appear in the analog output. Furthermore, aging and environmental changes (primarily, change in temperature) will change the values of circuit parameters, resistance in particular. This also will result in DAC error. These types of error, which are due to the imprecision of circuit parameters and variations of parameter values, are termed parametric errors. Effects of such errors can be reduced by several ways, including the use of compensation hardware (and perhaps software) and directly, by using precise and robust circuit components and employing good manufacturing practices. 5. Reference voltage variations: Since the analog output of a DAC is proportional to the reference voltage, vref ; any variations in the voltage supply will directly appear as an error. This problem can be overcome by using stabilized voltage sources with sufﬁciently low output impedance. 6. Monotonicity: Clearly, the output of a DAC should change by its resolution ðdy ¼ vref =2n Þ for each step of one LSB increment in the digital value. This ideal behavior might not exist in some real DACs due to errors such as those mentioned above. At least the analog output should not decrease as the value of the digital input increases. This is known as the monotonicity requirement that should be met by a practical DAC. 7. Nonlinearity: Suppose that the digital input to a DAC is varied from ½0 0…0 to ½1 1…1 in steps of one LSB. Ideally the analog output should increase in constant jumps of dy ¼ vref =2n ; giving a staircase-shaped analog output. If we draw the best linear ﬁt for this ideally montonic staircase response, it will have a slope equal to the resolution/step. This slope is known as the ideal scale factor. Nonlinearity of a DAC is measured by the largest deviation of the DAC output from this best linear ﬁt. Note that, in the ideal case, the nonlinearity is limited to half the resolution ð1=2Þdy:

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One cause of nonlinearity is clearly the faulty bit-transitions. Another cause is circuit nonlinearity in the conventional sense. Speciﬁcally, owing to nonlinearities in circuit elements such as opamps and resistors, the analog output will not be proportional to the value of the digital word dictated by the bitswitchings (faulty or not). This latter type of nonlinearity can be accounted for by using calibration.

2.5.2

Analog-to-Digital Conversion

Analog signals, which are continuously deﬁned with respect to time, have to be sampled at discrete time points and the sample values have to be represented in the digital form (according to a suitable code) to be read into a digital system such as a microcomputer. An ADC is used to accomplish this. For example, since response measurements of dynamic systems are usually available as analog signals, these signals have to be converted into the digital form before passing on to a signal analysis computer. Hence, the computer interface for the measurement channels should contain one or more ADCs. DACs and ADCs are usually situated on the same digital interface board. However, the ADC process is more complex and time consuming than the DAC process. Furthermore, many types of ADCs use DACs to accomplish the analog-to-digital conversion. Hence, ADCs are usually more costly than and their conversion rate is usually slower than that of DACs. Several types of ADCs are commercially available. The principle of operation varies depending on the type.

2.5.3

Analog-to-Digital Converter Performance Characteristics

For ADCs that use a DAC internally, the same error sources that were discussed previously for DACs apply. Code ambiguity at the output register is not a problem because the converted digital quantity is transferred instantaneously to the output register. Code ambiguity in the DAC register can still cause error in ADCs that use a DAC. Conversion time is a major factor as it is much larger for an ADC. In addition to resolution and dynamic range, quantization error will be applicable to an ADC. These considerations that govern the performance of an ADC are discussed below. 2.5.3.1

Resolution and Quantization Error

The number of bits, n; in an ADC register determines the resolution and dynamic range of the ADC. For an n-bit ADC, the output register size is n bits. Hence, the smallest possible increment of the digital output is one LSB. The change in the analog input that results in a change of one LSB at the output is the resolution of the ADC. The range of digital outputs is from 0 to 2n 2 1 for the unipolar (unsigned) case. This represents the dynamic range. Hence, as for a DAC, the dynamic range of an n-bit ADC is given by the ratio DR ¼ 2n 2 1

ð2:62Þ

DR ¼ 20 log10 ð2n 2 1Þ dB

ð2:63Þ

or, in decibels

The full-scale value of an ADC is the value of the analog input that corresponds to the maximum digital output. Suppose that an analog signal within the dynamic range of the ADC is converted. Since the analog input (sample value) has inﬁnitesimal resolution and the digital representation has a ﬁnite resolution (one LSB), an error is introduced in the ADC process. This is known as the quantization error. A digital number increments in constant steps of 1 LSB. If an analog value falls at an intermediate point within a single-LSB step, then there is a quantization error. Rounding of the digital output can be accomplished as follows. The magnitude of the error when quantized up is compared with that when quantized down, say, using two hold elements and a differential ampliﬁer. Then, we retain the digital value corresponding to the lower error magnitude. If the analog value is below the 1/2 LSB mark, then the corresponding digital value is represented by the value in the beginning of the step. If the analog value is above the 1/2

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LSB mark, then the corresponding digital value is the value at the end of the step. It follows that, with this type of rounding, the quantization error does not exceed 1/2 LSB. 2.5.3.2

Monotonicity, Nonlinearity, and Offset Error

Considerations of monotonicity and nonlinearity are important for an ADC as well as for a DAC. The input is an analog signal and the output is digital in the case of ADC. Disregarding quantization error, the digital output of an ADC will increase in constant steps in the shape of an ideal staircase when the analog input is increased from zero in steps of the device resolution ðdyÞ: This is the ideally monotonic case. The best straight-line ﬁt to this curve has a slope equal to 1=dy (LSB/V). This is the ideal gain or ideal scale factor. However, there will still be an offset error of 1/2 LSB because the best linear ﬁt will not pass through the origin. Adjustments can be made for this offset error. Incorrect bit-transitions can take place in an ADC due to various errors that might be present and due to circuit malfunctions. The best linear ﬁt under such faulty conditions will have a slope different from the ideal gain. The difference is the gain error. Nonlinearity is the maximum deviation of the output from the best linear ﬁt. It is clear that, with perfect bit transitions, in the ideal case, a nonlinearity of 1/2 LSB will be present. Nonlinearities larger than this result from incorrect bit-transitions. As in the case of DAC, another source of nonlinearity in an ADC is the existence of circuit nonlinearities that would deform the analog input signal before being converted into the digital form. 2.5.3.3

Analog-to-Digital Converter Conversion Rate

It is clear that ADC is much more time consuming than DAC. The conversion time is a very important factor because the rate at which conversion can take place governs many aspects of data acquisition, particularly in real-time applications. For example, the data sampling rate has to synchronize with the ADC conversion rate. This, in turn, will determine the Nyquist frequency (half the sampling rate) which is the maximum value of useful frequency present in the sampled signal. Furthermore, the sampling rate will dictate storage and memory requirements. Another important consideration related to the ADC conversion rate is the fact that a signal sample must be maintained at that value during the entire process of conversion into the digital form. This requires a hold circuit and this circuit should be able to perform accurately at the largest possible conversion time for the particular ADC unit. The total time taken to convert an analog signal will depend on other factors besides the time taken for conversion from sampled data to digital data. For example, in multiple-channel data acquisition, the time taken to select the channel has to be counted. Furthermore, time needed to sample the data and time needed to transfer the converted digital data into the output register have to be included. The conversion rate for an ADC is the inverse of the overall time needed for a conversion cycle. Typically, however, conversion rate depends primarily on the bit conversion time in the case of one comparison-type ADC and on the integration time in the case of an integration-type ADC. A typical time period for a comparison step or counting step in an ADC is Dt ¼ 5 msec: Hence, for an eight-bit successiveapproximation ADC the conversion time is 40 msec: The corresponding sampling rate is in the order of (less than) 1=40 £ 1026 ¼ 25 £ 103 samples=sec (or 25 kHz). The maximum conversion rate for an eightbit counter ADC is about 5 £ ð28 2 1Þ ¼ 1275 msec: The corresponding sampling rate would be of the order of 780 samples/sec. Note that this is considerably slow. The maximum conversion time for a dualslope ADC can be still larger (slower).

2.5.4

Sample-and-Hold Circuitry

In typical applications of data acquisition that use ADC, the analog input to ADC can be very transient. Furthermore, ADC is not instantaneous (conversion time is much larger than the DAC time). Speciﬁcally, the incoming analog signal might be changing at a rate higher than the ADC conversion rate. Then, the input signal value will vary during the conversion period and there will be an ambiguity as to the input value corresponding to a digital output value. Hence, it is necessary to sample the analog input signal and maintain the input to the ADC at this value until the ADC is completed. In other words,

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Sampling Rate Control (Timing)

Analog Input

Voltage Follower (Low Zout)

Solid-State (FET) Switch

C

FIGURE 2.16

Voltage Follower (High Zin)

S/H Output (Supply to ADC)

Holding Capacitor

A sample and hold circuit.

since we are typically converting analog signals that can vary at a high speed, it is often necessary to sample and hold (S/H) the input signal for each ADC cycle. Each data sample must be generated and captured by the S/H circuit on the issue of the “start conversion” (SC) control signal, and the captured voltage level has to be maintained constant until the “conversion complete” (CC) control signal is issued by the ADC unit. The main element in an S/H circuit is the holding capacitor. A schematic diagram of a S/H circuit is shown in Figure 2.16. The analog input signal is supplied through a voltage follower to a solid-state switch. The switch typically uses a ﬁeld-effect transistor (FET), such as the metal-oxide semiconductor ﬁeld effect transistor (MOSFET). The switch is closed in response to a “sample pulse” and is opened in response to a “hold pulse.” Both control pulses are generated by the control logic unit of the ADC. During the time interval between these two pulses, the holding capacitor is charged to the voltage of the sampled input. This capacitor voltage is then supplied to the ADC through a second voltage follower. The functions of the two voltage followers are now explained. When the FET switch is closed in response to a sample command (pulse), the capacitor must be charged as quickly as possible. The associated time constant (charging time constant) tc is given by

tc ¼ Rs C

ð2:64Þ

in which Rs ¼ source resistance C ¼ capacitance of the holding capacitor Since tc must be very small for fast charging and, since C is ﬁxed by the holding requirements (typically C is of the order of 100 pF where 1 pF ¼ 1 £ 10212 F), we need a very small source resistance. The requirement is met by the input voltage follower (which is known to have a very low output impedance), thereby providing a very small Rs : Furthermore, since a voltage follower has a unity gain, the voltage at the output of this input voltage follower is equal to the voltage of the analog input signal, as required. Next, once the FET switch is opened in response to a hold command (pulse), the capacitor should not discharge. This requirement is met due to the presence of the output voltage follower. Since the input impedance of a voltage follower is very high, the current through its leads is almost zero. Because of this, the holding capacitor has a virtually zero discharge rate under hold conditions. Furthermore, we like the output of this second voltage follower to be equal to the voltage of the capacitor. This condition is also satisﬁed due to the fact that a voltage follower has a unity gain. Hence, the sampling will be almost instantaneous and the output of the S/H circuit will be maintained (almost) constant during the holding period due to the presence of the two voltage followers. Note that the practical S/H circuits are zero-orderhold devices by deﬁnition.

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2.5.5

Vibration Monitoring, Testing, and Instrumentation

Digital Filters

A ﬁlter is a device that eliminates undesirable frequency components in a signal and passes only the desirable frequency components through. In analog ﬁltering, the ﬁlter is a physical, dynamic system, typically an electric circuit. The signal to be ﬁltered is applied (input) to this dynamic system. The output of the dynamic system is the ﬁltered signal. It follows that any physical dynamic system can be interpreted as an analog ﬁlter. An analog ﬁlter can be represented by a differential equation with respect to time. It takes an analog input signal, uðtÞ; that is deﬁned continuously in time, t; and generates an analog output, yðtÞ: A digital ﬁlter is a device that accepts a sequence of discrete input values (say, sampled from an analog signal at sampling period Dt). {uk } ¼ {u0 ; u1 ; u2 ; …}

ð2:65Þ

and generates a sequence of discrete output values: {yk } ¼ {y0 ; y1 ; y2 ; …}

ð2:66Þ

Hence, a digital ﬁlter is a discrete-time system and it can be represented by a difference equation. An nth order linear difference equation can be written in the form a0 yk þ a1 yk21 þ · · · þ an yk2n ¼ b0 uk þ b1 uk21 þ · · · þ bm uk2m

ð2:67Þ

This is a recursive algorithm in the sense that it generates one value of the output sequence using previous values of the output sequence and all values of the input sequence up to the present time point. Digital ﬁlters represented in this manner are termed recursive digital ﬁlters. There are ﬁlters that employ digital processing, in which a block (a collection of samples) of the input sequence is converted in a one-shot computation into a block of the output sequence. Such ﬁlters are not recursive ﬁlters. Nonrecursive ﬁlters usually employ digital Fourier analysis, the FFT algorithm in particular. We restrict our discussion below to recursive digital ﬁlters. Our intention in the present section is to give a brief (and nonexhaustive) introduction to the subject of digital ﬁltering. 2.5.5.1

Software Implementation and Hardware Implementation

In digital ﬁlters, signal ﬁltering is accomplished through digital processing of the input signal. The sequence of input data (usually obtained by sampling and digitizing the corresponding analog signal) is processed according to the recursive algorithm of the particular digital ﬁlter. This generates the output sequence. This digital output can be converted into an analog signal using a DAC if so desired. A recursive digital ﬁlter is an implementation of a recursive algorithm that governs the particular ﬁltering (e.g., low-pass, high-pass, band-pass, and band-reject). The ﬁlter algorithm can be implemented either by software or by hardware. In software implementation, the ﬁlter algorithm is programmed into a digital computer. The processor (e.g., the microprocessor) of the computer can process an input data sequence according to the run-time ﬁlter program stored in the memory (in machine code) to generate the ﬁltered output sequence. Digital processing of data is accomplished by means of logic circuitry that can perform basic arithmetic operations such as addition. In the software approach, the processor of a digital computer makes use of these basic logic circuits to perform digital processing according to the instructions of a software program stored in the computer memory. Alternatively, a hardware digital processor can be put together to perform a somewhat complex, yet ﬁxed, processing operation. In this approach, the program of computation is said to be in hardware. The hardware processor is then available as an IC chip whose processing operation is ﬁxed and cannot be modiﬁed. The logic circuitry in the IC chip is designed to accomplish the required processing function. Digital ﬁlters implemented by this hardware approach are termed hardware digital ﬁlters. The software implementation of digital ﬁlters has the advantage of ﬂexibility; the ﬁlter algorithm can be easily modiﬁed by changing the software program that is stored in the computer. If, on the other hand, a large number of ﬁlters of a particular (ﬁxed) structure are needed commercially, then it is economical to

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design the ﬁlter as an IC chip and replicate the chip in mass production. In this manner, very low-cost digital ﬁlters can be produced. A hardware ﬁlter can operate at a much faster speed than a software ﬁlter because, in the former case, processing takes place automatically through logic circuitry in the ﬁlter chip without having to access the processor, a software program, and various data items stored in the memory. The main disadvantage of a hardware ﬁlter is that its algorithm and parameter values cannot be modiﬁed, and the ﬁlter is dedicated to a ﬁxed function.

2.6

Bridge Circuits

A full bridge is a circuit having four arms which are connected in a lattice form. Four nodes are formed in this manner. Two opposite nodes are used for the excitation (voltage or current supply) of the bridge and the remaining two opposite nodes provide the bridge output. A bridge circuit is used to make some form of measurement. Typical measurements include change in resistance, change in inductance, change in capacitance, oscillating frequency, or some variable (stimulus) that causes these. There are two basic methods of making the measurement: 1. Bridge balance method 2. Imbalance output method A bridge is said to be balanced when the output voltage is zero. In the bridge-balance method, we start with a balanced bridge. Then, when one is preparing to make a measurement, the balance of the bridge will be upset due to the associated variation, resulting in a nonzero output voltage. The bridge can be balanced again by varying one of the arms of the bridge (assuming, of course, that some means is provided for the ﬁne adjustments that may be required). The change that is required to restore the balance provides the measurement. In this method, the bridge can be balanced precisely using a servo device. In the imbalance output method, we usually start with a balanced bridge. However, the bridge is not balanced again after undergoing the change due to the variable that is being measured. Instead, the output voltage of the bridge due to the resulting imbalance is measured and used as an indication of the measurement. There are many types of bridge circuits. If the supply to the bridge is DC, then we have a DC bridge. Similarly, an AC bridge has an AC excitation. A resistance bridge has only resistance elements in its four arms. An impedance bridge has impedance elements consisting of resistors, capacitors, and inductors in one or more of its arms. If the bridge excitation is a constant-voltage supply, we have a constant-voltage bridge. If the bridge supply is a constant current source, we have a constantcurrent bridge.

2.6.1

Wheatstone Bridge

The Wheatstone bridge is a resistance bridge with a constant DC voltage supply (i.e., a constant-voltage resistance bridge). A Wheatstone bridge is used in strain-gage measurements, and also in force, torque, and tactile sensors that employ strain-gage techniques. Since a Wheatstone bridge is used primarily in the measurement of small changes in resistance, it could be used in other types of sensing applications as well (for example, in resistance temperature detectors or RTDs). Consider the Wheatstone bridge circuit shown in Figure 2.17(a). The bridge output, vo ; may be expressed as

vo ¼

ðR1 R4 2 R2 R3 Þ v ðR1 þ R2 ÞðR3 þ R4 Þ ref

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ð2:68Þ

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Note that the bridge-balance requirement is R1 R ¼ 3 R2 R4

R2

R1

ð2:69Þ

R3

vref (a)

−

(Constant) Small i R2

R1

ð2:70Þ

+ vo

Note that the output is nonlinear in dR=R: If, however, dR=R is assumed to be small in comparison to 2, we have the linearized relationship.

R3

R4

RL

Load −

ð2:71Þ

The error due to linearization, which is a measure of nonlinearity, may be given as the percentage NP ¼ 100 1 2

Load

+

or

dvo dR ¼ vref 4R

RL

R4

B −

½ðR þ dRÞR 2 R2 v 20 ðR þ dR þ RÞðR þ RÞ ref dvo dR=R ¼ vref ð4 þ 2dR=RÞ

+ vo

Suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the beginning. The bridge is balanced according to Equation 2.69 and then R1 is increased by dR: For example, R1 may represent the only active strain gage and the remaining three elements in the bridge are identical, dummy elements. Then, in view of Equation 2.68, the change in output due to the change dR is given by dvo ¼

Small i

A

Linearized output % Actual output ð2:72Þ

(b)

iref (Constant)

FIGURE 2.17 (a) Wheatstone bridge (the constantvoltage resistance bridge); (b) the constant-current bridge.

Hence, from Equation 2.70 and Equation 2.71, we have NP ¼ 50

2.6.2

dR % R

ð2:73Þ

Constant-Current Bridge

When large resistance variations dR are required for a measurement, the Wheatstone bridge may not be satisfactory due to its nonlinearity, as indicated by Equation 2.70. The constant-current bridge has less nonlinearity and is preferred in such applications. However, it requires a current-regulated power supply, which is typically more costly than a voltage-regulated power supply. As shown in Figure 2.17(b), the constant-current bridge uses a constant-current excitation, iref ; instead of a constant-voltage supply. Note that the output equation for the constant-current bridge can be determined from Equation 2.68 simply by knowing the voltage at the current source. Suppose that this is the voltage, vref ; with the polarity as shown in Figure 2.17(a). Now, since the load current is assumed small (high-impedance load), the current through R2 is equal to the current through R1 and is

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given by vref ðR1 þ R2 Þ Similarly, current through R4 and R3 is given by vref ðR3 þ R4 Þ Accordingly, iref ¼

vref vref þ ðR1 þ R2 Þ ðR3 þ R4 Þ

vref ¼

ðR1 þ R2 ÞðR3 þ R4 Þ i ðR1 þ R2 þ R3 þ R4 Þ ref

or ð2:74Þ

Substituting Equation 2.74 in Equation 2.68, we have the output equation for the constant-current bridge; thus, vo ¼

ðR1 R4 2 R2 R3 Þ i ðR1 þ R2 þ R3 þ R4 Þ ref

ð2:75Þ

Note that the bridge-balance requirement is again given by Equation 2.69. To estimate the nonlinearity of a constant-current bridge, suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the beginning and R1 is changed by dR while the other resistors remain inactive. Again, R1 will represent the active element (the sensing element) and may correspond to an active strain gage. The change in output, dvo ; is given by dvo ¼

½ðR þ dRÞR 2 R2 i 20 ðR þ dR þ R þ R þ RÞ ref

or dvo dR=R ¼ Riref ð4 þ dR=RÞ

ð2:76Þ

By comparing the denominator on the RHS of this equation with Equation 2.70, we observe that the constant-current bridge is more linear. Speciﬁcally, using the deﬁnition given by Equation 2.72, the percentage nonlinearity may be expressed as Np ¼ 25

dR % R

ð2:77Þ

It is noted that the nonlinearity is halved by using a constant-current excitation instead of a constantvoltage excitation.

2.6.3

Bridge Ampliﬁers

The output from a resistance bridge is usually very small in comparison to the reference, and it has to be ampliﬁed in order to increase the voltage level to a useful value (for example, in system monitoring or data logging). A bridge ampliﬁer is used for this purpose. This is typically an instrumentation ampliﬁer or a differential ampliﬁer. The bridge ampliﬁer is modeled as a simple gain, Ka ; that multiplies the bridge output. 2.6.3.1

Half-Bridge Circuits

A half bridge may be used in some applications that require a bridge circuit. A half bridge has only two arms and the output is tapped from the mid-point of the two arms. The ends of the two arms are excited

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by a positive voltage and a negative voltage. +vref Rf Initially, the two arms have equal resistances so R1 that, nominally, the bridge output is zero. One of (Active the arms has the active element. Its change in Element) A − resistance results in a nonzero output voltage. It is Output R2 vo noted that the half-bridge circuit is somewhat + (Dummy) similar to a potentiometer circuit. A half-bridge ampliﬁer consisting of a resistance −vref half-bridge and an output ampliﬁer is shown in Figure 2.18. The two bridge arms have resistances R1 and R2 ; and the ampliﬁer uses a feedback FIGURE 2.18 A half bridge with an output ampliﬁer. resistance Rf : To obtain the output equation, we use the two basic facts for an unsaturated opamp; the voltages at the two leads are equal (due to high gain) and current in both leads is zero (due to high input impedance). Hence, voltage at node A is zero and the current balance equation at node A is vref ð2vref Þ v þ þ o ¼0 R1 R2 Rf This gives vo ¼ R f

1 1 2 v R2 R1 ref

ð2:78Þ

Now, suppose that initially R1 ¼ R2 ¼ R and the active element R1 changes by dR: The corresponding change in output is dvo ¼ Rf

1 1 2 v 20 R R þ dR ref

or dvo R dR=R ¼ f vref R ð1 þ dR=RÞ

ð2:79Þ

Note that Rf =R is the ampliﬁer gain. Now, in view of Equation 2.72, the percentage nonlinearity of the half-bridge circuit is Np ¼ 100

dR % R

ð2:80Þ

It follows that the nonlinearity of a half-bridge circuit is worse than that of the Wheatstone bridge.

2.6.4

Impedance Bridges

An impedance bridge contains general impedance elements, Z1 ; Z2 ; Z3 ; and Z4 , in its four arms, as shown in Figure 2.19(a). The bridge is excited by an AC supply, vref : Note that vref represents a carrier signal and the output, vo , has to be demodulated if a transient signal representative of the variation in one of the bridge elements is needed. Impedance bridges can be used, for example, to measure capacitances in capacitive sensors and changes of inductance in variable-inductance sensors and eddy-current sensors. Also, impedance bridges can be used as oscillator circuits. An oscillator circuit can serve as a constantfrequency source of a signal generator (in vibration testing) or it can be used to determine an unknown circuit parameter by measuring the oscillating frequency.

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Analyzing using frequency-domain concepts, it is seen that the frequency spectrum of the impedance-bridge output is given by ðZ1 Z4 2 Z2 Z3 Þ vo ðvÞ ¼ v ðvÞ ðZ1 þ Z2 ÞðZ3 þ Z4 Þ ref

Z2

Z1

Output vo

ð2:81Þ

Z3

Z4

This reduces to Equation 2.68 in the DC case of a Wheatstone bridge. The balanced condition is given by Z1 Z ¼ 3 Z2 Z4

~

ð2:82Þ (a)

This is used to measure an unknown circuit parameter in the bridge. Let us consider two examples. 2.6.4.1

vref (AC Supply) C1

R2

Owen Bridge

The Owen bridge shown in Figure 2.19(b) may be used to measure the inductance L4 or capacitance C3 ; by the bridge-balance method. To derive the necessary equation, note that the voltage –current relation for an inductor is v¼L

di dt

ð2:83Þ

dv dt

ð2:84Þ

Output vo

L4 R3

R4

C3

~ vref

(b)

and for a capacitor it is i¼C

R1 C4

It follows that the voltage/current transfer function (in the Laplace domain) for an inductor is vðsÞ ¼ Ls iðsÞ

R2

R3

ð2:85Þ

vo R4

C3

and that for a capacitor is vðsÞ 1 ¼ iðsÞ Cs

(c)

Accordingly, the impedance of an inductor element at frequency v is ZL ¼ jvL

~

ð2:86Þ

ð2:87Þ

vref

FIGURE 2.19 (a) General impedance bridge; (b) Owen bridge; (c) Wien-bridge oscillator.

and the impedance of a capacitor element at frequency v is Zc ¼

1 jvC

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ð2:88Þ

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Applying these results to the Owen bridge, we have Z1 ¼

1 jvC1

Z2 ¼ R2 Z3 ¼ R3 þ

1 jvC3

Z4 ¼ R4 þ jvL4 in which v is the excitation frequency. Now, from Equation 2.82, we have 1 1 ðR þ jvL4 Þ ¼ R2 R3 þ jvC1 4 jvC3 By equating the real parts and the imaginary parts of this equation, we obtain the two equations L4 ¼ R2 R3 C1 and R4 R ¼ 2 C1 C3 Hence, we have L4 ¼ C1 R2 R3

ð2:89Þ

and C3 ¼ C1

R2 R4

ð2:90Þ

It follows that L4 and C3 can be determined with the knowledge of C1 ; R2 ; R3 ; and R4 under balanced conditions. For example, with ﬁxed C1 and R2 ; an adjustable R3 could be used to measure the variable L4 ; and an adjustable R4 could be used to measure the variable C3 : 2.6.4.2

Wien-Bridge Oscillator

Now consider the Wien-bridge oscillator shown in Figure 2.19(c). For this circuit, we have Z 1 ¼ R1 Z 2 ¼ R2 Z3 ¼ R3 þ

1 jvC3

1 1 ¼ þ jvC4 Z4 R4 Hence, from Equation 2.82, the bridge-balance requirement is R1 1 ¼ R3 þ R2 jvC4

1 þ jvC4 R4

Equating the real parts, we obtain R1 R C ¼ 3 þ 4 R2 R4 C3

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ð2:91Þ

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and, by equating the imaginary parts, we obtain 0 ¼ vC4 R3 2 Hence,

1 vC3 R4

1 v ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ C3 C4 R3 R4

ð2:92Þ

Equation 2.92 tells us that the circuit is an oscillator whose natural frequency is given by this equation under balanced conditions. If the frequency of the supply is equal to the natural frequency of the circuit, large-amplitude oscillations will take place. The circuit can be used to measure an unknown resistance (e.g., in strain-gage devices) by ﬁrst measuring the frequency of the bridge signals at resonance (natural frequency). Alternatively, an oscillator that is excited at its natural frequency can be used as an accurate source of periodic signals (a signal generator).

2.7

Linearizing Devices

Nonlinearity is present in any physical device, to varying levels. If the level of nonlinearity in a system (component, device, or equipment) can be neglected without exceeding the error tolerance, then the system can be assumed to be linear. In general, a linear system is one that can be expressed as one or more linear differential equations. Note that the principle of superposition holds for linear systems. Speciﬁcally, if the system response to an input, u1 ; is y1 and the response to another input, u2 ; is y2 ; then the response to a1 u1 þ a2 u2 is a1 y1 þ a2 y2 : Nonlinearities in a system can appear in two forms: 1. Dynamic manifestation of nonlinearities 2. Static manifestation of nonlinearities The useful operating region of many systems can exceed the frequency range where the frequencyresponse function is ﬂat. The operating response of such a system is said to be dynamic. Examples include a typical dynamic system (e.g., automobile, aircraft, chemical process plant, robot), actuator (e.g., hydraulic motor), and controller (e.g., PID control circuitry). Nonlinearities of such systems can manifest themselves in a dynamic form such as the jump phenomenon (also known as the fold catastrophe), limit cycles, and frequency creation. Design changes, extensive adjustments, or reduction of the operating signal levels and bandwidths are generally necessary to reduce or eliminate these dynamic manifestations of nonlinearity. In many instances, such changes are not practical and we have to somehow manage with the presence of these nonlinearities under dynamic conditions. Design changes might involve replacing conventional gear drives with devices such as harmonic drives in order to reduce backlash, replacing nonlinear actuators with linear actuators, and using components that have negligible Coulomb friction and that make small motion excursions. A wide majority of sensors, transducers, and signal-modiﬁcation devices are expected to operate in the ﬂat region of the frequency-response function. The input/output relation of these types of devices, in the operating range, is expressed (modeled) as a static curve rather than a differential equation. Nonlinearities in these devices will manifest themselves in the static operating curve in many forms. These manifestations include saturation, hysteresis, and offset. In the ﬁrst category of systems (plants, actuators, and compensators), if a nonlinearity is exhibited in the dynamic form, proper modeling and control practices should be employed in order to avoid unsatisfactory degradation of the system performance. In the second category of systems (sensors, transducers and signal modiﬁcation devices), if nonlinearities are exhibited in the “static” operating curve, again the overall performance of the system will be degraded. Hence, it is important to “linearize” the output of such devices. Note that, in dynamic manifestations, it is not realistic to linearize the output because the response is in the dynamic form. The solution in that case is either to minimize nonlinearities

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through design modiﬁcations and adjustments so that a linear approximation would be valid, or to take the nonlinearities into account in system modeling and control. In the present section, we are not concerned with this aspect. Instead, we are interested in the linearization of devices in the second category, whose operating characteristics can be expressed by static input–output curves. Linearization of a static device can be attempted by making design changes and adjustments, as in the case of dynamic devices. However, since the response is static, and since we normally deal with an available (ﬁxed) device whose internal hardware cannot be modiﬁed, we should consider ways of linearizing the input–output characteristic by modifying the output itself. Static linearization of a device can be made in three ways: 1. Linearization using digital software 2. Linearization using digital (logic) hardware 3. Linearization using analog circuitry

Output y

(a)

0

Input u

Output y

(b)

0

Input u

FIGURE 2.20 (a) A general static nonlinear characteristic; (b) an offset nonlinearity.

In the software approach to linearization, the output of the device is read into a processor with software-programmable memory and the output is modiﬁed according to the program instructions. In the hardware approach, the device output is read by a device having ﬁxed logic circuitry that processes (modiﬁes) the data. In the analog approach, a linearizing circuit is directly connected at the output of the device so that the output of the linearizing circuit is proportional to the input of the device. We shall discuss these three approaches in the rest of this section, heavily emphasizing the analog-circuit approach. Hysteresis-type static nonlinearity characteristics have the property that the input–output curve is not one to one. In other words, one input value may correspond to more than one (static) output value, and one output value may correspond to more than one input value. Let us disregard these types of nonlinearities. Our main concern is the linearization of a device having a single-valued static response curve that is not a straight line. An example of a typical nonlinear input –output characteristic is shown in Figure 2.20(a). Strictly speaking, a straight-line characteristic with a simple offset, as shown in Figure 2.20(b), is also a nonlinearity. In particular, note that superposition does not hold for an input– output characteristic of this type, given by y ¼ ku þ c

ð2:93Þ

It is very easy, however, to linearize such a device because the simple addition of a DC component will convert the characteristic into the linear form given by y ¼ ku

ð2:94Þ

This method of linearization is known as offsetting. Linearization is more difﬁcult in the general case where the characteristic curve could be much more complex.

2.7.1

Linearization by Software

If the nonlinear relationship between the input and the output of a nonlinear device is known, the input can be “computed” for a known value of the output. In the software approach of linearization, system

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composed of a processor and memory that can be programmed using software (i.e., a digital computer) is used to compute the input using output data. Two approaches can be used. They are 1. Equation inversion 2. Table lookup In the ﬁrst method, the nonlinear characteristic of the device is known in the analytic (equation) form: y ¼ f ðuÞ

ð2:95Þ

in which u ¼ device input y ¼ device output Assuming that this is a one-to-one relationship, a unique inverse given by the equation u ¼ f 21 ðyÞ

ð2:96Þ

can be determined. This equation is programmed into the read-and-write memory (RAM) of the computer as a computation algorithm. When the output values, y; are supplied to the computer, the processor will compute the corresponding input values, u; using the instructions (executable program) stored in the RAM. In the table lookup method, a sufﬁciently large number of pairs of values ðy; uÞ are stored in the memory of the computer in the form of a table of ordered pairs. These values should cover the entire operating range of the device. When a value for y is entered into the computer, the processor scans the stored data to check whether that value is present. If so, the corresponding value of u is read and this is the linearized output. If the value of y is not present in the data table, then the processor will interpolate the data in the vicinity of the value and will compute the corresponding output. In the linear interpolation method, the area of the data table where the y value falls is ﬁtted with a straight line and the corresponding u value is computed using this straight line. Higher order interpolations use nonlinear interpolation curves such as quadratic and cubic polynomial equations (splines). Note that the equation inversion method is usually more accurate than the table lookup method and it does not need excessive memory for data storage. However, it is relatively slow because data are transferred and processed within the computer using program instructions that are stored in the memory and that typically have to be accessed in a sequential manner. The table lookup method is fast. Since the accuracy depends on the number of stored data values, this is a memory-intensive method. For better accuracy, more data should be stored. However, since the entire data table has to be scanned to check for a given data value, this increase in accuracy is derived at the expense of speed as well as memory requirements.

2.7.2

Linearization by Hardware Logic

The software approach to linearization is ﬂexible in the sense that the linearization algorithm can be modiﬁed (e.g., improved, changed) simply by modifying the program stored in the RAM. Furthermore, highly complex nonlinearities can be handled by the software method. As mentioned before, the method is relatively slow, however. In the hardware logic method of linearization, the linearization algorithm is permanently implemented in the IC form using appropriate digital logic circuitry for data processing, and memory elements (e.g., ﬂip-ﬂops). Note that the algorithm and numerical values of parameters (except input values) cannot be modiﬁed without redesigning the IC chip, because a hardware device typically does not have programmable memory. Furthermore, it will be difﬁcult to implement very complex linearization algorithms by this method and, unless the chips are mass produced for an extensive commercial market, the initial cost of chip development will make the production of linearizing chips economically infeasible. In bulk production, however, the per-unit cost will be very small. Furthermore, since the access of stored

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program instructions and extensive data manipulation are not involved, the hardware method of linearization can be substantially faster than the software method. A digital linearizing unit that has a processor and a read-only memory (ROM), whose program cannot be modiﬁed, also lacks the ﬂexibility of a programmable software device. Hence, such a ROM-based device also falls into the category of hardware logic devices.

2.7.3

Analog Linearizing Circuitry

Three types of analog linearizing circuitry can be identiﬁed: 1. Offsetting circuitry 2. Circuitry that provides a proportional output 3. Curve shapers We will describe each of these categories now. An offset is a nonlinearity that can be easily removed using an analog device. This is accomplished by simply adding a DC offset of equal value to the response, but in the opposite direction. Deliberate addition of an offset in this manner is known as offsetting. The associated removal of original offset is known as offset compensation. There are many applications of offsetting. Unwanted offsets such as those present in ADC and DAC results can be removed by analog offsetting. Constant (DC) error components such as steady-state errors in dynamic systems due to load changes, gain changes, and other disturbances can be eliminated by offsetting. Common-mode error signals in ampliﬁers and other analog devices can also be removed by offsetting. In measurement circuitry such as potentiometer (ballast) circuits, where the actual measurement signal is a change, dvo ; in a steady output signal, vo ; the measurement can be completely wiped out due to noise. To reduce this problem, ﬁrst the output should be offset by 2vo so that the net output is dvo and not vo þ dvo : This output is subsequently conditioned by ﬁltering and ampliﬁcation. Another application of offsetting is the additive change of scale of a measurement, for example from a relative scale (e.g., velocity) to an absolute scale. In summary, some of the applications of offsetting are: 1. Removal of unwanted offsets and DC components in signals (e.g., in ADC, DAC, signal integration). 2. Removal of steady-state error components in dynamic system responses (e.g., due to load changes and gain changes in Type 0 systems. Note: Type 0 systems are open-loop systems having no free integrators). 3. Rejection of common-mode levels (e.g., in ampliﬁers and ﬁlters). 4. Error reduction when a measurement is an increment of a large steady output level (e.g., in ballast circuits for strain-gage and RTD sensors). 5. Scale changes in an additive manner (e.g., conversion from relative to absolute units or from absolute to relative units). We can remove unwanted offsets in the simple manner as discussed above. Let us now consider more complex nonlinear responses that are nonlinear in the sense that the input–output curve is not a straight line. Analog circuitry can be used to linearize this type of response as well. The linearizing circuit used will generally depend on the particular device and the nature of its nonlinearity. Hence, often linearizing circuits of this type have to be discussed with respect to a particular application. For example, such linearization circuits are useful in transverse-displacement capacitative sensor. Several useful circuits are described below. Consider the type of linearization that is known as curve shaping. A curve shaper is a linear device whose gain (output/input) can be adjusted so that response curves with different slopes can be obtained. Suppose that a nonlinear device having an irregular (nonlinear) input– output characteristic is to be linearized. First, we apply the operating input simultaneously to the device and the curve shaper, and the gain of the curve shaper is adjusted such that it closely matches that of the device in a small range

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of operation. Now, the output of the curve shaper can be utilized for any task that requires the device output. The advantage here is that linear assumptions are valid with the curve shaper, which is not the case for the actual device. When the operating range changes, the curve shaper must be returned to the new range. Comparison (calibration) of the curve shaper and the nonlinear device can be done off line and, once a set of gain values corresponding to a set of operating ranges is determined in this manner for the curve shaper, it is possible to completely replace the nonlinear device with the curve shaper. Then the gain of the curve shaper can be adjusted depending on the actual operating range during system operation. This is known as gain scheduling. Note that we can replace a nonlinear device with a linear device (curve shaper) within a multicomponent system in this manner without greatly sacriﬁcing the accuracy of the overall system.

2.7.4

Offsetting Circuitry

Common-mode outputs and offsets in ampliﬁers R and other analog devices can be minimized by Input including a compensating resistor that can provide R B − Output vi ﬁne adjustments at one of the input leads. vo vref + A Furthermore, the larger is the feedback signal (DC Reference) Rc level in a feedback system, the smaller is the steadyRo state error. Hence, steady-state offsets can be reduced by reducing the feedback resistance (thereby increasing the feedback signal). Furthermore, since a ballast (potentiometer) circuit FIGURE 2.21 An inverting ampliﬁer circuit for offset provides an output of vo þ dvo and a bridge compensation. circuit provides an output of dvo ; the use of a bridge circuit can be interpreted as an offset compensation method. The most straightforward way of offsetting is by using a differential ampliﬁer (or a summing ampliﬁer) to subtract (or add) a DC voltage to the output of the nonlinear device. The DC level has to be variable so that various levels of offset can be provided using the same circuit. This is accomplished by using an adjustable resistance at the DC input lead of the ampliﬁer. An operational-ampliﬁer circuit for offsetting is shown in Figure 2.21. Since the input, vi ; is connected to the negative lead of the opamp, we have an inverting ampliﬁer, and the input signal will appear in the output, vo ; with its sign reversed. This is also a summing ampliﬁer because two signals can be added together by this circuit. If the input, vi ; is connected to the positive lead of the opamp, we will have a noninverting ampliﬁer. The DC voltage, vref ; provides the offsetting voltage. The resistor, Rc (compensating resistor), is variable so that different values of offset can be compensated using the same circuit. To obtain the circuit equation, we write the current balance equation for node A, using the usual assumption that the current through an input lead is zero for an opamp because of very high input impedance; thus vref 2 vA v ¼ A Rc Ro or vA ¼

Ro v ðRo þ Rc Þ ref

ðiÞ

Similarly, the current balance at node B gives vi 2 vB v 2 vB þ o ¼0 R R or vo ¼ 2vi þ 2vB

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ðiiÞ

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Vibration Monitoring, Testing, and Instrumentation

Since vA ¼ vB for the opamp (because of very high open-loop gain), we can substitute Equation i into Equation ii. Then, vo ¼ 2vi þ

2Ro v ðRo þ Rc Þ ref

ð2:97Þ

Note the sign of vi is reversed at the output (because this is an inverting ampliﬁer). This is not a problem because polarity can be reversed at input or output in connecting this circuit to other circuitry, thereby recovering the original sign. The important result here is the presence of a constant offset term on the RHS of Equation 2.97. This term can be adjusted by picking the proper value for Rc so as to compensate for a given offset in vi :

2.7.5

Proportional-Output Circuitry

An operational-ampliﬁer circuit may be employed Active Element to linearize the output of a capacitive transverseR1 displacement sensor. In constant-voltage and constant-current resistance bridges and in a R3 constant-voltage half bridge, the relation between B − DC Supply Output v the bridge output, dvo ; and the measurand (the ref vo A + change in resistance in the active element) is R4 RL Load nonlinear. The nonlinearity is least for the R2 constant-current bridge and it is highest for the half bridge. Since dR is small compared with R; however, the nonlinear relations can be linearized without introducing large errors. However, the FIGURE 2.22 A proportional-output circuit for an linear relations are inexact and are not suitable if active resistance element (strain gage). dR cannot be neglected in comparison to R: Under these circumstances, the use of a linearizing circuit would be appropriate. One way to obtain a proportional output from a Wheatstone bridge is to feed back a suitable factor of the bridge output into the bridge supply, vref : Another way is to use the opamp circuit shown in Figure 2.22. This should be compared with the Wheatstone bridge shown in Figure 2.17(a). Note that R represents the only active element (e.g., an active strain gage). First, let us show that the output equation for the circuit in Figure 2.22 is similar to Equation 2.68. Using the fact that the current through an input lead of an unsaturated opamp can be neglected, we have the following current balance equations for nodes A and B: vref 2 vA v ¼ A R4 R2 vref 2 vB v 2 vB þ o ¼0 R3 R1 Hence, vA ¼

R2 v ðR2 þ R4 Þ ref

vB ¼

R1 vref þ R3 vo ðR1 þ R3 Þ

and

Now, using the fact vA ¼ vB for an opamp, we obtain R1 vref þ R3 vo R2 ¼ v ðR1 þ R3 Þ ðR2 þ R4 Þ ref

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Accordingly, we have the circuit output equation vo ¼

ðR2 R3 2 R1 R4 Þ v R3 ðR2 þ R4 Þ ref

ð2:98Þ

Note that this relation is quite similar to the Wheatstone bridge equation (Equation 2.68). The balance condition (i.e., vo ¼ 0) is again given by Equation 2.69. Suppose that R1 ¼ R2 ¼ R3 ¼ R4 ¼ R in the beginning (the circuit is balanced), so vo ¼ 0: Then suppose that the active resistance R1 is changed by dR (say, owing to a change in strain in the strain gage R1 ). Then, using Equation 2.98, we can write an expression for the charge in circuit output as dvo ¼

½R2 2 RðR þ dRÞ vref 2 0 RðR þ RÞ

or dvo 1 dR ¼2 vref 2 R

ð2:99Þ

By comparing this result with Equation 2.71, we observe that the circuit output, dvo , is proportional to the measurand, dR: Furthermore, note that the sensitivity of the circuit in Figure 2.22 (1/2) is double that of a Wheatstone bridge that has one active element (1/4), which is a further advantage of the proportional-output circuit. The sign reversal is not a drawback because it can be accounted for by reversing the load polarity. 2.7.5.1

Curve-Shaping Circuitry

A curve shaper can be interpreted as an ampliﬁer Resistance Switching Circuit whose gain is adjustable. A typical arrangement for Rf a curve-shaping circuit is shown in Figure 2.23. The feedback resistance, Rf ; is adjustable by some Input R means. For example, a switching circuit with a A − Output vi bank of resistors (say, connected in parallel vo through solid-state switches as in the case of + weighted-resistor DAC) can be used to switch the feedback resistance to the required value. Automatic switching can be realized by using Zener diodes that will start conducting at certain voltage FIGURE 2.23 A curve-shaping circuit. levels. In both cases (external switching by switching pulses or automatic switching using Zener diodes), ampliﬁer gain is variable in discrete steps. Alternatively, a potentiometer can be used as Rf so that the gain can be continuously adjusted (manually or automatically). The output equation for the curve-shaping circuit shown in Figure 2.23 is obtained by writing the current balance at node A, noting that vA ¼ 0; thus vi v þ o ¼0 R Rf or vo ¼ 2

Rf v R i

It follows that the gain ðRf =RÞ of the ampliﬁer can be adjusted by changing Rf :

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ð2:100Þ

[L

uB

\)!\gd gd\g\dQ8 A\dQ8 !d: dbAd!\gd

\2A__!dAg \Qd!_ g:\K2!\gd \2\

d !::\\gd g YA \Qd!_ bg:\K2!\gd :A\2A :\2A: g J! \d Y\ 2Y!pA8 YAA !A b!d gYA pA gJ 2\2\ Y! !A A: Jg \Qd!_ bg:\K2!\gd !d: A_!A: !^u !bp_A !A pY!A Y\JA8 g_!QA[ g[JAzAd2 2gdAA8 JAzAd2[g[g_!QA 2gdAA8 g_!QA[g[2Ad 2gdA8 !d: pA!^[Yg_: 2\2\u YA g)]A2\A gJ YA pAAd A2\gd \ g :\2 )\AM AA!_ gJ 2Y b\2A__!dAg 2\2\ !d: 2gbpgdAd Y! !A AJ_ \d YA \dbAd!\gd gJ :d!b\2 Abu

uBuj

Y!A Y\JA

\dg\:!_ \Qd!_ Q\Ad ) h ! \dSK | *7

,

Sjj7

Y! YA Jg__g\dQ YAA ApAAd!\A p!!bAA7 ! h !bp_\:A K h JAzAd2 * h pY!A !dQ_A

«ÕÌ Û

,

,V

w

"ÕÌ«ÕÌ Û

³

gA Y! YA pY!A !dQ_A ApAAd YA \bA uO pY!A[Y\JA 2\2\u AJAAd2A r!\dQ pg\ds gJ YA \Qd!_u YA pY!A !dQ_A \ !d \bpg!d 2gd\:A!\gd gd_ YAd g g bgA \Qd!_ 2gbpgdAd !A 2gbp!A:u YA g\A pA2b gJ ! \Qd!_ \ pAAdA: ! \ !bp_\:A rb!Qd\:As !d: YA pY!A !dQ_A \Y ApA2 g YA JAzAd2u Y!A[Y\J\dQ 2\2\ Y!A b!d !pp_\2!\gdu YAd ! \Qd!_ p!A YgQY ! Ab8 \ pY!A !dQ_A 2Y!dQA :A g :d!b\2 2Y!!2A\\2 gJ YA Abu gdAzAd_8 YA pY!A 2Y!dQA pg\:A A AJ_ \dJgb!\gd !)g YA :d!b\2 2Y!!2A\\2 gJ YA Abu pA2\K2!__8 Jg ! _\dA! 2gd!d[2gAJK2\Ad Ab8 Y\ pY!A Y\J \ Az!_ g YA pY!A !dQ_A gJ YA JAzAd2[ApgdA Jd2\gd r JAzAd2[!dJA Jd2\gds gJ YA Ab ! Y! p!\2_! JAzAd2u Y\ pY!A[Y\J\dQ )AY!\g \8 gJ 2gA8 dg _\b\A: g A_A2\2!_ Ab !d: \ Az!__ AY\)\A: ) gYA pA gJ Ab \d2_:\dQ bA2Y!d\2!_ \)!\dQ Abu YA pY!A Y\J )AAAd g \Qd!_ 2!d )A :AAb\dA: ) 2gdA\dQ YA \Qd!_ \dg YA A_A2\2!_ Jgb r\dQ \!)_A !d:2As !d: Y\J\dQ YA pY!A !dQ_A gJ gdA \Qd!_ YgQY ^dgd !bgd8 \dQ ! pY!A[Y\J\dQ 2\2\8 d\_ YA g \Qd!_ !A \d pY!Au dgYA !pp_\2!\gd gJ pY!A Y\JA \ \d \Qd!_ :Abg:_!\gdu g A!bp_A8 gdA bAYg: gJ !bp_\:A :Abg:_!\gd \dg_A pg2A\dQ YA bg:_!A: \Qd!_ gQAYA \Y YA 2!\A \Qd!_u Y\8 YgAA8 Az\A YA bg:_!A: \Qd!_ !d: YA 2!\A \Qd!_ g )A \d pY!Au !__8 YgAA8 \d2A YA bg:_!A: \Qd!_ Y! !_A!: !db\A: YgQY A_A2\2!_ 2\2\ Y!\dQ \bpA:!d2A 2Y!!2A\\28 \ pY!A !dQ_A Y! 2Y!dQA:u YAd8 \ \ dA2A! g Y\J YA pY!A !dQ_A gJ YA 2!\A d\_ YA g \Qd!_ !A \d pY!A8 g Y! :Abg:_!\gd 2!d )A pAJgbA: !22!A_u Ad2A8 pY!A Y\JA !A A: \d :Abg:_!\dQ8 Jg A!bp_A8 YAd :Abg:_!\dQ :\p_!2AbAd[Adg gpu pY!A[Y\JA 2\2\8 \:A!__8 Yg_: dg 2Y!dQA YA \Qd!_ !bp_\:A Y\_A 2Y!dQ\dQ YA pY!A !dQ_A ) ! Az\A: !bgdu !2\2!_ pY!A Y\JA 2!d \dg:2A gbA :AQAA gJ !bp_\:A :\g\gd r\Y ApA2 g JAzAd2s ! A__u \bp_A pY!A[Y\JA 2\2\ 2!d )A 2gd2A: \dQ A\!d2A rs !d: 2!p!2\!d2A rs A_AbAdu A\!d2A g ! 2!p!2\g gJ 2Y !d 2\2\ \ b!:A KdA[!:]!)_A g ! g g)!\d ! !\!)_A pY!A Y\JAu d gp!bp[)!A: pY!A Y\JA 2\2\ \ Ygd \d \QA uOu A 2!d Yg Y! Y\ 2\2\ pg\:A ! pY!A Y\J \Yg :\g\dQ YA \Qd!_ !bp_\:Au YA 2\2\ Az!\gd \ g)!\dA: ) \\dQ YA 2Ad )!_!d2A Az!\gd ! dg:A !d: 8 dg\dQ8 ! !_8 Y! YA 2Ad YgQY YA gp!bp _A!:

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Signal Conditioning and Modiﬁcation

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can be neglected due to high input impedance; thus vi 2 vA dv ¼C A RC dt vi 2 vB v 2 vB þ o ¼0 R R On simplifying and introducing the Laplace variable, we obtain vi ¼ ðts þ 1ÞvA

ðiÞ

1 ðv þ vo Þ 2 i

ðiiÞ

and vB ¼ in which, the circuit time constant, t; is given by

t ¼ Rc C Since vA ¼ vB as a result of very high gain in the opamp, by substituting Equation ii into Equation i, we obtain vi ¼

1 ðts þ 1Þðvi þ vo Þ 2

It follows that the transfer function GðsÞ of the circuit is given by vo ð1 2 tsÞ ¼ GðsÞ ¼ ð1 þ tsÞ vi

ð2:102Þ

It is seen that the magnitude of the frequency-response function Gð jvÞ is pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ t 2 v2 ﬃ lGðjvÞl ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 1 þ t 2 v2 or lGð jvÞl ¼ 1

ð2:103Þ

and the phase angle of GðjvÞ is /Gð jvÞ ¼ 2tan21 tv 2 tan21 tv or /GðjvÞ ¼ 22 tan21 tv ¼ 22 tan21 Rc Cv

ð2:104Þ

As needed, the transfer function magnitude is unity, indicating that the circuit does not distort the signal amplitude over the entire bandwidth. Equation 2.104 gives the phase lead of the output, vo ; with respect to the input, vi : Note that this angle is negative, indicating that actually a phase lag is introduced. The phase shift can be adjusted by varying the resistance, Rc :

2.8.2

Voltage-to-Frequency Converter

A voltage-to-frequency converter (VFC) generates a periodic output signal whose frequency is proportional to the level of an input voltage. Since such an oscillator generates a periodic output according to the voltage excitation, it is also called a voltage-controlled oscillator (VCO). A common type of VFC uses a capacitor. The time needed for the capacitor to be charged to a ﬁxed voltage level will depend on the charging voltage (it is inversely proportional). Suppose that this voltage is governed by the input voltage. Then, if the capacitor is made to periodically charge and discharge, we have an output whose frequency (inverse of the charge –discharge period) is proportional to the charging voltage. The output amplitude will be given by the ﬁxed voltage level to which the capacitor is charged in

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Reference Level vs VoltageSensitive Switch

Input −vi

C

R

A

Oscillator Output vo

− +

(a) Output vo(t)

T =

RC (vs − vo(0)) vi

vs

vo(0) (b) FIGURE 2.25

0

T

2T

3T

4T

Time t

A voltage-to-frequency converter (voltage-controlled oscillator): (a) circuit; (b) output signal.

each cycle. Consequently, we have a signal with a ﬁxed amplitude and a frequency that depends on the charging voltage (input). A VFC (or VCO) circuit is shown in Figure 2.25(a). The voltage-sensitive switch closes when the voltage across it exceeds a reference level, vs ; and it opens again when the voltage across it falls below a lower limit, vo ð0Þ: The programmable unijunction transistor (PUT) is such a switching device. Note that the polarity of the input voltage, vi ; is reversed. Suppose that the switch is open. Then, current balance at node A of the opamp circuit gives vi dv ¼C o R dt As usual, vA ¼ voltage at positive lead ¼ 0 because the opamp has a very high gain, and current through the opamp leads ¼ 0 because the opamp has a very high input impedance. The capacitor charging equation can be integrated for a given value of vi : This gives vo ðtÞ ¼

1 v t þ vo ð0Þ RC i

The switch is closed when the voltage across the capacitor vo ðtÞ equals the reference level vs : Then, the capacitor will be immediately discharged through the closed switch. Hence, the capacitor charging time, T, is given by vs ¼

1 v T þ vo ð0Þ RC i

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Accordingly, T¼

RC ðv 2 vo ð0ÞÞ vi s

ð2:105Þ

The switch opens again when the voltage across the capacitor drops to vo ð0Þ; and the capacitor again begins to charge from vo ð0Þ up to vs : This charging and instantaneous discharge cycle repeats periodically. The corresponding output signal is as shown in Figure 2.25(b). This is a periodic (sawtooth) wave with period T: The frequency of oscillation of the output ð1=TÞ is given by f ¼

vi RCðvs 2 vo ð0ÞÞ

ð2:106Þ

It is seen that the oscillator frequency is proportional to the input voltage vi : The oscillator amplitude is vs ; which is ﬁxed. VCOs have many applications. One application is in analog-to-digital conversion. In the VCO type analog-to-digital converters, the analog signal is converted into an oscillating signal using a VCO. Then, the oscillator frequency is measured using a digital counter. This count, which is available in the digital form, is representative of the input analog signal level. Another application is in digital voltmeters. Here, the same method as for ADC is used. Speciﬁcally, the voltage is converted into an oscillator signal and its frequency is measured using a digital counter. The count can be scaled and displayed to provide the voltage measurement. A direct application of the VCO is apparent from the fact it is actually a frequency modulator, providing a signal whose frequency is proportional to the input (modulating) signal. Hence, the VCO is useful in applications that require frequency modulation. Also, a VCO can be used as a signal (wave) generator for variable-frequency applications; for example, it can be used for excitation inputs for shakers in vibration testing, excitations for frequency-controlled DC motors, and pulse signals for translator circuits of stepping motors.

2.8.3

Frequency-to-Voltage Converter

A frequency-to-voltage converter (FVC) generates an output voltage whose level is proportional to the frequency of its input signal. One way to obtain a FVC is to use a digital counter to count the signal frequency and then use a DAC to obtain a voltage proportional to the frequency. A schematic representation of this type of FVC is shown in Figure 2.26(a). Frequency Signal (a)

Digital Counter

DAC

Voltage Output

Charging Voltage vs Comparator Frequency Signal −

(b) FIGURE 2.26

Switching Circuit

Capacitor Circuit

Switching Circuit

Voltage Output

Threshold Signal Frequency-to-voltage converters: (a) digital counter method; (b) capacitor charging method.

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An alternative FVC circuit is schematically shown in Figure 2.26(b). In this method, the frequency signal is supplied to a comparator along with a threshold voltage level. The sign of the comparator output will depend on whether the input signal level is larger or smaller than the threshold level. The ﬁrst sign change (negative to positive) in the comparator output is used to trigger a switching circuit that will respond by connecting a capacitor to a ﬁxed charging voltage. This will charge the capacitor. The next sign change (positive to negative) of the comparator output will cause the switching circuit to short the capacitor, thereby instantaneously discharging it. This charging –discharging process will be repeated in response to the oscillator input. Note that the voltage level to which the capacitor is charged each time will depend on the switching period (charging voltage is ﬁxed), which is in turn governed by the frequency of the input signal. Hence, the output voltage of the capacitor circuit will be representative of the frequency of the input signal. Since the output is not steady due to the ramp-like charging curve and instantaneous discharge, a smoothing circuit is provided at the output to remove the noisy ripples. Applications of FVC include demodulation of frequency-modulated signals, frequency measurement in mechanical vibration applications, and conversion of pulse outputs in some types of sensors and transducers into analog voltage signals.

2.8.4

Voltage-to-Current Converters

Measurement and feedback signals are usually Input Output Current io transmitted as current levels in the range of 4 to Voltage R R vi 20 mA rather than as voltage levels. This is B + particularly useful when the measurement site is Load P RL not close to the monitoring room. Since the A − R measurement itself is usually available as a voltage, R it has to be converted into current by using a voltage-to-current converter (VCC). For example, pressure transmitters and temperature transmitFIGURE 2.27 A voltage-to-current converter. ters in operability testing systems provide current outputs that are proportional to the measured values of pressure and temperature. There are many advantages to transmitting current rather than voltage. In particular, the voltage level will drop due to resistance in the transmitting path, but the current through a conductor will remain uncharged unless the conductor is branched. Hence, current signals are less likely to acquire errors due to signal weakening. Another advantage of using current instead of voltage as the measurement signal is that the same signal can be used to operate several devices in series (for example, a display, a plotter, and a signal processor simultaneously), without causing errors through signal weakening due to the power lost at each device, because the same current is applied to all devices. AVCC should provide a current proportional to an input voltage without being affected by the load resistance to which the current is supplied. An operational-ampliﬁer-based voltage-to-current convert circuit is shown in Figure 2.27. Using the fact that the currents through the input leads of an unsaturated opamp can be neglected (due to very high input impedance), we write the current summation equations for the two nodes, A and B, thus: vp 2 vA vA ¼ R R and vi 2 vB v 2 vB þ P ¼ io R R Accordingly, we have 2vA ¼ vP

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and vi 2 2vB þ vP ¼ Rio

ðiiÞ

Now, using the fact that vA ¼ vB for the opamp (due to very high gain), we substitute Equation i into Equation ii. This gives v ð2:107Þ io ¼ i R in which io ¼ output current vi ¼ input voltage It follows that the output current is proportional to the input voltage, irrespective of the value of the load resistance, RL ; as required for a VCC.

2.8.5

Peak-Hold Circuits

Output Unlike a S/H circuit that holds every sampled value Input Voltage Follower of the signal, a peak-hold circuit holds only the Signal Diode + + Peak Value vi largest value reached by the signal during the − (Output) vo − monitored period. Peak holding is useful in a variety of applications. In signal processing for v Reset shock and vibration studies, what are known as Switch response spectra (e.g., a shock response spectrum) are determined by using a response spectrum analyzer that exploits a peak holding scheme. Suppose that FIGURE 2.28 A peak-holding circuit. a signal is applied to a simple oscillator (a singledegree-of-freedom second-order system with no zeros) and the peak value of the response (output) is determined. A plot of the peak output as a function of the natural frequency of the oscillator, for a speciﬁed damping ratio, is known as the response spectrum of the signal for that damping ratio. Peak detection is also useful in machine monitoring and alarm systems. In short, when just one representative value of a signal is needed in a particular application, the peak value is a leading contender. Peak detection of a signal can be conveniently done using digital processing. For example, the signal may be sampled and the previous sample value replaced by the present sample value if and only if the latter is larger than the former. By sampling and then holding one value in this manner, the peak value of the signal is retained. Note that, usually, the time instant at which the peak occurs is not retained. Peak detection can be done using analog circuitry as well. This is, in fact, the basis of analog spectrum analyzers. A peak-holding circuit is shown in Figure 2.28. The circuit consists of two voltage followers. The ﬁrst voltage follower has a diode at its output that is forward biased by the positive output of the voltage follower and reverse-biased by a low-leakage capacitor, as shown. The second voltage follower presents the peak voltage that is held by the capacitor to the circuit output at a low output impedance, without loading the previous circuit stage (capacitor and ﬁrst voltage follower). To understand the operation of the circuit, suppose that the input voltage, vi ; is larger than the voltage to which capacitor is charged (v). Since the voltage at the positive lead of the opamp is vi and the voltage at the negative lead is v; the ﬁrst opamp will be saturated. Since the differential input to the opamp is positive under these conditions, the opamp output will be positive. The output will charge the capacitor until the capacitor voltage, v; equals the input voltage, vi : This voltage (call it vo ) is in turn supplied to the second voltage follower which presents the same value to its output (gain ¼ 1 for a voltage follower), but at a very low impedance level. Note that the opamp output remains at the saturated value only for a very short time (the time taken by the capacitor to charge). Now, suppose that vi is smaller than v: Then, the differential input of the opamp will be negative, and the opamp output will be saturated at the negative saturation level. This will reverse bias the diode. Hence, the output of the ﬁrst opamp will be in open circuit, and as a result the voltage supplied to the output voltage follower

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would still be the capacitor voltage and not the output of the ﬁrst opamp. It follows that the voltage level of the capacitor (and hence the output of the second voltage follower) would always be the peak value of the input signal. The circuit can be reset by discharging the capacitor through a solid-state switch that is activated by an external pulse.

2.9

Signal Analyzers and Display Devices

Vibration signal analysis may employ both analog and digital procedures. Since signal analysis results in extracting various useful bits of information from the signal, it is appropriate to consider the topic within the present context of signal modiﬁcation as well. Here, we will introduce digital signal analyzers. Signal display devices also make use of at least some signal processing. This may involve ﬁltering and change of the signal level and format. More sophisticated signal display devices, particularly digital oscilloscopes, can carry out more complex signal analysis functions such as those normally available with digital signal analyzers. Oscilloscopes as well are introduced in the present section, though they may be treated under vibration instrumentation. Signal-recording equipment commonly employed in vibration practice includes digital storage devices such as hard drives, ﬂoppy disks, and CD-ROMs, analog devices like tape recorders, strip-chart recorders and X–Y plotters, and digital printers. Tape recorders are used to record vibration data (transducer outputs) that are subsequently reproduced for processing or examination. Often, tape-recorded waveforms are also used to generate (by replay) signals that drive vibration test exciters (shakers). Tape recorders use tapes made of a plastic material that has a thin coating of a specially treated ferromagnetic substance. During the recording process, magnetic ﬂux proportional to the recorded signal is produced by the recording head (essentially an electromagnet), which magnetizes the tape surface in proportion to the signal variation. Reproduction is the reverse process, whereby an electrical signal is generated at the reproduction head by electromagnetic induction in accordance with the magnetic ﬂux of the magnetized (recorded) tape. Several signal-conditioning circuitries are involved in the recording and reproducing stages. Recording by FM is very common in vibration testing. Strip-chart recorders are usually employed to plot time histories (that is, quantities that vary with time), although they also may be used to plot such data as frequency-response functions and response spectra. In these recorders, a paper roll unwinds at a constant linear speed, and the writing head moves across the paper (perpendicular to the paper motion) proportionally to the signal level. There are many kinds of strip-chart recorders, which are grouped according to the type of writing head that is employed. Graphic-level recorders, which use ordinary paper, employ such heads as ink pens or brushes, ﬁber pens, and sapphire styli. Visicoders are simple oscilloscopes that are capable of producing permanent records; they employ light-sensitive paper for this. Several channels of input data can be incorporated with a visicoder. Obviously, graphic-level recorders are generally limited by the number of writing heads possible (typically, one or two), but visicoders can have many more input channels (typically, 24). Performance speciﬁcations of these devices include paper speed, frequency range of operation, dynamic range, and power requirements. In vibration experimentation, X –Y plotters are generally employed to plot frequency data (for example, PSD, frequency-response functions, response spectra, transmissibility curves), although they also can be used to plot time-history data. Many types of X– Y plotter are available, most of them using ink pens and ordinary paper. There are also hard-copy units that use heat-sensitive paper in conjunction with a heating element as the writing head. The writing head in an X –Y plotter is moved in the X and Ydirections on the paper by two input signals that form the coordinates for the plot. In this manner, a trace is made on stationary plotting paper. Performance speciﬁcations of X –Y plotters are governed by such factors as paper size; writing speed (in./sec, cm/sec); dead band (expressed as a percentage of the full scale), which measures the resolution of the plotter head; linearity (expressed as a percentage of the full scale), which measures the accuracy of the plot; minimum trace separation (in., cm) for multiple plots on the same axes; dynamic range; input impedance; and maximum input (mV/in., mV/cm).

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Today, the most widespread signal recording device is in fact the digital computer (memory, storage) and printer combination. This and the other (analog) devices used in signal recording and display make use of some signal modiﬁcation to accomplish their functions. However, we will not discuss these devices in the present section.

2.9.1

Signal Analyzers

Modern signal analyzers employ digital techniques of signal analysis to extract useful information that is carried by the signal. Digital Fourier analysis using FFT is perhaps the single common procedure that is used in the vast majority of signal analyzers. As we have noted before, Fourier analysis will produce the frequency spectrum of a time signal. It should be clear, therefore, why the terms digital signal analyzer, FFT analyzer, frequency analyzer, spectrum analyzer, and digital Fourier analyzer are to some extent synonymous as used in the commercial instrumentation literature. A signal analyzer typically has two (dual) or more (multiple) input signal channels. To generate results such as frequency response (transfer) functions, cross spectra, coherence functions, and cross-correlation functions, we need at least two data signals and hence a dual-channel analyzer. In hardware analyzers, digital circuitry rather than software is used to carry out the mathematical operations. Clearly, these are very fast but less ﬂexible (in terms of programmability and functional capability) for this reason. Digital signal analyzers, regardless of whether they use the hardware or the software approach, employ some basic operations. These operations, carried out in sequence, are: 1. 2. 3. 4.

Antialias ﬁltering (analog) Analog-to-digital conversion (i.e., single sampling) Truncation of a block of data and multiplication by a window function FFT analysis of the block of data

We have noted the following facts. If the sampling period of the ADC is DT (i.e., the sampling frequency is 1=DT) then the Nyquist frequency fc ¼ 1=2DT: This Nyquist frequency is the upper limit of the useful frequency content of the sampled signal. The cutoff frequency of the antialiasing ﬁlter should be set at fc or less. If there are N data samples in the block of data that is used in the FFT analysis, the corresponding record length is T ¼ N·DT: Then, the spectral lines in the FFT results are separated at a frequency spacing of DF ¼ 1=T: In view of the Nyquist frequency limit, there will be only N=2 useful spectral lines of FFT result. Strictly speaking, a real-time signal analyzer should analyze a signal instantaneously and continuously as the signal is received by the analyzer. This is usually the case with an analog signal analyzer. However, in digital signal analyzers, which are usually based on digital Fourier analysis, a block of data (i.e., N samples of record length T) is analyzed together to produce N=2 useful spectral lines (at frequency spacing 1=T). This is, then, not a truly real-time analysis. For practical purposes, if the speed of analysis is sufﬁciently fast, the analyzer may be considered real time, which is usually the case with hardware analyzers and also modern, high-speed software analyzers. The bandwidth B of a digital signal analyzer is a measure of its speed of signal processing. Speciﬁcally, for an analyzer that uses N data samples in each block of signal analysis, the associated processing time may be given by Tc ¼

N B

ð2:108Þ

Note that the larger the B; the smaller the Tc : The analyzer is considered real-time if the analysis time ðTc Þ of the data record is less than the generation time ðT ¼ N·DTÞ of the data record. Hence, we need Tc , T

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or N ,T B or N , N·DT B or 1 ,B DT

ð2:109Þ

In other words, a real-time analyzer has a bandwidth greater than its sampling rate. A multichannel digital signal analyzer can analyze one or more signals simultaneously and generate (and display) results such as Fourier spectra, power spectral densities, cross spectral densities, frequencyresponse functions, coherence functions, autocorrelations, and cross correlations. They are able to perform high-resolution analysis on a small segment of the frequency spectrum of a signal. This is termed zoom analysis. Essentially, in this case, the spectral line spacing, DF; is decreased while keeping the number of lines (N), and hence the number of time data samples, the same. That means the record length ðT ¼ 1=DFÞ has to be increased in proportion, for zoom analysis.

2.9.2

Oscilloscopes

An oscilloscope is used to observe one or two signals separately or simultaneously. Amplitude, frequency, and phase information of the signals can be obtained using an oscilloscope. In this sense, the oscilloscope is a signal modiﬁcation as well as a measurement (monitoring) and display device. Both analog and digital oscilloscopes are available. A typical application of an oscilloscope is to observe (monitor) experimental data such as vibration signals of machinery as obtained from transducers. They are also useful in observing and examining vibration test results, such as frequency-response plots, PSD curves, and response spectra. Typically, only temporary records are available on an analog oscilloscope screen. The main component of an analog oscilloscope is the cathode-ray tube (CRT), which consists of an electron gun (cathode) that deﬂects an electron ray according to the input-signal level. The oscilloscope screen has a coating of electron-sensitive material, so that the electron ray that impinges on the screen leaves a temporary trace on it. The electron ray sweeps across the screen horizontally, so that waveform traces can be recorded and observed. Usually, two input channels are available. Each input may be observed separately, or the variations in one input may be observed against those of the other. In this manner, signal phasing can be examined. Several sensitivity settings for the input-signal-amplitude scale (in the vertical direction) and sweep-speed selections are available on the panel. 2.9.2.1

Triggering

The voltage level of the input signal deﬂects the electron gun proportionally in the vertical (y-axis) direction on the CRT screen. This alone will not show the time evolution of the signal. The true time variation of the signal is achieved by means of a sawtooth signal that is generated internally in the oscilloscope and used to move the electron gun in the horizontal (x-axis) direction. As the name implies, the sawtooth signal increases linearly in amplitude until a threshold value then suddenly drops to zero, and then repeats this cycle again. In this manner, the observed signal is repetitively swept across the screen and a trace of it can be observed as a result of the temporary retention of the illumination of the electron gun on the ﬂuorescent screen. The sawtooth signal may be controlled (triggered) in several ways. For example, the external trigger mode uses an external signal from another channel (not the observed channel) to generate and synchronize the sawtooth signal. In the line trigger mode, the sawtooth signal is synchronized with the AC line supply (60 or 50 Hz). In the internal trigger mode, the observed signal (which is used to deﬂect the electron beam in the y direction) itself is used to generate (synchronize) the

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sawtooth signal. Since the frequency and the phase of the observed signal and the trigger signal are perfectly synchronized in the last case, the trace on the oscilloscope screen will appear stationary. Careful observation of a signal can be made in this manner. 2.9.2.2

Lissajous Patterns

Suppose that two signals, x and y; are provided to the two channels of an oscilloscope. If they are used to deﬂect the electron beam in the horizontal and the vertical directions, respectively, a pattern known as Lissajous pattern will be observed on the oscilloscope screen. Useful information about the amplitude and phasing of the two signals may be observed by means of these patterns. Consider sine waves x and y: Several special cases of Lissajous patterns are given below. 1. Same frequency, same phase: Here, x ¼ xo sinðvt þ fÞ y ¼ yo sinðvt þ fÞ Then we have

x y ¼ xo yo

which gives a straight-line trace with a positive slope, as shown in Figure 2.29(a). y

y

x

x (a)

(b) y

y

x

x

(d)

(c) y

y

x

wy (e)

yo

yintercept

wx

=

2 1

y

x

wy 3 = wx 1

x

wy 3 = wx 2

FIGURE 2.29 Some Lissajous patterns: (a) equal frequency and in-phase; (b) equal frequency and 908 out-of-phase; (c) equal frequency and 1808 out-of-phase; (d) equal frequency and u out-of-phase; (e) integral frequency ratio.

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2. Same frequency, 908 out-of-phase: Here, x ¼ xo sinðvt þ fÞ y ¼ yo sinðvt þ f þ p=2Þ ¼ yo cosðvt þ fÞ Then we have x xo

2

þ

y yo

2

¼1

which gives an ellipse, as shown in Figure 2.29(b). 3. Same frequency, 1808 out-of-phase: Here, x ¼ xo sinðvt þ fÞ y ¼ yo sinðvt þ f þ pÞ ¼ 2yo sinðvt þ fÞ Hence, x y þ ¼0 xo yo which corresponds to a straight line with a negative slope, as shown in Figure 2.29(c). 4. Same frequency, u out-of-phase: x ¼ xo sinðvt þ fÞ y ¼ yo sinðvt þ f þ uÞ When vt þ f ¼ 0; y ¼ yintercept ¼ yo sin u: Hence, sin u ¼

yintercept yo

In this case, we obtain a tilted ellipse as shown in Figure 2.29(d). The phase difference u is obtained from the Lissajous pattern. 5. Integral frequency ratio:

vy Number of y-peaks ¼ Number of x-peaks vx Three examples are shown in Figure 2.29(e).

vy 2 ¼ ; vx 1

vy 3 ¼ ; vx 1

vy 3 ¼ vx 2

Note: The above observations are true for narrowband signals as well. Broadband random signals produce scattered (irregular) Lissajous patterns. 2.9.2.3

Digital Oscilloscopes

The basic uses of a digital oscilloscope are quite similar to those of a traditional analog oscilloscope. The main differences stem from the manner in which information is represented and processed “internally” within the oscilloscope. Speciﬁcally, a digital oscilloscope ﬁrst samples a signal that arrives at one of its input channels and stores the resulting digital data within a memory segment. This is essentially a typical ADC operation. This digital data may be processed to extract and display the necessary information. The sampled data and the processed information may be stored on a ﬂoppy disk, if needed, for further processing using a digital computer. Also, some digital oscilloscopes have the

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communication capability so that the information may be displayed on a video monitor or printed to provide a hard copy. A typical digital oscilloscope has four channels so that four different signals may be acquired (sampled) into the oscilloscope and displayed. Also, it has various triggering options so that the acquisition of a signal may be initiated and synchronized by means of either an internal or an external trigger. Apart from the typical capabilities that are possible with an analog oscilloscope, a digital oscilloscope can automatically provide other useful features such as the following: 1. Automatic scaling of the acquired signal 2. Computation of signal features such as frequency, period, amplitude, mean, root-mean-square (rms) value, and rise time 3. Zooming into regions of interest of a signal record 4. Averaging of multiple signal records 5. Enveloping of multiple signal records 6. FFT capability, with various window options and antialiasing These various functions are menu selectable. Typically, ﬁrst a channel of the incoming data (signal) is selected and then an appropriate operation on the data is chosen from the menu (through menu buttons).

Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, WileyInterscience, New York. Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ. Broch, J.T. 1980. Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum. de Silva, C.W. 1983. Dynamic Testing and Seismic Qualiﬁcation Practice, D.C. Heath and Co., Lexington, KY. de Silva, C.W., and Palusamy, S.S., Experimental modal analysis — a modeling and design tool, Mech. Eng., ASME, 106, 6, 56 –65, 1984. de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib. Dig., 18, 10, 3–10, 1986. de Silva, C.W. 1989. Control Sensors and Actuators, Prentice Hall, Englewood Cliffs, NJ. de Silva, C.W. 2005. Mechatronics — An Integrated Approach, Taylor & Francis, CRC Press, Boca Raton, FL. de Silva, C.W. 2006. Vibration — Fundamentals and Practice, 2nd ed., Taylor & Francis, CRC Press, Boca Raton, FL. de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control — application in the distribution qualiﬁcation of microcomputers, Shock Vib. Dig., 18, 3 –13, 1986. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, UK. Meirovitch, L. 1980. Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville, MD. Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark.

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3

Vibration Testing 3.1 3.2

3.3

3.4

Clarence W. de Silva The University of British Columbia

3.5

Introduction .......................................................................... 3-1 Representation of a Vibration Environment ...................... 3-3

Test Signals † Deterministic Signal Representation † Stochastic Signal Representation † Frequency-Domain Representations † Response Spectrum † Comparison of Various Representations

Pretest Procedures ................................................................. 3-24

Purpose of Testing † Service Functions † Information Acquisition † Test-Program Planning † Pretest Inspection

Testing Procedures ................................................................ 3-37

Resonance Search † Methods of Determining FrequencyResponse Functions † Resonance-Search Test Methods † Mechanical Aging † Test-Response Spectrum Generation † Instrument Calibration † Test-Object Mounting † Test-Input Considerations

Some Practical Information ................................................. 3-52

Random Vibration Test Example Control Systems

†

Vibration Shakers and

Summary Vibration testing involves application of a vibration excitation to a test object and monitoring the resulting response. The ﬁrst step in vibration testing is the generation of a test excitation according to some speciﬁcation or objective. The applied excitation and the corresponding responses are measured at designated locations of the test object. Analysis of the test data will generate useful information about the tested object, which may be applicable in design development, manufacture, and utilization of the object. This chapter presents the basics of the planning of vibration tests, test signal representation and generation, vibration testing, and test data acquisition.

3.1

Introduction

Vibration testing is usually performed by applying a vibratory excitation to a test object and monitoring the structural integrity of the object and its performance of its intended function. The technique may be useful in several stages: (1) design development, (2) production, and (3) utilization of a product. In the initial design stage, the design weaknesses and possible improvements can be determined through the vibration testing of a preliminary design prototype or a partial product. In the production stage, the quality of the workmanship of the ﬁnal product can be evaluated using both destructive and nondestructive vibrating testing. A third application termed product qualiﬁcation, is intended for determining the adequacy of a product of good quality for a speciﬁc application (e.g., the seismic qualiﬁcation of a nuclear power plant) or a range of applications. 3-1 © 2007 by Taylor & Francis Group, LLC

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Vibration Monitoring, Testing, and Instrumentation

The technology of vibration testing has evolved rapidly since World War II and the technique has been successfully applied to a wide spectrum of products ranging from small printed circuit boards and microprocessor chips to large missiles and structural systems. Until recently, however, much of the signal processing that was required in vibration testing was performed through analog methods. In these methods, the measured signal is usually converted into an electric signal, which in turn is passed through a series of electrical or electronic circuits to achieve the required processing. Alternatively, motion or pressure signals can be used in conjunction with mechanical or hydraulic (e.g., ﬂuidic) circuits to perform analog processing. Today’s complex test programs require the capability for the fast and accurate processing of a large number of measurements. The performance of analog signal analyzers is limited by hardware costs, size, data handling capacity and computational accuracy. Digital processing for the synthesis and analysis of vibration test signals and for the interpretation and evaluation of test results, began to replace the classical analog methods in late 1960s. Today, specialpurpose digital analyzers with real-time digital Fourier analysis capability are commonly used in vibration testing applications. The advantages of incorporating digital processing into vibration testing include: ﬂexibility and convenience with respect to the type of the signal that can be analyzed and the complexity of the nature of processing that can be handled; increased speed of processing, accuracy and reliability; reduction in operational costs; practically unlimited repeatability of processing; and reduction in the overall size and weight of the analyzer. Vibration testing is usually accomplished using a shaker apparatus, as shown by the schematic diagram in Figure 3.1. The test object is secured to the shaker table in a manner representative of its installation during actual use (service). In-service operating conditions are simulated while the shaker table is actuated by applying a suitable input signal. The shakers of different types, with electromagnetic, electromechanical, or hydraulic actuators, are available. The shaker device may depend on the test requirement, availability, and cost. More than one signal may be required to simulate three-dimensional characteristics of the vibration environment. The test input signal is either stored on an analog magnetic tape or generated in real-time by a signal generator. The capability of the test object or a similar unit to withstand a “predeﬁned” vibration environment is evaluated by monitoring the dynamic response (accelerations, velocities, displacements, strains, etc.) and functional operability variables (e.g., temperatures, pressures, ﬂow rates, voltages, currents). Analysis of the response signals will aid in detecting existing defects or impending failures in various components of the test equipment. The control sensor output is useful in several ways, particularly in feedback control of the shaker, frequencyband equalization in real-time of the excitation signal, and the synthesizing of future test signals.

Test Object

Mounting Fixtures Power Amplifier

Exciter

Response Sensor

Filter/ Amplifier

Control Sensor

Filter/ Amplifier

Digital Analog/ Signal Digital Recorder, Interface Analyzer, Display

Signal Generator and Exciter Controller Reference (Required) Signal (Specification)

FIGURE 3.1

A typical vibration-testing arrangement.

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Vibration Testing

3-3

The excitation signal is applied to the shaker through a shaker controller, which usually has a built-in power ampliﬁer. The shaker controller compares the “control sensor” signal from the shaker–test object interface with the reference excitation signal from the signal generator. The associated error is used to control the shaker motion so as to push this error to zero. This is termed “equalization.” Hence, a shaker controller serves as an equalizer as well. The signals that are monitored from the test object include test response signals and operability signals. The former category of signals provides the dynamic response of the test object, and may include velocities, accelerations, and strains. The latter category of signals are used to check whether the test object performs in-service functions (i.e., it operates properly) during the test excitation, and may include ﬂow rates, temperatures, pressures, currents, voltages, and displacements. The signals may be recorded in a computer or a digital oscilloscope for subsequent analysis. When using an oscilloscope or a spectrum analyzer, some analysis can be done on line and the results are displayed immediately. The most uncertain part of a vibration test program is the simulation of the test input. For example, the operating environment of a product such as an automobile is not deterministic and will depend on many random factors. Consequently, it is not possible to generate a single test signal that can completely represent all various operating conditions. As another example, in seismic qualiﬁcation of equipment, the primary difﬁculty stems from the fact that the probability of accurately predicting the recurrence of an earthquake at a given site during the design life of the equipment is very small and that of predicting the nature of the ground motions if an earthquake were to occur is even smaller. In this case, the best that one can do is to make a conservative estimate for the nature of the ground motions due to the strongest earthquake that is reasonably expected. The test input should have (1) amplitude, (2) phasing, (3) frequency content, and (4) damping characteristics comparable to the expected vibration environment if satisfactory representation is to be achieved. A frequency-domain representation of the test inputs and responses can, in general, provide better insight regarding their characteristics than can a time domain representation, namely, a time history. Fortunately, frequency-domain information can be derived from time domain data by using Fourier transform techniques. In vibration testing, Fourier analysis is used in three principal ways: ﬁrst, to determine the frequency response of the test object in prescreening tests; second, to represent the vibration environment by its Fourier spectrum or its power spectral density (PSD) so that a test input signal can be generated to represent it; and third, to monitor the Fourier spectrum of the response at key locations in the test object and at control locations of the test table and use the information diagnostically or in controlling the exciter. The two primary steps of a vibration testing scheme are: Step 1: Step 2:

3.2

Specify the test requirements; Generate a vibration test signal that conservatively satisﬁes the speciﬁcations of Step 1.

Representation of a Vibration Environment

A complete knowledge of the vibration environment in which a device will be operating is not available to the test engineer or the test program planner. The primary reason for this is that the operating environment is a random process. When performing a vibration test, however, either a deterministic or a random excitation can be employed to meet the test requirements. This is known as the test environment. Based on the vibration-testing speciﬁcations or product qualiﬁcation requirements, the test environment should be developed to have the required characteristics of (1) intensity (amplitude), (2) frequency content (effect on the test-object resonances and the like), (3) decay rate (damping), and (4) phasing (dynamic interactions). Usually, these parameters are chosen to represent conservatively the worst possible vibration environment that is reasonably expected during the design life of the test object. So long as this requirement is satisﬁed, it is not necessary for the test environment to be identical to the operating vibration environment.

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3-4

Vibration Monitoring, Testing, and Instrumentation

In vibration testing, the excitation input (test environment) can be represented in several ways. The common representations are (1) by time signal, (2) by response spectrum, (3) by Fourier spectrum, and (4) by PSD function. Once the required environment is speciﬁed by one of these forms, the test should be conducted either by directly employing them to drive the exciter or by using a more conservative excitation when the required environment cannot be exactly reproduced.

3.2.1

Test Signals

Vibration testing may employ both random and deterministic signals as test excitations. Regardless of its nature, the test input should conservatively meet the speciﬁed requirements for that test. 3.2.1.1

Stochastic vs. Deterministic Signals

Consider a seismic time-history record. Such a ground-motion record is not stochastic. It is true that earthquakes are random phenomena and the mechanism by which the time history was produced is a random process. Once a time history is recorded, however, it is known completely as a curve of response value versus time (a deterministic function of time). Therefore, it is a deterministic set of information. However, it is also a “sample function” of the original stochastic process, the earthquake, by which it was generated. Hence, valuable information about the original stochastic process itself can be determined by analyzing this sample function on the basis of the ergodic hypothesis (see Section 3.2.3). Some may think that an irregular time-history record corresponds to a random signal. It should be remembered that some random processes produce very smooth signals. As an example, consider the sine wave given by a sinðvt þ fÞ: Let us assume that the amplitude a and the frequency v are deterministic quantities and the phase angle f is a random variable. This is a random process. Every time this particular random process is activated, a sine wave is generated that has the same amplitude and frequency but, generally, a different phase angle. Nevertheless, the sine wave will always appear as smooth as a deterministic sine wave. In a vibration-testing program, if we use a recorded time history to derive the exciter, it is a deterministic signal, even if it was originally produced by a random phenomenon such as an earthquake. Also, if we use a mathematical expression for the signal in terms of completely known (deterministic) parameters, it is again a deterministic signal. If the signal is generated by some random mechanism (whether computer simulation or physical) in real time, however, and if that signal is used as the excitation in the vibration test simultaneously as it is being generated, then we have a truly random excitation. Also, if we use a mathematical expression (with respect to time) for the excitation signal for which some of the parameters are not known numerically and the values are assigned to them during the test in a random manner, we again have a truly random test signal.

3.2.2

Deterministic Signal Representation

In vibration testing, time signals that are completely predeﬁned can be used as test excitations. They should be capable, however, of subjecting the test object to the speciﬁed levels of intensity, frequency, decay rate, and phasing (in the case of simultaneous multiple test excitations). Deterministic excitation signals (time histories) used in vibration testing are divided into two broad categories: single-frequency signals and multifrequency signals. 3.2.2.1

Single-Frequency Signals

Single-frequency signals have only one predominant frequency component at a given time. For the entire duration, however, the frequency range covered is representative of the frequency content of the vibration environment. For seismic-qualiﬁcation purposes, for example, this range should be at least 1 to 33 Hz. Some typical single-frequency signals that are used as excitation inputs in vibration testing of equipment are shown in Figure 3.2. The signals shown in the ﬁgure can be expressed by simple mathematical expressions. This is not a requirement, however. It is acceptable to store a very complex signal in a storage device and subsequently use it in the procedure. In picking a particular time history, we should give

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3-5

Acceleration

Vibration Testing

Time

Acceleration

(a)

T1 Frequency = w1

Acceleration

(b)

Frequency = w 2

T1 Frequency = w1

Acceleration

(c)

Frequency = w1

Acceleration

(d)

(e)

T2

Frequency = w1

Frequency = w 3

T2 Frequency = w 2

T1

T 11

T3

Frequency = w2

Pause

T1

T3 Frequency = w 3

T2

T3

Frequency = w 3

Frequency = w 2

T12

T2 Pause

FIGURE 3.2 Typical single-frequency test signals: (a) sine sweep; (b) sine dwell; (c) sine decay; (d) sine beat; (e) sine beat with pause.

proper consideration to its ease of reproduction and the accuracy with which it satisﬁes the test speciﬁcations. Now, let us describe mathematically the acceleration signals shown in Figure 3.2. 3.2.2.2

Sine Sweep

We obtain a sine sweep by continuously varying the frequency of a sine wave. Mathematically, uðtÞ ¼ a sin½vðtÞt þ f

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ð3:1Þ

3-6

Vibration Monitoring, Testing, and Instrumentation

The amplitude, a, and the phase angle, f, are usually constants and the frequency, vðtÞ; is a function of time. Both linear and exponential variations of frequency over the duration of the test are in common usage, but exponential variations are more common. For the linear variation (see Figure 3.3), we have

vðtÞ ¼ vmin þ ðvmax 2 vmin Þ

t Td

ð3:2Þ

in which

vmin ¼ lowest frequency in the sweep vmax ¼ highest frequency in the sweep Td ¼ duration of the sweep For the exponential variation (see Figure 3.3), we have log

vðtÞ vmin

¼

v t log max vmin Td

ð3:3Þ

or

vðtÞ ¼ vmin

vmax vmin

t=Td

ð3:4Þ

This variation is sometimes incorrectly called logarithmic variation. This confusion arises because of its deﬁnition using Equation 3.3 instead of Equation 3.4. It is actually an inverse logarithmic (i.e., exponential) variation. Note that the logarithm in Equation 3.3 can be taken to any arbitrary base. If base ten is used, the frequency increments are measured in decades (multiples of ten); if base two is used, the frequency increments are measured in octaves (multiples of two). Thus, the number of decades in the frequency range from v1 to v2 is given by log10 ðv2 =v1 Þ; for example, with v1 ¼ 1 rad/sec and v2 ¼ 100 rad/sec, we have log10 ðv2 =v1 Þ ¼ 2; which corresponds to two decades. Similarly, the number of octaves in the range v1 to v2 is given by log2 ðv2 =v1 Þ: Then, with v1 ¼ 2 rad/sec and v2 ¼ 32 rad/sec we have log2(v2/v1) ¼ 4, a range of four octaves. Note that these quantities are ratios and have no physical units. The foregoing deﬁnitions can be extended for smaller units; for instance, one-third octave represents increments of 21/3. Thus, if we start with 1 rad/sec and increment the frequency successively by one-third octave, we obtain 1, 21/3, 22/3, 2, 24/3, 25/3, 22, and so on. It is clear, for example, that there are four one-third octaves in the frequency range from 22/3 to 22. Note that v is known as the angular frequency (or radian frequency) and is usually measured in units of radians per second (rad/sec). w (t) wmax

Frequency

Linear Sine Sweep

Exponential Sine Sweep Sine Dwell

w min O FIGURE 3.3

Time

Td

t

Frequency variation in some single-frequency vibration-test signals.

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Vibration Testing

3-7

The more commonly used frequency is the cyclic frequency which is denoted by f. This is measured in hertz (Hz), which is identical to cycles per second (cps). It is clear that f ¼

v 2p

ð3:5Þ

This is true because there are 2p radians in one cycle. So that all important vibration frequencies of the test object (or its model) are properly excited, the sine sweep rate should be as slow as is feasible. Typically, rates of one octave per minute or slower are employed. 3.2.2.3

Sine Dwell

Sine-dwell signal is the discrete version of a sine sweep. The frequency is not varied continuously but is incremented by discrete amounts at discrete time points. This is shown graphically in Figure 3.3. Mathematically, for the rth time interval, the dwell signal is uðtÞ ¼ a sinðvr t þ fr Þ; Tr21 # t # Tr

ð3:6Þ

in which vr ; a, and f are kept constant during the time interval ðTr21 ; Tr Þ: The frequency can be increased by a constant increment or the frequency increments can be made bigger with time (exponential-type increment). The latter procedure is more common. Also, the dwelling-time interval is usually made smaller as the frequency is increased. This is logical because, as the frequency increases, the number of cycles that occur during a given time also increases. Consequently, steady-state conditions may be achieved in a shorter time. Sine-dwell signals can be speciﬁed using either a graphical form (see Figure 3.3) or a tabular form, giving the dwell frequencies and corresponding dwelling-time intervals. The amplitude is usually kept constant for the entire duration ð0; Td Þ; but the phase angle, f, may have to be changed with each frequency increment in order to maintain the continuity of the signal. 3.2.2.4

Decaying Sine

Actual transient vibration environments (e.g., seismic ground motions) decay with time as the vibration energy is dissipated by some means. This decay characteristic is not present, however, in sine-sweep and sine-dwell signals. Sine-decay representation is a sine dwell with decay (see Figure 3.2). For an exponential decay, the counterpart of Equation 3.6 can be written as uðtÞ ¼ a expð2lr tÞ sinðvr t þ fr Þ; Tr21 # t # Tr

ð3:7Þ

The damping parameter (the inverse of the time constant), l, is typically increased with each frequency increment in order to represent the increased decay rates of a dynamic environment (or increased modal damping) at higher frequencies. 3.2.2.5

Sine Beat

When two sine waves having the same amplitude but different frequencies are mixed together (added or subtracted), a sine beat is obtained. This signal is considered to be a sine wave having the average frequency of the two original waves, which is amplitude-modulated by a sine wave of frequency equal to half the difference of the frequencies of the two original waves. The amplitude modulation produces a transient effect which is similar to that caused by the damping term in the sine-decay equation (Equation 3.7). The sharpness of the peaks becomes more prominent when the frequency difference of the two frequencies is made smaller. Consider two cosine wave having frequencies ðvr þ Dvr Þ and ðvr 2 Dvr Þ and the same amplitude a/2. If the ﬁrst signal is subtracted from the second (that is, it is added with a 1808 phase shift from the ﬁrst wave), we obtain uðtÞ ¼

a ½cosðvr 2 Dvr Þt 2 cosðvr þ Dvr Þt 2

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ð3:8Þ

3-8

Vibration Monitoring, Testing, and Instrumentation

By straightforward use of trigonometric identities, we obtain uðtÞ ¼ aðsin vr tÞðsin Dvr tÞ; Tr21 # t # Tr

ð3:9Þ

This is a sine wave of amplitude, a; and frequency, v, modulated by a sine wave of frequency Dvr : Sinebeat signals are commonly used as test excitation inputs in vibration testing. Usually, the ratio vr =Dvr is kept constant. A typical value used is 20, in which case we obtain 10 cycles per beat. Here, cycles refer to the cycles at the higher frequency, vr ; and a beat occurs at each half cycle of the smaller frequency, Dvr : Thus, a beat is identiﬁed by a peak of amplitude a in the modulated wave and the beat frequency is 2Dvr : As in the case of a sine dwell, the frequency, vr ; of a sine-beat excitation signal is incremented at discrete time points, Tr ; so as to cover the entire frequency interval of interest ðvmin ; vmax Þ: It is a common practice to increase the size of the frequency increment and decrease the time duration at a particular frequency, for each frequency increment, just as is done for the sine dwell. The reasoning for this is identical to that given for sine dwell. The number of beats for each duration is usually kept constant (typically at a value over seven). A sine-beat signal is shown in Figure 3.2(d). 3.2.2.6

Sine Beat with Pauses

If we include pauses between sine-beat durations, we obtain a sine-beat signal with pauses. Mathematically, we have ( uðtÞ ¼

aðsin vr tÞðsin Dvr tÞ;

for Tr21 # t # T 0r ;

0;

for T 0r # t # Tr

ð3:10Þ

This situation is shown in Figure 3.2(e). When a sine-beat signal with pauses is speciﬁed as a test excitation, we must give the frequencies, the corresponding time intervals, and the corresponding pause times. Typically, the pause time is also reduced with each frequency increment. The single-frequency signal relations described in this section are summarized in Table 3.1. 3.2.2.7

Multifrequency Signals

In contrast to single-frequency signals, multifrequency signals usually appear irregular and can have more than one predominant frequency component at a given time. Three common examples of multifrequency signals are aerodynamic disturbances, actual earthquake records, and simulated road disturbance signals used in automotive dynamic tests. TABLE 3.1

Typical Single-Frequency Signals Used in Vibration Testing

Single Frequency Acceleration Signal Sine sweep

Mathematical Expression uðtÞ ¼ a sin½vðtÞt þ f

vðtÞ ¼ vmin þ ðvmax 2 vmin Þt=Td (linear) vðtÞ ¼ vmin

vmax vmin

t=Td

ðexponentialÞ

Sine dwell

uðtÞ ¼ a sinðvr t þ fr Þ Tr21 # t # Tr ; r ¼ 1; 2; …; n

Decaying sine

uðtÞ ¼ a expð2lr tÞ sinðvr t þ fr Þ Tr21 # t # Tr , r ¼ 1; 2; …; n

Sine beat

uðtÞ ¼ aðsin vr tÞ ðsin Dvr tÞ Tr21 # t # Tr ; vr =Dvr ¼ constant

Sine beat with pauses

( uðtÞ ¼

r ¼ 1; 2; …; n;

aðsin vr tÞðsin Dvr tÞ; for Tr21 # t # Tr0 ¼ 0;

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for T 0r # t # Tr

Vibration Testing

3.2.2.8

3-9

Actual Excitation Records

Typically, actual excitation records such as overhead guideway vibrations are sample functions of random processes. By analyzing these deterministic records, however, characteristics of the original stochastic processes can be established, provided that the records are sufﬁciently long. This is possible because of the ergodic hypothesis. Results thus obtained are not quite accurate, because the actual excitation signals are usually nonstationary random processes and hence are not quite ergodic. Nevertheless, the information obtained by a Fourier analysis is useful in estimating the amplitude, phase, and frequency-content characteristics of the original excitation. In this manner, we can choose a past excitation record that can conservatively represent the design-basis excitation for the object that needs to be tested. Excitation time histories can be modiﬁed to make them acceptably close to a design-basis excitation by using spectral-raising and spectral-suppressing methods. In spectral-raising procedures, a sine wave of required frequency is added to the original time history to improve its capability of excitation at that frequency. The sine wave should be properly phased such that the time of maximum vibratory motion in the original time history is unchanged by the modiﬁcation. Spectral suppressing is achieved, essentially, by using a narrowband reject ﬁlter for the frequency band that needs to be removed. Physically, this is realized by passing the time history signal through a linearly damped oscillator that is tuned to the frequency to be rejected and connected in series with a second damper. The damping of this damper is chosen to obtain the required attenuation at the rejected frequency. 3.2.2.9

Simulated Excitation Signals

Random-signal-generating algorithms can be easily incorporated into digital computers. Also, physical experiments can be developed that have a random mechanism as an integral part. A time history from any such random simulation, once generated, is a sample function. If the random phenomenon is accurately programmed or physically developed so as to conservatively represent a design-basis excitation, a signal from such a simulation may be employed in vibration testing. Such test signals are usually available either as analog records on magnetic tapes or as digital records on a computer disk. Spectral-raising and spectral-suppressing techniques, mentioned earlier, also may be considered as methods of simulating vibration test excitations. Before we conclude this section, it is worthwhile to point out that all test excitation signals considered in this section are oscillatory. Though the single-frequency signals considered may possess little resemblance to actual excitations on a device during operation, they can be chosen to possess the required decay, magnitude, phase, and frequency-content characteristics. During vibration testing, these signals, if used as excitations, will impose reversible stresses and strains on the test object, whose magnitudes, decay rates, and frequencies are representative of those that would be experienced during actual operation during the design life of the test object.

3.2.3

Stochastic Signal Representation

To generate a truly stochastic signal, a random phenomenon must be incorporated into the signalgenerating process. The signal has to be generated in real time, and its numerical value at a given time is unknown until that time instant is reached. A stochastic signal cannot be completely speciﬁed in advance, but its statistical properties may be prespeciﬁed. There are many ways of obtaining random processes, including physical experimentation (for example, by tossing a coin at equal time steps and assigning a value to the magnitude over a given time step depending on the outcome of the toss), observation of processes in nature (such as outdoor temperature), and digital-computer simulation. The last procedure is the one commonly used in signal generation associated with vibration testing. 3.2.3.1

Ergodic Random Signals

A random process is a signal that is generated by some random (stochastic) mechanism. Generally, each time the mechanism is operated, a different signal (sample function) is generated. The likelihood of any two sample functions becoming identical is governed by some probabilistic law. The random process is

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3-10

Vibration Monitoring, Testing, and Instrumentation

denoted by XðtÞ; and any sample function by xðtÞ: It should be remembered that no numerical computations can be made on XðtÞ because it is not known for certain. Its Fourier transform, for instance, can be written as an analytical expression but cannot be computed. Once a sample function, xðtÞ; is generated, however, any numerical computation can be performed on it because it is a completely known function of time. This important difference may be somewhat confusing. At any given time, t1 ; Xðt1 Þ is a random variable that has a certain probability distribution. Consider a well-behaved function, f {Xðt1 Þ}; of this random variable (which is also a random variable). Its expected value (statistical mean) is denoted E½f {Xðt1 Þ} : This is also known as the ensemble average because it is equivalent to the average value at t1 of a collection (ensemble) of a large number of sample functions of XðtÞ: Now, consider the function f {xðtÞ} of one sample function xðtÞ of the random process. Its temporal (time) mean is expressed by lim

T!1

1 ðT f {xðtÞ}dt 2T 2T

Now, if E½f {Xðt1 Þ} ¼ lim

T!1

1 ðT f {xðtÞ}dt 2T 2T

ð3:11Þ

then the random signal is said to be ergodic. Note that the right-hand side of Equation 3.11 does not depend on time. Hence, the left-hand side also should be independent of the time point t1 : As a result of this relation (known as the ergodic hypothesis), we can obtain the properties of a random process merely by performing computations using one of its sample functions. The ergodic hypothesis is links the stochastic domain of expectations and uncertainties and the deterministic domain of real records and actual numerical computations. Digital Fourier computations, such as correlation functions and spectral densities, would not be possible for random signals if not for this hypothesis. 3.2.3.2

Stationary Random Signals

If the statistical properties of a random signal, XðtÞ; are independent of the time point considered, it is stationary. In particular, Xðt1 Þ will have a probability density that is independent of t1 ; and the joint probability of Xðt1 Þ and Xðt2 Þ will depend only on the time difference, t2 2 t1 : Consequently, the mean value E½XðtÞ of a stationary random signal is independent of t; and the autocorrelation function deﬁned by E½XðtÞXðt þ tÞ ¼ fxx ðtÞ

ð3:12Þ

which depends on t and not on t: Note that ergodic signals are always stationary, but the converse is not always true. Consider Parseval’s theorem: ð1 21

x2 ðtÞdt ¼

ð1 21

lXð f Þl2 df

ð3:13Þ

This can be interpreted as an energy integral, and its value is usually inﬁnite for random signals. An appropriate measure for random signals is its power. This is given by its root-mean-square (RMS) value E½XðtÞ2 : PSD Fð f Þ is the Fourier transform of the autocorrelation function fðtÞ and, similarly, the latter is the inverse Fourier transform of the former. Hence,

fxx ðtÞ ¼

ð1 21

Fxx ð f Þexpðj2pf tÞdf

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ð3:14Þ

Vibration Testing

3-11

Now, from Equation 3.12 and Equation 3.14, we obtain RMS value ¼ E½XðtÞ2 ¼ fxx ð0Þ ¼

ð1 21

Fxx ð f Þdf

ð3:15Þ

It follows that the RMS value of a stationary random signal is equal to the area under its PSD curve. 3.2.3.3

Independent and Uncorrelated Signals

Two random signals XðtÞ and YðtÞ are independent if their joint distribution is given by the product of the individual distributions. A special case is that of uncorrelated signals, which satisfy E½Xðt1 ÞYðt2 Þ ¼ E½Xðt1 Þ E½Yðt2 Þ

ð3:16Þ

Consider the stationary case, with mean values

mx ¼ E½XðtÞ

ð3:17Þ

my ¼ E½YðtÞ

ð3:18Þ

wxx ðtÞ ¼ E½{XðtÞ 2 mx }{Xðt þ tÞ 2 mx } ¼ fxx ðtÞ 2 m2x

ð3:19Þ

wyy ðtÞ ¼ E½{YðtÞ 2 my }{Yðt 2 tÞ 2 my } ¼ fyy ðtÞ 2 m2y

ð3:20Þ

The autocovariance functions are given by

and the cross-covariance function is given by

wxy ðtÞ ¼ E½{XðtÞ 2 mx }{Yðt 2 tÞ 2 my } ¼ fxy ðtÞ 2 mx my

ð3:21Þ

For uncorrelated signals (Equation 3.16)

fxy ðtÞ ¼ mx my

ð3:22Þ

wxy ðtÞ ¼ 0

ð3:23Þ

and from Equation 3.21 it follows that The correlation-function coefﬁcient is deﬁned by

wxy ðtÞ rxy ðtÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ wxx ð0Þwyy ð0Þ

ð3:24Þ

21 # rxy ðtÞ # 1

ð3:25Þ

which satisﬁes For uncorrelated signals, rxy ðtÞ ¼ 0: This function measures the degree of correlation of the two signals. The correlation of two random signals, XðtÞ and YðtÞ; is measured in the frequency domain by its ordinary coherence function

g 2xy ð f Þ ¼

lFxy ð f Þl2 Fxx ð f ÞFyy ð f Þ

ð3:26Þ

which satisﬁes the condition 2 0 # gxy ðfÞ # 1

3.2.3.4

ð3:27Þ

Transmission of Random Excitations

When the excitation input to a system is a random signal, the corresponding system response will also be random. Consider the system shown by the block diagram in Figure 3.4(a). The response of the system is

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3-12

Vibration Monitoring, Testing, and Instrumentation

Excitation

U(t)

(a) Excitations

Response

h(t) ^ H(s)

U1(t)

Y

^ H1(s) Y1 +

Ur (t)

^ Hr (s)

(b)

Combined Response Y

+

Yr

Delay Ur

e−ts Excitation Uf (t)

A1

A2

(c)

+ +

^ H (s)

Overall Response Y

FIGURE 3.4 Combined response of a system to various random excitations: (a) system excited by a single input; (b) response to several random excitations; (c) response to a delayed excitation.

given by the convolution integral YðtÞ ¼

ð1 21

hðt1 ÞUðt 2 t1 Þdt1

ð3:28Þ

in which the response PSD is given by the Fourier transform

Fyy ð f Þ ¼ I{E½YðtÞYðt þ tÞ }

ð3:29Þ

Now, by using Equation 3.28 in Equation 3.29, in conjunction with the deﬁnition of Fourier transform, we can write ð1 ð1 ð1 dt expð2j2pf tÞE Fyy ð f Þ ¼ dt1 hðt1 ÞUðt 2 t1 Þ dt2 hðt2 ÞUðt þ t 2 t2 Þ 21

21

21

which can be expressed as ð1 ð1 ð1 dt1 hðt1 Þ dt2 hðt2 Þ dt expð2j2pf tÞfuu ðt þ t1 2 t2 Þ Fyy ð f Þ ¼ 21

21

21

0

Now, by letting t ¼ t þ t1 2 t2 , we can write

Fyy ð f Þ ¼

ð1 21

hðt1 Þexpðj2pft1 Þdt1

ð1 21

hðt2 Þexpð2j2pft2 Þdt2

ð1 21

fuu ðt 0 Þexpð2j2pf t 0 Þdt 0

Note that UðtÞ is assumed to be stationary. Next, since the frequency-response function is given by the Fourier transform of the impulse response function, we obtain

Fyy ð f Þ ¼ H p ð f ÞHð f ÞFuu ð f Þ

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ð3:30Þ

Vibration Testing

3-13

in which H p ð f Þ is the complex conjugate of Hð f Þ: Alternatively, if lHð f Þl denotes the magnitude of the complex quantity, we can write

Fyy ð f Þ ¼ lHð f Þl2 Fuu ð f Þ

ð3:31Þ

By using Equation 3.30 or Equation 3.31, we can determine the PSD of the system response from the PSD of the excitation if the system-frequency-response function is known. In a similar manner, it can be shown that the cross-spectral density function may be expressed as

Fuy ð f Þ ¼ Hð f ÞFuu ð f Þ

ð3:32Þ

Now, consider r stationary, independent, random excitations, U1 ; U2 ; …; Ur ; (which are assumed to have zero-mean values, without loss of generality) applied to r subsystems, having transfer functions ^ 1 ðsÞ; H ^ 2 ðsÞ; …; H ^ r ðsÞ; as shown in Figure 3.4(b). The total response, Y; consists of the sum of individual H responses, Y1 ; Y2 ; …; Yr : It can be shown that Y1 ; Y2 ; …; Yr are also stationary, independent, zero-mean, random processes. By deﬁnition, we have

fyy ðtÞ ¼ E½{Y1 ðtÞ þ · · · þ Yr ðtÞ}{Y1 ðt þ tÞ þ · · · þ Yr ðt þ tÞ}

ð3:33Þ

Now, for independent, zero-mean Yi ; Equation 3.33 becomes

fyy ðtÞ ¼ E½Y1 ðtÞY1 ðt þ tÞ þ · · · þ E½Yr ðtÞYr ðt þ rÞ

ð3:34Þ

Since Yi are stationary, we have

fyy ðtÞ ¼ fy1 y1 ðtÞ þ · · · þ fyr yr ðtÞ

ð3:35Þ

On Fourier transformation, we obtain

Fyy ð f Þ ¼ Fy1 y1 ð f Þ þ · · · þ Fyr yr ð f Þ

ð3:36Þ

In view of Equation 3.31, it can be written

Fyy ð f Þ ¼

r X i¼1

lHi ð f Þl2 Fui ui ð f Þ

ð3:37Þ

from which the response PSD can be determined if the input PSDs are known. If all inputs, Ui ðtÞ; have identical probability distributions (for example, when they are generated by the same mechanism), the corresponding PSDs will be identical. Note that this does not imply that the inputs are equal. They could be dependent, independent, correlated, or uncorrelated. In this case, Equation 3.37 becomes " # r X lHi ð f Þl2 Fuu ð f Þ ð3:38Þ Fyy ð f Þ ¼ i¼1

in which Fuu ð f Þ is the common input PSD. Finally, consider the linear combination of two excitations, Uf ðtÞ and Ur ðtÞ; with the latter excitation delayed in time by t but otherwise identical to the former. This situation is shown in Figure 3.4(c). From Laplace transform tables, it is seen that the Laplace transforms of the two signals are related by Ur ðsÞ ¼ expð2tsÞUf ðsÞ

ð3:39Þ

From Equation 3.39, it follows that (see Figure 3.4(c)): YðsÞ ¼ ðA1 expð2tsÞ þ A2 ÞHðsÞUf ðsÞ

ð3:40Þ

Fyy ð f Þ ¼ lðA1 expð2j2pf tÞ þ A2 ÞHð f Þl2 Fuu ð f Þ

ð3:41Þ

Consequently, we have From this result, the net response can be determined when the phasing between the two excitations is known. This has applications, for example, in determining the response of a vehicle to road disturbances at the front and rear wheels.

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3.2.4

Vibration Monitoring, Testing, and Instrumentation

Frequency-Domain Representations

In this section, we shall discuss the Fourier spectrum method and the PSD method of representing a test excitation. These are frequency-domain representations. 3.2.4.1

Fourier Spectrum Method

Since the time domain and the frequency domain are related through Fourier transformation, a time signal can be represented by its Fourier spectrum. In vibration testing, a required Fourier spectrum may be given as the test speciﬁcation. Then, the actual input signal that is used to excite the test object should have a Fourier spectrum that envelops the required Fourier spectrum. The generation of a signal to satisfy this requirement might be difﬁcult. Usually, digital Fourier analysis of the control sensor signal is necessary to compare the actual (test) Fourier spectrum with the required Fourier spectrum. If the two spectra do not match in a certain frequency band, the error (i.e., the difference in the two spectra) is fed back to correct the situation. This process is known as frequency-band equalization. Also, the sample step of the time signal in the digital Fourier analysis should be adequately small to cover the frequency range of interest in that particular vibration testing application. Advantages of using digital Fourier analysis in vibration testing include ﬂexibility and convenience with respect to the type of the signal that can be analyzed, availability of complex processing capabilities, increased speed of processing, accuracy and reliability, reduction in the test cost, practically unlimited repeatability of processing, and reduction in the overall size and weight of the analyzer. 3.2.4.2

Power Spectral Density Method

The operational vibration environment of equipment is usually random. Consequently, a stochastic representation of the test excitation appears to be suitable for a majority of vibration-testing situations. One way of representing a stationary random signal is by its PSD. As noted before, the numerical computation of the PSD is not possible, however, unless the ergodicity is assumed for the signal. Using the ergodic hypothesis, we can compute the PSD of a random signal simply by using one sample function (one record) of the signal. Three methods of determining the PSD of a random signal are shown in Figure 3.5. From Parseval’s theorem (Equation 3.13), we notice that the mean square value of a random signal may be obtained from the area under the PSD curve. This suggests the method shown in Figure 3.5(a) for estimating the PSD of a signal. The mean square value of a sample of the signal in the frequency band, Df ; having a certain center frequency is obtained by ﬁrst extracting the signal components in the band and then squaring them. This is done for several samples and averaged to obtain a high accuracy. It is then divided by Df : By repeating this for a range of center frequencies, an estimate for the PSD is obtained.

Signal (a)

Tracking Filter Bandwidth ∆f

Signal ADC (b) Signal (c)

ADC

Square Law Circuit

Digital Correlation Function

FFT Processor

Averaging Network

Digital Fourier Transform

Averaging Software

1 ∆f

Approximate psd

Display/Recording Unit

Approximate psd Display/Recording Averaging Unit Software Approximate psd DAC

Display/Recording Unit

FIGURE 3.5 Some methods of PSD determination: (a) the ﬁltering, squaring, and averaging method; (b) using an autocorrelation function; (c) using direct FFT.

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FIGURE 3.6

Effect of ﬁlter bandwidth on PSD results.

In the second scheme, shown in Figure3.5(b), correlation function is ﬁrst computed digitally. Its Fourier transform (by fast Fourier transform, or FFT) gives an estimate of the PSD. In the third scheme, shown in Figure 3.5(c), the PSD is computed directly using FFT. Here, the Fourier spectrum of the sample record is computed and the PSD is estimated directly, without ﬁrst computing the autocorrelation function. In these numerical techniques of computing PSD, a single sample function will not give the required accuracy, and averaging results for a number of sample records is usually needed. In real-time digital analysis, the running average and the current estimate are normally computed. In the running average, it is desirable to give a higher weighting to the more recent estimates. The ﬂuctuations about the local average in the PSD estimate could be reduced by selecting a larger ﬁlter bandwidth, Df (see Figure 3.6), and a large record length T. A measure of this ﬂuctuation is given by 1 1 ¼ pﬃﬃﬃﬃﬃﬃ Df T

ð3:42Þ

It should be noted that increasing Df results in reduction of the precision of the estimates while improving the appearance. To offset this, T must be increased further, or averaging must be done for several sample records. Generating a test-input signal with a PSD that satisfactorily compares with the required PSD can be a tedious task if one attempts to do it manually by mixing various signal components. A convenient method is to use an automatic multiband equalizer. By this means, the mean amplitude of the signal in each small frequency band of interest can be made to approach the spectrum of the speciﬁed vibration environment (see Figure 3.7). Unfortunately, this type of random-signal vibration testing can be more costly than testing with deterministic signals.

3.2.5

Response Spectrum

Response spectra are commonly used to represent signals associated with vibration testing. A given signal has a certain ﬁxed response spectrum, but many different signals can have the same response spectrum. For this reason, as will be clear shortly, the original signal cannot be reconstructed from its response spectrum (unlike in the case of a Fourier spectrum). This is a disadvantage. However, the physical signiﬁcance of a response spectrum makes it a good representation for a test signal.

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Vibration Monitoring, Testing, and Instrumentation

Probability Distribution

Random Number Generator

Simulated Vibration Environment

Random Signal Constructor

Automatic Multiband Equalizer

Shaping (Intensity) Function

Filter Circuit

Reference Power Spectrum FIGURE 3.7

Generation of a speciﬁed random vibration environment.

If a given signal is applied to a single-degree-of-freedom (single-DoF) oscillator (of a speciﬁc natural frequency), and the response of the oscillator (mass) is recorded, we can determine the maximum (peak) value of that response. Suppose that we repeat the process for a number of different oscillators (having different natural frequencies) and then plot the peak response values thus obtained against the corresponding oscillator natural frequencies. This procedure is shown schematically in Figure 3.8. For an inﬁnite number of oscillators (or for the same oscillator with continuously variable natural frequency), we get a continuous curve, which is called the response spectrum of the given signal. It is obvious, however, that the original signal cannot be completely determined from the knowledge of its response spectrum alone. As shown in Figure 3.8, for instance, another signal, when passed through a given oscillator, might produce the same peak response. Note that we have assumed the oscillators to be undamped; the response spectrum obtained using undamped oscillators corresponds to z ¼ 0: If all the oscillators are damped, however, and have the same damping ratio, z; the resulting response spectrum will correspond to that particular z. It is, therefore, clear that z is also a parameter in the response-spectrum representation. We should specify the damping value as well when we represent a signal by its response spectrum.

FIGURE 3.8

Deﬁnition of the response spectrum of a signal.

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Vibration Testing

3.2.5.1

3-17

Displacement, Velocity, and Acceleration Spectra

It is clear that a motion signal can be represented by the corresponding displacement, velocity, or acceleration. First, consider a displacement signal, uðtÞ: The corresponding velocity signal is u_ ðtÞ and the acceleration is u€ ðtÞ: Now consider an undamped simple oscillator, which is subjected to a support displacement uðtÞ; as shown in Figure 3.9. As usual, assuming that the displacements are measured with respect to a static equilibrium conﬁguration, the gravity effect can be ignored. Then, the equation of motion is given by m€yd ¼ kðu 2 yd Þ

yd m

k

u(t)

ð3:43Þ

or y€ d þ

v2n yd

¼

v2n uðtÞ

FIGURE 3.9 Undamped simple oscillator subjected to a support excitation.

ð3:44Þ

where the (undamped) natural frequency is given by

sﬃﬃﬃﬃ k vn ¼ m

ð3:45Þ

Suppose that the support (displacement) excitation, uðtÞ; is a unit impulse dðtÞ: Then, the corresponding (displacement) response y is called the impulse-response function, and is denoted by hðtÞ: It is known that hðtÞ is the inverse Laplace transform (with zero initial condition) of the transfer function of the system (Equation 3.44), as given by HðsÞ ¼

v2n ðs þ v2n Þ 2

ð3:46Þ

The impulse-response function (to an impulsive support excitation) for an undamped, single-DoF oscillator having natural frequency vn is given by hðtÞ ¼ vn sin vn t

ð3:47Þ

The displacement response yd ðtÞ of this oscillator, when excited by the displacement signal uðtÞ; is given by the convolution integral ð1 uðtÞsin vn ðt 2 tÞdt yd ðtÞ ¼ vn ð3:48Þ 0

The “velocity” response of the same oscillator, when excited by the velocity signal, u_ ðtÞ; is given by ð1 u_ ðtÞ sin vn ðt 2 tÞdt ð3:49Þ yv ðtÞ ¼ vn 0

and the “acceleration” response when excited by the acceleration signal, u€ ðtÞ; is ð1 ya ðtÞ ¼ vn u€ ðtÞ sin vn ðt 2 tÞdt 0

ð3:50Þ

These results immediately follow from Equation 3.44. Speciﬁcally, differentiate Equation 3.44 once to obtain y€ v þ v2n yv ¼ v2n u_ ðtÞ

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ð3:51Þ

3-18

Vibration Monitoring, Testing, and Instrumentation

and differentiate again, to obtain y€ a þ v2n ya ¼ v2n u€ ðtÞ

ð3:52Þ

in which dyd dt

ð3:53Þ

d2 yd dy ¼ v dt 2 dt

ð3:54Þ

yv ¼ ya ¼

If the peak value of yd ðtÞ is plotted against vn ; we get the displacement– spectrum curve of the displacement signal, uðtÞ: If the peak value of yv ðtÞ is plotted against vn ; we get the velocity–spectrum curve of the displacement signal, uðtÞ: If the peak value of ya ðtÞ is plotted against vn ; we get the acceleration– spectrum curve of the displacement signal, uðtÞ: Now consider Equation 3.49. Integration by parts gives ð1 2 yv ðtÞ ¼ ½vn uðtÞsin vn ðt 2 tÞ 1 ð3:55Þ uðtÞcos vn ðt 2 tÞdt 0 þ vn 0

The initial and ﬁnal conditions for uðtÞ are assumed to be zero. It follows that the ﬁrst term in Equation 3.55 vanishes. The second term is vn ½yd ðt þ p=2vn 2 tÞ ; which is clear by noting that sin vn ðt þ p=2vn 2 tÞ is equal to cos vn ðt 2 tÞ;; thus yv ðtÞ ¼ 2vn yd t þ

p 2vn

ð3:56Þ

If we integrate Equation 3.50 by parts twice, and apply the end conditions as before, we obtain ya ðtÞ ¼ 2v2n yd ðtÞ

ð3:57Þ

By taking the peak values of response time histories, we see from Equation 3.56 and Equation 3.57 that vðvn Þ ¼ vn dðvn Þ

ð3:58Þ

aðvn Þ ¼ v2n dðvn Þ

ð3:59Þ

in which dðvn Þ; vðvn Þ; and aðvn Þ represent the displacement spectrum, the velocity spectrum, and the acceleration spectrum, respectively, of the displacement time history, uðtÞ: It follows from Equation 3.58 and Equation 3.59 that aðvn Þ ¼ vn vðvn Þ

3.2.5.2

ð3:60Þ

Response-Spectra Plotting Paper

Response spectra are usually plotted on a frequency – velocity coordinate plane or on a frequency – acceleration coordinate plane. Values are normally plotted in logarithmic scale, as shown in Figure 3.10. First, consider the axes shown in Figure 3.10(a). Obviously, constant velocity lines are horizontal for this coordinate system. From Equation 3.58, the constant-displacement line corresponds to vðvn Þ ¼ cvn By taking logarithms of both sides, we obtain log vðvn Þ ¼ log vn þ log c It follows that the constant-displacement lines have a slope of þ1 on the logarithmic frequency –velocity plane. Similarly, from Equation 3.60, the constant-acceleration lines correspond to

vn vðvn Þ ¼ c

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3-19

Hence,

Format 1 t tan ns o C t= en Velocity = Constant

log vðvn Þ ¼ 2log vn þ log c

n io

at er

em

=

lac

ta ns Co

sp Di

el nt

Velocity (Log)

cc

3.2.5.3

A

(a)

Frequency (Log)

Co = y cit Ve

lo

Acceleration = Constant

Dis

plac

eme

nt =

Con s

ns

tan t

tant

Format 2 Acceleration (Log)

It follows that the constant-acceleration lines have a slope of negative one on the logarithmic frequency –velocity plane. Similarly, it can be shown from Equation 3.59 and Equation 3.60 that, on the logarithmic frequency–acceleration plane (Figure 3.10(b)), the constant-displacement lines have a slope of þ2, and the constant-velocity lines have a slope of þ1. On the frequency – velocity plane, a point corresponds to a speciﬁc frequency and a speciﬁc velocity. The corresponding displacement at the point is obtained (Equation 3.58) by dividing the velocity value by the frequency value at that point. The corresponding acceleration at that point is obtained (Equation 3.60) by multiplying the particular velocity value by the frequency value. Any units may be used for displacement, velocity, and acceleration quantities. A typical logarithmic frequency –velocity plotting sheet is shown in Figure 3.11. Note that the sheet is already graduated on constant displacement, velocity, and acceleration lines. Also, a period axis (period ¼ 1/cyclic frequency) is given for convenience in plotting. A plot of this type is called a nomograph.

(b)

Frequency (Log)

FIGURE 3.10 Response-spectra plotting formats: (a) frequency – velocity plane; (b) frequency– acceleration plane.

Zero-Period Acceleration

Frequently, response spectra are speciﬁed in terms of accelerations rather than velocities. This is particularly true in vibration testing associated with product qualiﬁcation, because typical operational disturbance records are usually available as acceleration time histories. No information is lost because the logarithmic frequency –acceleration plotting paper can be graduated for velocities and displacements as well. It is, therefore, clear that an acceleration quantity (peak) on a response spectrum has a corresponding velocity quantity (peak), and a displacement quantity (peak). In vibration testing, however, the motion variable that is in common usage is the acceleration. Zero-period acceleration (ZPA) is an important parameter that characterizes a response spectrum. It should be remembered, however, that zero-period velocity or zero-period displacement can be similarly deﬁned. ZPA is deﬁned as the acceleration value (peak) at zero period (or inﬁnite frequency) on a response spectrum. Speciﬁcally, ZPA ¼ lim aðvn Þ vn !1

ð3:61Þ

Consider the damped simple oscillator equation (for support motion excitation): y€ þ 2zvn y_ þ v2n y ¼ v2n uðtÞ

ð3:62Þ

By differentiating Equation 3.62 throughout, either once or twice, it is seen, as in Equation 3.51 and Equation 3.52, that if u and y initially refer to displacements, then the same equation is valid when both of them refer to either velocities or accelerations. Let us consider the case in which u and y refer to input and response acceleration variables, respectively. Consider a sinusoidal

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 3.11

Response-spectra plotting sheet or nomograph (frequency– velocity plane).

signal, uðtÞ; given by uðtÞ ¼ A sin vt

ð3:63Þ

The resulting response, yðtÞ; neglecting the transient components (that is, the steady-state value), is given by

v2n yðtÞ ¼ A pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ sinðvt þ fÞ ðv2n 2 v2 Þ2 þ 4z2 v2n v2

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ð3:64Þ

Vibration Testing

3-21

Hence, the acceleration-response spectrum, given by aðvn Þ ¼ ½yðtÞ max ; for a sinusoidal signal of frequency, v; and amplitude, A; is

Acceleration Spectrum

v2n aðvn Þ ¼ A pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ðv2n 2 v2 Þ2 þ 4z2 v2n v2

a(wn)

ð3:65Þ

A plot of this response is shown in Figure 3.12. Note that að0Þ ¼ 0: Also, ZPA ¼ lim aðvn Þ ¼ A vn !1

A 2ζ

A

ZPA O

ð3:66Þ FIGURE 3.12

wn = w Oscillator Natural Frequency

wn

Response spectrum and ZPA of a sine

It is worth observing that at the point vn ¼ v signal. (i.e., when the excitation frequency, v; is equal to the natural frequencies, vn ; of the simple oscillator), we have aðvn Þ ¼ A=ð2zÞ; which corresponds to an ampliﬁcation by a factor of 1=ð2zÞ over the ZPA value. 3.2.5.4

Uses of Response Spectra

In vibration testing, response-spectra curves are employed to specify the dynamic environment to which the test object is required to be subjected. This speciﬁed response spectrum is known as the required response spectrum (RRS). In order to satisfy conservatively the test speciﬁcation, the response spectrum of the actual test input excitation, known as the test response spectrum (TRS), should envelop the RRS. Note that, when response spectra are used to represent excitation input signals in vibration testing, the damping value of the hypothetical oscillators used in computing the response spectrum has no bearing on the actual damping that is present in the test object. In this application, the response spectrum is merely a representation of the shaker-input signal and, therefore, does not depend on system damping. Another use of response spectra is in estimating the peak value of the response of a multi-DoF or distributed-parameter system when it is excited by a signal whose response spectrum is known. To understand this concept, we recall the fact that, for a multi-DoF or truncated (approximated) distributed-parameter system having distinct natural frequencies, the total response can be expressed as a linear combination of the individual modal responses. Speciﬁcally, the response yðtÞ can be written 2 3 r X 6 2zi vi t 7 yðtÞ ¼ ð3:67Þ ai aðvi Þ exp4 qﬃﬃﬃﬃﬃﬃﬃﬃ 5sinðvi t þ fi Þ i¼1 1 2 z2i in which the spectrum, aðvi Þ, is comprised of the amplitude contributions from each mode (simple oscillator equation), with “damped” natural frequency, vi : Hence, aðvi Þ corresponds to the value of the response spectrum at frequency vi : The linear combination parameters, ai ; depend on the modalparticipation factors and can be determined from system parameters. Since the peak values of all terms in the summation on the right-hand side of Equation 3.67 do not occur at the same time, we observe that ½yðtÞ

peak

,

r X i¼1

ai aðvi Þ

ð3:68Þ

It follows that the right-hand side of the inequality (Equation 3.68) is a conservative upper-bound estimate (i.e., the absolute sum) for the peak response of the multi-DoF system. Some prefer to make the estimate less conservative by taking the square root of sum of the squares (SRSS): " r #1=2 X 2 2 ½yðtÞ SRSS ¼ ð3:69Þ ai a ðvi Þ i¼1

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Vibration Monitoring, Testing, and Instrumentation

The latter method, however, has the risk of giving an estimate that is less than the true value. Note that, in this application, the damping value associated with the response spectrum is directly related to modal damping of the system. Hence, the response spectrum, aðvi Þ, should correspond to the same damping ratio as that of the mode considered within the summation of the inequality (Equation 3.68). If all modal damping ratios, zi ; are identical or nearly so, the same response spectrum could be used to compute all terms in the inequality 3.68. Otherwise, different responsespectra curves should be used to determine each quantity, aðvi Þ; depending on the applicable modal damping ratio, zi :

3.2.6

Comparison of Various Representations

In this section, we shall state some major advantages and disadvantages of the four representations of the vibration environment that we have discussed. Time-signal representation has several advantages. It can be employed to represent either deterministic or random vibration environment. It is an exact representation of a single excitation event. Also, when performing multiexcitation (multiple shaker) vibration testing, phasing between the various inputs can be conveniently incorporated simply by delaying each excitation with respect to the others. There are also disadvantages to time-signal representation. Since each time history represents just one sample function (a single event) of a random environment, it may not be truly representative of the actual vibration excitation. This can be overcome by using longer signals, which, however, will increase the duration of the test, which is limited by test speciﬁcations. If the random vibration is truly ergodic (or at least stationary), this problem will not be as serious. Furthermore, the problem does not arise when testing with deterministic signals. An extensive knowledge of the true vibration environment to which the test object is subjected is necessary, however, in order to conclude that it is stationary or that it could be represented by a deterministic signal. In this sense, time-signal representation is difﬁcult to implement. The response-spectrum method of representing a vibration environment has several advantages. It is relatively easy to implement. Since the peak response of a simple oscillator is used in its deﬁnition, it is representative of the peak response or structural stress of simple dynamic systems; hence, there is a direct relation to the behavior of the physical object. An upper bound for the peak response of a multi-DoF system can be conveniently obtained by the method outlined in Section 3.2.5.4. Also, by considering the envelope of a set of response spectra at the same damping value, it is possible to use a single response spectrum to conservatively represent more than one excitation event. The method also has disadvantages. It employs deterministic signals in its deﬁnition. Sample functions (single events) of random vibrations can be used, however. It is not possible to determine the original vibration signal from the knowledge of its response spectrum, because it uses the peak value of response of a simple oscillator (more than one signal can have the same response spectrum). Thus, a response spectrum cannot be considered a complete representation of a vibration environment. Also, characteristics such as the transient nature and the duration of the excitation event cannot be deduced from the response spectrum. For the same reason, it is not possible to incorporate information on excitation-signal phasing into the response-spectrum representation. This is a disadvantage in multiple excitation testing. Fourier spectrum representation also has advantages. Since the actual dynamic environment signal can be obtained by inverse transformation, it has the same advantages as for the time-signal representation. In particular, since a Fourier spectrum is generally complex, phasing information of the test excitation can be incorporated into Fourier spectra, in multiple excitation testing. Furthermore, by considering an envelope Fourier spectrum (like an envelope response spectrum), it can be employed to represent conservatively more than one vibration environment. Also, it gives frequency-domain information (such as information about resonances), which is very useful in vibration testing situations. The disadvantages of Fourier spectrum representation include the following. It is a deterministic representation but, as in the response-spectrum method, a sample function (a single event) of a random vibration can be represented by its Fourier spectrum. Transient effects and event duration are hidden in this representation. Also, it is

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3-23

somewhat difﬁcult to implement, because complex procedures of multiband equalization might be necessary in the signal synthesis associated with this representation. PSD representation has the following advantages. It takes the random nature of a vibration environment into account. As in response-spectrum and Fourier-spectrum representations, by taking an envelope PSD, it can be used to represent conservatively more than one environment. It can display important frequency-domain characteristics, such as resonances. Its disadvantages include the following. It is an exact representation only for truly stationary or ergodic random environments. In nonstationary situations, as in seismic ground motions, signiﬁcant error could result. Also, it is not possible to obtain the original sample function (dynamic event) from its PSD. Hence, the transient characteristics and duration of the event are not known from its PSD. Since mean square values, not peak values, are considered, PSD representation is not structural-stress-related. Furthermore, since PSD functions are real (not complex), we cannot incorporate phasing information into them. This is a disadvantage in multiple excitation testing situations, but this problem can be overcome by considering either the cross spectrum (which is complex) or the cross correlation in each pair of test excitations. Random vibration testing is compared with sine testing (single-frequency, deterministic excitations) in Box 3.1. A comparison of various representations of test excitations is given in Box 3.2.

Box 3.1 RANDOM TESTING VS . SINE TESTING Advantages of Random Testing: 1. More realistic representation of the true environment 2. Many frequencies are applied simultaneously 3. All resonances, natural frequencies, and mode shapes are excited simultaneously Disadvantages of Random Testing: 1. Needs more power for testing 2. Control is more difﬁcult 3. More costly Advantages of Sine Testing: Appropriate for: 1. Fatigue testing of products that operate primarily at a known speed (frequency) under in-service conditions 2. Detecting sensitivity of a device to a particular excitation frequency 3. Detecting resonances, natural frequencies, modal damping, and mode shapes 4. Calibration of vibration sensors and control systems Disadvantages of Sine Testing: 1. Usually not a good representation of the true dynamic environment 2. Because vibration energy is concentrated at one frequency, it can cause failures that would not occur in service (particularly single-resonance failures) 3. Since only one mode is excited at a time, it can hide multiple-resonance failures that might occur in service

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Box 3.2 COMPARISON OF TEST EXCITATION REPRESENTATIONS Property True representation of a deterministic environment? True representation of a random environment? Frequency–time reversible? Signal phasing possible for multiaxis testing? Good representation of peak amplitude/stress events? Explicit accounting for modal responses

Time Signal

Response Spectrum

Fourier Spectrum

Power Spectral Density

Yes

Yes

Yes

No

One sample function Yes Yes

One sample function No No

One sample function Yes Yes

Yes

Yes

Yes

No

No

No

Yes

Yes

Yes

No No

In practice, the generation of an excitation signal for vibration testing may not follow any one of the analytical procedures and may incorporate a combination of them. For example, a combination of sinebeat signals of different frequencies with random phasing is one practical approach to the generation of a multifrequency, pseudo-random excitation signal. This approach is summarized in Box 3.3.

3.3

Pretest Procedures

The selection of a test procedure for the vibration testing of an object should be based on technical information regarding the test object and its intended use. Vendors usually prefer to use more established, conventional testing methods and are generally reluctant to incorporate modiﬁcations and

Box 3.3 TEST SIGNAL GENERATION Steps: 1. Generate a set of sine beats at discrete frequencies of interest for the vibration test, having speciﬁed amplitudes. 2. Phase shift (time shift) the signal components from Step 1 according to a random number generator. 3. Sum the signal components from Step 2.

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improvements. This is primarily due to economic reasons, convenience, testing-time limitation, availability of the equipment and facilities (test-lab limitations), and similar factors. Regulatory agencies, however, usually modify their guidelines from time to time, and some of these requirements are mandatory. Before conducting a vibration test on a test object, it is necessary to follow several pretest procedures. Such procedures are necessary in order to conduct a meaningful test. Some important pretest procedures include the following: 1. 2. 3. 4. 5. 6. 7.

Understanding the purpose of the test Studying the service functions of the test object Acquiring information on the test object Planning the test program Conducting pretest inspection of the test object Resonance-searching to gather dynamic information about the test object Mechanically aging the test object

In the following sections, we shall discuss each of the ﬁrst ﬁve items of these procedures to emphasize how they can contribute to a meaningful test. The last two items will be considered separately, in Section 3.4 on Testing Procedures.

3.3.1

Purpose of Testing

As noted before, vibration testing is useful in various stages of (1) design and development, (2) production and quality assurance, and (3) qualiﬁcation and utilization of a product. Depending on the outcome of a vibration test, design modiﬁcations or corrective actions can be recommended for a preliminary design or a partial product. To determine the most desirable location (in terms of minimal noise and vibration), for the compressor in a refrigerator unit, for example, a resonance-search test could be employed. As another example, vibration testing can be employed to determine vibration-isolation material requirements in structures for providing adequate damping. Such tests fall into the ﬁrst category of system development tests. They are beneﬁcial for the designer and the manufacturer in improving the quality of performance of the product. Government regulatory agencies do not usually stipulate the requirements for this category of tests, but they sometimes stipulate minimal requirements for safety and performance levels of the ﬁnal product, which can indirectly affect the development-test requirements. Custom-made items are exceptions for which the customers could stipulate the design-test requirements. For special-purpose products, it is sometimes also necessary to conduct a vibration test on the ﬁnal product before its installation for service operation. For mass-produced items, it is customary to select representative samples form each batch of the product for these tests. The purpose of such test is to detect any inferiorities in the workmanship or in the materials used. These tests fall into the second category, quality-assurance tests. These usually consist of a standard series of routine tests that are well established for a given product. Distribution qualiﬁcation and seismic qualiﬁcation of devices and components are good examples of the use of the third category, qualiﬁcation tests. A high-quality product such as a valve actuator, for instance, which is thoroughly tested in the design-development stage and at the ﬁnal production stage, will need further dynamic tests or analysis if it is to be installed in a nuclear power plant. The purpose in this testing is to determine whether the product (valve actuator) will be crucial for system-safety-related functions. Government regulatory agencies usually stipulate basic requirements for qualiﬁcation tests. These tests are necessarily application-oriented. The vendor or the customer might employ more elaborate test programs than those stipulated by the regulatory agency, but at least the minimum requirements set by the agency should be met before commissioning the plant. The purpose of any vibration test should be clearly understood before incorporating it into a test program. A particular test might be meaningless under some circumstances. If it is known, for instance, that no resonances below 35 Hz exist in a particular piece of equipment that requires seismic

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qualiﬁcation, then it is not necessary to conduct a resonance search because the predominant frequency content of seismic excitations occurs below 35 Hz. If, however, the test serves a dual purpose, such as mechanical aging in addition to resonance detection, then it may still be conducted even if there are no resonances in the predominant frequency range of excitation. If testing is performed on one test item selected from a batch of products to assure the quality of the entire batch or to qualify the entire batch, it is necessary to establish that all items in the batch are of identical design. Otherwise, testing of all items in the batch might be necessary unless some form-design similarity can be identiﬁed. “Qualiﬁcation by similarity” is done in this manner. The nature of the vibration testing that is employed will be usually governed by the test purpose. Single-frequency tests, using deterministic test excitations, for example, are well suited for designdevelopment and quality-assurance applications. The main reason for this choice is that the test-input excitations can be completely deﬁned; consequently, a complete analysis can be performed with relative ease, based on existing theories and dynamic models. Random or multifrequency tests are more realistic in a qualiﬁcation test, however, because under typical service conditions, the dynamic environments to which an object is subjected are random and have multiple frequencies by nature (for example, seismic disturbances, ground-transit road disturbances, aerodynamic disturbances). Since random-excitation tests are relatively more expensive and complex in terms of signal generation and data processing, singlefrequency tests might also be employed in qualiﬁcation tests. Under some circumstances, singlefrequency testing could add excessive conservatism to the test excitation. It is known, for instance, that single-frequency tests are justiﬁed in the qualiﬁcation of line-mounted equipment (i.e., equipment mounted on pipe lines, cables, and similar “line” structures), which can encounter in-service disturbances that are ampliﬁed because of resonances in the mounting structure.

3.3.2

Service Functions

For product qualiﬁcation by testing, it is required that the test object remain functional and maintain its structural integrity when subjected to a certain prespeciﬁed dynamic environment. In seismic qualiﬁcation of equipment, for instance, the dynamic environment is an excitation that adequately represents the amplitude, phasing, frequency content, and transient characteristics (decay rate and signal duration) of the motions at the equipment-support locations, caused by the most severe seismic disturbance that has a reasonable probability of occurring during the design life of the equipment. Monitoring the proper performance of in-service functions (functional-operability monitoring) of a test object during vibration testing can be crucial in the qualiﬁcation decision. The intended service functions of the test object should be clearly deﬁned prior to testing. For active equipment, functional operability is necessary during vibration testing. For passive equipment, however, only structural integrity need be maintained during testing. 3.3.2.1

Active Equipment

Equipment that should perform a mechanical motion (for example, valve closure, relay contact) or that produces a measurable signal (for example, an electrical signal, pressure, temperature, ﬂow) during the course of performing its intended functions is termed active equipment. Some examples of active equipment are valve actuators, relays, motors, pumps, transducers, control switches, and data recorders. 3.3.2.2

Passive Equipment

Passive equipment typically performs containment functions and consequently should maintain a certain minimum structural strength or pressure boundary. Such equipment usually does not perform mechanical motions or produce measurable response signals, but it may have to maintain displacement tolerances. Some examples of passive equipment are piping, tanks, cables, supporting structures, and heat exchangers.

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3-27

Functional Testing

When deﬁning the intended functions of an object for test purposes, the following information should be gathered for each active component of the object that will be tested: 1. The maximum number of times a given function should be performed during the design life of the equipment 2. The best achievable precision (or monitoring tolerance) for each functional-operability parameter and the time duration for which a given precision is required 3. Mechanisms and states of malfunction or failure 4. Limits of the functional-operability parameters (electrical signals, pressures, temperatures, ﬂow rates, mechanical displacements and tolerances, relay chatter, and so forth) that correspond to a state of malfunction or failure It should be noted that, under a state of malfunction, the object will not perform the intended function properly. Under a failure state, however, the object will not perform its intended function at all. For objects consisting of an assembly of several crucial components, it should be determined how a malfunction or failure of one component could result in the malfunction or failure of the entire unit. In such cases, any hardware redundancy (that is, when component failure does not necessarily cause unit failure) and possible interactive and chain effects (such as failure in one component overloading another, which could result in subsequent failure of the second component, and so on) should be identiﬁed. In considering functional precision, it should be noted that high precision usually means increased complexity of the test procedure. This is further complicated if a particular level of precision is required at a prescribed instant. It is a common practice for the test object supplier (the customer) to deﬁne the functional test, including acceptance criteria and tolerances for each function, for the beneﬁt of the test engineer. This information eventually is used in determining acceptance criteria for the tests of active equipment. Complexity of the required tests also depends on the precision requirements for the intended functions of the test object. Examples of functional failure are sensor and transducer (measuring instrumentation) failure, actuator (motors, valves, and so forth) failure, chatter in relays, gyroscopic and electronic-circuit drift, and discontinuity of electrical signals because of short-circuiting. It should be noted that functional failures caused by mechanical excitation are often linked with the structural integrity of the test object. Such functional failures are primarily caused in two situations: (1) when displacement amplitude exceeds a certain critical value once or several times, or (2) when vibrations of moderate amplitudes occur for an extended period of time. Functional failures in the ﬁrst category include, for example, short-circuiting, contact errors, instabilities and nonlinearities (in relays, ampliﬁer outputs, etc.). Such failures are usually reversible, so that, when the excitation intensity drops, the system will function normally. Under the second category, slow degradation of components will occur because of aging, wear, and fatigue, which can cause drift, offset, etc., and subsequent malfunction or failure. This kind of failure is usually irreversible. We must emphasize that the ﬁrst category of functional failure can be better simulated using high-intensity single-frequency testing and shock testing, and the second category by multifrequency or broadband random testing and low-intensity single-frequency testing. For passive devices, a damage criterion should be speciﬁed. This could be expressed in terms of parameters such as cumulative fatigue, deﬂection tolerances, wear limits, pressure drops, and leakage rates. Often, damage or failure in passive devices can be determined by visual inspection and other nondestructive means.

3.3.3

Information Acquisition

In addition to information concerning service functions, as discussed in the previous section, and dynamic characteristics determined from a resonance search, as will be discussed later in this chapter, there are other characteristics of the test object that need to be studied in the development of a vibration

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testing program. In particular, there are characteristics that cannot be described in exact quantitative terms. In determining the value of a piece of equipment, for instance, the monetary value (or cost) might be relatively easy to estimate, whereas it can be very difﬁcult to assign a dollar value to its signiﬁcance under service conditions. One reason for this is that a particular piece of equipment alone may not determine the proper operation of a complex system. Interaction of a particular unit with other subsystems in a complex operation would determine the importance attached to it and, hence, its value. In this sense, the true value of a test object is a relatively complex consideration. The service function of the test object is also an important consideration in determining its value. The value of a test object is important in planning a test program, because the cost of a test program and the effort expended therein are governed mainly by this factor. Many features of a test object that are signiﬁcant in planning a test program can be deduced from manufacturer’s data for the particular object. The following information is representative: 1. Drawings (schematic or to scale when appropriate) of principal components and the whole assembly, with the manufacturer’s name, identiﬁcation numbers, and dimensions clearly indicated 2. Materials used, design strengths, fatigue life, and so on, of various components, and factors determining the structural integrity of the unit 3. Component weight and total weight of the unit 4. Design ratings, capacities, and tolerances for in-service operation of each crucial component 5. Description of the intended functions of each component and of the entire unit, clearly indicating the parameters that determine functional operability of the unit 6. Interface details (inter-component as well as for the entire assembly), including in-service mounting conﬁgurations and mounting details 7. Details of the probable operating site or operating environment (particularly with respect to the excitation events if product qualiﬁcation is intended) 8. Details of any previous testing or analysis performed on that unit or a similar one Scale drawings and component-weight information describe the size and geometry of the test object. This information is useful in determining the following: 1. The locations of sensors (accelerometers, strain gages, stroboscopes, and the like) for monitoring dynamic response of the test object during tests; 2. The necessary ratings for vibration test (shaker) apparatus (power, force, stroke, bandwidth, and so on); 3. The degree of dynamic interaction between the test object and the test apparatus; 4. The level of coupling between various DoFs and modal interactions in the test object; 5. The assembly level of the test object (for example, whether it can be treated as a single component, as a subsystem consisting of several components, or as an independent, stand-alone system) In general, as the size and the assembly level increase, the tests become increasingly complex and difﬁcult to perform. To test heavy, complex test objects, we require a large test apparatus with high power ratings and the capability of multiple excitation locations. In this case, the number of operability parameters that are monitored and the number of observation (sensor) locations also will increase. 3.3.3.1

Interface Details

The dynamics of a piece of equipment depend on the way the equipment is attached to its support structure. In addition to the mounting details, the dynamic response of equipments is affected by other interfacing linkages, such as wires, cables, conduits, pipes, and auxiliary instrumentation. In the vibration testing of equipment, such interface characteristics should be simulated appropriately. Dynamics of the test ﬁxture and the details of the test object –ﬁxture interface are very important considerations that affect the overall dynamics of the test object. If the interface characteristics are not properly represented during testing, a nonuniform test could result, in which case some parts of the test object are be overtested and

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FIGURE 3.13

Inﬂuence of test ﬁxture on the test excitation signal.

other parts undertested. This situation can bring about failures that are not representative of the failures that could take place in actual service. In effect, the testing could become meaningless if interface details are not simulated properly. The test ﬁxture is a structure attached to the shaker table and used to mount the test object (see Figure 3.13). Test-ﬁxture dynamics can signiﬁcantly modify the shaker-table motion before reaching the test object. Such modiﬁcations include ﬁltering of the shaker motion and the introduction of auxiliary (cross-axis) motions. In the test set-up shown in Figure 3.13, for example, the direct motion will be modiﬁed to some extent by ﬁxture dynamics. In addition, some transverse and rotational motion components will be transmitted to the test object by the test ﬁxture because of its overhang. To minimize interface-dynamic effects in vibration testing situations, an attempt should be made to (1) make the test ﬁxture as light and as rigid as is feasible; (2) simulate in-service mounting conditions at the test object– ﬁxture interface; and (3) simulate other interface linkages, such as cables, conduits, and instrumentation, to represent in-service conditions. Very often, the design of a proper test ﬁxture can be a costly and time-consuming process. A trade-off is possible by locating the control sensors (accelerometers) at the mounting locations of the test object, and then using the error between the actual and the desired excitations through feedback to control the mounting-location excitations during testing. 3.3.3.2

Effect of Neglecting Interface Dynamics

We shall consider a simpliﬁed model to study some important effects of neglecting interface dynamics. In the model shown in Figure 3.14, the equipment and the mounting interface are modeled separately as single-DoF systems. Capital letters are used to denote the equipment parameters (mass, M; stiffness, K; and damping coefﬁcient, C). When mounting-interface dynamics are included, the model appears as in Figure 3.14(a). When the mounting-interface dynamics are neglected, we obtain the single-DoF model shown in Figure 3.14(b). Note that, in the latter case, the shaker motion, uðtÞ, is directly applied to the equipment mounts whereas, in the former case, it is applied through the interface. If the equipment response in the two cases is denoted by y and y~ ; respectively, one can see, by considering the system~ vÞ=UðvÞ; that frequency transfer functions, YðvÞ=UðvÞ and Yð ~ vÞ Yð Ms2 ðCs þ KÞ ðms2 þ cs þ kÞ þ ¼ ðMs2 þ Cs þ KÞ ðcs þ kÞ ðcs þ kÞ YðvÞ with s ¼ jv: The following nondimensional parameters are deﬁned: m Mass ratio a ¼ M

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ð3:70Þ

ð3:71Þ

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Vibration Monitoring, Testing, and Instrumentation

y

M

Test K Object (Equipment)

C ~ y m m

Interface

k

K

c

C u(t)

u(t) (a)

(b)

FIGURE 3.14 A simpliﬁed model to study the effect of interface dynamics: (a) with interface dynamics; (b) without interface dynamics.

Natural frequency ratio b ¼

vn Vn

Normalized excitation frequency v ¼

ð3:72Þ

v Vn

in which the natural frequency (undamped) of the equipment is rﬃﬃﬃﬃﬃ K Vn ¼ M

ð3:73Þ

ð3:74Þ

and the mounting-interface natural frequency is sﬃﬃﬃﬃ k vn ¼ m

ð3:75Þ

Then, Equation 3.70 can be written ~ vÞ Yð ðb2 þ 2jzbv 2 v2 Þ vð1 þ 2jZ vÞ ¼ 2 YðvÞ aðb2 þ 2jzbvÞð1 þ 2jZ v 2 v2 Þ ðb2 þ 2jzbvÞ

ð3:76Þ

in which z and Z denote the damping ratios of the interface and the equipment, respectively. ~ vÞ=YðvÞ is representative of the equipment-response ampliﬁcation when interfaceThe ratio Yð dynamic effects are neglected for a harmonic excitation. Figure 3.15 shows eight curves, corresponding to Equation 3.76, for the parameter combinations given in Table 3.2. Interpretation of the results becomes easier when peak values of the response ratios are compared for various parameter combinations. Sample results are given Table 3.2. 3.3.3.3

Effects of Damping

By comparing the cases from Case 1 to Case 4 in Table 3.2 with the cases from Case 5 to Case 8, respectively (comparing Case 1 with Case 5 and so forth), we see that increasing the interface damping has reduced the peak response (a favorable effect), irrespective of the values of the interface mass and natural frequency (a and b values).

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40.0

z = 0.1 2

Magnitude Ratio

32.0 24.0 16.0 8.0 0.0

4

1

3 0.00

0.43

(a) 40.0

0.86 1.29 1.71 2.14 Normalized Frequency

2.57

3.00

2.57

3.00

z = 0.2

Magnitude Ratio

32.0 24.0 16.0

6

8.0 0.0

8 0.00

0.43

(b) FIGURE 3.15

3.3.3.4

75

0.86 1.29 1.71 2.14 Normalized Frequency

Response ampliﬁcation when interface dynamic interactions are neglected.

Effects of Inertia

By comparing the cases from Case 1 to Case 4 in Table 3.2 with the cases from Case 5 to Case 8, respectively, we see that the interface inertia has a favorable effect in decreasing dynamic interaction, irrespective of the interface damping and natural frequency. TABLE 3.2

Response Ampliﬁcation Caused by Neglecting Interface Dynamics

Case (Curve No.)

1 2 3 4 5 6 7 8

Parameter Combination

Peak Value of Response Ratio

z

a

b

Z

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2

2.0 0.5 0.5 2.0 2.0 0.5 0.5 2.0

2.0 0.5 2.0 0.5 2.0 0.5 2.0 0.5

0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2

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1.11 38.80 2.77 17.80 0.89 18.40 1.71 5.98

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Vibration Monitoring, Testing, and Instrumentation

Effect of Natural Frequency

By comparing the cases from Case 1 to Case 4 in Table 3.2 with the cases from Case 5 to Case 8, respectively, we see that increasing the interface natural frequency has a favorable effect in decreasing dynamic interactions, irrespective of the interface damping and inertia. 3.3.3.6

Effect of Excitation Frequency

All the response plots (see Figure 3.15) diverge to 1 as v increases. This indicates that, at very high excitation-input frequencies, dynamic-testing results can become meaningless because of the interactions with interface dynamics. It can be concluded that, to reduce dynamic interactions caused by a mechanical interface, we should (1) increase interface damping as much is feasible, (2) increase interface mass as much is feasible, (3) increase interface natural frequency as much as is feasible, and (4) avoid testing at relatively high frequencies of excitation. It should be noted that, in the foregoing analysis and discussion, the mechanical interface was considered to include test ﬁxtures and the shaker table as well. 3.3.3.7

Other Effects of Interface

The type of vibration test used sometimes depends on the mechanical interface characteristics. An example is the testing of line-mounted equipment. Single-frequency testing is preferred for such equipment so as to add a certain degree of conservatism because, as a result of interface resonances, line-mounted equipment could be subjected to higher levels of narrowband excitation through the support structure. In vibration testing of multicomponent equipment cabinets, it is customary to test the empty cabinet ﬁrst, with the components replaced by dummy weights, and then to test the individual components separately, using different test excitations depending on the component locations and their mounting characteristics. Mechanical interface details of individual components are important in such situations. As a result, interface information is an important constituent of the pretest information that is collected for a test object. Most of the interface data, particularly information related to size and geometry (for example, mass, dimensions, conﬁgurations, and locations), can be gathered simply by observing the test object and using scale drawings supplied by the manufacturer. Knowledge of the size and number of anchor bolts used or the weld thickness, for example, can be obtained in this manner. When analysis is also used to augment testing, however, it is often necessary to know the loads transmitted (forces, moments, and so on), relative displacements, and stiffness values at the mechanical interface under in-service conditions. These must be determined by tests, by analysis (static or dynamic) of a suitable model, or from manufacturer’s data.

3.3.4

Test-Program Planning

The test program to which a test object is subjected depends on several factors including the following: 1. The objectives and speciﬁc requirements of the test 2. In-service conditions, including equipment-mounting features, the vibration environment, and speciﬁcations of the test environment 3. The nature of the test object, including complexity, assembly level, and functional-operability parameters to be monitored 4. Test-laboratory capabilities, available testing apparatus, past experience, conventions, and established practices of testing Some of these factors are based on solid technical reasons, whereas others depend on economics, convenience, and personal likes and dislikes. Initially, it is not necessary to develop a detailed test procedure; this is required only at the stage of actual testing. In the initial stage it is only necessary to select the appropriate test method, based on factors such as those listed in the beginning of this section. Before conducting the tests, however, a test procedure should be prepared in sufﬁcient detail. In essence, this is a pretest requirement.

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Objectives and speciﬁc requirements of a test depend on such considerations as whether the test is conducted at the design stage, the quality-control stage or the utilization stage. The objective of a particular test could be to verify the outcome of a previously conducted test. In that case, it is necessary to assess the adequacy of one or a series of tests conducted at an earlier time (for instance, when the speciﬁcations and government regulations were less stringent). Often, this could be done by analysis alone. Some testing might be necessary at times, but it usually is not necessary to repeat the entire test program. If the previous tests were conducted for the frequency range 1 to 25 Hz, for example, and the present speciﬁcations require a wider range of 1 to 35 Hz, it might be adequate merely to demonstrate (by analysis or testing) that there are no signiﬁcant resonances in the test object in the 25 to 35 Hz range. If it is necessary to qualify the test object for several different dynamic environments, a generic test that represents (conservatively, but without the risk of overtesting) all these environments could be used. For this purpose, special test-excitation inputs must be generated, taking into account the variability of the excitation characteristics under the given set of environments. Alternatively, several tests might be conducted if the dynamic environments for which the test object is to be qualiﬁed are signiﬁcantly different. Operating-basis earthquake (OBE) tests and safe-shutdown earthquake (SSE) tests in seismic qualiﬁcation of nuclear power plant equipment, for example, represent two signiﬁcantly different test conditions. Consequently, they cannot be represented by a single test. When qualifying an equipment for several geographic regions or locations, however, we might be able to combine all OBE tests into a single test and all SSE tests into another single test. Another important consideration in planning a test program is the required accuracy for the test, including the accuracy for the excitation inputs, response and operability measurements, and analysis. This is related to the value of the test object and the objectives of the test. When it is required to evaluate or qualify a group of equipment by testing a sample, it is ﬁrst necessary to establish that the selected sample unit is truly representative of the entire group. When the items in the batch are not identical in all respects, some conservatism could be added to the tests to minimize the possibility of an incorrect qualiﬁcation decision. It might be necessary to test more than one sample unit in such situations. When planning a test procedure, we should clearly identify the standards, government regulations, and speciﬁcations that are applicable to a particular test. The pertinent sections of the applicable documents should be noted, and proper justiﬁcation should be given if the tests deviate from regulatory-agency requirements. The excitation input that is employed in a vibration test depends on the in-service vibration environment of the test object. The number of tests needed will also depend on this to some extent. Test orientation depends mainly on the mounting features and the mechanical interface details of the test object under in-service conditions. Mounting features might govern the nature of the test excitations used for a particular test. Two distinct mounting types can be identiﬁed for most equipment: (1) line-mounted equipment and (2) ﬂoor-mounted equipment. Line-mounted equipment is equipment that is mounted upright or hanging from pipelines, cables, or similar line structures that are not rigid. Generally, devices such as valves, nozzles, valve actuators, and transducers are to be considered line-mounted equipment. Any equipment that is not line-mounted is considered ﬂoor-mounted. The supporting structure is considered to be relatively rigid in this case. Examples of such mounting structures are ﬂoors, walls, and rigid frames. Typical examples of ﬂoor-mounted equipment include motors, compressors, and cabinets of relays and switchgear. Wide-band ﬂoor disturbances are ﬁltered by line structures. Consequently, the environmental disturbances to which line-mounted equipment is subjected are generally narrowband disturbances. Accordingly, vibration testing of line-mounted equipment is best performed using narrowband random test excitations or single-frequency deterministic test excitations. Higher test intensities can be necessary for line-mounted equipment, because any low-frequency resonances that are be present

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in the mounting structure (which is relatively ﬂexible in this case) can amplify the excitations before reaching the equipment. Floor-mounted equipment often requires relatively wide-band random test excitations. As an example, consider a pressure transducer mounted on (1) a rigid wall, (2) a rigid I-section frame, (3) a pressurized gasline, or (4) a cabinet. In cases (1) and (2), wide-band random excitations with response spectra approximately equal to the ﬂoor-response spectra can be employed for vibration testing of the pressure transducer. For cases (3) and (4), however, ﬂexibility of the support structure should be taken into consideration in developing the RRS speciﬁcations for vibration testing. In case (3), a single-frequency deterministic test, such as a sine-beat test or a sine-dwell test, can be employed, giving sufﬁcient attention to testing at the equipment-resonant frequencies. In case (4), single-frequency tests can also be employed if the cabinet is considerably ﬂexible and not rigidly attached to a rigid structure (a ﬂoor or a wall). Alternatively, a wide-band test on the cabinet itself, with the pressure transducer mounted on it, could be used. Size, complexity, assembly level, and related features of a test object can signiﬁcantly complicate and extend the test procedure. In such cases, testing the entire assembly might not be practical and testing of individual components or subassemblies might not be adequate because, in the in-service dynamic environment, the motion of a particular component could be signiﬁcantly affected by the dynamics of other components in the assembly, the mounting structure, and other interface subsystems. Functional-operability parameters to be monitored during testing should be predetermined. They depend on the purpose of the test, the nature of the test object, and the availability and characteristics of the sensors that are required to monitor these parameters. Malfunction or failure criteria should be related in some way to the monitored operability parameters; that is, each operability parameter should be associated with one or several components in the test object that are crucial to its operation. The decision of whether to perform an active test (for example, whether a valve should be cycled during the test) and determination of the actuation time requirements (for example, the number of times the valve is cycled and at what instants during the test) should be made at this stage. The loading conditions for the test (that is, in-service loading simulation) also should be deﬁned. An important nontechnical factor that determines the nature of a vibration test is the availability of hardware (test apparatus) in the test laboratory. This is especially true when nonconventional vibration tests are required. Some speciﬁcations require three-DoF test inputs, for example, but most test laboratories have only one-DoF or two-DoF test machines. When two-DoF or one-DoF tests are used in place of three-DoF tests, it is ﬁrst required to determine what additional orientations of the test object should be tested in order to add the required conservatism. Additionally, it should be veriﬁed by analysis or testing that the modiﬁed series of tests does not cause signiﬁcant undertesting or overtesting of certain parts of the test object. Otherwise, some other form of justiﬁcation should be provided for replacing the test. Test plans prepared in the pretest stage should include an adequate description of the following important items: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

Test purpose Test-object details Test environment, speciﬁcations, and standards Functional-operability parameters and failure or malfunction criteria Pretest inspection Aging requirements Test outline Instrumentation requirements Data-processing requirements Methods of evaluation of the test results

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3.3.4.1

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Testing of Cabinet-Mounted Equipment

In the vibration testing of cabinet-mounted or panel-mounted equipment, the following is a standard procedure. Step 1: Test the cabinet or panel with the equipment replaced by a dummy weight. Step 2: Obtain the cabinet response at equipment-mounting locations and, based on these observations, develop the required vibration environment for testing (the RRS) of the equipment. Step 3: Test the equipment separately, using the excitations developed in Step 2. This procedure may not be satisfactory if there is a considerable degree of dynamic interaction between the equipment and the mounting cabinet. This could be illustrated by using a simpliﬁed model to represent cabinet-mounted equipment. The cabinet and the equipment are represented separately by single-DoF systems, as shown in Figure 3.16. Cabinet parameters are represented by upper-case letters and equipment parameters by lower-case letters. The cabinet responds when the equipment is replaced by a dummy weight of equal mass, which is denoted by y~ ðtÞ: The test excitation applied to the cabinet base is denoted by uðtÞ: It can be shown that the frequencyresponse ratio in the two cases is given by ~ vÞ Yð ½ms2 ðcs þ kÞ =ðms2 þ cs þ kÞ þ MS2 þ Cs þ K ¼ YðvÞ ðM þ mÞs2 þ Cs þ K

ð3:77Þ

with s ¼ jv: Using the nondimensional parameters deﬁned by the equations from Equation 3.71 to Equation 3.75, we obtain ~ vÞ Yð ðjvÞ2 að2zbjv þ b2 Þ þ ðjvÞ2 þ 2Zjv þ 1 ¼ YðvÞ ð1 þ aÞðjvÞ2 þ 2Zjv þ 1

ð3:78Þ

in which z denotes the equipment damping ratio and Z denotes the cabinet damping ratio. ~ vÞ=YðvÞ represents the ampliﬁcation in the cabinet response when the equipment is The ratio Yð replaced by a dummy weight for a harmonic excitation. Figure 3.17 shows eight curves obtained for the z; a; b; and Z combinations, as given in Table 3.2. Notice that the best response is obtained in Curve 6. It can be concluded that a dummy test procedure for cabinet-mounted equipment is satisfactory when x

m Equipment

c

k

Dummy Weight y

M Cabinet C

K

C u(t)

u(t)

FIGURE 3.16 system.

~ y

M Cabinet

K

(a)

m

(b)

A simpliﬁed model for (a) an equipment cabinet test system; (b) a dummy-weight cabinet test

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z = 0.1 1

Magnitude Ratio

16.0 12.0 8.0

3 4

4.0

2 0.0 0.00

0.43

0.86

(a) 20.0

1.29 1.71 2.14 Normalized Frequency

2.57

3.00

1.29 1.71 2.14 Normalized Frequency

2.57

3.00

z = 0.2

Magnitude Ratio

16.0 12.0 5

8.0 4.0

8 7

6

0.0 0.00

0.43

(b)

FIGURE 3.17

0.86

Cabinet response ampliﬁcation in dummy-weight tests.

the equipment inertia and natural frequency are small in comparison to the values for the cabinet. Also, increasing the damping level has a favorable effect on test results.

3.3.5

Pretest Inspection

Pretest inspection of a test object is important at least for two major reasons. First, if the equipment supplied for testing is different from the piece of equipment or the group of equipment that is required to be qualiﬁed, then these differences must be carefully observed and recorded in sufﬁcient detail. In particular, deviations in the model number, mounting features and other details of the mechanical interface, geometry, size, and signiﬁcant dynamic features should be recorded. Second, before testing, the test object should be inspected for any damage, deﬁciencies, or malfunctions. Structural integrity usually can be determined by visual inspection alone. To determine malfunctions by operability monitoring, however, the test object must be actuated and the operating environment should be simulated. If the equipment supplied for testing is not identical to that required to be tested, adequate justiﬁcation must be provided for the differences to guarantee that the objectives of the test can be achieved by testing the equipment that is supplied. Otherwise, the test should be abandoned pending the arrival of the correct test object.

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If any structural failure or operation malfunction is noted during pretest inspection, no corrective action should be taken by the test laboratory personnel unless those actions are notiﬁed to and fully authorized by the supplier of the test object. Otherwise, the test should be abandoned and the customer should be promptly notiﬁed of the anomalies. It is important that the functional-operability pretest inspection is performed in the same functional environment as that is experienced under normal in-service conditions. When monitoring functionaloperability parameters, it is necessary to guarantee that the monitoring instrumentation meets the required accuracy. Instrumentation data should be provided to the customer for review. This assures that the observed malfunction is real and not merely apparent, caused by a malfunction in the monitoring instrumentation and channels. The monitoring-equipment accuracy should be higher than that required for the operability parameter itself. Justiﬁcation is needed if some components in the test object are not actuated and monitored during pretest inspection. Also, the warm-up period and the total time of actuation should be justiﬁed. In particular, if the proper operation of the equipment is governed by the continuity of a parameter (such as an electrical signal), the time duration of monitoring should be noted. If, however, the proper function is governed by a change of state (such as the opening or closing of a valve, a switch, or a relay), the number of cycles of actuation is important.

3.4

Testing Procedures

Vibration testing may involve pretesting prior to the main tests. The objectives of pretesting may be (1) exploratory, in order to obtain dynamic information such as natural frequencies, mode shapes and damping about the test object; (2) preconditioning, in order to age or pass the “infant-mortality” stage so that the main test will be realistic and correspond to normal operating conditions. In the present section, we will describe both pretesting and main testing in an integrated manner.

3.4.1

Resonance Search

Vibration test programs usually require a resonance-search pretest. This is typically carried out at a lower excitation intensity than that used for the main test in order to minimize the damage potential (overtesting). The primary objective of a resonance-search test is to determine resonant frequencies of the test object. More elaborate tests are employed, however, to determine mode shapes and modal damping ratios in addition to resonant frequencies. Such frequency-response data on the test object are useful in planning and conducting the main test. Frequency-response data usually are available as a set of complex frequency-response functions. There are tests that determine the frequency-response functions of a test object, and simpler tests are available to determine resonant frequencies alone. Some of the uses of frequency-response data are given below. 1. A knowledge of the resonant frequencies of the test object is important in conducting the main test. More attention should be given, for example, when performing a main test in the vicinity of resonant frequencies. In the resonance neighborhoods, lower sweep rates should be used if sine sweep is used in the main test, and larger dwell periods should be used if a sine dwell is part of the main test. Frequency-response data give the most desirable frequency range for conducting main tests. 2. From frequency-response data, it is possible to determine the most desirable test excitation directions and the corresponding input intensities. 3. The degree of nonlinearity and the time variance in the system parameters of the test object can be estimated by conducting more than one frequency-response test at different excitation levels. If the deviation in the frequency-response functions thus obtained is

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sufﬁciently small, then a linear, time-invariant dynamic model is considered satisfactory in the analysis of the test object. 4. If no resonances are observed in the test object over the frequency range of interest, as determined by the operating environment for a given application, then a static analysis will be adequate to qualify the test object. 5. A set of frequency-response functions can be considered a dynamic model for the test specimen. This model can be employed in further studies of the test specimen by analytical means.

3.4.2

Methods of Determining Frequency-Response Functions

Three methods of determining frequency-response functions are outlined here. 3.4.2.1

Fourier Transform Method

If yðtÞ is the response at location B of the test object, when a transient input, uðtÞ; is applied at location A; then the frequency-response function, Hð f Þ; between locations A and B is given by the ratio of the Fourier integral transforms of the output, yðtÞ; and the input, uðtÞ: Hð f Þ ¼

Yð f Þ Uð f Þ

ð3:79Þ

In particular, if uðtÞ is a unit impulse, then Uð f Þ ¼ 1 and, hence, Hð f Þ ¼ Yð f Þ: 3.4.2.2

Spectral Density Method

If the input excitation is a random signal, the frequency-response function between the input point and the output point can be determined as the ratio of the cross-spectral density, Fuy ð f Þ; of the input, uðtÞ; and the output, yðtÞ; and the PSD, Fuu ð f Þ; of the input: Hð f Þ ¼ 3.4.2.3

Fuy ð f Þ Fuu ð f Þ

ð3:80Þ

Harmonic Excitation Method

If the input signal is sinusoidal (harmonic) with frequency, f ; the output also will be sinusoidal with frequency, f ; at steady state but with a change in the phase angle. Then, the frequency-response function is obtained as a magnitude function and a phase-angle function. The magnitude, lHð f Þl; is equal to the steady-state ampliﬁcation of the output signal, and the phase angle, /Hð f Þ; is equal to the steady-state phase lead of the output signal. This pair of curves, the magnitude plot and the phase angle plot, is called a Bode plot or Bode diagram.

3.4.3

Resonance-Search Test Methods

There are three basic types of resonance-search test methods. They are categorized according to the nature of the excitation used in the test; speciﬁcally, (1) impulsive excitation, (2) initial displacement, or (3) forced vibration. The ﬁrst two categories are free-vibration tests; that is, response measurements are made on free decay of the test object following a momentary (initial) excitation. Typical tests belonging to each of these categories are described in the following sections. 3.4.3.1

Hammer (Bump) Test and Drop Test

In a resonance search using the impulsive-excitation method, an impulsive force (a large magnitude of force acting over a very short duration) is applied at a suitable location of the test object, and the resulting transient response of the object is observed, preferably at several locations. This is

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FIGURE 3.18

Schematic diagram of a hammer test arrangement.

equivalent to applying an initial velocity to the test object and letting it vibrate freely. By Fourier analysis of the response data, it is possible to obtain the resonant frequencies, corresponding mode shapes, and modal damping. Hammer tests and drop tests belong to the impulsive-excitation category. A schematic diagram of the hammer-test arrangement is shown in Figure 3.18. A schematic diagram of the drop-test arrangement is shown in Figure 3.19. The angle of swing of the hammer or the drop height of the object determines the intensity of the applied impulse. Alternatively, the impulse can be generated using explosive cartridges (for relatively large structures) located suitably in the test object, or by ﬁring small projectiles at the test object. The response is monitored at several locations of the test object. The response at the point of application of the impulse is always monitored. Response analysis can be done in real time, or the response can be recorded for subsequent analysis. A major concern in these tests is making sure that all signiﬁcant resonances in the required frequency range are excited under the given excitation. Several tests for different conﬁgurations of the test object might be necessary to achieve this. Proper selection of the response-monitoring locations is also important in obtaining meaningful test results. By changing the impulsive-force intensity and repeating the test, any signiﬁcant nonlinear (or time-variant-parameter) behavior of the test object can be determined. A common practice is to monitor the impulsive-force signal during impact. In this way, poor impacts (for example, low-intensity impacts

FIGURE 3.19

Schematic diagram of a drop test arrangement.

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or multiple impacts caused by the bouncing back of a hammer) can be detected and the corresponding test results can be rejected. 3.4.3.2

Pluck Test

A resonance search on a test object can be performed by applying a displacement initial condition (rather than a velocity initial condition, as in impulsive tests) to a suitably mounted test object and measuring its subsequent response at various locations as it executes free vibrations. By properly selecting the locations and the magnitudes of the initial displacements, it is sometimes possible to excite various modes of vibration, provided that these modes are reasonably uncoupled. The pluck test is the most common test that uses the initial-displacement method. A schematic diagram of the test set-up is shown in Figure 3.20. The test object is initially deﬂected by pulling it with a cable. When the cable is suddenly released, the test object will undergo free vibrations about its staticequilibrium position. The response is observed for several locations of the test object and analyzed to obtain the required parameters. In Figure 3.18 to Figure 3.20, the frequency-response function between two locations (A and B, for example) is obtained by analyzing the corresponding two signals, using either the Fourier transform method (Equation 3.79) or the spectral-density method (Equation 3.80). These frequency-domain techniques will automatically provide the natural-frequency and modal-damping information. Alternatively, modal damping can be determined using time-domain methods, for example, by evaluating the logarithmic decrement of the response after passing it through a ﬁlter having a center frequency adjusted to the predetermined natural frequency of the test object for that mode. The accuracy of the estimated modal-damping value can be improved signiﬁcantly by such ﬁltering methods. Often, the most difﬁcult task in a natural-frequency search is the excitation of a single a mode. If two natural frequencies are close together, modal interactions of the two invariably will be present in the response measurements. Because of the closeness of the frequencies, the response curve will display a beat phenomenon, as shown in Figure 3.21, which makes it difﬁcult to determine damping by the logarithmic-decrement method. It is difﬁcult to distinguish between decay caused by damping and rapid drop-off caused by beating. In this case, one of the frequency components must be ﬁltered out, using a very narrowband-pass ﬁlter, before computing damping. The required testing time for the impulsive-excitation and initial-displacement test methods is relatively small in comparison with forced-vibration test durations. For this reason, these former (free-vibration) tests are often preferred in preliminary (exploratory) testing before the main tests. The directions and locations of impact or initial displacements should be properly chosen, however, so that as many signiﬁcant modes as possible will be excited in the desired frequency range. If the

FIGURE 3.20

Diagram of a pluck test arrangement.

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Vibration Testing

FIGURE 3.21

3-41

Beat phenomenon resulting from interaction of closely spaced modes.

impact is applied at a node point, (of a particular mode, for instance) it will be virtually impossible to detect that mode from the response data. Sometimes, a large number of monitoring locations are necessary to accurately determine mode shapes of the test object. This depends primarily on the size and dynamic complexity of the test object and the particular mode number. This, in turn, necessitates the use of more sensors (accelerometers and the like) and recorder channels. If a sufﬁcient number of monitoring channels is not available, the test will have to be repeated, each time using a different set of monitoring locations. Under such circumstances, it is advisable to keep one channel (monitoring location) unchanged and to use it as the reference channel. In this manner, any deviations in the test-excitation input can be detected for different tests and properly adjusted or taken into account in subsequent analysis (for example, by normalizing the response data). 3.4.3.3

Shaker Tests

A convenient method of resonance search is by using a continuous excitation. A forced excitation, which typically is a sinusoidal signal or a random signal, is applied to the test object by means of a shaker, and the response is continuously monitored. The test set-up is shown schematically in Figure 3.22. For sinusoidal excitations, signal ampliﬁcation and phase shift over a range of excitations will determine the frequency-response function. For random excitations, Equation 3.80 may be used to determine the frequency-response function. FIGURE 3.22 Schematic diagram of a shaker test for One or several portable exciters (shakers) or a resonance search. large shaker table similar to that used in the main vibration test can be employed to excite the test object. The number and the orientations of the shakers and the mounting conﬁgurations and monitoring locations of the test object should be chosen depending on the size and complexity of the test object, the required accuracy of the resonance-search results, and the modes of vibration that need to be excited. The shaker-test method has the advantage of being able to control the nature of the test-excitation input (for example, frequency content, intensity, and sweep rate), although it might be more complex and costly. The results from shaker tests are more accurate and more complete. Test objects usually display a change in resonant frequencies when the shaker amplitude is increased. This is caused by inherent nonlinearities in complex structural systems. Usually, the

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change appears as a spring-softening effect, which results in lower resonant frequencies at higher shaker amplitudes. If this nonlinear effect is signiﬁcant, the resonant frequencies for the main test level cannot be accurately determined using a resonance search at low intensity. Some form of extrapolation of the test results, or analysis using an appropriate dynamic model, might be necessary in this case to determine the resonant-frequency information that might be required to perform the main test.

3.4.4

Mechanical Aging

Before performing a qualiﬁcation test, it is usually necessary to age the test object to put it into a condition that represents its state following its operation for a predetermined period under in-service conditions. In this manner, it is possible to reduce the probability of burn-in failure (infant mortality) during testing. Some tests, such as design-development tests and quality-assurance tests, might not require prior aging. The nature and degree of aging that is required depends on such factors as the intended function of the test object, the operating environment, and the purpose of the dynamic test. In qualiﬁcation tests, it may be necessary to demonstrate that the test object still has adequate capability to withstand an extreme dynamic environment toward the end of its design life (that is, the period in which it can be safely operated without requiring corrective action). In such situations, it is necessary to age the test object to an extreme deterioration state, representing the end of the design life of the test object. Test objects are aged by subjecting them to various environmental conditions (for example, high temperatures, radiation, humidity, and vibrations). Usually, it is not practical to age the equipment at the same rate as it would age under a normal service environment. Consequently, accelerated aging procedures are used to reduce the test duration and cost. Furthermore, the operating environment may not be fully known at the testing stage. This makes the simulation of the true operating environment virtually impossible. Usually, accelerated aging is done sequentially, by subjecting the test equipment to the various environmental conditions one at a time. Under in-service conditions, however, these effects occur simultaneously, with the possibility of interactions between different effects. Therefore, when sequential aging is employed, some conservatism should be added. The type of aging used should be consistent with the environmental conditions and operating procedures of the speciﬁc application of the test object. Often these conditions are not known in advance, in which case, standardized aging procedures should be used. Our main concern in this section is mechanical aging, although other environmental conditions can signiﬁcantly affect the dynamic characteristics of a test object. The two primary mechanisms of mechanical aging are material fatigue and mechanical wearout. The former mechanism plays a primary role if in-service operation consists of cyclic loading over relatively long periods of time. Wearout, however, is a long-term effect caused by any type of relative motion between components of the test object. It is very difﬁcult to analyze component wearout, even if only the mechanical aspects are considered (that is the effects of corrosion, radiation, and the like are neglected). Some mechanical wearout processes resemble fatigue aging; however, they depend simultaneously on the number of cycles of load applications and the intensity of the applied load. Consequently, only the cumulative damage phenomenon, which is related to material fatigue, is usually treated in the literature. Although mechanical aging is often considered a pretest procedure (for example, the resonance-search test), it actually is part of the main test. In a dynamic qualiﬁcation program, if the test object malfunctions during mechanical aging, this amounts to failure in the qualiﬁcation test. Furthermore, exploratory tests, such as resonance-search tests, are sometimes conducted at higher intensities than what is required to introduce mechanical aging into the test object. 3.4.4.1

Equivalence for Mechanical Aging

It is usually not practical to age a test object under its normal operating environment, primarily because of time limitations and the difﬁculty in simulating the actual operating environment. Therefore, it may

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be necessary to subject the test object to an accelerated aging process in a dynamic environment of higher intensity than that present under normal operating conditions. Two aging processes are said to be equivalent if the ﬁnal aged condition attained by the two processes is identical. This is virtually impossible to realize in practice, particularly when the object and the environment are complex and the interactions of many dynamic causes have to be considered. In this case, a single most severe aging effect is used as the standard for comparison to establish the equivalence. The equivalence should be analyzed in terms of both the intensity and the nature of the dynamic excitations used for aging. 3.4.4.2

Excitation-Intensity Equivalence

A simpliﬁed relationship between the dynamic-excitation intensity, U; and the duration of aging, T; that is required to attain a certain level of aging, keeping the other environmental factors constant, may be given as T¼

c Ur

ð3:81Þ

in which c is a proportionality constant and r is an exponent. These parameters depend on such factors as the nature and sequence of loading and characteristics of the test object. It follows from Equation 3.81 that, by increasing the excitation intensity by a factor n, the aging duration can be reduced by a factor of nr : In practice, however, the intensity –time relationship is much more complex, and caution should be exercised when using Equation 3.81. This is particularly true if the aging is caused by multiple dynamic factors of varying characteristics that are acting simultaneously. Furthermore, there is usually an acceptable upper limit to n: It is unacceptable, for example, to use a value that will produce local yielding or any such irreversible damage to the equipment that is not present under normal operating conditions. It is not necessary to monitor functional operability during mechanical aging. Furthermore, it can happen that, during accelerated aging, the equipment malfunctions but, when the excitation is removed, it operates properly. This type of reversible malfunction is acceptable in accelerated aging. The time to attain a given level of aging is usually related to the stress level at a critical location of the test object. Since this critical stress can be related, in turn, to the excitation intensity, the relationship given by Equation 3.81 is justiﬁed. 3.4.4.3

Dynamic-Excitation Equivalence

The equivalence of two dynamic excitations that have different time histories can be represented using methods employed to represent dynamic excitations (for example, response spectrum, Fourier spectrum, and PSD). If the maximum (peak) excitation is the factor that primarily determines aging in a given system under a particular dynamic environment, then response-spectrum representation is well suited for establishing the equivalence of two excitations. If, however, the frequency characteristics of the excitation are the major determining factor for mechanical aging, then Fourier spectrum representation is favored for establishing the equivalence of two deterministic excitations, and PSD representation is suited for random excitations. When two excitation environments are represented by their respective PSD functions, F1 ðvÞ and F2 ðvÞ; if the signiﬁcant frequency range for the two excitations is ðv1 ; v2 Þ; then the degree of aging under the two excitations may be compared using the ratio ðv2 A1 v ¼ ðv12 A2 v1

F1 ðvÞdv F2 ðvÞdv

ð3:82Þ

in which A denotes a measure of aging. If the two excitations have different frequency ranges of interest, a range consisting of both ranges might be selected for the integrations in Equation 3.82.

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3.4.4.4

Vibration Monitoring, Testing, and Instrumentation

Cumulative Damage Theory

Miner’s linear cumulative damage theory may be used to estimate the combined level of aging resulting from a set of excitation conditions. Consider m excitations acting separately on a system. Suppose that each of these excitations produces a unit level of aging in N1 ; N2 ; …; Nm loading cycles, respectively, when acting separately. If, in a given dynamic environment, n1 ; n2 ; …; nm loading cycles, respectively, from the m excitations actually have been applied to the system (possibly all excitations were acting simultaneously), the level of aging attained can be given by A¼

m X ni N i i¼1

ð3:83Þ

The unit level of aging is achieved, theoretically, when A ¼ 1: Equation 3.83 corresponds to Miner’s linear cumulative damage theory. Because of various interactive effects produced by different loading conditions, when some or all of the m excitations act simultaneously, it is usually not necessary to have A ¼ 1 under the combined excitation to attain the unit level of aging. Furthermore, it is extremely difﬁcult to estimate Ni ; i ¼ 1; …; m: For such reasons, the practical value of A in Equation 3.83 for using in attaining a unit level of aging could vary widely (typically, from 0.3 to 3.0).

3.4.5

Test-Response Spectrum Generation

A vibration test may be speciﬁed by a RRS. In this case, the response spectrum of the actual excitation signal, that is, the TRS, should envelop the RRS during testing. It is customary for the purchaser (the owner of the test object) to provide the test laboratory with a multichannel FM tape or some form of signal storage device containing the components of the excitation input signal that should be used in the test. Alternatively, the purchaser may request that the test laboratory generate the required signal components under the purchaser’s supervision. If sine beats are combined to generate the test excitations, each FM tape should be supplemented by tabulated data giving the channel number, the beat frequencies (Hz) in that channel, and the amplitude ðgÞ of each sine-beat component. The RRS curve that is enveloped by the particular input should also be speciﬁed. The excitation signal that is applied to the shaker-table actuator is generated by combining the contents of each channel in an appropriate ratio so that the response spectrum of the excitation that is actually felt at the mounting locations of the test object (the TRS) satisfactorily matches the RRS supplied to the test laboratory. Matching is performed by passing the contents of each channel through a variablegain ampliﬁer and mixing the resulting components according to variable proportions. These operations are performed by a waveform mixer. The adjustment of the ampliﬁer gains is done by trial and error. The phase of the individual signal components should be maintained during the mixing process. Each channel may contain a single-frequency component (such as sine beat) or a multifrequency signal of ﬁxed duration (for example, 20 sec). If the RRS is complex, each channel may have to carry a multifrequency signal to achieve close matching of the TRS with the RRS. Also, a large number of channels might be necessary. The test excitation signal is generated continuously by repetitively playing the FM tape loop of ﬁxed duration. In product qualiﬁcation, response spectra are usually speciﬁed in units of acceleration due to gravity ðgÞ: Consequently, the contents in each channel of the test-input FM tape represent acceleration motions. For this reason, the signal from the waveform mixer must be integrated twice before it is used to drive the shaker table. The actuator of the exciter is driven by this displacement signal, and its control may be done by feedback from a displacement sensor. However, if the control sensor is an accelerometer, as is typical, double integration of that signal will be needed as well. In typical test facilities, a double integration unit is built into the shaker system. It is then possible to use any type of signal (displacement, velocity, acceleration) as the excitation input and to decide

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3-45

simultaneously on the number of integrations that are necessary. If the input signal is a velocity time history, for example, one integration should be chosen and so on. The tape speed should be speciﬁed (for example, 7.5, 15 in./s) when the signals recorded on tapes are provided to generate input signals for vibration testing. This is important to ensure that the frequency content of the signal is not distorted. The speeding up of the tape has the effect of scaling up of each frequency component in the signal. It has also the effect, however, of ﬁltering out very high-frequency components in the signal. If the excitation signals are available as digital records, then a DAC is needed to convert them into analog signals.

3.4.6

Instrument Calibration

The test procedure normally stipulates accuracy requirements and tolerances for various critical instruments that are used in testing. It is desirable that these instruments have current calibration records that are agreeable to an accepted standard. Instrument manufacturers usually provide these calibration records. Accelerometers, for example, may have calibration records for several temperatures (for example, 2 65, 75, 3508F) and for a range of frequencies (such as 1 to 1000 Hz). Calibration records for accelerometers are given in both voltage sensitivity (mV/g) and charge sensitivity (pC/g), along with percentage-deviation values. These tolerances and peak deviations for various test instruments should be provided for the purchaser’s review before they are used in the test apparatus. From the tolerance data for each sensor or transducer, it is possible to estimate peak error percentages in various monitoring channels in the test set-up, particularly in the channels used for functionaloperability monitoring. The accuracy associated with each channel should be adequate to measure expected deviations in the monitored operability parameter. It is good practice to calibrate sensor or transducer units, such as accelerometers and associated auxiliary devices, daily or after each test. These calibration data should be recorded under different scales when a particular instrument has multiple scales, and for different instrument settings.

3.4.7

Test-Object Mounting

When a test object is being mounted on a shaker table, care should be taken to simulate all critical interface features under normal installed conditions for the intended operation. This should be done as accurately as is feasible. Critical interface requirements are those that could signiﬁcantly affect the dynamics of the test object. If the mounting conditions in the test set-up signiﬁcantly deviate from those under installed conditions for normal operation, adequate justiﬁcation should be provided to show that the test is conservative (that is, the motions produced under the test mounting conditions are more severe than in in-service conditions). In particular, local mounting that would not be present under normal installation conditions should be avoided in the test set-up. In simulating in-service interface features, the following details should be considered as a minimum: 1. Test orientation of the test object should be its in-service orientation, particularly with respect to the direction of gravity (vertical), available DoF, and mounting locations. 2. Mounting details at the interface of the test object and the mounting ﬁxture should represent inservice conditions with respect to the number, size, and strength of welds, bolts, nuts, and other hold-down hardware. 3. Additional interface linkages, including wires cables, conduits, pipes, instrumentation (dials, meters, gauges, sensors, transducers, and so on), and the supporting brackets of these elements, should be simulated at least in terms of mass and stiffness, and preferably in terms of size as well. 4. Any dynamic effects of adjacent equipment cabinets and supporting structures under in-service conditions should be simulated or taken into account in analysis. 5. Operating loads, such as those resulting from ﬂuid ﬂow, pressure forces, and thermal effects, should be simulated if they appear to signiﬁcantly affect test object dynamics. In particular, the nozzle loads (ﬂuid) should be simulated in magnitude, direction, and location.

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Vibration Monitoring, Testing, and Instrumentation

The required mechanical interface details of the test object are obtained by the test laboratory at the information-acquisition stage. Any critical interface details that are simulated during testing should be included in the test report. At least three control accelerometers should be attached to the shaker table near the mounting location of the test object. One control accelerometer measures the excitation-acceleration component applied to the test object in the vertical direction. The other two measure the excitation-acceleration components in two horizontal directions at right angles. The two horizontal (control) directions are chosen to be along the two major freedom-of-motion directions (or dynamic principal axes) of the test object. Engineering judgment may be used in deciding these principal directions of high response in the test object. Often, geometric principal axes are used. The control accelerometer signals are passed through a response-spectrum analyzer (or a suitably programmed digital computer) to compute the TRS in the vertical and two horizontal directions that are perpendicular. Vibration tests generally require monitoring of the dynamic response at several critical locations of the test object. In addition, the tests may call for the determining of mode shapes and natural frequencies of the test object. For this purpose, a sufﬁcient number of accelerometers should be attached to various key locations in the test object. The test procedure (document) should contain a sketch of the test object, indicating the accelerometer locations. Also, the type of accelerometers employed, their magnitudes and directions of sensitivity, and the tolerances should be included in the ﬁnal test report.

3.4.8

Test-Input Considerations

In vibration testing, a signiﬁcant effort goes into the development of test excitation inputs. Not only the nature but also the number and the directions of the excitations can have a signiﬁcant effect on the outcomes of a test. This is so because the excitation characteristics determine the nature of a test. 3.4.8.1

Test Nomenclature

We have noted that a common practice in vibration testing is to apply synthesized vibration excitation to a test object that is appropriately mounted on a shaker table. Customarily, only translatory excitations as generated by linear actuators, are employed. Nevertheless the resulting motion of the test object usually consists of rotational components as well. A typical vibration environment may consist of threedimensional motions, however. The speciﬁcation of a three-dimensional test environment is a complex task, even after omitting the rotational motions at the mounting locations of the test object. Furthermore, practical vibration environments are random and they can be represented with sufﬁcient accuracy only in a probabilistic sense. Very often the type of testing that is used is governed mainly by the capabilities of the test laboratory to which the contract is granted. Test laboratories conduct tests using their previous experience and engineering judgment. Making extensive improvements to existing tests can be very costly and timeconsuming, and this is not warranted from the point of view of the customer or the vendor. Regulatory agencies usually allow simpler tests if sufﬁcient justiﬁcation can be provided indicating that a particular test is conservative with respect to regulatory requirements. The complexity of a shaker-table apparatus is governed primarily by the number of actuators that are employed and the number of independent directions of simultaneous excitation that it is capable of producing. Terminology for various tests is based on the number of independent directions of excitation used in the test. It would be advantageous to standardize this terminology to be able to compare different test procedures. Unfortunately, the terminology used to denote different types of tests usually depends on the particular test laboratory and the speciﬁc application. Attempts to standardize various test methods have become tedious, partly because of the lack of a universal nomenclature for dynamic testing. A justiﬁable grouping of test conﬁgurations is presented in this section. Figure 3.23 illustrates the various test types.

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z

z

y

z z

z

y

y

z

Rectilinear Biaxial

z

y

x Two DoF Triaxial

y x

Rectilinear Triaxial FIGURE 3.23

x Two DoF Biaxial

y x

x Principal Axes

y

x Rectilinear Uniaxial

x Three DoF

Vibration-test conﬁgurations.

In test nomenclature, the DoF refers to the number of directions of independent motions that can be generated simultaneously by means of independent actuators in the shaker table. According to this concept, three basic types of tests can be identiﬁed: 1. Single-DoF (or rectilinear) testing is that in which the shaker table employs only one exciter (actuator), producing test-table motions along the axis of that actuator. The actuator may not necessarily be in the vertical direction. 2. Two-DoF testing is that in which two independent actuators, oriented at right angles to each other, are employed. The most common conﬁguration consists of a vertical actuator and a horizontal actuator. Theoretically, the motion of each actuator can be speciﬁed independently. 3. Three-DoF testing is that in which three actuators, oriented at mutually right angles, are employed. A desirable conﬁguration consists of a vertical actuator and two horizontal actuators. At least theoretically, the motion of each actuator can be speciﬁed independently. It is common practice to specify the directions of excitation with respect to the geometric principal axes of the test object. This practice is somewhat questionable, primarily because it does not take into account the ﬂexibility and inertia distributions of the object. Flexibility and inertia elements in the test object have a signiﬁcant inﬂuence on the level of dynamic coupling present in a given pair of directions. In this respect, it is more appropriate to consider dynamic principal axes rather than geometric principal axes of the test object. One useful deﬁnition is in terms of eigenvectors of an appropriate three-dimensional, frequency-response function matrix that takes into account the response at every critical location in the test object. The only difﬁculty in this method is that prior frequency-response testing or analysis is needed to determine the test input direction. For practical purposes, the vertical axis (the direction of gravity) is taken as one principal axis. The single-DoF (rectilinear) test conﬁguration has three subdivisions, based on the orientation of the vibration exciter (actuator) with respect to the principal axes of the test object. It is assumed that one principal axis of the test object is the vertical axis and that the three principal axes are

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Vibration Monitoring, Testing, and Instrumentation

mutually perpendicular. The three subdivisions are as follows: 1. Rectilinear uniaxial testing, in which the single actuator is oriented along one of the principal axes of the test object 2. Rectilinear biaxial testing, in which the single actuator is oriented on the principal plane containing the vertical and one of the two horizontal principal axes (the actuator is inclined to both principal axes in the principal plane) 3. Rectilinear triaxial testing, in which the single actuator is inclined to all three orthogonal principal axes of the test object The two-DoF test conﬁguration has two subdivisions, based on the orientation of the two actuators with respect to the principal axes of the test object, as follows: 1. Two-DoF biaxial testing, in which one actuator is directed along the vertical principal axis and the other along one of the two horizontal principal axes of the test object 2. Two-DoF triaxial testing, in which one actuator is positioned along the vertical principal axis and the other actuator is horizontal but inclined to both horizontal principal axes of the test object 3.4.8.2

Testing with Uncorrelated Excitations

Simultaneous excitations in three orthogonal directions often produce responses (accelerations, stresses, etc.) that are very different from that which is obtained by vectorially summing the responses to separate excitations acting one at a time. This is primarily because of the nonlinear, time-variant nature of test specimens and test apparatus, their dynamic coupling, and the randomness of excitation signals. If these effects are signiﬁcant, it is theoretically impossible to replace a three-DoF test, for example, with a sequence of three single-DoF tests. In practice, however, some conservatism can be incorporated into two-DoF and single-DoF tests to account for these effects. These tests with added conservatism may be employed when three-DoF testing is not feasible. It should be clear by now that rectilinear triaxial testing is generally not equivalent to three-degree-freedom testing, because the former merely applies an identical excitation in all three orthogonal directions, with scaling factors (direction cosines). One obvious drawback of rectilinear triaxial testing is that the input excitation in a direction at right angles to the actuator is theoretically zero, and the excitation is at its maximum along the actuator. In three-DoF testing using uncorrelated random excitations, however, no single direction has a zero excitation at all times, and also the probability is zero that the maximum excitation occurs in a ﬁxed direction at all times. Three-DoF testing is mentioned infrequently in the literature on vibration testing. A major reason for the lack of three-DoF testing might be the practical difﬁculty in building test tables that can generate truly uncorrelated input motions in three orthogonal directions. The actuator interactions caused by dynamic coupling through the test table and mechanical constraints at the table supports are primarily responsible for this. Another difﬁculty arises because it is virtually impossible to synthesize perfectly uncorrelated random signals to drive the actuators. Two-DoF testing is more common. In this case, the test must be repeated for a different orientation of the test object (for example, with a 908 rotation about the vertical axis), unless some form of dynamic-axial symmetry is present in the test object. Test programs frequently specify uncorrelated excitations in two-DoF testing for the two actuators. This requirement lacks solid justiﬁcation, because two uncorrelated excitations applied at right angles do not necessarily produce uncorrelated components in a different pair of orthogonal directions, unless the mean square values of the two excitations are equal. To demonstrate this, consider the two uncorrelated excitations, u and v; shown Figure 3.24. The components u0 and v0 ; in a different pair of orthogonal directions obtained by rotating the original coordinates through an angle u in the counterclockwise direction, are given by u0 ¼ u cos u þ v sin u

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ð3:84Þ

Vibration Testing

3-49

v 0 ¼ 2u sin u þ v cos u

ð3:85Þ

Without loss of generality, we can assume that u and v have zero means. Then, u 0 and v 0 also will have zero means. Furthermore, since u and v are uncorrelated, we have EðuvÞ ¼ EðuÞEðvÞ ¼ 0

v

v′

ð3:86Þ

From Equation 3.84 and Equation 3.85, we obtain Eðu 0 v 0 Þ ¼ E½ðu cos u þ v sin uÞ ð2u sin u þ v cos uÞ

u′

θ θ

u

This, when expanded and substituted with Equation 3.86, becomes

FIGURE 3.24 correlation.

Effect of coordinate transformation on

Eðu0 v0 Þ ¼ sin u cos u½Eðv2 Þ 2 Eðu2 Þ 0

ð3:87Þ

0

Since u is any general angle, the excitation components u and v become uncorrelated if and only if Eðv2 Þ ¼ Eðu2 Þ

ð3:88Þ

This is the required result. Nevertheless, a considerable effort, in the form of digital Fourier analysis, is expended by vibration-testing laboratories to determine the degree of correlation in test signals employed in two-DoF testing. 3.4.8.3

Symmetrical Rectilinear Testing

Single-DoF (rectilinear) testing that is performed with the test excitation applied along the line of symmetry with respect to an orthogonal system of three principal axes of the test object mainframe is termed symmetrical rectilinear testing. In product qualiﬁcation literature,pthis ﬃﬃ test pﬃﬃis often pﬃﬃ referred to as the 458 test. The direction cosines of the input orientation are ð1= 3; 1= 3 ; 1 3Þ for this test pﬃﬃ conﬁguration. The single-actuator input intensity is ampliﬁed by a factor of 3 in order to obtain the required excitation intensity in the three principal directions. Note that symmetrical rectilinear testing falls into the category of rectilinear triaxial testing, as deﬁned earlier. This is one of the widely used testing conﬁgurations in seismic qualiﬁcation, for example. 3.4.8.4

Geometry vs. Dynamics

In vibration testing the emphasis is on the dynamic behavior rather than the geometry of the equipment. For a simple three-dimensional body that has homogeneous and isotropic characteristics, it is not difﬁcult to correlate its geometry to its dynamics. A symmetrical rectilinear test makes sense for such systems. The equipment we come across is often much more complex, however. Furthermore, our interest is not merely in determining the dynamics of the mainframe of the equipment. We are more interested in the dynamic reliability of various critical components located within the mainframe. Unless we have some previous knowledge of the dynamic characteristics in various directions of the system components, it is not possible to draw a direct correlation between the geometry and the dynamics of the tested equipment. 3.4.8.5

Some Limitations

In a typical symmetrical rectilinear test, we deal with “black-box” equipment whose dynamics are completely unknown. The excitation is applied along the line of symmetry of the principal axes of the mainframe. A single test of this type does not guarantee excitation of all critical components located inside the equipment. Figure 3.25 illustrates this further. Consider the plane perpendicular to the direction of excitation. The dynamic effect caused by the excitation is minimal along any line on

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this plane. (Any dynamic effect on this plane is Perpendicular caused by dynamic coupling among different body Plane axes.) Accordingly, if there is a component (or several components) inside the equipment whose direction of sensitivity lies on this perpendicular plane, the single excitation might not excite that component. Since we deal with a black box, we do not know the equipment dynamics beforehand. Hence, there is no way of identifying the Direction of existence of such unexcited components. When Excitation the equipment is put into service, a vibration of sufﬁcient intensity may easily overstress this FIGURE 3.25 Illustration of the limitation of a single component along its direction of sensitivity and rectilinear test. may bring about component failure. It is apparent P that at least three tests, performed in three orthogonal directions, are necessary to guarantee A excitation of all components, regardless of their 45° direction of sensitivity. A second example is given in Figure 3.26. O O Consider a dual-arm component with one arm sensitive in the O – O direction and the second arm sensitive in the P – P direction. If component failure occurs when the two arms are in contact, a single excitation in either the O – O direction or the P – P direction will not bring about component failure. If the component is located inside a black box, such that either the O – O direction or the A P – P direction is very close to the line of symmetry of the principal axes of the mainframe, a single P symmetrical rectilinear test will not result in system malfunction. This may be true, because FIGURE 3.26 Illustrative example of the limitation of we do not have a knowledge of component several rectilinear tests. dynamics in such cases. Again, under service conditions, a vibration of sufﬁcient intensity can produce an excitation along the A – A direction, subsequently causing system malfunction. A further consideration in using rectilinear testing is dynamic coupling between the directions of excitation. In the presence of dynamic coupling, the sum of individual responses of the test object resulting from four symmetrical rectilinear tests is not equal to the response obtained when the excitations are applied simultaneously in the four directions. Some conservatism should be introduced when employing rectilinear testing for objects having a high level of dynamic coupling between the test directions. If the test-object dynamics are restrained to only one direction under normal operating conditions, however, then rectilinear testing can be used without applying any conservatism. 3.4.8.6

Testing Black Boxes

When the equipment dynamics are unknown, a single rectilinear test does not guarantee proper testing of the equipment. To ensure excitation of every component within the test object that has directional intensities, three tests should be carried out along three independent directions. The ﬁrst test may be carried out with a single horizontal excitation, for example. The second test could then be performed with the equipment rotated through 908 about its vertical axis, and using the same horizontal excitation. The last test would be performed with a vertical excitation.

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Vibration Testing

Alternatively, if symmetrical rectilinear tests are preferred, four such tests should be performed for four equipment orientations (for example, an original test, a 908 rotation, a 1808 rotation, and a 2708 rotation about the vertical axis). These tests also ensure excitation of all components that have directional intensities. This procedure might not be very efﬁcient, however. The shortcoming of this series of four tests is that some of the components will be overtested. It is clear from Figure 3.27, for example, that the vertical direction is excited by all four tests. The method has the advantage, however, of simplicity of performance. 3.4.8.7

3-51

2

3

1

4

FIGURE 3.27 Directions of excitation in a sequence of four rectilinear tests.

Phasing of Excitations

D The main purpose of rotating the test orientation in rectilinear testing is to ensure that all comv ponents within the equipment are excited. Phasing of different excitations also plays an important role, however, when several excitations are used simultaneously. To explore this concept further, it u A B –u should be noted that a random input applied in the A – B direction or in the B – A direction has the –v same frequency and amplitude (spectral) characC teristics. This is clear because the PSD of u ¼ PSD of ð2uÞ; and the autocorrelation of u ¼ FIGURE 3.28 Signiﬁcance of excitation phasing in autocorrelation of ð2uÞ: Hence, it is seen that, if two-DoF testing. the test is performed along the A – B direction, it is of no use to repeat the test in the B – A direction. It should be understood, however, that the situation is different when several excitations are applied simultaneously. The simultaneous action of u and v is not the same as the simultaneous action of 2u and v (see Figure 3.28). The simultaneous action of u and v is the same, however, as the simultaneous action of 2u and 2v: Obviously, this type of situation does not arise when there are no simultaneous excitations, as in rectilinear testing.

3.4.8.8

Testing a Gray or White Box

When some information regarding the true dynamics of the test object is available, it is possible to reduce the number of necessary tests. In particular, if the equipment dynamics are completely known, then a single test would be adequate. The best direction for excitation of the system in Figure 3.26, for example, is A – A: (Note that A – A may be lined up in any arbitrary direction inside the equipment housing. In such a situation, knowledge of the equipment dynamics is crucial.) This also indicates that it is very important to accumulate and use any past experience and data on the dynamic behavior of similar equipment. Any test that does not use some previously known information regarding the equipment is a blind test, and it cannot be optimal in any respect. As more information is available, better tests can be conducted. 3.4.8.9

Overtesting in Multitest Sequences

It is well known that increasing the test duration increases aging of the test object because of prolonged stressing and load cycling of various components. This is the case when a test is repeated one or more times at the same intensity as that prescribed for a single test. The symmetrical rectilinear test requires four separate tests at the same excitation intensity as that prescribed for a single test. As a result, the

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equipment becomes subjected to overtesting, at least in certain directions. The degree of overtesting is small if the tests are performed in only three orthogonal directions. In any event, a certain amount of dynamic coupling is present in the test-object’s structure and, to minimize overtesting in these sequential tests, a smaller intensity than that prescribed for a single test should be employed. The value of the intensity-reduction factor clearly depends on the characteristics of the test object, the degree of reliability expected, and the intensity value itself. More research is necessary to develop expressions for intensityreduction factors for various test objects.

3.5

Some Practical Information

Some useful practical information on vibration testing of products is given here. TABLE 3.3

Random Vibration Tests for a Product Development Application

Vibration Test

Root-Mean-Square Value of Excitation (g)

Peak Value of the Excitation PSD (g2/Hz)

Minimum Times the Random Vibration is Applied

Minimum Duration of Vibration (min)

A

2.7

0.01

1

60

B

6.0

0.05

2

30

C D E F

3.2 5.8 4.9 6.3

0.01 0.02 0.01 0.04

1 1 1 2

15 15 15 5

TABLE 3.4

Vibration Axes

Major horizontal axis Major horizontal axis All three All three All three All three

Capabilities of Five Commercial Control Systems for Vibration-Test Shakers

System

A

B

C

D

E

Random test Sine test Transient and shock tests Hydraulic shaker Preprogrammed test set-ups Amplitude scheduling

Yes Yes Yes

Yes Yes Yes

Yes Optional Optional

Yes No No

Yes Yes Yes

OK Max. 63

OK Max. 25

OK Max. 99

OK 10 per disk

OK Not given

32 Levels and duration

10 Levels over 60 dB

Yes

Yes

0.5 dB steps; can pick no. of steps and rate No

No

On-line reference modiﬁcation Use of measured spectra as reference Transmissibility

Min. start: 225 dB; min. step: 0.25 dB; can pick step durations No

Coherence

Yes

Correlation Shock response spectrum Sine on random Random on random

No

Yes

Yes (measurement– pass feature)

Yes

No

No

Yes

Yes

No

Yes

No

No

Yes

Yes Yes

Measurement option Measurement option No Yes

No Optional

No No

Yes Yes

Yes Yes

Sine bursts No

Optional Optional

No No

No No

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TABLE 3.5

Important Hardware Characteristics of Five Systems for Shaker Control

System Reference spectrum break points Spectrum resolution (number of spectral lines) Nature or random drive signal Measured signal averaging Operator interface Output devices Memory Mass storage Number of measurements (control input) channels Number of controller output channels

A

B

C

D

E

40

32

50

10

45

Can pick 100, 200, 400, 600, 800 lines

Can pick 64, 128, 256, 512 lines (optional 1024 lines)

Can pick 100, 200, 400, 800 lines

200 Lines 10 Hz spacing

Not given

Gaussian, periodic pseudo-random Arithmetic peak-hold

Gaussian

Gaussian

Pick any number: 10 to 1000 lines (optional 2048 lines) Pseudo-random

True power

Peak pick

No

Keyboard, push button, dialog, set-up Standard or graphics terminal, X – Y record printer, digital plot 64K One ﬂoppy drive, 256K

Keyboard, push button, dialog, menu-driven Graphics terminal, video hard copier, digital plotter, X – Y recorder 128K Hard þ ﬂoppy 8, 20, 30B

Keyboard 10 soft keys, dialog Like IBM PC, Monochrome-900 Epson printer

Keyboard Graphics terminal printer, hard copy, X – Y plotter

2 standard; 4 optional, multiplexer optional

1 standard; 16, 31 optional

64K Two ﬂoppy drives 360K each 1 standard; 4 optional

32K std, 64K option Two ﬂoppy drives 256K each Not given

One

One

One

One

RMS, peak-hold Keyboard, menudriven RS 232 CRT screen, hard copy, video print, digital plot 128K Floppy drive 0.5 MB; hard drive 10 MB 2 standard; 16 optional

One

3-53

© 2007 by Taylor & Francis Group, LLC

3-54

Speciﬁcations of Five Shaker Control Systems

System Accelerometer signal (controller input) Controller output signal Input frequency ranges

Control loop time Equalization time for 10 dB range Resolution Dynamic range Control accuracy

Sine sweep rate a

A ^125 mV to ^8 V full scale 2.4 V RMS (random) 20 V peak to peak (sine and random) Random: DC to 200, 500 Hz, 1, 2, 3, 4, 8 kHz;. Sine: 1–8 Hz; shock: 10–125 Hz, 312 Hz,…, 5kHz 2.1 sec (2 kHz, 200 lines) Within ^3 dB in two loops 12 bit 65 dB ^1 dB at Q ¼ 30; ^2 dB at Q ¼ 50 (100 Hz Resonance at 1 octave/min OK

RMS, root-mean-square.

© 2007 by Taylor & Francis Group, LLC

B a

10 mV RMS to ^8 V max. typically .500 mV RMS 20 V P – P max. 50 mA max.

C Max. 10 V peak, 3.5 V, RMS

D

E

20 V P – P max.

1 to 1000 mV/g user picked 10 V peak 3 V RMS

Not given Not given

Seven ranges max. freq.: 500 Hz– 5 kHz, min. freq ¼ 1 line

100, 500 Hz, 1, 2, 4, 5, 10 kHz

10–2000 Hz

10 –5000 Hz

0.3 sec, 64 lines; 0.9 sec, 256 lines, 3 sec, 1024 lines (2kHz) 2 or 3 loops

4 sec, 100 lines, 8 sec, 200 lines (2 kHz) Within ^1 dB in one loop

2 sec Within ^1 dB in 6 sec

2.5 sec for 256 lines at 2 kHz Not given

12 bit 65 dB ^1 dB (at 90% conﬁdence)

72 dB ^1dB over 72 dB

60 dB ^1 dB

12 bit — ^1 dB (at 95% conﬁdence)

0.1– 100 oct/min (log) 1 Hz–100 kHz/min (linear)

0.1–100 oct/min max.; 0.1 Hz–6 kHz/min

N/A

0.001 –10 oct/min; 1–6000 Hz/min

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TABLE 3.6

Vibration Testing

3.5.1

3-55

Random Vibration Test Example

Table 3.3 lists several random vibration tests in the frequency range of 0 to 500 Hz in an application related to product development.

3.5.2

Vibration Shakers and Control Systems

Table 3.4 lists capabilities of ﬁve commercial control systems that may be used for shaker control in random vibration testing of products. Table 3.5 summarizes important hardware characteristics of the ﬁve control systems. Table 3.6 gives some important speciﬁcations of the ﬁve control systems.

Bibliography Broch, J.T. 1980. Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum, Denmark. Buzdugan, G., Mihaiescu, E., and Rades, M. 1986. Vibration Measurement, Martinus Nijhoff Publishers, Dordrecht, The Netherlands. de Silva, C.W. and Palusamy, S.S., Experimental modal analysis—a modeling and design tool, Mech. Eng., ASME, 106, 56 –65, 1984. de Silva, C.W. 1983. Dynamic Testing and Seismic Qualiﬁcation Practice, D.C. Heath and Co., Lexington, MA. de Silva, C.W., A dynamic test procedure for improving seismic qualiﬁcation guidelines, J. Dyn. Syst. Meas. Control, Trans. ASME, 106, 143 –148, 1984. de Silva, C.W., Hardware and software selection for experimental modal analysis, Shock Vib. Digest 16, 3– 10, 1984. de Silva, C.W., Matrix eigenvalue problem of multiple-shaker testing, J. Eng. Mech. Div., Trans. ASCE, 108, EM2, 457 –461, 1982. de Silva, C.W., Optimal input design for the dynamic testing of mechanical systems, J. Dyn. Syst. Meas. Control, Trans. ASME, 109, 111–119, 1987. de Silva, C.W., Seismic qualiﬁcation of electrical equipment using a uniaxial test, Earthquake Eng. Struct. Dyn., 8, 337–348, 1980. de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib. Digest, 18, 3– 10, 1986. de Silva, C.W., Sensory information acquisition for monitoring and control of intelligent mechatronic systems, Int. J. Inf. Acquisit., 1, 1, 89 –99, 2004. de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control—application in the distribution qualiﬁcation of microcomputers, Shock Vib. Digest, 18, 3– 13, 1986. de Silva, C.W., Loceff, F., and Vashi, K.M., Consideration of an optimal procedure for testing the operability of equipment under seismic disturbances, Shock Vib. Bull., 50, 149 –158, 1980. de Silva, C.W. 2005. MECHATRONICS—An Integrated Approach, Taylor & Franciss, CRC Press, Boca Raton, FL. de Silva, C.W., Singh, M., and Zaldonis, J., Improvement of response spectrum speciﬁcations in dynamic testing, J Eng. Industry, Trans. ASME, 112, 4, 384 –387, 1990. de Silva, C.W. 2006. VIBRATION—Fundamentals and Practice, Taylor & Franciss, CRC Press, Boca Raton, FL. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd, Letchworth, U.K. McConnell, K.G. 1995. Vibration Testing, Wiley, New York. Meirovitch, L. 1980. Computational Methods in Structural Dynamics, Sijthoff & Noordhoff, Rockville, MD. Randall, R.B. 1977. Application of B&K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark.

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4

Experimental Modal Analysis 4.1 4.2 4.3 4.4

Clarence W. de Silva The University of British Columbia

Introduction .......................................................................... 4-1 Frequency-Domain Formulation ......................................... 4-2 Transfer-Function Matrix

†

Principle of Reciprocity

Experimental Model Development ..................................... 4-8 Extraction of the Time-Domain Model

Curve Fitting of Transfer Functions .................................... 4-10

Problem Identiﬁcation † Single- and Multi-Degree-of-Freedom Techniques † Single-Degree-of-Freedom Parameter Extraction in the Frequency Domain † Multi-Degree of Freedom Curve Fitting † A Comment on Static Modes and Rigid-Body Modes † Residue Extraction

4.5

Laboratory Experiments ....................................................... 4-18

4.6

Commercial EMA Systems .................................................. 4-24

Lumped-Parameter System

†

Distributed-Parameter System

System Conﬁguration

Summary In experimental modal analysis (EMA), ﬁrst the modal information (natural frequencies, modal damping ratios, and mode shapes) of a test object is determined through experimentation, and this information is then used to determine a model for the test object. Once an “experimental model” is obtained in this manner, it may be used in a variety of practical uses including system analysis, fault detection and diagnosis, design, and control. This chapter presents some standard techniques and procedures associated with EMA.

4.1

Introduction

Experimental modal analysis (EMA) is basically a procedure of “experimental modeling.” The primary purpose here is to develop a dynamic model for a mechanical system, using experimental data. In this sense, EMA is similar to “model identiﬁcation” in control system practice, and may utilize somewhat related techniques of “parameter estimation.” It is the nature of the developed model, which may distinguish EMA from other conventional procedures of model identiﬁcation. Speciﬁcally, EMA produces a modal model as the primary result, which consists of: 1. Natural frequencies 2. Modal damping ratios 3. Mode shape vectors Once a modal model is known, standard results of modal analysis may be used to extract an inertia (mass) matrix, a damping matrix, and a stiffness matrix, which constitute a complete dynamic model for the experimental system, in the time domain. Since EMA produces a modal model (and in some cases a complete time-domain dynamic model) for a mechanical system from test data of the system, its uses can be 4-1 © 2007 by Taylor & Francis Group, LLC

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extensive. In particular, EMA is useful in mechanical systems, primarily with regard to vibration, in: 1. Design 2. Diagnosis 3. Control In the area of design, the following three approaches that utilize EMA should be mentioned: 1. Component modiﬁcation 2. Modal response speciﬁcation 3. Substructuring In component modiﬁcation, we modify (i.e., add, remove, or vary) inertia (mass), stiffness, and damping parameters in a mechanical system and determine the resulting effect on the modal response (natural frequencies, damping ratios, and mode shapes) of the system. In modal response speciﬁcation, we establish the best changes, from the design point of view, in system parameters (inertia, stiffness, and damping values and their degrees of freedom (DoF), in order to give a “speciﬁed” (prescribed) change in the modal response. In substructuring, two or more subsystem models are combined using dynamic interfacing components, and the overall model is determined. Some of the subsystem models used in this manner can be of analytical origin (e.g., ﬁnite element models). Diagnosis of problems (faults, performance degradation, component deterioration, impending failure, etc.) of a mechanical system requires condition monitoring of the system, and analysis and evaluation of the monitored information. Often, analysis involves extraction of modal parameters using monitored data. Diagnosis may involve the establishment of changes (both gradual and sudden), patterns, and trends in these system parameters. Control of a mechanical system may be based on modal analysis. Standard and well-developed techniques of modal control are widely used in mechanical system practice. In particular, vibration control, both active and passive, can use modal control. In this approach, the system is ﬁrst expressed as a modal model, then control excitations, parameter adaptations, and so on are established that result in a speciﬁed (derived) behavior in various modes of the system. Of course, techniques of EMA are commonly used here, both in obtaining a modal model from test data and in establishing modal excitations and parameter changes that are needed to realize a prescribed behavior in the system. The standard steps of EMA are as follows: 1. Obtain a suitable (admissible) set of test data, consisting of forcing excitations and motion responses for various pairs of DoF of the test object. 2. Compute the frequency transfer functions (the frequency response functions) of the pairs of test data, using Fourier analysis. Digital Fourier analysis using Fast Fourier Transform (FFT) is the standard way of accomplishing this. Either software-based (computer) equipment or hardwarebased instrumentation may be used. 3. Curve ﬁt analytical transfer functions to the computed transfer functions. Determine natural frequencies, damping ratios, and residues for various modes in each transfer function. 4. Compute mode shape vectors. 5. Compute inertia (mass) matrix M, stiffness matrix K, and damping matrix C. Some variations of these steps is possible in practice, and Step 5 is omitted in some situations. In the present chapter, we will study some standard techniques and procedures associated with the process of EMA.

4.2

Frequency-Domain Formulation

Frequency-domain analysis of vibrating systems is very useful in a wide variety of applications. The analytical convenience of frequency-domain methods results from the fact that differential equations in the time domain become algebraic equations in the frequency domain. Once the

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4-3

necessary analysis is performed in the frequency domain, it is often possible to interpret the results without having to transform them back to the time domain through inverse Fourier transformation. In the context of the present chapter, frequency-domain representation is particularly important because it is the frequency transfer functions that are used for extracting the necessary modal parameters. For the convenience of notation, we shall develop the frequency-domain results using the Laplace variable, s: As usual, the straightforward substitution of s ¼ jv or s ¼ j2pf gives the corresponding frequency-domain results.

4.2.1

Transfer-Function Matrix

Let us consider a linear mechanical system that is represented by M€y þ C_y þ Ky ¼ fðtÞ

ð4:1Þ

where f(t) ¼ forcing excitation vector (nth order column) y ¼ displacement response vector (nth order column) M ¼ mass (inertia) matrix ðn £ nÞ C ¼ damping (linear viscous) matrix ðn £ nÞ K ¼ stiffness matrix ðn £ nÞ If the assumption of proportional damping is made, the coordinate transformation y ¼ Cq

ð4:2Þ

decouples Equation 4.1 into the canonical form of modal equations Mq€ þ C_q þ Kq ¼ CT fðtÞ

ð4:3Þ

where C ¼ modal matrix ðn £ nÞ of n independent modal vector vectors ½c1 ; c2 ; …; cn ¯ ¼ diagonal matrix of modal masses Mi M ¯ ¼ diagonal matrix of modal damping constants Ci C ¯ ¼ diagonal matrix of modal stiffnesses Ki K Speciﬁcally, we have M ¼ CT MC

ð4:4Þ

C ¼ CT CC

ð4:5Þ

K ¼ CT KC

ð4:6Þ

If the modal vectors are assumed to be M-normal, then we have Mi ¼ 1 Ki ¼ v2i and furthermore, we can express Ci in the convenient form Ci ¼ 2zi vi where

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By Laplace transformation of the response canonical equations of modal motion (Equation 4.3), assuming zero initial conditions, we obtain 2 2 3 s þ 2zv1 s þ v21 0 6 7 6 7 s2 þ 2zv2 s þ v22 6 7 6 7 ð4:7Þ 6 7QðsÞ ¼ CT FðsÞ . 6 7 . 6 7 . 4 5 2 2 0 s þ 2zvn s þ vn Laplace transforms of the modal response (or generalized coordinate) vector, qðtÞ; and the forcing excitation vector, fðtÞ; are denoted by the column vectors, QðsÞ and FðsÞ; respectively. The square matrix on the left-hand side of Equation 4.7 is a diagonal matrix. Its inverse is obtained by inverting the diagonal elements. Consequently, the following modal transfer relation results: 2 3 G1 0 6 7 6 7 G2 6 7 6 7 T QðsÞ ¼ 6 ð4:8Þ 7C FðsÞ . 6 7 . 6 7 . 4 5 0 Gn in which the diagonal elements are the damped simple-oscillator transfer functions Gi ðsÞ ¼

1 s2 þ 2zi vi s þ v2i

for i ¼ 1; 2; …; n

ð4:9Þ

Note that vi ; the ith undamped natural frequency (in the time domain), is only approximately equal to the frequency of the ith resonance of the transfer function (in the frequency domain), as given by qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð4:10Þ vri ¼ 1 2 2z2i vi As we have discussed before, and as is clear from Equation 4.10, the approximation improves for decreasing modal damping. Consequently, in most applications of EMA, the resonant frequency is taken to be equal (approximately) to the natural frequency for a given mode. From the time-domain coordinate transformation (Equation 4.2), the Laplace domain coordinate transformation relation is obtained as YðsÞ ¼ CQðsÞ Substitute Equation 4.8 into Equation 4.11; thus 2 G1 6 6 G2 6 6 YðsÞ ¼ C6 6 6 4 0

ð4:11Þ 0

..

.

3 7 7 7 7 T 7C FðsÞ 7 7 5

ð4:12Þ

Gn

Equation 4.12 is the excitation –response (input –output) transfer relation. It is clear that the n £ n transfer function matrix, G, for the n-DoF system is given by 2 3 G1 0 6 7 6 7 G2 6 7 6 7 T GðsÞ ¼ C6 ð4:13Þ 7C .. 6 7 6 7 . 4 5 0 Gn

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Notice in particular that GðsÞ is a symmetric matrix; speciﬁcally GT ðsÞ ¼ GðsÞ

ð4:14Þ T

T T

T

which should be clear from the matrix transposition property, ðABCÞ ¼ C B A : An alternative version of Equation 4.13 that is extensively used in EMA can be obtained by using the partitioned form (or assembled form) of the modal matrix in Equation 4.13. Speciﬁcally, we have 2 32 T 3 G1 0 c1 6 76 7 6 7 6 T7 G2 6 76 c2 7 6 76 7 GðsÞ ¼ ½c1 ; c2 ; …; cn 6 ð4:15Þ 76 . 7 .. 6 76 . 7 6 7 6 7 . . 4 54 5 0 Gn cTn On multiplying out the last two matrices on the right-hand side of Equation 4.15 term by term, the following intermediate result is obtained: 2 3 G1 cT1 6 7 6 G cT 7 6 2 27 6 7 GðsÞ ¼ ½c1 ; c2 ; …; cn 6 . 7 6 . 7 6 . 7 4 5 T Gn cn Note that Gi are scalars while ci are column vectors. The two matrices in this product can be multiplied out now to obtain the matrix sum GðsÞ ¼ G1 c1 cT1 þ G2 c2 cT2 þ · · · þ Gn cn cTn ¼

n X r¼1

Gr cr cTr

ð4:16Þ

in which cr is the rth modal vector that is normalized with respect to the mass matrix. Notice that each term cr cTr in the summation (Equation 4.16) is an n £ n matrix with the element corresponding to its ith row and kth column being ðci ck Þr : The ikth element of the transfer matrix GðsÞ is the transfer function Gik ðsÞ; which determines the transfer characteristics between the response location, i; and the excitation location, k: From Equation 4.16, this is given by Gik ðsÞ ¼

n X r¼1

Gr ðci ck Þr ¼

n X r¼1

ðci ck Þr s2 þ 2zr vr s þ v2r

ð4:17Þ

with s ¼ jv ¼ j2pf in the frequency domain. Note that ðci Þr is the ith element of the rth modal vector, and is a scalar quantity. Similarly, ðci ck Þr is the product of the ith element and the kth element of the rth modal vector, and is also a scalar quantity. This is the numerator of each modal transfer function within the right-hand side summation of Equation 4.17, and is the residue of the pole (eigenvalue) of that mode. Equation 4.17 is useful in EMA. Essentially, we start by determining the residues ðci ck Þr of the poles in an admissible set of measured transfer functions. We can determine the modal vectors in this manner. In addition, by analyzing the measured transfer functions, the modal damping ratios, zi ; and the natural frequencies, vi ; can be estimated. From these results, an estimate for the time-domain model (i.e., the matrices M, K, and C) can be determined.

4.2.2

Principle of Reciprocity

By the symmetry of transfer matrix, as given by Equation 4.14, it follows that Gik ðsÞ ¼ Gki ðsÞ

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ð4:18Þ

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Vibration Monitoring, Testing and Instrumentation

This fact is further supported by Equation 4.17. This symmetry can be interpreted as Maxwell’s principle of reciprocity. To understand this further, consider the complete set of transfer relations given by Equation 4.12 and Equation 4.13: Y1 ðsÞ ¼ G11 ðsÞF1 ðsÞ þ G12 ðsÞF2 ðsÞ þ · · · þ G1n ðsÞFn ðsÞ Y2 ðsÞ ¼ G21 ðsÞF1 ðsÞ þ G22 ðsÞF2 ðsÞ þ · · · þ G2n ðsÞFn ðsÞ .. .

ð4:19Þ

Yn ðsÞ ¼ Gn1 ðsÞF1 ðsÞ þ Gn2 ðsÞF2 ðsÞ þ · · · þ Gnn ðsÞFn ðsÞ Note that the diagonal elements, G11 ; G22 ; …; Gnn ; are driving-point transfer functions (or autotransfer functions) and the rest are cross-transfer functions. Suppose that a single excitation, Fk ðsÞ; is applied at the kth DoF with all the other excitations set to zero. The resulting response at the ith DoF is given by Yi ðsÞ ¼ Gik ðsÞFk ðsÞ

ð4:20Þ

Similarly, when a single excitation, Fi ðsÞ; is applied at the ith DoF, the resulting response at the kth DoF is given by Yk ðsÞ ¼ Gki ðsÞFi ðsÞ

ð4:21Þ

In view of the symmetry that is indicated by Equation 4.18, it follows from Equation 4.20 and Equation 4.2.1 that if the two separate excitations, Fk ðsÞ and Fi ðsÞ; are identical then the corresponding responses, Yi ðsÞ and Yk ðsÞ; are also identical. In other words, the response at the ith DoF due to a single force at the kth DoF is equal to the response at the kth DoF when the same single force is applied at the ith DoF. This is the frequency-domain version of the principle of reciprocity.

Example 4.1 Consider the two-DoF system shown in Figure 4.1. Assume that the excitation forces, f1 ðtÞ and f2 ðtÞ; act at the y1 and y2 DoFs, respectively. The equations of motion are given by " # " # " # m 0 c 0 2k 2k y€ y_ þ y ¼ fðtÞ ðiÞ 0 m 0 c 2k 2k This system has proportional damping (speciﬁcally, it is clear that C is proportional to M) and hence possesses the same real modal vectors as does the undamped system. Let us ﬁrst obtain the transfer matrix in the direct manner. By taking the Laplace transform (with zero initial conditions) of the equations of motion (i), we have " 2 # ms þ cs þ 2k 2k YðsÞ ¼ FðsÞ ðiiÞ 2k ms2 þ cs þ 2k y2

c y1

k c m k FIGURE 4.1

f1(t)

m

f2(t)

k A vibrating system with proportional damping.

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Hence, in the relation YðsÞ ¼ GðsÞFðsÞ; the transfer matrix G is given by " 2 #21 ms þ cs þ 2k 2k GðsÞ ¼ 2k ms2 þ cs þ 2k " 2 # ms þ cs þ 2k k 1 ¼ ðms2 þ cs þ 2kÞ2 2 k2 k ms2 þ cs þ 2k

ðiiiÞ

The characteristic polynomial DðsÞ of the system is DðsÞ ¼ ðms2 þ cs þ 2kÞ2 2 k2 ¼ ðms2 þ cs þ kÞðms2 þ cs þ 3kÞ

ðivÞ

and is common to the denominator of all four transfer functions in the matrix. Speciﬁcally, G11 ðsÞ ¼ G22 ðsÞ ¼

ms2 þ cs þ 2k DðsÞ

ðvÞ

G12 ðsÞ ¼ G21 ðsÞ ¼

k DðsÞ

ðviÞ

This result implies that the characteristic equation characterizes the entire system (particularly, the natural frequencies and damping ratios) and, no matter what transfer function is measured (or analyzed), the same natural frequencies and modal damping are obtained. We can put these transfer functions into the partial fraction form. For example, ms2 þ cs þ 2k A1 s þ A2 A3 s þ A4 ¼ þ 2 2 2 ðms þ cs þ kÞðms þ cs þ 3kÞ ðms þ cs þ kÞ ðms2 þ cs þ 3kÞ

ðviiÞ

By comparing the numerator coefﬁcients, we ﬁnd that A1 ¼ A3 ¼ 0 (this is the case when the modes are real; with complex modes, A1 – 0 and A3 – 0 in general) and A2 ¼ A4 ¼ 1=2: These results are summarized below:

with

G11 ðsÞ ¼ G22 ðsÞ ¼

1=ð2mÞ 1=ð2mÞ þ 2 s2 þ 2z1 v1 s þ v21 s þ 2z2 v2 s þ v22

ðviiiÞ

G12 ðsÞ ¼ G21 ðsÞ ¼

1=ð2mÞ 1=ð2mÞ 2 2 s2 þ 2z1 v1 s þ v21 s þ 2z2 v2 s þ v22

ðixÞ

pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃ v1 ¼ k=m; v2 ¼ 3k=m; z1 ¼ c 4mk; and z2 ¼ c 12mk

By comparing the residues (numerators) of these expressions with the relation expressed in Equation 4.17, we can determine the M-normal modal vectors. Speciﬁcally, by examining G11 : ðc 21 Þ1 ¼

1 ; 2m

ðc 21 Þ2 ¼

1 2m

and by examining at G12 : ðc1 c2 Þ1 ¼

1 ; 2m

ðc1 c2 Þ2 ¼ 2

1 2m

We need consider only two admissible transfer functions (e.g., G11 and G12 ; or G11 and G21 ; or G12 and G22 ; or G21 and G22 ) in order to completely determine the modal vectors. Speciﬁcally, we obtain " # 2 pﬃﬃﬃﬃ 3 " # 2 pﬃﬃﬃﬃ 3 c1 c1 1= 2m 1= 2m ¼ 4 pﬃﬃﬃﬃ 5 and ¼4 pﬃﬃﬃﬃ 5 c2 1 c2 2 1= 2m 21= 2m Note that the modal masses are unity for these modal vectors. Also, there is an arbitrariness in the sign. As usual, we have overcome this problem by making the ﬁrst element of each modal vector positive.

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Vibration Monitoring, Testing and Instrumentation

4.3

Experimental Model Development

We have noted that the process of extracting modal data (natural frequencies, modal damping, and mode shapes) from measured excitation –response data is termed experimental modal analysis. Modal testing and the analysis of test data are the two main steps of EMA. Information obtained through EMA is useful in many applications, including the validation of analytical models for dynamic systems, fault diagnosis in machinery and equipment, in situ testing for requaliﬁcation to revised regulatory speciﬁcations, and design development of mechanical systems. In the present development, it is assumed that the test data are available in the frequency domain as a set of transfer functions. In particular, suppose that an admissible set of transfer functions is available. The actual process of constructing or computing these frequency transfer functions from measured excitation –response (input –output) test data (in the time domain) is known as model identiﬁcation in the frequency domain. This step should precede the actual modal analysis in practice. Numerical analysis (or curve ﬁtting) is the basic tool used for this purpose, and it will be discussed in a later section. The basic result used in EMA is Equation 4.17 with s ¼ jv or s ¼ j2pf for the frequency-transfer functions. For convenience, however, the following notation is used: Gik ðvÞ ¼ Gik ðf Þ ¼

n X r¼1

v2r

ðci ck Þr 2 v2 þ 2jzr vr v

ð4:22Þ

ðci ck Þr 2 f 2 þ 2jzr fr f

ð4:23Þ

n X r¼1

4p2

f12

where v and f are used in place of jv and j2pf in the function notation Gð Þ: As already observed in Example 4.1, it is not necessary to measure all n2 transfer functions in the n £ n transfer function matrix, G, in order to determine the complete modal information. Owing to the symmetry of G it follows that at most only 1=2nðn þ 1Þ transfer functions are needed. In fact, it can be “shown by construction” (i.e., in the process of developing the method itself) that only n transfer functions are needed. These n transfer functions cannot be chosen arbitrarily, however, even though there is a wide choice for the admissible set of n transfer functions. A convenient choice is to measure any one row or any one column of the transfer function matrix. It should be clear from the following development that any set of transfer functions that spans all n DoF of the system would be an admissible set provided that only one autotransfer function is included in the minimal set. Hence, for example, all the transfer functions on the main diagonals or on the main cross diagonal of G, do not form an admissible set. Suppose that the kth column ðGik ; i ¼ 1; 2; …; nÞ of the transfer function matrix is measured by applying a single forcing excitation at the kth DoF and measuring the corresponding responses at all n DoF in the system. The main steps in extracting the modal information from this data are given below: 1. Curve ﬁt the (measured) n transfer functions to expressions of the form given by Equation 4.22. In this manner determine the natural frequencies vr ; the damping ratios zr ; and the residues ðci ck Þr ; for the set of modes r ¼ 1; 2; and so on. 2. The residues of a diagonal transfer function (i.e., point transfer functions or autotransfer function), Gkk ; are ðc2k Þ1 ; ðc2k Þ2 ; …; ðc2k Þn : From these, determine the kth row of the modal matrix; ðck Þ1 ; ðck Þ2 ; …; ðck Þn : Note that M-normality is assumed. However, the modal vectors are arbitrary up to a multiplier of 2 1. Hence, we may choose this row to have all positive elements. 3. The residues of a nondiagonal transfer function, that is, a cross-transfer function, Gkþi;k are ðckþi ck Þ1 ; ðckþi ck Þ2 ; …; ðckþi ck Þn : By substituting the values obtained in Step 2 into these values, determine the k þ ith row of the modal matrix; ðckþi Þ1 ; ðckþi Þ2 ; …; ðckþi Þn : The complete modal matrix C is obtained by repeating this step for i ¼ 1; 2; …; n 2 k and i ¼ 21; 22; …; 2k þ 1: Note that the associated modal vectors are M-normal. The procedure just outlined for determining the modal matrix veriﬁes, by construction, that only n transfer functions are needed to extract the complete modal information. It further reveals that it is not

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essential to perform the transfer function measureAn Admissible Set ments in a row fashion or column fashion. A A Minimal Set An Inadmissible Set diagonal element (i.e., a point transfer function, or an autotransfer function) should always be measured. The remaining n 2 1 transfer functions must be off diagonal but otherwise can be chosen arbitrarily, provided that all n DoF are spanned G = either as an excitation point or as a measurement location (or both). This guarantees that no symmetric transfer function elements are included. This deﬁnes a minimal set of transfer FIGURE 4.2 A nonminimal admissible set, a minimal function measurements. An admissible set of more set, and inadmissible set of possible transfer function than n transfer functions can be measured in measurements. practice so that redundant measurements are available in addition to the minimal set that is required. Such redundant data are useful for checking the accuracy of the modal estimates. Examples for an admissible (nonminimal) set, a minimal set, and an inadmissible set of transfer functions matrix elements are shown schematically in Figure 4.2. Note that the inadmissible set in this example contains 11 transfer function measurements but the sixth DoF is not covered by this set. On the other hand, a minimal set requires only six transfer functions.

4.3.1

Extraction of the Time-Domain Model

Once the complete modal information is extracted by modal analysis, it is possible, at least in theory, to determine a time-domain model (M, K, and C matrices) for the system. To obtain the necessary equations, ﬁrst premultiply by ðCT Þ21 and postmultiply by C Equation 4.4, Equation 4.5, and Equation 4.6 to obtain M ¼ ðCT Þ21 MC21

ð4:24Þ

K ¼ ðCT Þ21 KC21

ð4:25Þ

C ¼ ðCT Þ21 CC21

ð4:26Þ

where M ¼ I ¼ identity matrix

Since the modal matrix C is nonsingular because M is assumed nonsingular in the dynamic models that we use (i.e., each DoF has an associated mass, or the system does not possess static modes), the inverse transformations given by the equations from Equation 4.24 to Equation 4.26 are feasible. It appears, however, that two matrix inversions are needed for each result. Since M, K, and C matrices are diagonal, their inverse is given by inverting the diagonal elements. This fact can be used to obtain each result through just one matrix inversion. Equation 4.24, Equation 4.25, and Equation 4.26 are written as M ¼ ðCM21 CT Þ21 21

ð4:27Þ

T 21

K ¼ ðCK C Þ

ð4:28Þ

C ¼ ðCC21 CT Þ21

ð4:29Þ

M21 ¼ I

ð4:30Þ

Note that for the present M-normal case K

21

¼

diag½1=v21 ; 1=v22 ; …; 1=v2n

C21 ¼ diag½1=ð2z1 v1 Þ; 1=ð2z2 v2 Þ; …; 1=ð2zn vn Þ

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ð4:31Þ ð4:32Þ

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Vibration Monitoring, Testing and Instrumentation

By substituting the equations from Equation 4.30 to Equation 4.32 into the equations from Equation 4.27 to Equation 4.29, we obtain the relations that can be used in computing the time-domain model: 0 2 B 6 B 6 B 6 B 6 K ¼ BC6 B 6 B 6 @ 4 0 2 B 6 B 6 B 6 B 6 C ¼ BC6 B 6 B 6 @ 4

M ¼ ðCCT Þ21 1=v21

0 1=v22 ..

. 1=v2n

0 1=ð2z1 v1 Þ

3

7 C 7 C 7 C 7 TC 7C C 7 C 7 C 5 A 0

1=ð2z2 v2 Þ ..

ð4:33Þ

121

.

ð4:34Þ

3

121

7 C 7 C 7 C 7 TC 7C C 7 C 7 C 5 A

ð4:35Þ

1=ð2zn vn Þ

0

Alternatively, only one matrix inversion (that of C) is needed if we use the fact that ðCT Þ21 ¼ ðC21 ÞT Then, M ¼ ðC21 ÞT MC21 21 T

ð4:36Þ

21

K ¼ ðC Þ KC

ð4:37Þ

C ¼ ðC21 ÞT CC21

ð4:38Þ

The main steps of EMA are summarized in Box 4.1. In practice, frequency-response data are less accurate at higher resonances. Some of the main sources of error are as follows: (1) Aliasing distortion in the frequency domain, due to ﬁnite sampling rate of data, will distort highfrequency results during digital computation. (2) Inadequate spectral-line resolution (or frequency resolution) and frequency coverage (bandwidth) can introduce errors at high-frequency resonances. The frequency resolution is ﬁxed both by the signal record length ðTÞ and the type of time window used in digital Fourier analysis, but the resonant peaks are sharper for higher frequencies. Frequency coverage depends on the data sampling rate. (3) Low signal-to-noise ratio (SNR) at high frequencies, in part due to noise and poor dynamic range of equipment and in part due to low signal levels, will result in data measurement errors. Signal levels are usually low at high frequencies because inertia in a mechanical system acts as a low-pass ﬁlter 1=ðmv2 Þ: (4) Computations involving high order matrices (multiplication, inversion, etc.) will lead to numerical errors in complex systems with many DoF. It is customary, therefore, to extract modal information only for the ﬁrst several modes. In that case, it is not possible to recover the mass, stiffness, and damping matrices. Even if these matrices are computed, their accuracy is questionable due to their sensitivity to the factors listed above.

4.4

Curve Fitting of Transfer Functions

Parameter estimation in vibrating systems can be interpreted as a technique of experimental modeling. This process requires experimental data in a suitable form, preferably excitation –response data, and is often represented as a set of transfer functions in the frequency domain. Parameter estimation using

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Experimental Modal Analysis

4-11

Box 4.1 MAIN STEPS OF EXPERIMENTAL MODAL ANALYSIS 1. Measure an admissible set of excitation ðuÞ and response ð yÞ signals. (Cover all DoF; one response measurement should be for the excitation location.) 2. Group the signals, assign windows, and ﬁlter the signals. 3. Compute transfer functions using FFT and the spectral density method: Guy ¼ Fuy =Fuu 4. Compute ordinary coherence functions

g2uy ¼

lFuy l2 Fuu Fyy

and choose the accurate transfer functions on this basis (guy close to 1 ) accept; guy close to 0 ) reject). 5. Curve ﬁt n admissible transfer functions to expressions: n X ðci ck Þr Gik ¼ 2 2 v v 2 þ 2jzr vr v r r¼1 Hence, extract: Residues ðci ck Þr ) mode shapes vectors cr which are M-normal; Natural frequencies (undamped), vr ; Modal damping ratios (viscous), zr : 6. Form the modal matrix C ¼ ½c1 ; c2 ; …; cn ; Compute C21 : 7. Modal mass matrix M ¼ I; Modal stiffness matrix K ¼ diag v21 ; v22 ; …; v2n ; Modal damping matrix C ¼ diag½2z1 v1 ; 2z2 v2 ; …; 2zn vn ; 8. Compute the system model: Mass matrix M ¼ ðC21 ÞC21 ; Stiffness matrix K ¼ ðC21 ÞT KC21 ; Damping matrix C ¼ ðC21 ÞCC21 . measured response data is termed model identiﬁcation or, simply, identiﬁcation in the literature on systems and control. We shall present a parameter estimation procedure that involves frequency transfer functions, which is particularly useful in EMA.

4.4.1

Problem Identiﬁcation

Transfer functions that are computed from measured time histories using digital Fourier analysis (e.g., FFT) cannot be directly used in the modal analysis computations. The data must be available as analytical transfer functions. Therefore, it is important to represent the computed transfer functions with suitable analytical expressions. This is done, in practice, either by curve ﬁtting

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4-12

Vibration Monitoring, Testing and Instrumentation

a suitable transfer function model into the computed data or by simpliﬁed methods such as “peak picking.” Accordingly, this conversion of data is an experimental modeling technique. Identiﬁcation of transfer function models from measured data is an essential step in EMA. Apart from that, it has other important advantages. In particular, analytical transfer function plots clearly identify system resonances and generate numerical values for the corresponding parameters (resonant frequencies, damping, phase angles, and magnitudes) in a convenient manner. This form represents a signiﬁcant improvement over the crude transfer function plots, which are normally far less presentable and rather difﬁcult to interpret.

4.4.2

Single- and Multi-Degree-of-Freedom Techniques

Several single-DoF techniques exist for extracting analytical parameters from experimental transfer functions. In particular, the methods of curve ﬁtting (circle ﬁtting) and peak picking are considered here. In a single-DoF method, only one resonance is considered at a time. Single-DoF curve ﬁtting, or more correctly, single-resonance curve ﬁtting is the term used to denote any curve ﬁtting procedure that ﬁts a quadratic (second-order) transfer function into each resonance in the measured transfer function, one at a time. In the case of closely spaced modes (or closely spaced resonances), the associated error can be very large. The accuracy is improved if expressions of a higher order than quadratic are used for this purpose, but unacceptable errors can still exist. In peak picking, each resonance of experimental transfer function data is examined individually; the resonant frequency and the damping constant corresponding to that resonance are determined by comparing with an analytical single-DoF transfer function. In multi-DoF curve ﬁtting, or more appropriately, multiresonance curve ﬁtting, all resonances (or modes) of importance are considered simultaneously and ﬁtted into an analytical transfer function of suitable order. This method is generally more accurate but computationally more demanding than the single-resonance method. In choosing between the single-resonance and multiresonance methods, the required accuracy should be weighted against the cost and speed of computation.

4.4.3 Single-Degree-of-Freedom Parameter Extraction in the Frequency Domain The theory of curve ﬁtting by a circle (i.e., circle ﬁtting) for each resonance of an experimentally determined transfer function is presented ﬁrst. Next, the peak picking method will be described. 4.4.3.1

Circle-Fit Method

It can be shown that the mobility transfer function (velocity/force) of a single-DoF system with linear viscous damping, when plotted on the Nyquist plane of real axis and imaginary axis for the frequency transfer function, is a circle. Similarly, it can be shown that the receptance or dynamic ﬂexibility or compliance transfer function (displacement/force) of a single-DoF system with hysteretic damping, when plotted on the Nyquist plane, is also a circle. Note that, for hysteretic damping, the damping constant (in the time domain) is not actually a constant but is inversely proportional to the frequency of motion. However, in this case, in the frequency domain, the damping term will be independent of frequency. The fact that such circle representations are possible for transfer functions of a single-DoF system may be used in ﬁtting a circle to a transfer function that is computed from experimental data. This will lead to determining the analytical parameters for the transfer function. This approach is illustrated now through analytical development.

Case of Viscous Damping Consider a single-DoF system with linear, viscous damping, as given by m€y þ c_y þ ky ¼ f ðtÞ

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ð4:39Þ

Experimental Modal Analysis

4-13

where m; k; and c are the mass, stiffness, and damping constants of the system elements, respectively, f ðtÞ is the excitation force, and y is the displacement response. Equation 4.39 may be expressed in the standard form: 1 y€ þ 2zvn y_ þ v2n y ¼ f ðtÞ ð4:40Þ m Receptance YðsÞ 1 ¼ FðsÞ m s2 þ 2zvn s þ v2n

with s ¼ jv

ð4:41Þ

Mobility VðsÞ sYðsÞ s ¼ ¼ FðsÞ FðsÞ m s2 þ 2zvn s þ v2n

with s ¼ jv

Consider the mobility (velocity/force) transfer function given by s GðsÞ ¼ 2 s þ 2zvn s þ v2n

ð4:42Þ

ð4:43Þ

where the constant parameter, m, in Equation 4.42 has been omitted, without loss of generality. In the frequency domain ðs ¼ jvÞ, we have Gð jvÞ ¼

v2n

jv 2 v þ 2jzvn v

ð4:44Þ

2

Multiply the numerator and the denominator of GðjvÞ in Equation 4.44 by the complex conjugate of the denominator (i.e., v2n 2 v2 2 2jzvn v). Then, the denominator is converted to the square of its original magnitude, as given by D ¼ ðv2n 2 v2 Þ2 þ ð2zvn vÞ2

ð4:45Þ

and the frequency transfer function (Equation 4.44) is converted into the form Gð jvÞ ¼

jv 2 v 2 v2 2 2jzvn v D n

Gð jvÞ ¼ Re þ j Im where Re ¼

2zvn v2 D

and

ð4:46Þ Im ¼

v 2 ðv 2 v2 Þ D n

ð4:47Þ

Now, we can write Re 2

1 8z2 v2n v2 2 4z2 v2n v2 2 ðv2n 2 v2 Þ2 4z2 v2n v2 2 ðv2n 2 v2 Þ2 ¼ ¼ 4zvn D 4zvn D 4zvn

Hence, in view of Equation 4.47 we have " #2 2 v2 ðv2n 2 v2 Þ2 1 4z2 v2n v2 2 ðv2n 2 v2 Þ2 2 þIm ¼ þ Re 2 4zvn D D2 4zvn ¼ ¼

16z 4 v4n v4 2 8z2 v2n v2 ðv2n 2 v2 Þ2 þ ðv2n 2 v2 Þ4 þ 16z2 v2n v2 ðv2n 2 v2 Þ2 16z2 v2n D2 4z2 v2n v2 þ ðv2n 2 v2 Þ2 16z2 v2n D2

2

¼

D2 1 ¼ R2 2 2 2 ¼ 16z2 v2n 16z vn D

It follows that the transfer function, GðjvÞ; represents a circle in the real– imaginary plane, with the following properties: Circle radius R ¼

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1 4zvn

ð4:48Þ

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Vibration Monitoring, Testing and Instrumentation

Circle center ¼

1 ;0 4zvn

Im

ð4:49Þ

Now, we may reintroduce the constant parameter, m; back into the transfer function, as in Equation 4.42. Then, we have 1 Circle radius R ¼ ; 4zvn m ð4:50Þ 1 ;0 Center ¼ 4zvn m

1 Re 2zwnm

(a) Im

G( jw) Plane

0 Re

−

Case of Hysteretic Damping Consider a single-DoF system with hysteretic damping. The equation motion is given by h y_ þ ky ¼ f ðtÞ v with f ðtÞ ¼ f0 sin vt

1 4zwnm

0

A sketch of this circle is shown in Figure 4.3(a). As mentioned before, the plane formed by the real and imaginary parts of GðjvÞ as the Cartesian x and y axes, respectively, is the Nyquist plane. The plot of GðjvÞ on this plane is the Nyquist diagram. It follows that the Nyquist diagram of the mobility function (Equation 4.42 or Equation 4.44) is a circle.

m€y þ

G( jw) Plane

1 2h

(b) FIGURE 4.3 (a) Circle ﬁt of a mobility function with viscous damping; (b) circle ﬁt of a receptance function with hysteretic damping.

ð4:51Þ

Note the frequency dependent damping constant, with the hysteretic damping parameter, h; in the time domain. The receptance function, Gð jvÞ; is given by Gð jvÞ ¼

1 ðk 2 mv2 Þ þ jh

Note that the damping term, jh; is independent of frequency in the frequency domain for this case. As for the case of viscous damping, we can easily show that the Nyquist plot of this transfer function is a circle with Radius ¼

1 2h

and Center ¼ 0; 2

1 2h

ð4:52Þ

A sketch of the resulting circle is shown in Figure 4.3(b). In general, for a multi-DoF viscous-damped system we have the “mobility” function Gik ð jvÞ ¼ jv

n X r¼1

ðci ck Þr v2r 2 v2 þ 2jzr vr v

ð4:53Þ

If the resonances are not closely spaced we can assume that, near each resonance ðrÞ Gik ¼ constant offset ðcomplexÞ þ single DoF mobility function ¼ constant offset þ

jvðci ck Þr v2r 2 v2 þ 2jzr vr v

ð4:54Þ

We can curve ﬁt each resonance r to a circle this way and thereby extract the ðci ck Þr value (the residue) from the radius of the circle ﬁt. Note: This method cannot be used if the resonances are closely spaced and consequently if signiﬁcant modal interactions are present.

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Experimental Modal Analysis

4.4.3.2

4-15

Peak-Picking Method

The peak-picking method is also a single-DoF method in view of the fact that each resonance of an experimentally determined transfer function is considered separately. The approach is to compare the resonance region with an analytical transfer function of a damped single-DoF system. One of three types of transfer functions, receptance, mobility, and accelerance as listed in Table 4.1, can be used for this purpose. Note that, when the level of damping is small, it can be assumed (approximately) that the pﬃﬃﬃﬃﬃ resonance is at the undamped natural frequency vn ¼ k=m: Substituting this value for v in each of the frequency transfer functions, we can determine the transfer function value at resonance, denoted by Gpeak ð jvÞ: It is noted from Table 4.1 that this function value in general depends on the damping constant and the natural frequency. Since vn is known directly from the peak location of the transfer function, it is possible to compute c (or the damping ratio, z) by ﬁrst determining the corresponding peak magnitude. Speciﬁcally, from Table 4.1, it is clear that we should pick the imaginary part of the frequency transfer function for receptance or accelerance data and the real part of the transfer function for mobility data. Then, we pick the peak value of the chosen part of the transfer function and the frequency at the peak. Table 4.2 gives normalized expressions for the three frequency transfer functions, receptance ¼ displacement/force; mobility ¼ velocity/force; accelerance ¼ acceleration/force, in the frequency domain, in the case of a single-DoF mechanical system with (1) viscous damping and (2) hysteretic damping. It may be veriﬁed that their Nyquist diagrams are circles (either exactly or approximately), thereby enabling one to use the circle-ﬁt method. Note that r ¼ v=vn ; where v is the excitation frequency and vn is the undamped natural frequency; z is the damping ratio in the case of viscous damping; and d ¼ h=k where h is the hysteretic damping parameter and k is the system stiffness. Peak picking is good in cases where modes are well separated and lightly damped. It does not work when the system is highly damped (or overdamped) or when the damping is zero (inﬁnite peak). It is a quick approach that is appropriate for initial evaluations and trouble shooting. TABLE 4.1

Some Frequency Transfer Functions Used in Peak Picking Single-DoF system

m€y þ c_y þ ky ¼ f ðtÞ

Receptance (dynamic ﬂexibility, compliance) Displacement y Gr ð jvÞ ¼ Force f Mobility Gm ð jvÞ ¼

Velocity v Force f

Accelerance Ga ð jvÞ ¼

1 with s ¼ jv ms2 þ cs þ k s with s ¼ jv ms2 þ cs þ k

Acceleration a Force f

ms2

Resonant peaks Gpeak ð jvÞ (occur approximately at v ¼ vn for light damping)

s2 with s ¼ jv þ cs þ k j Grpeak ¼ 2 c vn 1 Gm peak ¼ c Gapeak ¼

TABLE 4.2

jvn c

Normalized Frequency Response Functions for Single-DoF Curve Fitting

Frequency Response Function Receptance Mobility Accelerance

With Viscous Damping

With Hysteretic Damping

1 1 2 r 2 þ 2jzr jr 1 2 r 2 þ 2jzr

1 1 2 r 2 þ jd jr 1 2 r 2 þ jd

2r 2 1 2 r 2 þ 2jzr

2r 2 1 2 r 2 þ jd

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4-16

4.4.4

Vibration Monitoring, Testing and Instrumentation

Multi-Degree of Freedom Curve Fitting

We shall now discuss a general multiresonance curve ﬁtting method. The corresponding single-resonance method should also be clear from this general procedure. Note that many different versions of problem formulation and algorithm development are possible for least squares curve ﬁtting, but the results should be essentially the same. The method presented here is a frequency-domain method as we are dealing with experimentally determined frequency transfer functions. In a comparable time-domain method, a suitable analytical expression of the complex exponential form is ﬁtted into the experimental impulse response function obtained by the inverse Fourier transformation of measured transfer function. That method acquires additional error due to truncation (leakage) and ﬁnite sampling rate (aliasing) during the inverse FFT. 4.4.4.1

Formulation of the Method

The objective of the present multiresonance (multi-DoF) curve ﬁtting procedure is to ﬁt the computed (measured) transfer function data into an analytical expression of the form GðsÞ ¼

b0 þ b1 s þ · · · þ bm sm a0 þ a1 s þ · · · þ ap21 sp21 þ sp

for m # p

ð4:55Þ

The data for curve ﬁtting are the N complex transfer function values ½G1 ; G2; …; GN computed at discrete frequencies ½v1 ; v2 ; …; vN : Typically, if 1024 samples of time history were used in the FFT computations to determine the transfer function, we would have 512 valid spectral lines of transfer function data. However, near the high-frequency end, these data values become excessively distorted due to the aliasing error; only a part of the 512 spectral lines will be usable, typically the ﬁrst 400 lines. In that case, we have N ¼ 400: This value can be doubled by doubling the FFT block size (to 2048 words in the buffer), thereby doubling the record length or the sampling rate. It is acceptable to leave out part of the computed transfer function, not for poor accuracy but because that part falls outside the frequency band of interest in that particular modal analysis problem. A less wasteful practice is to pick the sampling rate of the measured time-history data to reﬂect the highest frequency of interest in the modal analysis. The (complex) error in the estimated value at each frequency point (spectral line), vi ; is given by e~i ¼ Gðvi Þ 2 Gi ¼

b0 þ b1 si þ · · · þ bm sm i p21

a0 þ a1 si þ · · · þ ap21 s i

þ sp

2 Gi

ð4:56Þ

with s ¼ jv: The characteristic equation of the dynamic model is given by

DðsÞ ¼ a0 þ a1 s þ · · · þ ap21 sp21 þ sp ¼ 0

ð4:57Þ

Its roots are the eigenvalues of the system. For damped systems, they occur in complex conjugates with negative real parts (note: p ¼ 2 £ number of DoFs, in typical cases). For systems with rigid-body modes, zero eigenvalues will also be present. However, since there is some damping in the system and since the lowest frequency that is tested and analyzed is normally greater than zero even for systems with rigidbody modes, we have Dð jvÞ – 0 ð4:58Þ in the frequency range of interest. Hence, the estimation error given by Equation 4.56 can be expressed as p21 p si

ei ¼ b0 þ b1 si þ · · · þ bm sm i 2 Gi a0 þ a1 si þ · · · þ ap21 si

ð4:59Þ

with s ¼ jv: The quadratic error function is given by the sum of the squares of magnitude error for all discrete frequency points used in modal analysis; thus J¼

N X i¼1

lei l2 ¼

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N X i¼1

epi ei

ð4:60Þ

Experimental Modal Analysis

FIGURE 4.4

4-17

An example of multi-DoF curve ﬁtting on experimental data.

Note that [ ]p denotes the complex conjugate. Complex conjugation is achieved by simply replacing ðjvÞ with ð2jvÞ in Equation 4.59. It follows that Equation 4.60 can be written as J¼

N Dh X i¼1

b0 þ b1 ð2jvi Þ þ · · · þ bm ð2jvi Þm 2 Gpi {a0 þ a1 ð2jvi Þ þ · · · þ ap21 ð2jvi Þp21

ih iE þð2jvi Þp } b0 þ b1 ð jvi Þ þ · · · þ bm ð jvi Þm 2 Gi {a0 þ a1 ð jvi Þ þ · · · þ ap21 ð jvi Þp21 þ ð jvi Þp } ð4:61Þ The basis of the least squares curve ﬁtting method of parameter estimation is to pick the transfer function parameters bi ; i ¼ 0; 1; …; m and ai ; i ¼ 0; …; p 2 1; such that the quadratic error function, J; is a minimum. Analytically, this requires

›J ¼0 dbk ›J ¼0 dak

k ¼ 0; 1; …; m

ð4:62Þ

k ¼ 0; 1; …; p 2 1

ð4:63Þ

Note that Equation 4.62 and Equation 4.63 correspond to m þ p þ 1 linear equations in the m þ p þ 1 unknowns bi ; i ¼ 0; 1; …; m and ai ; i ¼ 0; 1; …; p 2 1: A well-deﬁned solution exists to this set of nonhomogeneous equations provided that the equations are linearly independent, which is guaranteed if the determinant of the coefﬁcients of the unknown parameters does not vanish. It is a good practice to check for linear independence of the set of m þ p þ 1 equations using this determinant condition prior to performing further computations to solve the equations. The solution approach itself is primarily computational in nature and is not presented here. Figure 4.4 shows a result of multi-DoF curve ﬁtting on an experimental frequency transfer function, as collected from a civil engineering structure. Note the close match of the magnitude but not the phase angle. This analysis resulted in the resonant frequency and damping ratio values that are given in Table 4.3.

4.4.5

A Comment on Static Modes and Rigid-Body Modes

Some test systems may possess static modes, and rigid-body modes under rare circumstances. Static modes arise in analytical models if we fail to assign an inertia (mass) element for each DoF. Rigid-body modes arise in analytical models if proper restraints are not provided for the inertia elements. In practice,

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4-18

Vibration Monitoring, Testing and Instrumentation TABLE 4.3 Extracted Parameters in an Example of Experimental Modal Analysis Mode No.

Resonant Frequency (Hz)

Damping Ratio (viscous)

2

1 2 3 4 5 6

1.170 £ 1022 8.149 £ 1023 6.033 £ 1023 5.931 £ 1023 4.580 £ 1023 3.676 £ 1023

1.773 £ 10 3.829 £ 102 6.145 £ 102 7.018 £ 102 9.839 £ 102 1.190 £ 103

however, static modes arise if a coordinate is assigned to a DoF that actually does not exist, or if some parts of the physical system are relatively light with stiff restraints (i.e., very high natural frequencies), and rigid-body modes arise in the presence of relatively heavy components restrained by very ﬂexible elements (i.e., very low natural frequencies). Note that the assumed transfer function (Equation 4.55) allows for both these extremes. Speciﬁcally, if static modes are present it is necessary that the transfer function can be expressed as a sum of a constant term (static mode) and an ordinary transfer function (without a static mode). Hence, it will approach a nonzero constant value as the frequency, v; increases. This requires m ¼ p: If rigid-body modes are present, the characteristic polynomial DðsÞ of the model should have a factor s2 : This corresponds to a0 ¼ a1 ¼ 0:

4.4.6

Residue Extraction

The estimated transfer functions as given by Equation 4.55 is in the form of the ratio of two polynomials; the rational fraction form. This has to be converted into the partial fraction form given by Equation 4.17 in order to extract the residues ðci ck Þr that are needed for determining the mode shapes. For this, the natural frequencies, vr ; and the modal damping ratios, zr ; should be computed ﬁrst. These are given by the roots of the characteristic Equation 4.57 as qﬃﬃﬃﬃﬃﬃﬃﬃ ð4:64Þ lr ; lpr ¼ zr vr ^ j 1 2 z2r vr ; r ¼ 1; 2; …; n Once these eigenvalues are known, by solving Equation 4.57 using the estimated values for a0 ; a1 ; …; ap21 ; it is a straightforward task to compute the quadratic factors

Dr ðsÞ ¼ s2 þ 2zr vr s þ v2r

r ¼ 1; 2; …; n

ð4:65Þ

Note that, from Equation 4.17 ðci cr Þ ¼ ½Gik ðsÞDr ðsÞ

s¼lr

ð4:66Þ

assuming distinct eigenvalues. This is true because, when the partial fraction form is multiplied by Dr ðsÞ; it will cancel out with the denominator of the partial fraction corresponding to the rth mode, leaving its residue. Then, when s is set equal to Dr ; all the remaining partial fraction terms will vanish due to the fact that Dr ðlr Þ ¼ 0; provided that the eigenvalues are distinct. Since Gik ðsÞ are known from the estimated transfer functions, the residues can be computed using Equation 4.66. Finally, the mode shapes are determined using the procedure outlined earlier. Some curve ﬁtting approaches are summarized in Box 4.2.

4.5

Laboratory Experiments

Testing and analysis are important in the practice of mechanical vibration and are integral in EMA. In this section, we will describe two experiments in the category of modal testing. One experiment deals with a lumped-parameter system and the other with a distributed-parameter (or continuous) system.

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4-19

Box 4.2 CURVE FITTING OF TRANSFER FUNCTION DATA Single-resonance curve ﬁtting: (a) Viscous damping: 1. Compute mobility (velocity/force) function near resonance. 2. Scale the data. 3. Curve ﬁt to circle in the Nyquist plane (Argand diagram). (b) Hysteretic damping: 1. Compute receptance (displacement/force) function near resonance. 2. Scale the data. 3. Curve ﬁt to circle in the Nyquist plane (Argand diagram). Multiresonance curve ﬁtting: 1. Compute transfer function over the entire frequency range. 2. Scale the data. 3. Curve ﬁt to a general polynomial ratio with static and rigid-body modes.

Both experiments have direct practical implications and have been used in an established undergraduate course in mechanical vibrations.

4.5.1

Lumped-Parameter System

A schematic representation of a prototype unit that is used in laboratory for modal testing is shown in Figure 4.5. A view of the experimental system is shown in Figure 4.6. The system is a crude representation of an engine unit that is supported on ﬂexible mounts and subjected to unbalance forces and moments. The test object is assumed to consist of lumped elements of inertia, stiffness, and damping. The rectangular metal box, which represents the engine housing, is mounted on four springs and damping elements at the four corners. Inside the box are two pairs of identical and meshed gears, which are driven by a single DC motor. Each gear has two slots at diametrically opposite locations in order to place the eccentric masses. Various types of unbalance excitations can be generated by placing the four eccentric masses at different combinations of locations on the gear wheels. The drive motor is operated by a DC power supply with a speed control knob. The motor speed (and hence the gear speed) is measured using an optical encoder that is mounted on the drive shaft. It generates pulses as the encoder disk rotates with the shaft, in proportion to the angle of rotation. The pulse frequency of the encoder determines the shaft speed. A pair of accelerometers with magnetic bases is mounted on the top of the engine box. The locations that are used for this purpose are indicated in Figure 4.5. Figure 4.6 shows, from left to right, the following components of the experimental system: 1. Digital spectrum analyzer 2. A combined instrument panel consisting of a vibration meter, a tunable band-pass ﬁlter, and a unit consisting of a conditioning ampliﬁer and a phase meter

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4-20

Vibration Monitoring, Testing and Instrumentation

Digital Oscilloscope

Vibration Meter

Spectrum Analyzer

Charge Amplifiers & Phase Meter

Tunable Filter

A or B

Eccentric Masses

C or D

Speed Controller and Power Supply

Optical Encoder (Tachometer) Engine Box

Engine Mount

Drive Motor A

Counter-rotating Gear Pairs

Accelerometer Locations B Top View of Engine Box

FIGURE 4.5

3. 4. 5. 6.

D

C

Schematic diagram of an experimental setup for modal testing in laboratory.

Power supply for the instrument panel, placed on top of the panel Engine unit with two accelerometers mounted on top surface of the housing Digital oscilloscope placed on a shelf top immediately above the engine unit DC power supply and speed controller combination for the drive motor

The phase meter measures the phase difference between two input signals. The tunable ﬁlter is a band-pass ﬁlter and it can be tuned by a ﬁne-adjustment dial so that a signal in a very narrow band (i.e., harmonic signal) can be ﬁltered and measured. The vibration meter measures the magnitude (peak or rms value) of a signal. The choice of a displacement value (i.e., double integration), a velocity value (a single integration), or an acceleration value (no integration) is available, and can be selected using a knob. By placing the eccentric masses at various locations on the gear wheels, different modes can be excited. For example, the placement of all four eccentric masses at the vertical radius location above the rotating axis will generate a net harmonic force in the vertical direction, as the motor is driven. This will excite the heave (up and down) mode of the engine box. If the two masses on a meshed pair are placed at the vertical radius location below the rotating axis while the masses on the other meshed

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Experimental Modal Analysis

FIGURE 4.6

4-21

A view of an experimental setup for modal testing at the University of British Columbia.

pair are placed vertically above the rotating axis, then it will result a net moment (pitch) about a central horizontal axis of the engine box. This will excite the pitch mode, and so on. For a given arrangement of eccentric masses, two tests can be carried out, one in the frequency domain and the other in the time domain. 4.5.1.1

Frequency-Domain Test

Choose the “displacement” setting of the vibration meter. Start the motor and maintain the speed at a low value, say 4 Hz. Tune the ﬁlter, using its dial, until the vibration meter reading becomes the largest. The tuned frequency will be, in the ideal case, equal to the motor speed. Record the motor speed (i.e., the excitation frequency) and the magnitude of the displacement response. Increase the motor speed in 1-Hz steps and repeat the measurements, until reaching a reasonably high frequency, covering at least one resonance, say 25 Hz. Reduce the speed in steps of 1 Hz and repeat the measurements. Take some more measurements in the neighborhood of each resonance using smaller frequency steps. Plot the data as a frequency spectrum after compensating for the fact that the amplitude of the excitation force increases with the square of the drive speed (hence, divide the vibration magnitudes by square of the frequency). This experiment can be used, for example, to measure mode shapes, resonant frequencies, and damping ratios (by the half-power bandwidth method). 4.5.1.2

Time-Domain Tests

A test can be conducted to determine the damping ratio corresponding to a particular mode by the logarithmic decrement method. Here, ﬁrst pick the eccentric mass arrangement so as to excite the desired mode. Next increase the motor speed and then ﬁne-tune the operation at the desired resonance. Maintain the speed steady at this condition and observe the accelerometer signal using the oscilloscope, while making sure that at least ten complete cycles can be viewed on the screen. Suddenly, turn off the motor and record the decay of the acceleration signal using the oscilloscope. Another test that can be carried out is an impact (hammer) test. Here, use the spectrum analyzer to record and analyze a vibration response of the engine box through an accelerometer. Gently tap the engine box in different critical directions (e.g., at points A, B, C, and D in the vertical direction, in Figure 4.5, or in the horizontal direction on the side of the engine box in the neighborhood of these points) and acquire the vibration signal using the spectrum analyzer. Process the signal using the spectrum analyzer, obtain the resonant frequencies, and compare them with those obtained from sine testing.

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4-22

4.5.2

Vibration Monitoring, Testing and Instrumentation

Distributed-Parameter System

All real vibrating systems have continuous components. Often, however, we make distributed-parameter assumptions depending on the properties and the operating frequency range of the vibrating system. When a lumped-parameter approximation is not adequate, a distributed-parameter analysis will be needed. Modal testing and comparison with analytic results can validate an analytical model. The response of a distributed-parameter system will depend on the boundary conditions (the supporting conditions) as well as the initial conditions. For forced excitations, the response will depend on the nature of the excitation as well. Natural frequencies and mode shapes are system characteristics and depend on the boundary conditions, but not on the initial conditions and forcing excitations. Consider the experimental setup schematically represented in Figure 4.7. A view of the setup is shown in Figure 4.8. The device that is tested is a ski. For analytical purposes, it may be approximated as a thin beam. The objective of the test is to determine the natural frequencies and mode shapes of the ski. Since the signiﬁcant frequency range of the excitation forces on a ski, during use, is below 15 Hz, it is advisable to determine the modal information in the frequency range of about double the operating range (i.e., 0 to 30 Hz). In particular, in the design of a ski, natural frequencies below 15 Hz should be avoided, while keeping the unit as light and strong as possible. These are conﬂicting design requirements. It follows that modal testing can play an important role in the design development of a ski. Consider the experimental setup that is sketched in Figure 4.7. The ski is ﬁrmly supported at its middle on the electrodynamic shaker. Two accelerometers are mounted on either side of the support and are movable along the ski. The accelerometer signals are acquired and conditioned using charge ampliﬁers. The two signals are observed in the x –y mode of the digital oscilloscope so that both the amplitudes and the phase difference can be measured. The sine-random signal generator is set to the sine mode so that a harmonic excitation is generated at the shaker head. The shape of the motion can be observed in slow motion by illuminating the ski with the hand-held stroboscope, with the strobe frequency set to within about ^1 Hz of the excitation frequency. In the experimental system shown in Figure 4.8, we observe, from left to right, the following components: 1. Electrodynamic shaker with the ski mounted on its exciter head (two accelerometers are mounted on the ski) 2. Hand-held stroboscope, placed beside the shaker ch 1 ch 2

Charge Amplifiers ch 1

ch 3 ch 2

Ski

Electrodynamic Vibration Exciter

Accelerometer Power Amplifier

Hand-held Stroboscope

FIGURE 4.7

Digital Oscilloscope

Sine-Random Signal Generator

Power Supply Schematic diagram of a laboratory setup for modal testing of a ski.

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Experimental Modal Analysis

FIGURE 4.8

3. 4. 5. 6. 7.

4-23

A view of the experimental system for modal testing of a ski at the University of British Columbia.

Power ampliﬁer for driving the shaker, placed on top of the side table Two charge ampliﬁers placed on top of the power ampliﬁer and connected to the accelerometers Sine-random signal generator, placed on the table top, next to the ampliﬁer Power supply for the shaker Static load– deﬂection measurement device for determining the modulus of rigidity (EI) of the ski

Prior to modal testing, the modulus of rigidity of the ski is determined by supporting it on the two smooth end pegs of the loading structure, and loading at the midspan using incremental steps of 500-gm weights up to 4.0 kg, placed on a scale pan that is suspended at the midspan of the ski. The midspan deﬂection of the ski is measured using a spring-loaded dial gage that is mounted on the loading structure. If the midspan stiffness (force/deﬂection) as measured in this manner is k; it is known that the modulus of rigidity is EI ¼

kl3 48

ð4:67Þ

where l is the length of the ski. Note that this formula is for a simply supported ski, which is the case in view of the smooth supporting pegs. Also, weigh the ski and then compute m ¼ mass per unit length. With this information, the natural frequencies and mode shapes can be computed for various end conditions. In particular, compute this modal information for the following supporting conditions: 1. Free–free 2. Clamped at the center Next, perform modal testing using the experimental setup and compare the results with those computed using the analytical formulation. The natural frequencies (actually, the resonant frequencies, which are almost equal to the natural frequencies in the present case of light damping) can be determined by increasing the frequency of excitation in small steps using the sine generator and noting the frequency values at which the amplitudes of the accelerometer signals reach local maxima, as observed on the oscilloscope screen. A mode shape is measured as follows. First, detect the corresponding natural frequency as above. While maintaining the shaker excitation at this frequency, place the accelerometer near the shaker head, and then move the other accelerometer from one end of the ski to the other in small steps of displacement and observe the amplitude ratio and the phase difference of the two accelerometer signals using the oscilloscope. Note that in-phase signals mean the motions of the two points are in the same direction,

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4-24

Vibration Monitoring, Testing and Instrumentation

and the out-of-phase signals mean the motions are in the opposite directions. The mode shapes can be veriﬁed by observing the modal vibrations in slow motion using the stroboscope, as indicated before. Node points are the vibration-free points. They can be detected from the mode shapes. In particular, a tiny piece of paper will remain stationary at a node while making large jumps on either side of the node. Also, the phase angle of the vibration signal, as measured by an accelerometer, will jump by 1808 if the accelerometer is carefully moved across the node point.

4.6

Commercial EMA Systems

Commercially available EMA systems typically consist of an FFT analyzer, a modal analysis processor, a graphics terminal, and a storage device. Digital plotters, channel selectors, hard copy units, and other accessories can be interfaced, and the operation of the overall system can be coordinated through a host computer to enhance its capability. The selection of hardware for a particular application should address speciﬁc objectives as well as hardware capabilities. Software selection is equally important. Proper selection of an EMA system is difﬁcult unless the underlying theory is understood. In particular, the determination of transfer functions via FFT analysis; extraction of natural frequencies, modal damping ratios, and mode shapes from transfer function data; and the construction of mass, stiffness, and damping matrices from modal data should be considered. We have already presented the underlying theory. In the present section, we will describe the features of a typical EMA system.

4.6.1

System Conﬁguration

The extraction of modal parameters from dynamic test data is essentially a two-step procedure consisting of: 1. FFT analysis 2. Modal analysis In the ﬁrst step, appropriate frequency transfer functions are computed and stored. These raw transfer functions form the input data for the subsequent modal analysis, yielding modal parameters (natural frequencies, damping ratios, and mode shapes) and a linear differential equation model for the dynamic system (test object). 4.6.1.1

FFT Analysis Options

The basic hardware conﬁguration of a commercial modal analysis system is shown in Figure 4.9. Notice that the FFT analyzer forms the front end of the system. The excitation signal and the response measurements can be transmitted on line to the FFT analyzer (through charge ampliﬁers for piezoelectric sensors); many signals can be transmitted simultaneously in the multiple-channel case. Alternatively, all measurements may ﬁrst be recorded on a multiple-track FM tape and subsequently fed into the analyzer through a multiplexer. In the ﬁrst case, it is necessary to take the FFT analyzer to the test site; an FM tape recorder is needed at the test site in the second case. Through advances in microelectronics and LSI technology, the FFT analyzer has rapidly evolved into a powerful yet compact instrument that is often smaller in size than the conventional tape recorder used in vibration data acquisition; either device can be used in the ﬁeld with equal convenience. On-site FFT analysis, however, allows one to identify and reject unacceptable measurements (e.g., low signal levels and high noise components) during data acquisition, so that alternative data that might be needed for a complete modal identiﬁcation can be collected without having to repeat the test at another time. The main advantage of the FM tape method is that data are available in analog form, free of quantization error (digital word-size dependent), aliasing distortion (data sampling-rate dependent), and signal truncation error (data block-size dependent). Sophisticated analog ﬁltering is often necessary, however,

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Experimental Modal Analysis

Sensors and Transducers from Test Object (or Tape Player)

4-25

CRT Screen

Memory

Multi-Channel FFT Analyzer

Modal Analysis Processor

Data Storage (Hard Drive, CD ROM)

Input/Output Devices (Keyboard, Graphics Monitor, Printer, etc.) FIGURE 4.9

The conﬁguration of a commercial experimental modal analysis system.

to remove extraneous noise entering from the recording process (e.g., line noise and tape noise), as well as from the measurement process (e.g., sensor and ampliﬁer noise). The analog-to-digital converter (ADC) is normally an integral part of the analyzer. The raw transfer functions, once computed, are stored on a ﬂoppy disk or hard disk as the “transfer function ﬁle.” This constitutes the input data ﬁle for modal extraction. Some analyzers, instead, compute power spectral densities with respect to the excitation signal and store these in the data ﬁle. From these data, it is possible instantly to compute coherence functions, transfer functions, and other spectral information using keyboard commands. Another procedure has been to compute Fourier spectra of all signals and store them as raw data, from which other spectral functions can be conveniently computed. Most analyzers have small CRT screens to display spectral results. Low-coherent transfer functions are detected by analytical or visual monitoring and are automatically discarded. In principle, the same processor can be used for both FFT analysis and modal analysis. Some commercial modal analysis systems use a plug-in programmable FFT card in a common processor cage. Historically, however, the digital FFT analyzer was developed as a stand-alone hardware unit to be used as a powerful measuring instrument in a wide variety of applications, rather than just as a data processor. Uses include measurement of resonant frequencies and damping in vibration isolation applications, measurement of phase lag between two signals, estimation of signal noise levels, identiﬁcation of the sources of noise in measured signals, and measurement of correlation in a pair of signals. Because of this versatility, most modal analysis systems do come with a standard FFT analyzer unit as the front end and a separate computer for modal analysis. 4.6.1.2

Modal Analysis Components

In addition to the transfer function ﬁle, the modal analysis processor needs geometric information about the test object, typically coordinates of the mass points and directions of the DoF. This information is stored in a “geometry ﬁle.” The results of modal analysis are usually stored in two separate ﬁles: the “parameter ﬁle” containing natural frequencies, modal damping ratios, mass matrix, stiffness matrix, and damping matrix; and the “mode shape ﬁle,” containing mode shape vectors that are used for graphics display and printout. Individual modes can be displayed on the CRT screen of the graphics monitor either as a static traces or in animated (dynamic) form. The graphics monitor and printer are standard components of the system. The entire system may be interfaced with other peripheral I/O devices using an IEEE-488 interface bus or the somewhat slower serial RS-232 interface. For example, the overall operation can be coordinated, and further processing done, using a host computer. A desktop (personal) computer may substitute for the modal analysis processor, graphics monitor, and storage devices in the standard system, resulting in a reasonable reduction of the overall cost. An alternative conﬁguration that

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4-26 TABLE 4.4

Vibration Monitoring, Testing and Instrumentation Comparative Data for Four Modal Analysis Systems Description

System A

System B

System C

System D

Number of weighting window options available Analyzer data channels Maximum DoFs per analysis Maximum number of modes analyzed Multi-DoFs curve ﬁtting FFT Resolution (usable spectral lines/512) Zoom analysis capability in FFT Statistical error-band analysis Static mode-shape extremes Animated graphics capability Color graphics capability Hidden-line display Color printing Structural mass and stiffness matrices Approximate cost

3 2 750 @ 20 modes 50 @ 250 DoF Yes 400 Yes No Yes Yes No No No No $30,000

10 2 450 20 Yes 400 Yes No Yes Yes Yes No No No $20,000

5 2 725 @ 5 modes 10 (typical) No 400 Yes No Yes Yes No No Yes No $25,000

4 2 750 64 Yes 400 Optional No Yes Yes No No No Yes $50,000

is particularly useful in data transfer and communication from remote test sites uses a voice-grade telephone line and a modem coupler to link the FFT analyzer to the main processor. The ﬁrst step in selecting a modal analysis system for a particular application is to understand the speciﬁc needs of that application. For industrial applications of modal testing, the following requirements are typically adequate: 1. Acceptance of a wide range of measured signals having a variety of transient and frequency band characteristics 2. Capability to handle up to 300 DoF of measured data in a single analysis 3. FFT with frequency resolution of at least 400 spectral lines per 512 4. Zoom analysis capability 5. Capability to perform statistical error-band analysis 6. Static display and plot of mode-shape extremes 7. Animated (dynamic) display of mode shapes 8. Color graphics 9. Hidden-line display 10. Color printing with high line resolution 11. Capability to generate an accurate time-domain model (mass, stiffness, and damping matrices) The capabilities of four representative modal analysis systems are summarized in Table 4.4.

Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, WileyInterscience, New York. Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ. de Silva, C.W., Seismic qualiﬁcation of electrical equipment using a uniaxial test, Earthquake Eng. Struct. Dyn., 8, 337– 348, 1980. de Silva, C.W., Matrix eigenvalue problem of multiple-shaker testing, J. Eng. Mech. Div., Trans. ASCE, 108, 457– 461, 1982. de Silva, C.W. 1983a. Testing and Seismic Qualiﬁcation Practice, D.C. Heath and Co., Lexington, MA. de Silva, C.W., Selection of shaker speciﬁcations in seismic qualiﬁcation tests, J. Sound Vib., 91, 21–26, 1983b. de Silva, C.W., A Dynamic test procedure for improving seismic qualiﬁcation guidelines, J. Dyn. Syst. Meas. Control, Trans. ASME, 106, 143 –148, 1984a.

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Experimental Modal Analysis

4-27

de Silva, C.W., Hardware and software selection for experimental modal analysis, Shock Vib. Dig., 16, 3– 10, 1984b. de Silva, C.W., On the modal analysis of discrete vibratory systems, Int. J. Mech. Eng. Educ., 12, 35 –44, 1984c. de Silva, C.W., The digital processing of acceleration measurements for modal analysis, Shock Vib. Dig., 18, 3– 10, 1986. de Silva, C.W., Optimal input design for the dynamic testing of mechanical systems, J. Dyn. Syst. Meas. Control, Trans. ASME, 109, 111–119, 1987. de Silva, C.W. 2005. MECHATRONICS — An Integrated Approach, Taylor & Francis, CRC Press, Boca Raton, FL. de Silva, C.W. 2006. VIBRATION — Fundamentals and Practice, 2nd ed., Taylor & Francis, CRC Press, Boca Raton, FL. de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control — application in the distribution qualiﬁcation of microcomputers, Shock Vib. Dig., 18, 3 –13, 1986. de Silva, C.W., Loceff, F., and Vashi, K.M., Consideration of an optimal procedure for testing the operability of equipment under seismic disturbances, Shock Vib. Bull., 50, 149 –158, 1980. de Silva, C.W. and Palusamy, S.S. Experimental modal analysis — a modeling and design tool, Mech. Eng., ASME, 106, 56 –65, 1984. de Silva, C.W., Singh, M., and Zaldonis, J., Improvement of response spectrum speciﬁcations in dynamic testing, J. Eng. Ind., Trans. ASME, 112, 384 –387, 1990.

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5

Mechanical Shock 5.1

5.2 5.3

5.4 5.5 5.6 5.7 5.8

5.9

Deﬁnitions .............................................................................. 5-2

Shock † Simple (or Perfect) Shock † Half-Sine Shock Versed-Sine (or Haversine) Shock † Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock † Rectangular Shock

†

Description in the Time Domain ........................................ 5-3 Shock Response Spectrum .................................................... 5-4

Need † Shock Response Spectrum Deﬁnition † Response of a Linear One-Degree-of-Freedom System † Deﬁnitions † Standardized Shock Response Spectrum † Choice of Damping † Shock Response Spectra Domains † Algorithms for Calculation of the Shock Response Spectra † Choice of the Digitization Frequency of the Signal † Use of Shock Response Spectra for the Study of Systems with Several Degrees of Freedom

Pyroshocks ............................................................................. 5-17 Use of Shock Response Spectra ............................................ 5-18

Severity Comparison of Several Shocks † Test Speciﬁcation Development from Real Environment Data

Standards ................................................................................ 5-24

Types of Standards † Installation Conditions of Test Item † Uncertainty Factor † Bump Test

Damage Boundary Curve ..................................................... 5-26

Deﬁnition

†

Analysis of Test Result

Shock Machines ..................................................................... 5-28

Main Types † Impact Shock Machines † Shock Simulators (Programmers) † Limitations † Pneumatic Machines † High Impact Shock Machines † Speciﬁc Test Facilities

Generation of Shock Using Shakers .................................... 5-44

Principle Behind the Generation of a Simple Shape Signal versus Time † Main Advantages † Pre- and Postshocks † Limitations of Electrodynamic Shakers † The Use of Electrohydraulic Shakers

5.10 Control by a Shock Response Spectrum ............................. 5-52 Principle † Principal Shapes of Elementary Signals † Comparison of WAVSIN, SHOC Waveforms, and Decaying Sinusoid † Criticism of Control by a Shock Response Spectrum

5.11 Pyrotechnic Shock Simulation ............................................. 5-58

Christian Lalanne Engineering Consultant

Simulation Using Pyrotechnic Facilities † Simulation Using Metal-to-Metal Impact † Simulation Using Electrodynamic Shakers † Simulation Using Conventional Shock Machines

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5-2

Vibration Monitoring, Testing, and Instrumentation

Summary Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (in material handling, transportation, etc.). This kind of environment, although of extremely short duration (from a fraction of a millisecond to a few dozen milliseconds), is often severe and cannot be neglected. What is presented in this chapter is summarized here. After a brief recapitulation of the shock shapes most widely used in tests, the shock response spectrum (SRS) is presented with its numerous deﬁnitions and calculation methods. The main properties of the spectrum are described, showing that important characteristics of the original signal can be drawn from it, such as its amplitude or the velocity change associated with the movement during a shock. The SRS is the ideal tool for comparing the severity of several shocks and for drafting speciﬁcations. Recent standards require writing test speciﬁcations from real environment measurements associated with the life proﬁle of the material (test tailoring). The process that makes it possible to transform a set of recorded shocks into a speciﬁcation of the same severity is detailed. Packages must protect the equipment contained within them from various forms of disturbance related to handling and possible free fall drop and impact onto a ﬂoor. A method to characterize the shock fragility of the packaged product, using the “damage boundary curve” (DBC), and to choose the characteristics of the cushioning material constituting the package is described. The principle of shock machines that are currently most widely used in laboratories is described. To reduce costs by restricting the number of changes in test facilities, speciﬁcations expressed in the form of a simple shock (halfsine, rectangle, sawtooth with a ﬁnal peak) can occasionally be tested using an electrodynamic exciter. The problems encountered, which stress the limitations of such means, are set out together with the consequences of modiﬁcations, that have to be made to the shock proﬁle on the quality of the simulation. Determining a simple shape shock of the same severity as a set of shocks on the basis of their response spectrum is often a delicate operation. Thanks to progress in computerization and control facilities, this difﬁculty can sometimes be overcome by expressing the speciﬁcation in the form of a response spectrum and by controlling the exciter directly from that spectrum. In practical terms, as the exciter can only be driven with a signal that is a function of time, the software of the control rack determines a time signal with the same spectrum as the speciﬁcation displayed. The principles of composition of the equivalent shock are described with the shapes of the basic signals commonly used, while their properties and the problems that can be encountered, both in the generation of the signal and with respect to the quality of the simulation obtained, are emphasized. Pyrotechnic devices or equipment (cords, valves, etc.) are frequently used in satellite launchers due to the very high degree of accuracy that they provide in operating sequences. Shocks induced in structures by explosive charges are extremely severe, with very speciﬁc characteristics. It is shown that they cannot be correctly simulated in the laboratory by conventional means and that their simulation requires speciﬁc tools.

5.1 5.1.1

Deﬁnitions Shock

Shock occurs when a force, a position, a velocity, or an acceleration is abruptly modiﬁed and creates a transient state in the system considered. The modiﬁcation is normally regarded as abrupt if it occurs in a time period that is short compared with the natural period concerned (AFNOR, 1993). Shock is deﬁned as a vibratory excitation having a duration between the natural period of the excited mechanical system and two times that period (Figure 5.1).

5.1.2

Simple (or Perfect) Shock

A shock whose signal can be represented exactly in simple mathematical terms is called a simple (or perfect) shock. Standards generally specify one of the three following: half-sine (approached by a versed sine waveform), terminal peak sawtooth, and rectangular shock (approached by a trapezoidal waveform).

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Mechanical Shock

5.1.3

5-3

Half-Sine Shock

This is a simple shock for which the acceleration – time curve has the form of a half-period (part positive or negative) of a sinusoid.

Acceleration (m/s2)

150

5.1.4 Versed-Sine (or Haversine) Shock This is a simple shock for which the acceleration – time curve has the form of one period of the curve representative of the function ½1 2 cosð Þ ; with this period starting from the zero value of this function. It is thus a signal ranging between two minima of a sine wave.

100 50 0 −50 −100 −150

0

2

4

6 8 10 Time (ms)

12

14

FIGURE 5.1 Example of a shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

5.1.5 Terminal Peak Sawtooth Shock or Final Peak Sawtooth Shock This is a simple shock for which the acceleration –time curve has the shape of a triangle, where acceleration increases linearly up to a maximum value and then instantly decreases to zero.

5.1.6

Rectangular Shock

This is a simple shock for which the acceleration–time curve increases instantaneously up to a given value, remains constant throughout the signal, and decreases instantaneously to zero. In practice, what is carried out are trapezoidal shocks.

5.1.6.1

Trapezoidal Shock

This is a simple shock for which the acceleration–time curve grows linearly up to a given value, remains constant during a certain time period, after which it decreases linearly to zero.

5.2

Description in the Time Domain

Three parameters are necessary to describe a shock in the time domain: its amplitude, its duration, t; and its form. The physical parameter expressed in terms of time is generally an acceleration, x€ ðtÞ, but can be also a velocity, vðtÞ; a displacement, xðtÞ, or a force, FðtÞ: In the ﬁrst case, which we will consider in particular in this chapter, the velocity change corresponding to the shock movement is equal to (Table 5.1)

DV ¼

ðt 0

x€ ðtÞdt

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ð5:1Þ

5-4

Vibration Monitoring, Testing, and Instrumentation TABLE 5.1

Main Simple Shock Waveforms (Amplitude, xm ; Duration, t; Velocity Change, DV) Waveform

5.3 5.3.1

Function

Half-sine

xðtÞ ¼ xm sin

Versed-sine

xðtÞ ¼

Rectangle

xðtÞ ¼ xm

Terminal peak sawtooth

xðtÞ ¼ xm

p t t

xm 2p 1 2 cos t t 2

DV 2 x t p m 1 x t 2 m xm t

t t

1 x t 2 m

Shock Response Spectrum Need

Very often, the problem is to evaluate the relative severity of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a speciﬁcation, etc.). A shock is an excitation of short duration, which induces transitory dynamic stress in structures. These stresses are a function of the following: *

*

The characteristics of the shock (amplitude, duration, and shape). The dynamic properties of the structure (resonant frequencies, Q factors; see Chapter 19).

The severity of a shock can thus be estimated only according to the characteristics of the system that undergoes it. The evaluation of this severity requires in addition the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are as follows: *

*

The exceeding of a value threshold of the stress in a mechanical part can lead to either a permanent deformation (acceptable or not) or a fracture, or at any rate, a functional failure. If the shock is repeated many times (e.g., the shock recorded on the landing gear of an aircraft, the operation of an electromechanical contactor), the fatigue damage accumulated in the structural elements can lead in the long term to fracture (Lalanne, 2002c).

The comparison would be difﬁcult to carry out if one used a ﬁne model of the structure, and in any case this is not always available, particularly at the stage of the development of the speciﬁcation of dimensioning. One searches for a method of general nature, which leads to results that can be extrapolated to any structure.

5.3.2

Shock Response Spectrum Deﬁnition

In a thesis on the study of earthquakes’ effects on buildings, Biot (1932) proposed a method consisting of applying the shock under consideration to a “standard” mechanical system, which thus does not claim to be a model of the real structure. It is composed of a support and of N linear one-degree-offreedom (one-DoF) resonators, comprising each one are a mass, mi a spring of stiffness, p kiﬃﬃﬃﬃﬃ and ﬃ a damping device, ci ; chosen such that the fraction of critical damping (damping ratio) j ¼ ci 2 ki mi is the same for all N resonators. A model for the shock response spectrum (SRS) is shown in Figure 5.2 (also see Chapter 3). When the support is subjected to the shock,pﬃﬃﬃﬃﬃﬃ each mass, mi ; has a speciﬁc movement response according to its natural frequency, f0i ¼ ð1=2pÞ ki =mi and to the chosen damping ratio, j; while a

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Mechanical Shock

5-5

z(t) t f1

x

f2

x

fN−1

x

fN

x .. x(t) t

FIGURE 5.2 permission.)

Model of the SRS. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With

stress, si ; is induced in the elastic element. The analysis consists of seeking the largest stress, smi ; observed at each frequency in each spring. For applications deviating from the assumptions of deﬁnition of the SRS (linearity, only one DoF), it is desirable to observe a certain prudence if one wishes to estimate quantitatively the response of a system starting from the spectrum (Bort, 1989). The response spectra are more often used to compare the severity of several shocks. It is known that the tension static diagram of many materials comprises a more-or-less linear arc on which the stress is proportional to the deformation. In dynamics, this proportionality can be allowed within certain limits for the peaks of the deformation. If a mass –spring–damper system is supposed to be linear, it is then appropriate to compare two shocks by the maximum response stress, sm , that they induce or by the maximum relative displacement, zm ; that they generate. This occurs since it is supposed

sm ¼ Kzm

ð5:2Þ

zm is a function only of the dynamic properties of the system, whereas sm is also a function, via K; of the properties of the materials which constitute it. The curve giving the largest relative displacement, zsup multiplied by v20 (v0 ¼ 2p f0 ) according to the natural frequency, f0 ; for a given damping ratio j; is the SRS.

5.3.3 5.3.3.1

Response of a Linear One-Degree-of-Freedom System Shock Deﬁned by a Force

Consider a mass – spring – damping system subjected to a force, FðtÞ; applied to the mass (Figure 5.3). The differential equation of the movement is written as d2 z dz m 2 þc þ kz ¼ FðtÞ dt dt

Force Mass m

ð5:3Þ

where zðtÞ is the relative displacement of the mass, m; relative to its support in response to the shock, FðtÞ: This equation can be expressed in the form (Lalanne, 2002b): d2 z dz FðtÞ þ 2jv0 þ v20 z ¼ ð5:4Þ dt 2 dt m pﬃﬃﬃﬃ where pﬃﬃﬃﬃﬃ j ¼ c=2 km (damping ratio) and v0 ¼ k=m (natural frequency).

Damping constant c

Stiffness k

Fixed support FIGURE 5.3 Linear one-Dof system subjected to a force. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-6

Vibration Monitoring, Testing, and Instrumentation

5.3.3.2

Shock Deﬁned by an Acceleration

Let us set x€ ðtÞ as an acceleration applied to the base of a linear one-DoF mechanical system, with y€ðtÞ the absolute acceleration response of the mass, m; and zðtÞ the relative displacement of the mass, m; with respect to the base (Figure 5.4). The equation of the movement is written as above: m

d2 y dy dx ¼ 2kðy 2 xÞ 2 c 2 dt 2 dt dt

Mass m

Damping constant c

Stiffness k

Moving base

ð5:5Þ

that is d2 y dy dx þ 2jv0 þ v20 y ¼ v20 xðtÞ þ 2jv0 2 dt dt dt ð5:6Þ or while setting zðtÞ ¼ yðtÞ 2 xðtÞ d2 z dz d2 x þ v20 z ¼ 2 2 þ 2jv0 2 dt dt dt

Absolute reference FIGURE 5.4 Linear one-DoF system subjected to acceleration. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

ð5:7Þ

The differential equation (Equation 5.7) can be integrated by parts or by using the Laplace transformation. If the excitation is an acceleration of the support, the response relative displacement is given, for zero initial conditions, by an integral called Duhamel’s integral: qﬃﬃﬃﬃﬃﬃﬃﬃ ðt 21 pﬃﬃﬃﬃﬃﬃﬃﬃ ð5:8Þ zðtÞ ¼ x€ ðaÞe2jv0 ðt2aÞ sin v0 1 2 j2 ðt 2 aÞda v 0 1 2 j2 0 where a is an integration variable homogeneous with time. The absolute acceleration of the mass is given by qﬃﬃﬃﬃﬃﬃﬃﬃ ðt v0 x€ ðaÞe2jv0 ðt2aÞ ½ð1 2 2j2 Þsin v0 1 2 j2 ðt 2 aÞ y€ ðtÞ ¼ pﬃﬃﬃﬃﬃﬃﬃﬃ 1 2 j2 0 qﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃ þ 2j 1 2 j2 cos v0 1 2 j2 ðt 2 aÞ da

5.3.4 5.3.4.1

ð5:9Þ

Deﬁnitions Response Spectrum

This is a curve representative of the variations of the largest response of a linear one-DoF system subjected to a mechanical excitation, plotted against its natural frequency, f0 ¼ v0 =2p; for a given value of its damping ratio (see Chapter 3). 5.3.4.2

Absolute Acceleration Shock Response Spectrum

In the most usual cases where the excitation is deﬁned by an absolute acceleration of the support or by a force applied directly to the mass, the response of the system can be characterized by the absolute acceleration of the mass (which can be measured using an accelerometer ﬁxed to this mass). The response spectrum is then called the absolute acceleration SRS. 5.3.4.3

Relative Displacement Shock Spectrum

In similar cases, we often calculate the relative displacement of the mass with respect to the displacement of the base of the system. This displacement is proportional to the stress created in the

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Mechanical Shock

5-7

spring (since the system is regarded as linear). In practice, one generally expresses in ordinates the quantity v20 zsup ; which is called the equivalent static acceleration (Biot, 1941). This product has the dimensions of acceleration, but does not represent the absolute acceleration of the mass, except when damping is zero. However, when damping is close to the current values observed in mechanics, and in particular when j ¼ 0:05; as a ﬁrst approximation one can assimilate v20 zsup to the absolute acceleration y€ sup of the mass, m (Lalanne, 1975, 2002b). The quantity v20 zsup is termed pseudo-acceleration. In the same way, one terms the product v0 zsup pseudo-velocity. The spectrum giving v20 zsup vs. the natural frequency is named the relative displacement shock spectrum. In each of these two important categories, the response spectrum can be deﬁned in various ways according to how the largest response at a given frequency is characterized. 5.3.4.4 Primary Positive Shock Response Spectrum or Initial Positive Shock Response Spectrum This is the highest positive response observed during the shock. 5.3.4.5

Primary (or Initial) Negative Shock Response Spectrum

This is the highest negative response observed during the shock. 5.3.4.6

Secondary (or Residual) Shock Response Spectrum

This is the largest response observed after the end of the shock. Here also, the spectrum can be positive or negative.

Example An example giving standardized primary and residual relative displacement SRS curves for a half-sine pulse is shown in Figure 5.5. 5.3.4.7

Positive (or Maximum Positive) Shock Response Spectrum

This is the largest positive response due to the shock, without reference to the duration of the shock. It thus corresponds to the envelope of the positive primary and residual spectra. Half-sine (1 m/s2 - 1 s)

2.0

Primary positive spectrum

w 20 zsup (m/s2)

1.5 1.0

Residual positive spectrum

0.5 0.0 −0.5

Primary negative spectrum

−1.0 −1.5 0.0

Residual negative spectrum 0.5

1.0

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

x = 0.05 4.0

4.5

5.0

FIGURE 5.5 Standardized primary and residual relative displacement SRS of a half-sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-8

5.3.4.8

Vibration Monitoring, Testing, and Instrumentation

Negative (or Maximum Negative) Shock Response Spectrum

This is the largest negative response due to the shock, without reference to the duration of the shock. As before, it corresponds to the envelope of the negative primary and residual spectra. 5.3.4.9

Maximax Shock Response Spectrum

This is the envelope of the absolute values of the positive and negative spectra. 5.3.4.10

Choice of Shock Response Spectrum

Which spectrum must be used? Absolute acceleration SRS can be useful when absolute acceleration is the parameter easiest to compare with a characteristic value (as in a study of the effects of a shock on a man, a comparison with the speciﬁcation of an electronics component, etc.). In practice, it is very often the stress (and thus the relative displacement) which seems the most interesting parameter. The spectrum is primarily used to study the behavior of a structure, to compare the severity of several shocks (the stress created is a good indicator), to write test speciﬁcations (as it is also a comparison between the real environment and the test environment), or to dimension a suspension (relative displacement and stress are then useful). The damage is assumed to be proportional to the largest value of the response, i.e., to the amplitude of the spectrum at the frequency considered, and it is of little importance for the system whether this maximum, zm ; takes place during or after the shock. The most interesting spectra are thus the positive and negative spectra that are most frequently used in practice, with the maximax spectrum. The distinction between positive and negative spectra must be made each time the system, if dissymmetrical, behaves differently, for example under different tension and compression. It is, however, useful to know these various deﬁnitions so as to be able to correctly interpret the curves published.

The Shock Response Spectrum is a curve representative of the variations of the largest response of a linear one-DoF system subjected to a mechanical excitation, plotted against its natural frequency, for a given value of its damping ratio. The response can be deﬁned by the pseudo-acceleration, v20 zsup (relative displacement shock spectrum) or by the absolute acceleration of the mass (absolute acceleration SRS). For the usual values of Q; the spectra are very close. The most interesting spectra are the positive and negative spectra, which are most frequently used in practice, with the maximax spectrum.

The relation between the various types of SRS that have been discussed here is shown in Figure 5.6.

Primary (initial) positive SRS Primary (initial) negative SRS

Positive SRS

Secondary (residual) positive SRS

Negative SRS

Maximax SRS

Secondary (residual) negative SRS FIGURE 5.6

Relation between the different types of SRS.

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Mechanical Shock

Standardized Shock Response Spectrum

5.3.5.1

Deﬁnition

For a given shock, the spectra plotted for various values of the duration and the amplitude are similar in shape. It is thus useful, for simple shocks, to have a standardized or reduced spectrum plotted in dimensionless coordinates, while plotting on the abscissa the product f0 t (instead of f0 ) or v0 t and on the ordinate the spectrum/shock pulse amplitude ratio, v20 zm =€xm ; which, in practice, amounts to tracing the spectrum of a shock of duration equal to 1 sec and amplitude 1 m/sec2. This is shown in Figure 5.7. These standardized spectra can be used for two purposes: *

*

1.4 1.2 SRS / xm

5.3.5

5-9

1.0 0.8 0.6 0.4 Q = 10

0.2 0.0 0.0

1.0

2.0

f0t

3.0

4.0

5.0

FIGURE 5.7 Standardized positive SRS of a terminal peak sawtooth pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Plotting of the spectrum of a shock of the same form, but of arbitrary amplitude and duration. Investigating the characteristics of a simple shock of which the spectrum envelope is a given spectrum (resulting from measurements from the real environment).

5.3.5.2

Standardized Shock Response Spectra of Simple Shocks

Figure 5.8 to Figure 5.15 give the reduced SRSs for various pulse forms, with unit amplitude and unit duration, for several values of damping. To obtain the spectrum of a particular shock of arbitrary amplitude, x€ m ; and duration, t (different from 1) from these spectra, it is enough to regraduate the scales as follows: *

*

For the amplitude, multiply the reduced values by x€ m : For the abscissae (x-axis values), replace each value fð¼ f0 tÞ by f0 ¼ f=t:

We will see later on how these spectra can be used for the calculation of test speciﬁcations.

2.0 1.5

Half-sine (1 m/s2 - 1 s) 0

0.05

1.0 w 20 zsup (m/s2)

0.025

0.1 0.25 0.5

Positive spectra

0.5 0.5

0.0 −0.5

0.25

−1.0

0.1

−1.5 −2.0 0.0

0

0.5

0.05 0.025

1.0

1.5

Negative spectra 2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.8 Standardized positive and negative relative displacement SRS of a half-sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-10

Vibration Monitoring, Testing, and Instrumentation

Half-sine (1 m/s2 - 1 s)

Absolute acceleration ysup (m/s2)

2.0

0.025

0

1.5

0.1

1.0

0.5

0.05

Positive spectra

0.25

0.5 0.0

0.5

−0.5

0.25

−1.0

0.1 0.05

−1.5

0

−2.0 0.0

Negative spectra

0.025

0.5

1.0

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.9 Standardized positive and negative absolute acceleration SRS of a half-sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

5.3.5.2.1 Half-Sine Pulse Figure 5.8 and Figure 5.9 show the standardized SRS curves in this case. 5.3.5.2.2 Versed Sine Pulse Figure 5.10 shows the standardized SRS curves in this case. 5.3.5.2.3 Terminal Peak Sawtooth Pulse Figure 5.11 and Figure 5.12 show the standardized SRS curves for terminal peak sawtooth (TPS) pulse. 5.3.5.2.4 Rectangular Pulse Figure 5.13 gives the standardized SRS curves for a rectangular pulse shock. 5.3.5.2.5 Trapezoidal Pulse Figure 5.14 presents the standardized SRS curve for a trapezoidal pulse. A comparison of various SRS curves is given in Figure 5.15. Versed-sine (1 m/s2 - 1 s)

2.0 1.5

0.025

0

0.05

1.0 w 20 zsup (m/s2)

Positive spectra

0.1 0.25 0.5

0.5 0.0

0.5

−0.5

0.25

−1.0

0.1 0.05

−1.5 −2.0 0.0

0

0.5

1.0

Negative spectra

0.025

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.10 Standardized positive and negative relative displacement SRS of a versed sine pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Mechanical Shock

5-11

1.5

T.P.S. (1 m/s2 - 1 s) 0.025 0.05

0

w 20 zsup (m/s2)

1.0 0.5

0.1

0.25 0.5

0.0

0.5 0.25

−0.5

0.1 0.05

−1.0 −1.5 0.0

Positive spectra

Negative spectra

0.025

0

0.5

1.0

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.11 Standardized positive and negative relative displacement SRS of a TPS pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

5.3.6

Choice of Damping

The choice of damping should be carried out according to the structure subjected to the shock. When this is not known, or studies are being carried out with a view to comparison with other already calculated spectra, the outcome is that one plots the shock response spectra with a damping ratio equal to 0.05 (i.e., Q ¼ 10; see Chapter 19). It is an approximately average value for the majority of structures. Unless otherwise speciﬁed, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping, this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram.

5.3.7

Shock Response Spectra Domains

Three domains can be schematically distinguished for shock spectra. 1. An impulse domain at low frequencies, in which the amplitude of the spectrum (and thus of the response) is lower than the amplitude of the shock: The system reduces the effects of the shock. It is thus

1.5

T.P.S. (1 m/s2−1s−td = 0.1s) 0

w 20 zsup (m/s2)

1.0 0.5

0.1 0.25 0.5

0.0

Positive spectra

0.5 0.25

−0.5

0.1 0.05

−1.0

Negative spectra

0.025

0

−1.5 0.0

0.025 0.05

0.5

1.0

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.12 Standardized positive and negative relative displacement SRS of a TPS pulse with nonzero decay time. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Vibration Monitoring, Testing, and Instrumentation

w 20 zsup (m/s2)

5-12

Rectangle (1 m / s2 - 1s)

2.5 2.0 1.5 1.0 0.5 0.0 −0.5 −1.0 −1.5 −2.0 −2.5 0.0

0

0.025

0.05

Positive spectra

0.1 0.25 0.5

0.5 0.25 0.1 0.05 0.025

0

0.5

1.0

Negative spectra

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.13 Standardized positive and negative relative displacement SRS of a rectangular pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

in this impulse region that it would be advisable to choose the natural frequency of an isolation system to the shock, from which we can deduce the stiffness envisaged of the insulating material: k ¼ mv20 ¼ 4p2 f02 m ð5:10Þ with m being the mass of the material to be protected. The shock here is of very short duration with respect to the natural period of the system. In this impulse region ð0 # f0 t # 0:2Þ: *

*

The form of the shock has little inﬂuence on the amplitude of the spectrum. Only (for a given damping value) the velocity change DV associated with the shock, equal to the algebraic surface under the curve x€ ðtÞ is important. The slope p at the origin of the spectrum plotted for zero damping in linear scales is proportional to the velocity change DV corresponding to the shock pulse (Lalanne, 2002b): dðv20 zsup Þ p¼ ¼ 2pDV ð5:11Þ df0 This relation is approximate if damping is small. Trapezoid (1 m/s2−1 s−t r = 0.1 s−t d = 0.1 s)

2.5

0

2.0

0.025

w 02 zsup (m/s2)

1.5

Positive spectra

0.05 0.1 0.25 0.5

1.0 0.5 0.0

0.5

−0.5

0.25

−1.0

0.1

−1.5 −2.0 −2.5 0.0

0.05 0.025

0

0.5

1.0

Negative spectra

1.5

2.0 2.5 3.0 Frequency (Hz)

3.5

4.0

4.5

5.0

FIGURE 5.14 Standardized positive and negative relative displacement SRS of a trapezoidal pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Mechanical Shock

5-13

2.0 1.5

w 20 zsup (m/s2)

1.0 ξ = 0.05

0.5 0.0 −0.5 −1.0 −1.5 −2.0 0.0

FIGURE 5.15 *

1.0

2.0 3.0 Frequency (Hz)

4.0

5.0

Comparison between the SRS of the three main simple shock waveforms.

The positive and negative spectra are in general the residual spectra (it is sometimes necessary that the frequency of spectrum is very small, and there can be exceptions for certain long shocks in particular). They are nearly symmetrical so long as damping is small.

2. A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping: All occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), as the natural period of the system is small compared with the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. Real shocks having necessarily a rise time different from zero, this restriction remains theoretical. 3. An intermediate domain in which there is dynamic ampliﬁcation of the effects of the shock, the natural period of the system being close to the duration of the shock: This ampliﬁcation, which is more or less signiﬁcant depending on the shape of the shock and the damping of the system, does not exceed 1.77 for shocks of traditional simple shape (half-sine, versed sine, TPS). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid. Various domains of an SRS are illustrated in Figure 5.16.

Example Consider a half-sine shock pulse, amplitude x€ m ¼ 50 m=sec2 ; duration t ¼ 11 msec; positive SRS (relative displacement) for a damping ratio j ¼ 0:05: The slope p of the SRS (Figure 5.16) at the origin is equal to p ¼ 30:6=15 ¼ 2:04 m=sec; yielding DV <

p 2:04 ¼ < 0:325 m=sec 2p 2p

a value to be compared with the surface under the half-sine shock pulse (Table 5.1): DV ¼

5.3.8

2 2 x€ t ¼ £ 50 £ 11 £ 1023 < 0:35 m=sec p m p

Algorithms for Calculation of the Shock Response Spectra

Various algorithms have been developed to solve the second-order differential equation (Equation 5.7; O’Hara, 1962; Gaberson, 1980; Smallwood, 1981; Cox, 1983; Hughes and Belytschko, 1983; Irvine, 1986;

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5-14

Vibration Monitoring, Testing, and Instrumentation 90

Q factor : 10

Impulse 80 Domain 70

Static Domain

m/s2

60

30.6

Intermediate Domain

50

Shock amplitude

40

Slope p ≈ 2 π ∆V

30 20 10 0

0

15

50

100

FIGURE 5.16

150

200

250

300

350

400

450

500

Hz Shock response spectra domains.

Dokainish and Subbaraj, 1989; Colvin and Morris, 1990; Hale and Adhami, 1991; Mercer and Lincoln, 1991; Seipel, 1991; Merritt, 1993; Grivelet, 1996). Very reliable results are obtained in particular with those of Cox (1983) and Smallwood (1986, 2002).

5.3.9 Choice of the Digitization Frequency of the Signal The SRS is obtained by considering the largest peak of the response of a one-DoF system. This response is in general calculated by the algorithms with the same temporal step as that of the shock signal. First of all, the digitization (sampling) frequency must be sufﬁcient to correctly represent the signal itself and in particular not to truncate its peaks. Two cases are shown in Figure 5.17 and Figure 5.18. When the natural frequency of the one-DoF system is lower than the smallest shock frequency, the detection of peaks of the response can be carried out accurately even if the signal digitization (sampling) frequency is insufﬁcient for correctly describing the shock (Figure 5.17). The error on the SRS is then only related to the poor digitization (sampling) of shock and results in an inaccuracy on the velocity change associated with the shock, i.e., on the SRS slope at low frequency. Even if the sampling frequency allows a good representation of the shock, it can be insufﬁcient for the response when the natural frequency of the system is higher than the maximum frequency of

Shock pulse Response

FIGURE 5.17 Sampling frequency sufﬁcient for the response and too low for the shock pulse (error on the slope of SRS at low frequency).

Response Shock pulse

FIGURE 5.18 Sampling frequency sufﬁcient for the shock pulse and too low for the response (error on the SRS at high frequency).

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Mechanical Shock

5-15

the signal (Figure 5.18). The error is here related to the detection of the largest peak of the response, which occurs throughout shock (primary spectrum). Figure 5.19 shows the error made in the more stringent case when the points surrounding the peak are symmetrical with respect to the peak. If we set

Sampled data Response (sinusoid)

Sample frequency SRS maximum frequency

it can be shown that, in this case, the error made according to the sampling factor, SF ; is equal to (Sinn and Bosin, 1981; Wise, 1983) eS ¼ 100 1 2 cos

p SF

FIGURE 5.19 Error made in measuring the amplitude of the peak. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

ð5:12Þ

The sampling frequency must be higher than 16 times the maximum frequency of the spectrum so that the error made at high frequency is lower than 2% (23 times the maximum frequency for an error lower than 1%). The rule of thumb often used to specify a sampling factor equal to ten can lead to an error of about 5%. Percentage error as a function of the sampling factor is plotted in Figure 5.20. Also see Table 5.2. Algorithms use generally the same sampling frequency for the shock input and the response of the one-DoF system. This choice led to deﬁne the sampling frequency according to the highest SRS frequency. In order to decrease the computing time, it could be interesting to determine a sampling frequency varying with each natural frequency (Smallwood, 2002).

5

4

Error (%)

SF ¼

Error

3

2

1

0

5

10

15 20 Sampling factor

25

30

FIGURE 5.20 Error made in measuring the amplitude of the peak plotted against sampling factor. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Note: The sampling frequency must be higher than 16 times the maximum frequency of the spectrum so that the error made at high frequency is lower than 2%.

TABLE 5.2 Some Sampling Factors with Corresponding Error on the SRS SRS Maximum Frequency Multiplied by

Error (%)

23 16 10

1 2 5

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5-16

Vibration Monitoring, Testing, and Instrumentation

5.3.10 Use of Shock Response Spectra for the Study of Systems with Several Degrees of Freedom By deﬁnition, the response spectrum gives the largest value of the response of a linear single-DoF system subjected to a shock. If the real structure is comparable to such a system, the SRS can be used to evaluate this response directly. This approximation is often possible, with the displacement response being mainly due to the ﬁrst mode. In general, however, the structure comprises several modes, which are simultaneously excited by the shock. The response of the structure consists of the algebraic sum of the responses of each excited mode. The maximum response of each one of these modes can be read on the SRS, but the following apply. *

*

One does not have any information concerning the moment of occurrence of these maxima. The phase relationships between the various modes are not preserved and the exact way in which the modes are combined cannot be known simply. The SRS is plotted for a given constant damping over all the frequency range, whereas this damping varies from one mode to another in the structure.

It thus appears difﬁcult to use an SRS to evaluate the response of a system presenting more than one mode. However, it happens that this is the only possible means. The problem is to know how to combine these “elementary” responses so as to obtain the total response and to determine, if need be, any suitable participation factors dependent on the distribution of the masses of the structure, of the shapes of the modes, etc. When there are several modes, several proposals have been made to limit the value of the total response of the mass j of the one of the DoF starting from the values read on the SRS, as follows. *

*

*

*

*

*

Add the values with the maxima of the responses of each mode, without regard to the phase (Benioff, 1934). Sum the absolute values of the maximum modal responses (Biot, 1932). As it is not probable that the values of the maximum responses take place all at the same moment with the same sign, the real maximum response is lower than the sum of the absolute values. This method gives an upper limit of the response and thus has a practical advantage: the errors are always on the side of safety. However, it sometimes leads to excessive safety factors (Shell, 1966). Perform an algebraic sum of the maximum responses of the individual modes. A study showed that, in the majority of the practical problems, the distribution of the modal frequencies and the shape of the excitation are such that the possible error remains probably lower than 10% (Rubin, 1958; Fung and Barton, 1958). Add to the response of the ﬁrst mode a ﬁxed percentage of the responses of the other modes, or increase in the response of the ﬁrst mode by a constant percentage (Clough, 1955). Combine the responses of the modes by taking the square root of the sum of the squares to obtain an estimate of the most probable value (Merchant and Hudson, 1962). This criterion gives values of the total response lower than the sum of the absolute values and provides a more realistic evaluation of the average conditions (Ostrem and Rumerman, 1965; Ridler and Blader, 1969). Average the sum of the absolute values and the square root of the sum of the squares (Jennings, 1958). One can also choose to deﬁne positive and negative limiting values starting from a system of weighted averages. For example, the relative displacement response of the mass j is estimated by ﬃ sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ n n X X 2 zm þ p lzm li max lzj ðtÞl ¼ t$0

i¼1

i

pþ1

i¼1

ð5:13Þ

where the terms lzm li are the absolute values of the maximum responses of each mode and p is a weighting factor (Merchant and Hudson, 1962).

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Mechanical Shock

5.4

5-17

Pyroshocks

The aerospace industry uses many pyrotechnic devices such as explosive bolts, squib valves, jet cord, and pin pushers. During their operation, these devices generate shocks which are characterized by very strong acceleration levels at very high frequencies that can be sometimes dangerous for the structures, but especially for the electric and electronic components involved. An example of a pyroshock is given in Figure 5.21. Pyroshock intensity is often classiﬁed according to the distance from the point of detonation of the device. Agreement on classifying intensity according to this criterion is not unanimous. Two ﬁelds are generally considered (Table 5.3): *

*

The near-ﬁeld, close to the source (material within about 15 cm of point of detonation FIGURE 5.21 Example of a pyroshock. (Source: of the device, or about 7.5 cm for less Lalanne, Chocs Mecaniques, Hermes Science Publiintense pyrotechnic devices), in which the cations. With permission.) effects of the shocks are primarily related to the propagation of a stress wave in the material. The far-ﬁeld (material beyond about 15 cm for intense pyrotechnic devices, or beyond 7.5 cm for less intense devices) in which the shock is then propagated whilst attenuating in the structure and from which the effects of this wave combine with a damped oscillatory response of the structure at its frequencies of resonance (or the structural response only).

Three ﬁelds are sometimes suggested: the near-ﬁeld, the mid-ﬁeld (same deﬁnition as the far-ﬁeld above, between 15 and 60 cm, or 7.5 and 15 cm for the less intense shocks), and the far-ﬁeld, where only the structural response effect persists. An investigation by Moening (1986) showed that the causes of observed failures on the American launchers between 1960 and 1986 (63 due to pyroshocks) are mainly the difﬁculty in evaluating these shocks a priori, especially the lack of consideration of these excitations during design and the absence of rigorous test speciﬁcations. Such shocks have the following general characteristics. *

The levels of acceleration are very important; the shock amplitude is not simply related to the quantity of explosive used (Hughes, 1983). The quantity of metal cut by a jet cord is, for example, a more signiﬁcant factor than the mass of the explosive.

TABLE 5.3

Characteristics of Each Pyroshock Intensity Domain Field

Near ﬁeld (stress wave propagation effect) Far-ﬁeld (stress wave propagation effect þ structural response effect)

Distance from the Source (cm)

Intense Pyrotechnic Devices (cm)

,7.5

,15

.7.5

.15

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

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Shock Amplitude Frequency Content .5000g, up to 300,000g above 100,000 Hz 1000 to 5000g above 10,000 Hz

5-18

Vibration Monitoring, Testing, and Instrumentation

*

*

*

The signals assume an oscillatory shape. The shocks have very close components according to three axes; their positive and negative response spectra are curves that are roughly symmetrical with respect to the axis of the frequencies. They begin at zero frequency with a very small slope at the origin, grow with the frequency until a maximum located at some kHz, even a few tens of kHz, is reached, and then tend according to the rule towards the amplitude of the temporal signal. Due to their contents at high frequencies, such shocks can damage electric or electronic components. The a priori estimate of the shock levels is neither easy nor precise.

These characteristics make pyroshocks difﬁcult to measure, requiring sensors that are able to accept amplitudes of 100,000g, frequencies being able to exceed 100 kHz, with important transverse components. They are also difﬁcult to simulate. The dispersions observed in the response spectra of shocks measured under comparable conditions are often important, 3 dB with more than 8 dB compared with the average value, according to the authors (Smith, 1984, 1986). The reasons for this dispersion are in general related to inadequate instrumentation and the conditions of measurement (Smith, 1986): *

*

*

*

The ﬁxing of the sensors on the structure using insulated studs or wedge which act like mechanical ﬁlters. Zero shift, due to the fact that high accelerations make the crystal of the accelerometer work in a temporarily nonlinear ﬁeld (this shift can affect the calculation of the SRS). Saturation of the ampliﬁers. Resonance of the sensors.

With correct instrumentation, the results of measurements carried out under the same conditions are actually very close. The spectrum does not vary with the tolerances of manufacture and the assembly tolerances.

5.5

Use of Shock Response Spectra

5.5.1

Severity Comparison of Several Shocks

A shock, A, is regarded as more severe than a shock, B, if it induces in each resonator a larger stress. One then carries out an extrapolation, which is certainly open to criticism, by supposing that, if shock A is more severe than shock B when it is applied to all the standard resonators, it is also more severe with respect to an arbitrary real structure (which cannot be linear nor have a single DoF).

5.5.2

Test Speciﬁcation Development from Real Environment Data

5.5.2.1

Synthesis of Spectra

Let us consider the most complex case where the real environment, described by curves of acceleration against time, is supposed to be composed of p different events (handling shock, inter-stage cutting shock on a satellite launcher, etc.), with each one of these events itself characterized by ri successive measurements. These ri measurements allow a statistical description of each event. The following procedure holds for each one (Lalanne, 2002b; see Figure 5.22). *

Calculate the SRS of each signal recorded with the damping ratio of the principal mode of the structure if this value is known, if not, use the conventional value 0.05. In the same way, the

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Mechanical Shock

5-19

Event # 1 Handling shock r1 measured data

Calculation of r1 S.R.S.

Mean and standard deviation spectra or envelope

k (m + 3 s) or k × env.

Event # 2 Loading shock r2 measured data

Calculation of r2 S.R.S.

Mean and standard deviation spectra or envelope

k (m + 3 s) or k × env.

Event # p Ignition shock rp measured data

Calculation of rp S.R.S.

Mean and standard deviation spectra or envelope

k (m + 3 s) or k × env.

Envelope × Test factor FIGURE 5.22

*

Envelope

Process of developing a speciﬁcation from real shocks measurements.

frequency band of analysis will have to envelop the principal resonant frequencies of the structure (known or foreseeable frequencies). If the number of measurements is sufﬁcient, calculate the mean spectrum, m (mean of the points at each frequency) as well as the standard deviation spectrum (s), then the mean spectrum þ a standard deviations, according to the frequency; if it is insufﬁcient, make the envelope of the spectra.

The value of a can be either arbitrary (for example 2.5 or 3) or the result of a statistical calculation. It is often considered that the SRS amplitudes obey to a log-normal distribution. If yi is the logarithm of the SRS amplitude, yj ¼ log10 SRSj ; the real environment envelope (for a given probability P0 of not exceeding at the conﬁdence level p0 ) can be deﬁned by SRSEnv ¼ 10my þasy

ð5:14Þ

where my and sy are, respectively, the mean and the standard deviation of the yj values: my ¼

ri 1 X y ri j¼1 j

vﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ u ri uX u ðy 2 my Þ2 u u j¼1 j sy ¼ t ri 2 1

ð5:15Þ

ð5:16Þ

The number of standard deviations, a; is given in Table 5.4 for different values of ri ; P0 ; and p0 : SRSEnv can also be deﬁned as the upper one-sided normal tolerance interval for which 100 P0 % of the values will lie below the limit with 100p0 % conﬁdence. *

Apply the mean spectrum or the mean spectrum þ a standard deviations a statistical uncertainty coefﬁcient (Lalanne, 2002d), calculated for a probability of tolerated maximum failure (taking into account the uncertainties related to the dispersion of the real environment and of the mechanical strength), or contractual (if one uses the envelope).

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5-20

Vibration Monitoring, Testing, and Instrumentation

TABLE 5.4 Number of Standard Deviations Corresponding to a Given Probability of not Exceeding, P0 ; at the Conﬁdence Level, p0 p0 ¼ 0:75

ri \P0

3 4 5 6 7 8 9 10 15 20 30 40 50

p0 ¼ 0:90

p0 ¼ 0:95

0.75

0.90

0.95

0.99

0.75

0.90

0.95

0.99

0.75

0.90

0.95

0.99

1.464 1.256 1.152 1.087 1.043 1.010 0.984 0.964 0.899 0.865 0.825 0.803 0.788

2.501 2.134 1.961 1.860 1.791 1.740 1.702 1.671 1.577 1.528 1.475 1.445 1.426

3.152 2.680 2.463 2.336 2.250 2.190 2.141 2.103 1.991 1.933 1.869 1.834 1.811

4.396 3.726 3.421 3.243 3.126 3.042 2.977 2.927 2.776 2.697 2.613 2.568 2.538

2.602 1.972 1.698 1.540 1.435 1.360 1.302 1.257 1.119 1.046 0.966 0.923 0.894

4.258 3.187 2.742 2.494 2.333 2.219 2.133 2.065 1.866 1.765 1.657 1.598 1.560

5.310 3.957 3.400 3.091 2.894 2.755 2.649 2.568 2.329 2.208 2.080 2.010 1.965

7.340 5.437 4.666 4.242 3.972 3.783 3.641 3.532 3.212 3.052 2.884 2.793 2.735

3.804 2.619 2.149 1.895 1.732 1.617 1.532 1.465 1.268 1.167 1.059 0.999 0.961

6.158 4.416 3.407 3.006 2.755 2.582 2.454 2.355 2.068 1.926 1.778 1.697 1.646

7.655 5.145 4.202 3.707 3.399 3.188 3.031 2.911 2.566 2.396 2.220 2.126 2.065

10.552 7.042 5.741 5.062 4.641 4.353 4.143 3.981 3.520 3.295 3.064 2.941 2.863

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

Each event thus being synthesized in only one spectrum, one proceeds to an envelope of all the spectra obtained to deduce from it an SRS covering the totality of the shocks of the life proﬁle. After multiplication by a test factor, which takes account of the number of tests performed to demonstrate the resistance of the equipment (Lalanne, 2002d), this spectrum will be used as reference “real environment” for the determination of the speciﬁcation. The reference spectrum can consist of the positive and negative spectra or the envelope of their absolute value (maximax spectrum). In this last case, the speciﬁcation will have to be applied according to the two corresponding half-axes of the test item. 5.5.2.2

Nature of the Speciﬁcation

There is an inﬁnity of shocks having a given response spectrum. This property is related to the very great loss of information in computing the SRS, since one retains only the largest value of the response according to the time to constitute the SRS at each natural frequency. According to the characteristics of the spectrum and available means, the speciﬁcation can be expressed in the forms given below. *

It can be a simple shape signal according to the time realizable on the usual shock machines (half-sine, TPS, rectangular pulse).

One can thus try to ﬁnd a shock of simple form, in which the spectrum is closed to the reference spectrum, characterized by its form, its amplitude, and its duration. It is in general desirable that the positive and negative spectra of the speciﬁcation, respectively, cover the positive and negative spectra of the ﬁeld environment. If this condition cannot be obtained by application of only one shock (owing to the particular shape of the spectra, and the limitations of the facilities), the speciﬁcation will be made up of two shocks, one on each half-axis. The envelope must be approaching the reference SRS as well as possible, if possible on all the spectrum in the frequency band retained for the analysis, if not in a frequency band surrounding the resonant frequencies of the test item (if they are known). *

It can be a SRS. In this case, the speciﬁcation is directly the reference SRS.

5.5.2.3

Choice of Shape

The choice of the shape of a shock is carried out by a comparison of the shapes of the positive and negative spectra of the real environment with those of the spectra of the usual shocks of simple shape (half-sine, TPS, rectangle; Figure 5.23).

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Mechanical Shock

5.5.2.4

5-21

Amplitude

The amplitude of a shock is obtained by plotting the horizontal straight line that closely envelops the positive reference SRS at high frequency (Figure 5.24). This line cuts the y-axis at a point which gives the amplitude sought (here, one uses the property of the spectrum at high frequencies, which tends in this zone towards the amplitude of the signal in the time domain).

S.R.S.

Half- sine

f

T.P.S.

S.R.S.

f

5.5.2.5

Duration

S.R.S.

Rectangle The shock duration is given by the coincidence of a particular point of the reference spectrum f (Figure 5.24) and the reduced spectrum of the simple shock selected above. Q = 10 One in general considers the abscissa, f01 , of the ﬁrst point which reaches the value of the asymptote at the high frequencies (amplitude FIGURE 5.23 Shapes of the SRS of the realizable of shock) as shown in Figure 5.25. Table 5.5 shocks on the usual machines. (Source: Lalanne, Chocs joins together some values of this abscissa for the Mecaniques, Hermes Science Publications. With permost usual simple shocks according to the Q mission.) factor (Lalanne, 2002b). Another possibility is to use the coordinates of the ﬁrst (higher) peak of the SRS, as given in Table 5.6. Notes:

1. If the calculated duration must be rounded (in milliseconds), the higher value should always be considered, so that the spectrum of the speciﬁed shock remains always higher or equal to the reference spectrum. 500

S.R.S. (m/s2)

340

400

Envelope at the high frequencies

300 Real environment

200 100 0 −100 −200 −300 −400

x = 0.05 0

49.5

100

200 Frequency (Hz)

300

400

FIGURE 5.24 Determination of the amplitude and duration of the speciﬁcation. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-22

Vibration Monitoring, Testing, and Instrumentation

2. It is in general difﬁcult to carry out shocks of duration lower than 2 msec on standard shock machines (except for very light equipment).

Example

1.0 S.R.S. / xm

One will validate the speciﬁcation by checking that the positive and negative spectra of the shock thus determined will envelop the respective reference spectra and one will verify, if the resonant frequencies of the test item are known, that one does not overtest exaggeratedly at these frequencies.

T.P.S. dimensionless S.R.S.

1.5

:

0.5 0.0

−0.5 −1.0 −1.5 0.0

x= 0.05 0.415

1.0

2.0 f0t

3.0

4.0

As an example, let us consider the positive and negative spectra characterizing the real environ- FIGURE 5.25 Determination of the shock duration. ment plotted in Figure 5.24, which is a result of a (Source: Lalanne, Chocs Mecaniques, Hermes Science true synthesis. It is noted that the negative Publications. With permission.) spectrum preserves a signiﬁcant level throughout the entire frequency domain (the beginning of the spectrum being excluded). The most suitable simple shock shape is the TPS. The shock amplitude, whatever its waveform, is equal to 340 m/sec2. The abscissa, f01 , of the ﬁrst point that reaches the value of the asymptote is equal to 0.415. From this value, f0 ¼ 49:5 Hz; the duration is given by t ¼ 0:415=49:5 ¼ 0:0084 sec: The duration of the shock will thus be (rounding up) t ¼ 9 msec; which slightly moves the spectrum towards the left and makes it possible to better cover the low frequencies. Figure 5.26 shows the spectra of the environment and those of the TPS pulse thus determined. The main steps of deriving a shock test speciﬁcation from the SRS of a real environment are outlined in Table 5.7.

TABLE 5.5 Values of the Dimensionless Frequency Corresponding to the First Passage of the SRS by the Amplitude Unit j

Q

3 4 5 6 7 8 9 10p 15 20 30 40 50 1

0.1667 0.1250 0.1000 0.0833 0.0714 0.0625 0.0556 0.0500 0.0333 0.0250 0.0167 0.0125 0.0100 0.0000

f01 Half-sine

TPS

Rectangle

0.358 0.333 0.319 0.310 0.304 0.293 0.295 0.293 0.284 0.280 0.276 0.274 0.272 0.267

0.564 0.499 0.468 0.449 0.437 0.427 0.421 0.415 0.400 0.392 0.385 0.382 0.379 0.371

0.219 0.205 0.197 0.192 0.188 0.185 0.183 0.181 0.176 0.174 0.172 0.170 0.170 0.167

(p) Conventional value

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Mechanical Shock TABLE 5.6 the SRS

5-23

Values of the Dimensionless Frequency and Amplitude Corresponding to the First Peak of j

Q

3 4 5 6 7 8 9 10p 15 20 30 40 50 1

Half-sine

0.1667 0.1250 0.1000 0.0833 0.0714 0.0625 0.0556 0.0500 0.0333 0.0250 0.0167 0.0125 0.0100 0.0000

TPS

Rectangle

f0 peak

SRS Peak

f0 peak

SRS Peak

f0 peak

SRS Peak

0.875 0.86 0.845 0.840 0.830 0.83 0.830 0.826 0.820 0.820 0.815 0.810 0.813 0.810

1.4249 1.4958 1.5425 1.5757 1.600 1.6194 1.6346 1.6470 1.6854 1.7054 1.7258 1.7363 1.7426 1.7685

0.723 0.700 0.688 0.681 0.676 0.672 0.669 0.667 0.661 0.658 0.656 0.654 0.653 0.650

1.0462 1.0894 1.1182 1.1387 1.1541 1.1660 1.1755 1.1832 1.2073 1.2199 1.2328 1.2393 1.2433 1.2596

0.508 0.504 0.503 0.502 0.502 0.501 0.501 0.501 0.501 0.501 0.501 0.501 0.501 0.500

1.5880 1.6731 1.7292 1.7690 1.7985 1.8214 1.8396 1.8545 1.9005 1.9244 1.9490 1.9615 1.9691 2.000

(p) Conventional value

500 x = 0.05

T.P.S.

400

S.R.S. (m/s2)

300 200

Real environment

100 0 −100 −200 −300 −400

0

100

200 Frequency (Hz)

300

400

FIGURE 5.26 SRS of the speciﬁcation and of the real environment. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

5.5.2.6

Difﬁculties

The response spectra of shocks measured in the real environment often have a complicated shape which is impossible to envelop by the spectrum of a shock of simple shape realizable with the usual test facilities of the drop table type. This problem arises in particular when the spectrum presents an important peak (Smallwood and Witte, 1972). The spectrum of a shock of simple shape will be (see Figure 5.27): either an envelope of the peak, which will lead to signiﬁcant overtesting compared with the other frequencies, or envelope of the spectrum except the peak, consequently leading to undertesting at the frequencies close to the peak, if one knows that the material does not have any resonance in the frequency band around the peak. The simulation of shocks of pyrotechnic origin leads to this kind of situation.

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5-24

Vibration Monitoring, Testing, and Instrumentation TABLE 5.7 Deriving a Shock Test Speciﬁcation from the SRS of the Real Environment Speciﬁcation

Determined by

Shock waveform

Comparing the shape of the SRS of the real environment (reference) to that of the SRS of the simple shape shocks (half-sine, TPS or rectangular waveform) The SRS amplitude at high frequency Writing that the abscissa of the ﬁrst point which reaches the value of the asymptote at high frequencies (amplitude of shock) is the same for the reduced SRS of the chosen simple shock and the reference SRS

Shock amplitude Shock duration

Under-test for 1

Over-test

Over-test for 2

2 1

f

fp

f

FIGURE 5.27 Examples of SRS that are difﬁcult to envelop with the SRS of a simple shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Shock pulses of simple shape (half-sine, TPS) have, in logarithmic scales, a slope of 6 dB/octave (i.e., 458) at low frequencies incompatible with those larger ones, of spectra of pyrotechnic shocks (.9 dB/octave). When the levels of acceleration do not exceed the possibilities of the shakers, simulation with control using spectra is of interest (Section 5.10). Note: In general, it is not advisable to choose a simple shock shape as a speciﬁcation when the real shock is oscillatory in nature. In addition to overtesting at low frequencies (the oscillatory shock is with very small velocity change), the amplitude of the simple shock thus calculated is more sensitive to the value of the Q factor in the intermediate frequency range.

5.6 5.6.1

Standards Types of Standards

There are two types of standards: (1) those which specify arbitrary shock pulses (IEC, ISO, MIL STD 810 C, …), and (2) those which require test tailoring (GAM EG 13, 1986; DEF STAN, 1999; MIL STD 810 F, 2000; NATO, 2000). For the ﬁrst case, the most frequently speciﬁed shock shapes are the half-sine, the TPS, and the rectangular (or trapezoidal) waveforms. In these standards, a table proposes several values of levels and durations, with preferred combinations (for example, 30g, 18 msec or 50g, 11 msec). To take account of the limitations of test facilities and unavoidable signal distortions, the shock carried out is regarded as acceptable if the time acceleration signal lies between two tolerance limits. Two shocks included within these limits can, however, have very different effects (which can be evaluated with the SRS; Lalanne, 2002b).

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Mechanical Shock

Some examples of standards are given in Figure 5.28 and Figure 5.29. In the second case, the test speciﬁcation is preferably written from real measurements corresponding to the life proﬁle of the material. The data to be used can be any of the following, in the preferential order: *

*

*

*

Functional real environment time history measurements of the material. Data measured under similar conditions and estimated to be representative. Data issued from prediction or calculation. Default values (fallback levels), obviously more arbitrary in character, to be used if measured data not available (classical pulse shock or SRS).

Derivation of the test speciﬁcation and subsequent test should be carried out, in the preferential order as follows: *

*

*

*

5-25

Integration time

1.5 D

1.2 A A 0.8 A Nominal pulse

0.4 D

0.1 D D D 2.4 D = T1 6 D = T2

Limits of tolerances

FIGURE 5.28 Half-sine pulse (NATO Stanag 4370, AECTP 403). T1: minimum time during which the pulse shall be monitored for shocks produced using a conventional shock-testing machine; T2: minimum time during which the pulse shall be monitored for shocks produced using a vibration generator.

1.15 A Ideal Sawtooth Pulse Tolerance Limits

For measured data of the same event, if the 0.07 D A measured pulse shapes are very similar, use 0.15 A 0.02 A direct reproduction of the measured data 0.05 A under shaker waveform control (if poss0.15 A ible). If the measured shock shapes are very D 0.3 D 0.05 A 0.3 A different, use the following method. For measured data of different shock events use a synthesis of measurements using FIGURE 5.29 TPS pulse (MIL STD 810 F). D: duration of nominal pulse; A: peak acceleration of SRS (see Section 5.5.2). Test on shaker nominal pulse. with SRS control if possible. If not possible, test on shock machine with a classical pulse having the same SRS. If there is no measured data of the real shock, but measured data under similar conditions, use the method as above. If there are no measured data, fallback levels and provisional values are to be replaced by results of measurement as soon as possible.

The transformation shock spectrum-signal has an inﬁnite number of solutions, and very different signals can have identical response spectra. Standards often require specifying in addition to the spectrum other complementary data such as the duration of the signal time, the velocity change during the shock or the number of cycles (less easy), in order to deal with the spectrum and the couple amplitude/duration of the signal at the same time (see Section 5.10.4). It is not correct to decompose a SRS into two separate domains in order to be able to meet a shock requirement (a low frequency component and a high frequency component). If the specimen has no signiﬁcant low natural frequency, it is permissible to allow the low frequency domain of the SRS to fall out of tolerance in order to satisfy the high frequency part of the requirement. The tolerance on the SRS amplitude should be, for example (MIL STD 810 F), 2 1.5 dB, þ3 dB over the speciﬁed frequency range; a tolerance of þ3 dB, þ 6 dB being permissible over a limited frequency range. It is generally required to determine the positive and negative spectra (absolute acceleration or relative displacement) at Q ¼ 10; at at least 1/12-octave frequency intervals.

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5-26

Vibration Monitoring, Testing, and Instrumentation

In the absence of accurate information on the number of shocks which the material will undergo in its service life, a minimum is often required of three shocks in both directions along each of the three orthogonal axes, a total of 18 shocks.

5.6.2 *

*

*

*

*

*

5.6.3

Installation Conditions of Test Item The test item should be mechanically fastened to the shock machine, directly by its normal means of attachment or by means of a ﬁxture. The mounting conﬁguration should enable the test item to be subjected to shocks along the various axes and directions as speciﬁed. External connections necessary for measuring purposes should add minimum restraint and mass. The ﬁxture should not modify the dynamic behavior of the test item. Material intended for use with isolators should be tested with its isolators. The direction of gravity or any loading factors (mechanisms, shock isolators, etc.) must be taken into account by compensation or by suitable simulation.

Uncertainty Factor

An uncertainty factor may be added to the resulting envelope if conﬁdence in the data is low or in order to take account of the dispersion of levels in the real environment when the data set is small. This factor can be arbitrary, of the order of 3 to 6 dB, for example, or determined from a reliability computation, taking account of the statistical distributions of the real environment and of the material strength (Lalanne, 2002d). It is important that all uncertainties be clearly deﬁned and that uncertainties are not superimposed upon estimates that already account for uncertainties. Note: The purpose of the test is to demonstrate that the equipment has at least the speciﬁed strength at the time of its design. However, for obvious reasons of cost, this demonstration is generally conducted only on one specimen. To take into account the variability of the strength of the material, it is possible to increase the test severity by applying a “test factor.” This second factor depends on the number of tests to be conducted and on the coefﬁcient of variation of the material strength (Lalanne, 2002d).

5.6.4

Bump Test

A bump test is a test in which a simple shock is repeated many times (DEF STAN, 1999; IEC, 1987b; AFNOR, 1993). Standardized severities are proposed. For example, half-sine, 10 g, 16 ms, 3000 bumps (shock) per axis, 3 bumps a second. The purpose of this test is not to simulate any speciﬁc service condition. It is simply considered that it could be useful as a general ruggedness test to provide some conﬁdence in the suitability of equipment for transportation in wheeled vehicles. It is intended to produce in the specimen effects similar of those resulting from repetitive shocks likely those encountered during transportation. In this test, the equipment is always fastened (with its isolators if it is normally used with isolators) to the bump machine during conditioning.

5.7 5.7.1

Damage Boundary Curve Deﬁnition

Products are placed in a package to be protected from possible free-fall drops and impacts onto a ﬂoor or a shipping platform during transport or handling. This packaging is often made up of a cushioning material (for example, honeycomb or foam) which absorbs the impact energy (related to the impact velocity) either by inelastic deformation, and which generates a shock at the entry of the material, whose

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Mechanical Shock

5-27

Product Package Product Cushion Material

Vi

Input Shock FIGURE 5.30

Shock transmitted to product during the crushing of package.

5.7.2

Critical Velocity

Acceleration

shape is often comparable to a rectangular or a trapezoid pulse (Figure 5.30). Alternatively, it can be made of an elastic material, which produces at the material entry a shock with a near half-sine waveform. After determination of the shock environment, a statistical analysis allows one to specify the design drop height, with a given percentage of loss tolerated. To choose the characteristics of the cushioning material constituting the package, it is ﬁrst of all necessary to determine the shock fragility of the product that would be subjected to a shock with one of these two forms. It can be considered that the severity of a shock is related to its amplitude and to its associated velocity change (we saw that these two parameters intervene in the SRS). We thus determine the largest acceleration and the largest velocity change that the unpackaged product subjected to these shocks can support. At the time of two series of tests carried out on a shock machine, we note, for a given acceleration, the critical velocity change or, for a given velocity change, the critical maximum shock acceleration that leads to a damage on the material (deformation, fracture, faulty operation after the shock, etc.). Results are expressed on a diagram of the acceleration –velocity change by a curve deﬁned as the damage boundary curve (DBC; ASTM D3332), as shown in Figure 5.31. Variable velocity change tests begin with a short-duration shock, then the duration is increased (by preserving constant acceleration) until the appearance of damage (functional or physical). The critical velocity change is equal to the velocity change just lower than that producing damage (ASTM, 1994). The variable acceleration tests are performed on a new material, starting with a small acceleration level and with a rather large velocity change (at least 1.5 times the critical velocity change previously determined). The tests should be carried out in the more penalizing impact conﬁguration (unit orientation). Damage

Analysis of Test Results

Damage can occur if the acceleration and the velocity change are together higher than the critical acceleration and the critical velocity change. From the critical velocity change, the critical drop height can be calculated. If Vi is the impact velocity, VR is the rebound velocity, and a is the

Acr

Product Fragility ∆Vcr

FIGURE 5.31 shock pulse).

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Velocity Change Damage boundary curve (rectangular

Vibration Monitoring, Testing, and Instrumentation

rate of rebound ðVR ¼ 2aVi Þ; the velocity change DV is equal to DV ¼ VR 2 Vi ¼ 2ðVi þ aVi Þ ¼ 2ð1 þ aÞVi

ð5:17Þ

Acceleration

5-28

and the free-fall drop height Hcr to V2 DV 2 Hcr ¼ i ¼ 2g 2gð1 þ aÞ2

Half-sine shock pulse Rectangle shock pulse

ð5:18Þ

Velocity change

If this critical height is lower than the design FIGURE 5.32 DBC comparison of half-sine and height deﬁned from the real use conditions of the rectangular shock pulses. product, it is necessary to use a package with a medium cushioning and then to deﬁne its characteristics (crush stress, thickness) so that maximum acceleration at the time of impact is lower than the critical acceleration. If not, no protection is necessary. Tests are in general carried out with a rectangular shock waveform, for two reasons. *

*

As the rectangular shock is most severe (see SRSs), the result is conservative, as seen in Figure 5.32. The DBC is made up only of two lines, which makes it possible to determine the curve from only two set of tests (saving time) by destroying only two specimens. A much more signiﬁcant number of sets of tests would be necessary to determine the curve from a half-sine shock waveform.

Note: If, for cost reasons, the same product is used to determine the critical velocity change or the critical acceleration, it undergoes several shocks before failure. The test result is usable only if the product fails in a brittle mode. If the material is ductile, each shock damages the product by an effect of fatigue, which should be taken into account (Burgess, 1996, 2000).

5.8 5.8.1

Shock Machines Main Types

A shock machine, whatever its standard, is primarily a device allowing modiﬁcation over a short time period of the velocity of the material to be tested (also, see Chapter 1). Two principal categories are usually distinguished (Lalanne, 2002b): *

*

The ﬁrst category is that of impulse machines, which increase the velocity of the test item during the shock. The initial velocity is in general zero. The air gun, which creates the shock during the setting of the velocity in the tube, is an example. The second category is that of impact machines, which decrease the velocity of the test item throughout the shock and/or which change its direction.

The test facilities now used are classiﬁed as follows: *

*

*

In free fall machines, the impact is made on a shock simulator (in American literature these devices are termed “shock programmers”) adapted to the shape of the speciﬁed shock (elastomer discs, conical or cylindrical lead pellets, pneumatic shock simulators, etc.). To increase the impact velocity, which is limited by the drop height, that is, by the height of the guide columns, the fall can be accelerated by the use of bungee cords. In pneumatic machines, the velocity is derived from a pneumatic actuator. In electrodynamic exciters, the shock is speciﬁed either by the shape of a temporal signal, its amplitude and its duration, or by a SRS.

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Mechanical Shock

*

5-29

Exotic machines are designed to carry out shocks that are nonrealizable by the preceding methods, generally because their amplitude and duration characteristics are not compatible with the performances from these means. The desired shapes, not being normal, are not possible with the shock simulators delivered by the manufacturers.

We will try to show in the following sections how mechanical shocks could be simulated on materials in the laboratory. The facilities described are the most current, but the list is far from being exhaustive. Many other processes were or are still used to satisfy particular needs (Nelson and Prasthofer, 1974; Powers, 1974, 1976; Conway et al., 1976).

5.8.2

Impact Shock Machines

Most machines with free or accelerated drop testing belong to the category of impact shock machines. The machine itself allows the setting of velocity of the test item. The shock is carried out by impact, with the help of the shock simulator (programmer), which formats the acceleration of braking according to the desired shape. The impact can be without rebound when the velocity is zero at the end of the shock, or with rebound when the velocity changes sign during the movement. Laboratory machines of this type consist of two vertical guide rods on which the table carrying the test item slides (Figure 5.33). The impact velocity is obtained by gravity, after the dropping of the table from a certain height or using bungee cords allowing one to obtain a larger impact velocity. Let us consider a free fall shock machine for which the friction of the shock table on the guidance system can be neglected. The necessary drop height, H; to obtain the desired impact velocity, vi , is given by H¼

vi2

2g

ð5:19Þ

FIGURE 5.33 Elements of a shock-test machine. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

where g is the acceleration of gravity (9.81 m/sec2). These machines are limited by the possible drop height, that is, by the height of the columns and the height of the test item when the machine is provided with a gantry. It is difﬁcult to increase the height of the machine due to overcrowding and problems with guiding the table. However, the impact velocity can be increased using a force complementary to gravity by means of bungee cords tended before the test and exerting a force generally directed downwards. The acceleration produced by the cords is in general much higher than gravity, which then becomes negligible. This idea was used to design horizontal (Lonborg, 1963) and vertical machines (Marshall et al., 1965; La Verne Root and Bohs, 1969), this last conﬁguration being less cumbersome. During impact, the velocity of the table changes quickly and forces of great amplitude appear between the table and machine bases. To generate a shock of a given shape, it is necessary to control the amplitude of the force throughout the stroke during its velocity change. This is carried out using a shock simulator (programmer).

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5-30

5.8.2.1

Vibration Monitoring, Testing, and Instrumentation

Universal Shock-Test Machines

5.8.2.1.1 Impact Mode As an example, the MRLw Company (Monterey Research Laboratory) has marketed a machine allowing the carrying out of shocks according to two modes: impulse and impact (Bresk and Beal, 1966). In the two test conﬁgurations, the test item is installed on the upper face of the table. The table is guided by two rods that are ﬁxed at a vertical frame. To carry out a test according to the impact mode (the general case), one raises the table by the height required by means of a hoist attached to the top of the frame, using the intermediary assembly for raising and dropping (see Figure 5.34). By opening the blocking system in a high position, the table falls under the effect of gravity or owing to the relaxation of elastic cords if the fall is accelerated. After rebound, as seen on the shock simulator (programmer), the table is again blocked to avoid a second impact. 5.8.2.1.2 Impulse Mode The impulse mode shocks (see Figure 5.35) are obtained while placing the table on the piston of the shock simulator (used for the realization of initial peak sawtooth shock pulses). The piston of this hydropneumatic shock simulator (programmer) propels the table upward according to an appropriate force proﬁle to produce the speciﬁed acceleration signal. The table is stopped in its stroke to prevent its falling down for a second time on the shock simulator (programmer).

5.8.3 Shock Simulators (Programmers)

FIGURE 5.34 MRL universal shock-test machine (impact mode). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

We will describe only the most frequently used shock simulators to carry out half-sine, TPS and trapezoid shock pulses. 5.8.3.1

Half-Sine Pulse

These shocks are obtained using an elastic material interposed between the table and the solid mass reaction. 5.8.3.1.1 Shock Duration The shock duration is calculated by supposing that the table and the shock simulator (programmer), for this length of time, constitute a linear mass– spring system with only one DoF. From the differential equation of the movement (valid only during the elastomeric material compression and its relaxation, so long as there is contact between the table and the shock simulator, i.e., during a half-period) m

d2 x þ kx ¼ 0 dt 2

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ð5:20Þ

Mechanical Shock

5-31

the shock duration t can be deduced: rﬃﬃﬃﬃ m t¼p k

ð5:21Þ

where m is the mass of the moving assembly (table þ ﬁxture þ test item) and k is the stiffness constant of the shock simulator (programmer). This expression shows that, theoretically, the duration can be regarded as a function alone of the mass, m; and of the stiffness of the target. It is, in particular, independent of the impact velocity. The mass, m; and the duration, t; being known, we deduce from it the stiffness constant, k; of the target: k¼m

p2 t2

ð5:22Þ

5.8.3.1.2 Impact Velocity Let us set vi as the impact velocity of the table and vR as the velocity of rebound. The elastomeric shock simulators often have a coefﬁcient of restitution, a ðvR ¼ 2avi Þ; of about 50%. In a ﬁrst approximation, we will consider that the rebound is perfect ða ¼ 1Þ: The impact velocity is then equal to DV=2; where DV is the velocity change given by Table 5.1: DV ¼

2 x€ t p m

5.8.3.1.3 Maximum Deformation of the Shock Simulator (Programmer) If xm is the maximum deformation of the shock simulator (programmer) during the shock, it becomes, by equalizing the kinetic loss of energy and the deformation energy during the compression of the shock simulator (programmer)

yielding

FIGURE 5.35 MRL universal shock-test machine (impulse mode). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

1 2 1 2 mv ¼ kxm 2 i 2

ð5:23Þ

rﬃﬃﬃﬃ m x m ¼ vi k

ð5:24Þ

5.8.3.1.4 Shock Amplitude From Equation 5.20, one has, in absolute terms, m€xm ¼ kxm ; yielding x€ m ¼ xm ðk=mÞ and, according to Equation 5.24 sﬃﬃﬃﬃ k x€ m ¼ vi ð5:25Þ m

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5-32

Vibration Monitoring, Testing, and Instrumentation

where the impact velocity, vi is equal to

pﬃﬃﬃﬃﬃﬃ vi ¼ 2gH

ð5:26Þ

2

where g is the acceleration of gravity (1g ¼ 9.81 m/sec ) and H is the drop height. This relation, established theoretically for perfect rebound, remains usable in practice as long as the rebound velocity remains higher than approximately 50% of the impact velocity. Having determined k from m and t; it is enough to act on the impact velocity, that is, on the drop height, to obtain the required shock amplitude. 5.8.3.1.5 Characteristics of the Target For a cylindrical shock simulator (programmer), we have k¼

ES L

.. x(t)

ð5:27Þ

where S and L are, respectively, the cross section and the height of the shock simulator (programt t mer) and where E is Young’s modulus of material in compression. Depending on the materials available, that is the possible values of E; one chooses the values of L and S that lead to a realizable shock simulator (by avoiding too large a height-to-diameter ratio FIGURE 5.36 High frequencies at impact. (Source: to eliminate the risks from buckling). When the Lalanne, Chocs Mecaniques, Hermes Science Publitable has a large surface, it is possible to place cations. With permission.) four shock simulators to distribute the effort. The cross section of each shock simulator (programmer) is then calculated starting from the value of S determined above and divided by four. If the surface of impact is planar, a wave created at the time of the impact is propagated in the cylinder and makes several up and down FIGURE 5.37 Impact module with conical impact excursions (Figure 5.36). From it, at the face (open module). (Source: Lalanne, Chocs Mecaniques, beginning of the signal the appearance of a Hermes Science Publications. With permission.) high frequency oscillation that distorts the desired half-sine pulse results. To avoid this phenomenon, the front face of the shock simulator (programmer) is designed to be slightly conical (Figure 5.37) in order to insert the load material gradually (open module). The shock thus created is between a half-sine and a versed-sine pulse. In addition, a good empirical rule is to limit the maximum dynamic deformation of the shock simulator (programmer) from 10 to 15% of its initial thickness to avoid distortion of the half-sine due to damping of the material. If this limit is exceeded, the shape obtained risks nonlinear tendencies.

Example Consider the realization of a half-sine shock 340 m/sec2, 9 msec. It is supposed that the mass of the moving assembly (table þ ﬁxture þ test item) is equal to 470 kg (test item þ ﬁxture mass: 150 kg). From Equation 5.22 k¼m

p2 p2 < 5:727 £ 107 N=m 2 ¼ 470 ð0:009Þ2 t

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Mechanical Shock

5-33

The impact velocity is calculated from Equation 5.25: rﬃﬃﬃﬃ t m 0:009 < 0:974 m=sec vi ¼ x€ m ¼ x€ m ¼ 340 p p k which leads to the drop height, H ¼ ðvi2 =2gÞ < 48 £ 1023 m: During the impact, the elastomeric target will be deformed to a height equal to Equation 5.24: rﬃﬃﬃﬃ m t xm ¼ vi ¼ x€ m k p

2

¼ 300

0:009 p

2

< 2:79 £ 1023 m

The velocity change during the shock is equal to DV ¼ ð2=pÞ€xm t ¼ ð2=pÞ340 £ 0:009 < 1:91 m=sec: It is checked that DV ¼ 2vi : With L being the height of the target, its diameter D is calculated from k ¼ ES=L: D2 ¼

4 k L p E

If the target is an elastomer of Young’s modulus, E ¼ 5 £ 107 N=m2 yielding, if L ¼ 0:015 m; D < 0:148 m: It remains to check that the stress in the material does not exceed the acceptable value. The manufacturers provide cylindrical modules made up of an elastomer sandwiched between two metal plates. The shock simulator (programmer) is composed of stacked modules of various stiffnesses (Figure 5.38). A relatively low number of different modules allows the covering of a broad range of shock durations by combinations of these elements (Brooks, 1966; Brooks and Mathews, 1966; Gray, 1966; Bresk, 1967). The modules are in general distributed between the bottom of the table and the top of the solid mass of reaction to regularly distribute the load at the time of the shock in the lower part of the table. One thus avoids exciting its bending mode at lower frequency and amplifying the vibrations due to resonance of the table. The shock simulators for very short duration shock are made up of a high strength and high Young’s modulus thermoplastic material. The selected plastic is highly resilient and very hard. It is used within its yield stress and can thus be useful almost indeﬁnitely. Reproducibility is very good. The shock simulator (programmer) is composed of a cylinder of this material attached to a planar circular plate screwed to the lower part of the table of the shock machine.

Drop table Modules of different thickness and hardness bolted together. Addition of modules increases duration. Removing modules decreases duration. Open face shaped modules

Machine base FIGURE 5.38 Distribution of the modules (half-sine shock pulse). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-34

Vibration Monitoring, Testing, and Instrumentation

5.8.3.2

Terminal Peak Sawtooth Shock Pulse

To generate a TPS shock pulse, any target made up of an inelastic material (crushable material) with a curve dynamic deﬂection-load which follows a cubic law is thus appropriate (McWhirter, 1961). This curve is approximated by using shock simulators of conical shape. The material generally used is lead or honeycomb. The cones can be calculated as follows: *

Crushed length: xm ¼

*

x€ m t2 3

ð5:28Þ

yielding the height of the cone, h . 1:2 xm (to allow material to become deformed to the necessary height). Force maximum: Fm ¼ Sm scr ¼ m€xm

ð5:29Þ

where Sm is the cross section of the cone at height xm and scr is the crush stress of material constituting the target yielding: Sm ¼

m€xm scr

ð5:30Þ

When all the kinetic energy of the table is dissipated by the crushing of the lead, acceleration decreases to zero. The shock machine must have a very rigid solid mass of reaction, so that the time of decay to zero is not too long and satisﬁes the speciﬁcation. The speed of this decay to zero is a function of the mass of reaction and of the mass of the table: if the solid mass of reaction has a nonnegligible elasticity, the time, already nonzero because of the imperfections inherent in the shock simulator (programmer), can become too long and unacceptable. For lead, the order of magnitude of scr is 760 kg/cm2 (7.6 £ 107 N/m2 ¼ 76 MPa). The range of possible durations lies between approximately 2 and 20 msec. For each machine and each shock, it is necessary to carry out preliminary tests to check that the shock simulator (programmer) is well calculated. The shock simulators are destroyed with each test. It is thus a relatively expensive method. One prefers to use, if possible, a Universal Programmer (Section 5.8.3.4).

Example TPS shock pulse, 340 m/sec2, 9 msec (example of Section 5.5.2.5) Unit table mass: 320 kg Fixture þ test item mass: 150 kg Conical lead shock simulator (scr ¼ 760 kg=cm2 ¼ 7:6 £ 107 N=m2 Þ The impact velocity is calculated from Table 5.1: DV ¼ vi ¼

x€ m t 340 £ 0:009 ¼ < 1:53 m=sec 2 2

which leads to the theoretical drop height H ¼ ðvi2 =2gÞ < 0:119 m: During the impact, the target will be deformed to a height equal to Equation 5.28: xm ¼

x€ m t2 340 £ 0:0092 ¼ < 9:18 £ 1023 m 3 3

yielding the height of the target h $ 1:2xm < 1:2 £ 9:18 £ 1023 < 0:011 m:

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Mechanical Shock

5-35

The cross section of the cone at height xm is equal to Equation 5.30 Sm ¼

m€xm 470 £ 340 ¼ < 2:1 £ 1023 m2 scr 7:6 £ 107

corresponding to a diameter (at height xm ) D < 0:052 m: 5.8.3.3

Rectangular Pulse – Trapezoidal Pulse

This test is carried out by impact. A cylindrical shock simulator (programmer) consists of a material which is crushed with constant force (lead, honeycomb) or using the Universal Shock simulator (programmer). In the ﬁrst case, the characteristics of the shock simulator (programmer) can be calculated as follows: *

The cross section is given according to the shock amplitude to be realized using the relation Fm ¼ m€xm ¼ Sscr

ð5:31Þ

yielding S¼

*

m€xm scr

ð5:32Þ

Starting from the dynamics of the impact without rebound, the length of crushing is equal to xm ¼

x€ m t 2 2

ð5:33Þ

and that of the shock simulator (programmer) must be at least equal to 1:4xm in order to allow a correct crushing of the matter with constant force. The shock amplitude is controlled by the cross section of the shock simulator (programmer), the crush stress of material, and the mass of the total carriage mass. The duration is affected only by the impact velocity. This method produces relatively disturbed signals, because of the impact between two plane surfaces. They are adapted only for shocks of short duration, because of the limits of deformation. A long duration requires a plastic deformation over a big length, but it is difﬁcult to maintain the constant force of resistance on such a stroke. The honeycombs lend themselves better to the realization of a long duration shock (Gray, 1966).

Example Rectangular shock pulse, 340 m/sec2, 9 msec Unit table mass: 320 kg Fixture þ test item mass: 150 kg Cylindrical lead shock simulator ðscr ¼ 760 kg=cm2 ¼ 7:6 £ 107 N=m2 Þ The impact velocity is calculated from Table 5.1: DV ¼ vi ¼ x€ m t ¼ 340 £ 0:009 ¼ 3:06 m=sec which leads to the theoretical drop height H ¼ ðvi2 =2gÞ < 0:477 m: During the impact, the target will be deformed to a height equal to Equation 5.33: xm ¼

x€ m t 2 340 £ 0:0092 ¼ < 13:8 £ 1023 m 2 2

yielding the height of the target h $ 1:4xm < 1:4 £ 13:8 £ 1023 < 0:019 m:

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5-36

Vibration Monitoring, Testing, and Instrumentation

TABLE 5.8

Characteristics of the Target for the Half-Sine, Sawtooth, and Rectangular Pulses Half-Sine

TPS

Maximum deformation of the shock simulator (programmer)

x€ t 2 xm ¼ m 2 p

x€ t 2 xm ¼ m 3

Shock simulator (programmer) cross section at height xm

S p 2m ¼ h Et 2

Sm ¼

Shock simulator (programmer) height

m€xm scr

x€ m t 2 3 x€ m t vi ¼ 2

h $ 1:2

Impact velocity

vi ¼

x€ m t p

Free fall height

H¼

vi2 2g

Rectangle xm ¼ S¼

x€ m t 2 2

m€xm scr

h $ 1:4

x€ m t 2 2

vi ¼ x€ m t

The cross section of the cylinder is equal to Equation 5.32 Sm ¼

m€xm 470 £ 340 ¼ < 2:1 £ 1023 m2 scr 7:6 £ 107

corresponding to a diameter D < 0:052 m: Table 5.8 recapitulates the main relations allowing the predimensioning of targets for generating halfsine, sawtooth and rectangular shock pulses. 5.8.3.4

Universal Shock Simulator (Programmer)

w1

MTS has manufactured a shock simulator (programmer), known as Universal, still used in many laboratories, to produce half-sine, TPS, and trapezoidal shock pulses after various adjustments. This shock simulator (programmer) consists of a cylinder ﬁxed under the table of the machine, ﬁlled with a gas under pressure, and, in the lower part of a piston, a rod and a head (Figure 5.39). 5.8.3.4.1 Generating a Half-Sine Shock Pulse The chamber is put under sufﬁcient pressure so that, during the shock, the piston cannot move (Figure 5.39). The shock pulse is thus formatted only by the compression of the stacking of elastomeric cylinders (modular shock simulators), placed under the piston head. One is thus brought back to the case of Section 5.8.3.1.

5.8.3.4.2 Generating a Terminal Peak Sawtooth Shock Pulse The gas pressure (nitrogen) in the cylinder is selected so that, after compression of elastomer during duration, t; the piston, assembled in the cylinder as indicated in Figure 5.40, is suddenly released for a force corresponding to the required maximum acceleration, x€ m : The pressure that was exerted before separation over the whole area of the piston applies only after separation to one area equal to that of the rod, producing a negligible resistant force. Acceleration thus passes very quickly from x€ m to zero, as shown in Figure 5.41. The rise phase is not perfectly linear, but corresponds rather to an arc of versed-sine (since if the pressure were sufﬁciently strong, one would obtain a versed-sine by compression of the elastomer alone). 1

Registered trademark of MTS Systems Corporation.

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Mechanical Shock

5-37

FIGURE 5.39 MTSw Universal shock simulator (half-sine and rectangle pulse conﬁguration). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

5.8.3.4.3 Trapezoidal Shock Pulse The assembly here is the same as that of the half-sine pulse (Figure 5.39). At the time of the impact, there is: *

*

*

Compression of the elastomer until the force exerted on the piston balances the compressive force produced by nitrogen (this phase gives the ﬁrst part [rise] of the trapezoid). Up and down displacement of the piston in the part of the cylinder of smaller diameter, approximately with constant force, since volume varies little (this phase corresponds to the horizontal part of the trapezoid). Relaxation of elastomer: decay to zero acceleration.

The rise and decay parts are not perfectly linear for the same reason as in the case of the TPS pulse.

5.8.4 5.8.4.1

Limitations Limitations of the Shock Machines

The limitations are often represented graphically by straight lines plotted in logarithmic scales, delimiting the domain of realizable shocks (amplitude, duration). The shock machine is

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 5.40 MTSw Universal shock simulator (TPS pulse conﬁguration). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

limited by (IMPAC 6060F Operating Manual; Figure 5.42):

.. x(t)

The allowable maximum force on the table. To carry out a shock of amplitude x€ m ; the force generated on the table, given by

.. xm

*

T.P.S.

F ¼ ½mtable þ mprogrammer þ mfixture þ mtest item x€ m

Versed-sine

t

ð5:34Þ

must be lower than or equal to the acceptable maximum force, Fmax : Knowing the total carriage mass, the relation (Equation 5.34) allows calculation of the possible maximum acceleration under the test conditions

t

FIGURE 5.41 Realization of a TPS shock pulse. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

ð€xm Þmax ¼ Fmax =½mtable þ mprogrammer þ mfixture þ mtest item

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ð5:35Þ

Mechanical Shock

This limitation is represented on the abacus by a horizontal line constant, x€ m : This emphasizes maximum free fall height, H; or the maximum impact velocity, that is the velocity change, DV of the shock pulse. If vR is the rebound velocity, equal to a percentage a of the impact velocity, we have DV ¼ vR 2 vi ¼ 2ð1 þ aÞvi pﬃﬃﬃﬃﬃﬃ ¼ 2ð1 þ aÞ 2gH ðt ¼ x€ ðtÞdt 0

Maximum acceleration (m/s2)

*

5-39

Velocity change limit Stroke limit Force limit on elastomer Shock duration (ms)

ð5:36Þ

FIGURE 5.42 Abacus of the limitations of a shock machine. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

yielding H¼

Table acceleration limit

DV 2 2gð1 þ aÞ2

ð5:37Þ

where a is a function of the shape of the shock and of the type of shock simulator (programmer) used. In practice, there are losses of energy by friction during the fall and especially in the shock simulator (programmer) during the realization of the shock. Taking account of these losses is difﬁcult to calculate analytically and so one can set: H¼b

DV 2 2g

ð5:38Þ

where b takes into account at the same time losses of energy and rebound. As an example, the manufacturer of machine IMPAC 60 £ 60 (MRL) gives Table 5.9, according to the type of shock simulators (IMPAC 6060F Operating Manual). The limitation related to the drop height can be represented by parallel straight lines on a diagram giving the velocity change, DV; as a function of the drop height in logarithmic scales. The velocity change being, for all simple shocks, proportional to the product x€ m t; we have pﬃﬃﬃﬃﬃﬃﬃﬃ DV ¼ lx€ m t ¼ 2gH=b

ð5:39Þ

Some typical values for the amplitude £ duration product are given in Table 5.10. On logarithmic scales ð€xm ; tÞ; the limitation relating to the velocity change is represented by parallel, inclined straight lines (Figure 5.42).

TABLE 5.9

Loss Coefﬁcient b

Shock Simulator (Programmer)

Value of b

Elastomer (half-sine pulse) Lead (rectangle pulse) Lead (TPS pulse)

0.556 0.2338 1.544

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

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5-40

Vibration Monitoring, Testing, and Instrumentation TABLE 5.10

Amplitude £ Duration Limitation

Waveform

Shock Simulator (Programmer)

Half-sine TPS

Elastomer Lead cone Universal shock simulator (programmer) Universal shock simulator (programmer)

Rectangle

ð€xm t_Þmax (m/sec) 17.7 10.8 7.0 9.2

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

5.8.4.2

Limitations of Shock Simulators

Elastomeric materials are used to generate shocks of: *

*

Half-sine shape (or versed-sine with a conical frontal module to avoid the presence of high frequencies). TPS and rectangular shapes, in association with a Universal shock simulator.

Elastomer shock simulators are limited by the allowable maximum force, a function of Young’s modulus, and their dimensions (Figure 5.42). This limitation is in fact related to the need to maintain the stress lower than the yield stress of the material, so that the target can be regarded as a pure stiffness. The maximum stress, smax ; developed in the target at the time of the shock can be expressed according to Young’s modulus, E; to the maximum deformation, xm ; and to the thickness, h; of the target according to x smax ¼ E m ð5:40Þ h with, for an impact with perfect rebound, xm ¼ x€ m t 2 =p 2 : It is necessary that, if Re is the elastic ultimate stress E€xm t 2 , Re hp 2

ð5:41Þ

that is h.

E€xm t 2 Re p 2

ð5:42Þ

Taking into account the mass of the carriage assembly, this limitation can be transformed into maximum acceleration ðFm ¼ m€xm Þ: With four shock simulators used simultaneously, the maximum acceleration is naturally multiplied by four. This limitation is represented on the abacus of Figure 5.42 by the straight lines of greater slope. The Universal shock simulator is limited (MRL 2680 Operating Manual): *

*

By the acceptable maximum force. By the stroke of the piston: for each waveform, the displacement during the shock is always proportional to the product x€ m t 2 :

This limitation is provided by the manufacturer. In short, the domain of the realizable shock pulses is limited on this diagram by straight lines representative of the conditions given in Table 5.11. TABLE 5.11

Summary of Limitations on the Domain of Realizable Shock Pulses

x€ m ¼ constant x€ m t ¼ constant x€ m t2 ¼ constant x€ m t4 ¼ constant

Acceptable force on the table or on the Universal shock simulator Drop height (DV) Piston stroke of the Universal shock simulator Acceptable force for elastomers

Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.

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Mechanical Shock

5.8.5

Pneumatic Machines

Pneumatic machines in general consist of a cylinder separated into two parts by a plate bored to let pass the rod of a piston located lower down (Figure 5.43). The rod crosses the higher cylinder, comes out of the cylinder, and supports a table receiving the test item. The surface of the piston subjected to the pressure is different according to whether it is on the higher face or the lower face, as long as it is supported in the higher position on the Teﬂonw2 seat (Thorne, 1964). Initially, the moving piston, rod, and table rose by ﬁlling the lower cylinder (reference pressure). The higher chamber is then inﬂated to a pressure of approximately ﬁve times the reference pressure. When the force exerted on the higher face of the piston exceeds the force induced by the pressure of reference, the piston releases. The useful surface area of the higher face increases quickly and the piston is subjected in a very short time to a signiﬁcant force exerted towards the bottom. It involves the table, which compresses the shock simulators (elastomers, lead cones, etc.) placed on the top of the body of the jack. This machine is assembled on four rubber bladders ﬁlled with air to uncouple it from the ﬂoor of the building. The body of the machine is used as a solid mass of reaction. The interest behind this lies in its performance and its compactness. An industrial pneumatic shock machine is shown in Figure 5.44.

5.8.6

5-41

FIGURE 5.43 The principle of pneumatic machines. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

High Impact Shock Machines

The ﬁrst machine was developed in 1939 to simulate the effects of underwater explosions (mines) on the equipment onboard military ships. Such explosions, which generally occur at large distances from the ships, create shocks that are propagated in all the structures. The procedure consisted of specifying the machine to be used, the method of assembly, the adjustment of the machine and so on, and not of a SRS or a simple shape shock. Two models of machines of this type were built to test lightweight and medium weight equipment. 5.8.6.1

Lightweight High Impact Shock Machine

The lightweight high impact shock machine, the ﬁrst built, consists of a welded frame of standard steel sections and two hammers, one sliding vertically, the other describing an arc of a circle in a vertical plane, according to a pendular motion (Figure 5.45). 2

Registered trademark of E.I. du Pont de Nemours & Company, Inc., Wilmington, DE.

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5-42

Vibration Monitoring, Testing, and Instrumentation

A target plate carrying the test item can be placed to receive one or the other of the hammers. The combination of the two movements and the two positions of the target makes it possible to deliver shocks according to three perpendicular directions without disassembling the test item. Each hammer weighs approximately 200 kg and can fall a maximum height of 1.50 m (Conrad, 1952). The target is a plate of steel of 86 cm £ 122 cm £ 1.6 cm, reinforced and stiffened on its back face by I-beams. In each of the three impact positions of the hammer, the target plate is assembled on springs in order to absorb the energy of the hammer with a limited displacement (38 mm to the maximum). Rebound of the hammer is prevented. Several intermediate standardized plates simulate various conditions of assembly of the equipment on board. These plates are inserted between the target and the equipment tested to provide certain insulation at the time of impact and to restore a shock considered comparable with the real shock. The mass of the equipment tested on this machine should not exceed 100 kg. For ﬁxed test conditions (direction of impact, equipment mass, intermediate plate), the shape of the shock obtained is not very sensitive to the drop height. The duration of the produced shocks is about 1 msec and the amplitudes range between 5000 and 10,000 m/sec2. 5.8.6.2 Medium Weight High Impact Shock Machine This machine was designed to test equipment whose mass, including the ﬁxture, is less than 2500 kg (Figure 5.46). It consists of a hammer weighing 1360 kg that swings through an arc of a circle at an angle greater than 1808 and comes to strike an anvil at its lower face. Under the impact, this anvil, ﬁxed under the table carrying the test item, moves vertically upwards. The movement of this unit is limited to approximately 8 cm at the top and 4 cm at the bottom (Vigness, 1947, 1961a; Conrad, 1951; Lazarus, 1967) by stops that halt it and reverse its movement. The equipment being tested is ﬁxed on the table via a group of steel channel beams (and not directly to the rigid anvil structure), so that the natural frequency of the test item on this support metal structure is about 60 Hz.

FIGURE 5.44 Benchmarkw SM 105 pneumatic shock machine. SM 105 and Benchmark are registered trademarks owned by Benchmark Electronics, Hunstville, Inc. (courtesy Benchmark Electronics).

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Mechanical Shock

5-43

FIGURE 5.45 High impact shock machine for lightweight equipment. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

The shocks obtained are similar to those produced with the machine for light equipment (Lazarus, 1967). The shocks carried out on all these facilities are not very reproducible and are sensitive to the ageing of the machine and the assembly (the results can differ after dismantling and reassembling the equipment on the machine under identical conditions, in particular at high frequencies; Vigness, 1961b). These machines can also be used to generate simple shape shocks such as half-sine or TPS pulses (Vigness, 1963), while inserting either an elastic or a plastic material between the hammer and the anvil carrying the test item. One thus obtains durations of about 10 msec at 20 msec for the half-sine pulse and 10 msec for the TPS pulse.

FIGURE 5.46 High impact machine for medium weight equipment. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-44

Vibration Monitoring, Testing, and Instrumentation

5.8.7

Speciﬁc Test Facilities

When the impact velocity of standard machines is insufﬁcient, one can use other means to obtain the desired velocity. For example: *

*

*

One can use drop testers, equipped with two vertical (or inclined) guide cables (McWhirter, 1963; Lalanne, 1975). The drop height can reach a few tens of meters. FIGURE 5.47 Inclined plane impact tester (CONBUR It is wise to make sure that the guidance is tester). (Source: Lalanne, Chocs Mecaniques, Hermes correct and, in particular, and that friction Science Publications. With permission.) is negligible. It is also desirable to measure the impact velocity (using photoelectric cells or any other device). One can also use gas guns, which initially use the expansion of a gas (often air) under pressure in a tank to propel a projectile carrying the test item towards a target equipped with a shock simulator ﬁxed at the extremity of a gun on a solid reaction mass (McWhirter, 1961; McWhirter, 1963; Yarnold, 1965; Lazarus, 1967; Lalanne, 1975). One ﬁnds the impact mode to be as above. It is necessary that the shock created at the time of the velocity setting in the gun is of low amplitude with regard to the speciﬁed shock carried out at the time of the impact. Another operating mode consists of using the phase of the velocity setting to program the speciﬁed shock, the projectile then being braked at the end of the gun by a pneumatic device, with a small acceleration with respect to the principal shock. A major disadvantage of guns is related to the difﬁculty of handling cables instrumentation, which must be wound or unreeled in the gun, in order to follow the movement of the projectile. Alternatively, one can use inclined-plane impact testers (Vigness, 1961a; Lazarus, 1967). These were especially conceived to simulate shocks undergone during too severe handling operations or in trains. They are made up primarily of a carriage on which the test item is ﬁxed, traveling on an inclined rail and coming to run up against a wooden barrier (Figure 5.47).

The shape of the shock can be modiﬁed by using elastomeric “bumpers” or springs. Tests of this type are often named “CONBUR tests.”

5.9

Generation of Shock Using Shakers

In about the mid-1950s, with the development of electrodynamic exciters for the realization of vibration tests, the need for a realization of shocks on this facility was quickly felt. This simulation on a shaker, when possible, indeed presents a certain number of advantages (Coty and Sannier, 1966).

5.9.1

Principle Behind the Generation of a Simple Shape Signal versus Time

The objective is to carry out on the shaker a shock of simple shape (half-sine, triangle, rectangle, etc.) of given amplitude and duration similar to that made on the classical shock machines. This technique was mainly developed during the years 1955 to 1965 (Wells and Mauer, 1961). The transfer function between the electric signal of the control applied to the coil and acceleration to the input of the test item is not constant. It is thus necessary to calculate the signal of control according to this transfer function and the signal to be realized.

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Mechanical Shock

5-45

The process is as follows (Favour et al., 1969; Magne, 1971): *

*

*

*

Measurement of the transfer function of the installation (including the ﬁxture and the test item) using a calibration signal; the measurement of the transfer function of the installation can be made using a calibration signal of the shock type, random vibration or sometimes by fast swept sine (Favour, 1974; Lalanne, 2002a). Calculation of the Fourier transform of the signal speciﬁed at the input of the test item. By division of this transform by the transfer function, calculation of the Fourier transform of the signal of control. Calculation of the control signal vs. time by inverse transformation.

In all cases, the procedure consists of measurement and calculation of the signal of control to 2n dB (212, 29, 26, and/or 23). The speciﬁed level is applied only after several adjustments on a lower level. These adjustments are necessary because of the sensitivity of the transfer function to the amplitude of the signal (nonlinearities). The development can be carried out using a dummy item representative of the mass of the specimen. However, particularly if the mass of the test specimen is signiﬁcant (with respect to that of the moving element), it is deﬁnitely preferable to use the real test item or a model with dynamic behavior very near to it. If random vibration is used as the calibration signal, its root-mean-square (rms) value is calculated in order to be lower than the amplitude of the shock (but not too distant in order to avoid the effects of any nonlinearities). This type of signal can result in application to the test item of many substantial peaks of acceleration compared with the shock itself.

5.9.2

Main Advantages

The realization of the shocks on shakers has very interesting advantages: *

*

*

*

*

*

Possibility of obtaining very diverse shocks shapes. Use of the same means for the tests with vibrations and shocks, without disassembly (saving time) and with the same ﬁxtures (Wells and Mauer, 1961; Hay and Oliva, 1963). Possibility of a better simulation of the real environment, in particular by direct reproduction of a signal of measured acceleration (or of a given shock spectrum). Better reproducibility than on the traditional shock machines. Very easy realization of the test on two directions of an axis. No need to use a shock machine.

In practice, however, one is rather quickly limited by the possibilities of the exciters, which therefore do not make it possible to generalize their use for shock simulation.

5.9.3

Pre- and Postshocks

5.9.3.1

Requirements

5.9.3.2

Solutions

Ð The velocity change, DV ¼ t0 x€ ðtÞdt (t ¼ shock duration), associated with shocks of simple shape (half-sine, rectangle, TPS, etc.) is different from zero. At the end of the shock, the velocity of the table of the shaker must, however, be zero. It is thus necessary to devise a method to satisfy this need.

One way of bringing back the variation of velocity associated with the shock to zero can be the addition of a negative acceleration to the principal signal so that the area under the pulse has the same value on the

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5-46

Vibration Monitoring, Testing, and Instrumentation

Pre-shock alone

3

2

1

Pre-and post -shocks

Pre-shock alone

FIGURE 5.48 Possibilities for pre- and postshock positioning. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Displacement

Acceleration Velocity

1

Displacement

Velocity

Displacement 2

Velocity

3

FIGURE 5.49 Kinematics of the movement with preshock alone 1 ; symmetrical pre and postshocks and postshock alone 3 . (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

2

side of positive accelerations and on the side of negative accelerations (Lalanne, 2002b). Various solutions are possible a priori (Figure 5.48): *

*

*

A preshock alone. A postshock alone. Pre- and postshocks, possibly of equal durations.

Preshock alone 1 ; which requires a less powerful power ampliﬁer, thus seems preferable to postshock alone 3 : The use of symmetrical pre- and postshocks is however better, because of a certain number of additional advantages (Magne and Leguay, 1972; Figure 5.49). *

*

*

The ﬁnal displacement is minimal. If the speciﬁed shock is symmetrical (with respect to the vertical line t=2), this residual displacement is zero (Young, 1964). For the same duration, t; of the speciﬁed shock and for the same value of maximum velocity, the possible maximum level of acceleration is twice as big. The maximum force is provided at the moment when acceleration is maximum, that is, when the velocity is zero (one will be able to thus have the maximum current). The solution with symmetrical pre- and postshocks requires minimal electric power.

Another parameter is the shape of these pre- and postshocks, the most used shapes being the triangle, the half-sine, and the rectangle (Figure 5.50).

FIGURE 5.50 Shapes of pre- and postshock pulses. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Mechanical Shock

5-47

Owing to discontinuities at the ends of the pulse, the rectangular compensation is seldom satisfactory (Smallwood, 1985). One often prefers a versed-sine applied to the entire signal (Hanning window), which has the advantages of being zero and smoothed at the ends (ﬁrst zero derivative) and presenting symmetrical pre- and postshocks. In all the cases, the amplitude of pre- and postshocks must remain small with respect to that of the principal shock (preferably lower than approximately 10%), in order not to deform too much the temporal signal and consequently, the shock spectrum. For a given shape of pre- and postshock, this choice thus imposes the duration. 5.9.3.2.1 Optimized Pre- and Postpulses Another method was developed (Fandrich, 1981) in order to take into account the tolerances on the shape of the signal allowed by the standards (R.T. Fandrich refers to standard MIL-STD 810 C) and to best use the possibilities of the shaker. The solution suggested consists of deﬁning the following: 1. One must deﬁne a preshock made up of the ﬁrst two terms of the development in a Fourier series of a rectangular pulse (with coefﬁcients modiﬁed after a parametric analysis). The table being in equilibrium in a median position before the test, the objective of this preshock is twofold: To give to a velocity, just before the principal shock, having a value close to one of the two limits of the shaker so that, during the shock, the velocity can use the entire range of variation permitted by the machine. To place, in the same way, the table as close as possible to one of the thrusts so that the moving element can move during the shock in the entire space between the two thrusts (limitation in displacement equal, according to the machines, to 2.54 or 5.08 cm). 2. One must also deﬁne a postshock composed of one period of a signal of the shape, Kt y sinð2pf1 tÞ; where the constants K; y; and f1 are evaluated in order to cancel the acceleration, the velocity, and the displacement at the end of the movement of the table. *

*

Acceleration, Velocity, Displacement

The frequency and the exponent are selected in order to respect the ratio of the velocity to the displacement at the end of the principal shock. The amplitude of the postshock is adjusted to obtain the desired velocity change. Figure 5.51 shows the total signal obtained in the case of a principal shock half-sine 30g, 11 msec. This methodology has been improved to provide a more general solution (Lax, 2001). Note: The realization of shocks on free or accelerated fall machines imposes de facto preshocks and/or postshocks, the existence of which the user is not always aware, but that can modify the shock severity at Half-sine 30 g, 11ms ACCELERATION PRE-SHOCK DISPLACEMENT

VELOCITY

0

50

100 150 Time (ms)

POST-SHOCK

200

250

FIGURE 5.51 Overall movement in a half-sine shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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5-48

Vibration Monitoring, Testing, and Instrumentation

low frequencies. The movement of shock starts with dropping the table from the necessary height to produce the speciﬁed shock and ﬁnishes with stopping the table after rebound on the shock simulator (Lalanne, 2002b). The preshock takes place during the fall of the table, the postshock during its rebound (Figure 5.52). These pre- and postshocks modify the SRS low frequency and can lead to an unexpected behavior of material when its natural frequency is low. 5.9.3.3

x··m

g

τ

ti

tR

FIGURE 5.52 Shock performed. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Incidence on the Shock Response Spectra

Figure 5.53 shows the response, v20 zðtÞ; of a one-DoF system ðf0 ¼ 4 Hz; j ¼ 0:05Þ to a TPS shock (the example of Section 5.5.2.5): *

*

*

For z0 ¼ z_0 ¼ 0 (conditions of the response spectrum). In the case of a shock with impact (free fall). In the case of a shock on shaker (half-sine symmetrical pre and postshocks with amplitude equal to 34 m/sec2).

We observed in this example the differences between the theoretical response at 4 Hz and the responses actually obtained on the shaker and shock machine. According to the test facility used, the shock applied can undertest or overtest the material. For the estimate of shock severity, one must take account of the whole of the signal of acceleration. In Figure 5.54, for j ¼ 0:05; is the SRS of: *

*

*

The nominal shock, calculated under the usual conditions of the spectra ðz0 ¼ z_0 ¼ 0Þ: The realizable shock on shaker, with its pre- and postshocks. The realizable shock by impact, taking of account of the fall and rebound phases.

One notes in this example that for: *

*

*

f0 # 6 Hz; the spectrum of the shock by impact is lower than the nominal spectrum, but higher than the spectrum of the shock on the shaker. 6 Hz , f0 , 30 Hz; the spectrum of the shock on the shaker is much overestimated. f0 , 30 Hz; all the spectra are superimposed.

w 20 z(t) (m/s2)

100 80 60 40 20

Response to a half-sine shock 500 m/s2 10 ms Theoretical Shaker

−0 −20 −40 −60 −80 −100 0.00

f0 = 5 Hz x = 0.05 0.05

0.10

0.15

0.20 0.25 Time (s)

Impact 0.30

0.35

0.40

FIGURE 5.53 Inﬂuence of the realization mode of a TPS shock on the response of a one-DoF system. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Mechanical Shock

5-49

TPS shock 340 m/s2 9 ms

450 400

x = 0.05

w 20 zsup (m/s2)

350 300

Shaker, with pre-and post-shocks

250 200 150 100

Theoretical

50

Impact

0 −1 10

100

101 Frequency f 0 (Hz)

102

104

FIGURE 5.54 Inﬂuence of the realization mode of a TPS shock on the SRS. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

This result appears logical when we remember that the slope of the shock spectrum at the origin is, for zero damping, proportional to the velocity change associated with the shock. The compensation signal, added to bring the velocity change back to zero, thus makes the slope of the spectrum at the origin zero. In addition, the response spectrum of the compensated signal can be larger than the spectrum of the theoretical signal close to the frequency corresponding to the inverse of the duration of the compensation signal. It is thus advisable to make sure that the variations observed are not in a range that includes the resonant frequencies of the test item.

It is necessary to add pre- and/or postshock to the speciﬁed shock in order to bring back the velocity of the shaker table to zero at the end of the shock pulse. The use of pre- and postshocks is best. Their amplitude must remain small with respect to that of the principal shock (lower than approximately 10%). The realization of shocks on free or accelerated fall machines also imposes pre- and postshocks. These pre- and postshocks lead to differences at low frequency between the spectrum of the speciﬁed shock and the spectrum of the shock actually carried out on the test facility.

5.9.4

Limitations of Electrodynamic Shakers

5.9.4.1

Mechanical Limitations

Performances of electrodynamic shakers (see Chapter 1) are limited in the following ﬁelds (Miller, 1964; Magne and Leguay, 1972). *

They are limited in terms of the maximum stroke of the coil-table unit (according to the machines being used, 25.4 to 75 mm peak-to-peak). At the time of the realization of a usual simple shock on shaker, the displacement starts from the equilibrium position (rest) of the coil, passes through a maximum, then returns to the initial position. In fact only half of the available stroke is used. For better use of the capacities of the machine, it is possible to shift the rest position from the central value towards one of the extreme values (Figure 5.55; Miller, 1964; McClanahan and Fagan, 1966; Smallwood and Witte, 1972).

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5-50

Vibration Monitoring, Testing, and Instrumentation

Rest Rest Maximum displacement during the shock movement

FIGURE 5.55 Displacement of the coil of the shaker. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

*

*

The maximum velocity is also limited (Young, 1964): 1.5 to 3 m/sec in sine mode (in shock, one can admit a larger velocity with nontransistorized ampliﬁers (electronic tubes), because these ampliﬁers can generally accept a very short overvoltage). During the movement of the moving element in the air-gap of the magnetic coils, there is an electromotive force (emf) produced which is opposed to the voltage supply. The velocity must thus have a value such that this emf is lower than the acceptable maximum output voltage of the ampliﬁer. The velocity must in addition be zero at the end of the shock movement (Smallwood and Witte, 1972; Galef, 1973). There is a limit to maximum acceleration, related to the maximum force. McClanahan and Fagan (1965) consider that the realizable maxima shock levels are approximately 20% below the vibratory limit levels in velocity and in displacement. The majority of authors agree that the limits in force are, for the shocks, larger than those indicated by the manufacturer (in sine mode). The determination of the maximum force and the maximum velocity is based, in vibration, on considerations of the fatigue of the shaker mechanical assembly. Since the number of shocks that the shaker will carry out is very much lower than the number of cycles of vibrations that it will undergo during its life, the parameter maximum force can be, for the shock applications, increased considerably.

Another reasoning consists of considering the acceptable maximum force, given by the manufacturer in random vibration mode, expressed by its rms value. Knowing that one can observe random peaks being able to reach 4.5 times this value (limitation of control system), one can admit the same limitation in shock mode. One ﬁnds other values in the literature, such as: *

*

# 4 times the maximum force in sine mode, with the proviso of not exceeding 300g on the armature assembly (Hug, 1972). . 8 times the maximum force in sine mode in certain cases (very short shocks; 0.4 msec, for example; Gallagher and Adkins, 1966). Dinicola (1964) and Keegan (1973) give a factor of about ten for the shocks of duration lower than 5 msec.

The limits of velocity, displacement, and force are not affected by the mass of the specimen. 5.9.4.1.1 Abacuses For a given shock and for given pre- and postshocks shapes, the velocity and the displacement can be calculated as a function of time by integration of the expressions of the acceleration, as well as the maximum values of these parameters, in order to compare them with the characteristics of the facilities. From these data, abacuses can be established allowing quick evaluation of the possibility of realization of a speciﬁed shock on a given test facility (characterized by its limits of velocity and

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Mechanical Shock

5-51

10 5

Shock amplitude (m/s2)

FORCE LIMITATION

A

10 4

C

VELOCITY LIMITATION

D

p = 0.05 E 0.10 0.25 0.50

10 3

10 2

F

A′

DISPLACEMENT LIMITATION

10 1 −4 10

10−3

G

C′ 10−2

1.00 F′ G′ D′ E′ 10−1

Shock duration (s) FIGURE 5.56 Abacus of the realization domain of a shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

of displacement). These abacuses are made up of straight-line segments on logarithmic scales (Figure 5.56). *

*

AA0 corresponding to the limitation of velocity: the condition vm # vL (vL ¼ acceptable maximum velocity on the facility considered) results in a relationship of the form x€ m t # constant (independent of p; the ratio of the absolute values of the pre- and postshocks amplitude and of the principal shock amplitude). CC, DD, and so on, the greater slope corresponding to the limitation in displacement for various values of p ( p ¼ 0.05, 0.10, 0.25, 0.50, and 1.00).

A particular shock will be thus realizable on the shaker only if the point of coordinates t; x€ m (duration and amplitude of the shock considered) is located under these lines, this useful domain increasing when p increases.

Example

103

The TPS shock pulse (340 m/sec2, 9 msec) is realizable on this shaker with p ¼ 0:05:

0.25

340

0.5 1.0

m/s2

TPS shock pulse, 340 m/sec , 9 msec (example of Section 5.5.2.5) Unit table mass: 192 kg Shaker: 135 kN (maximum velocity: 1.78 m/sec, maximum stroke: ^12.7 mm; see Figure 5.57) Test item þ ﬁxture mass: 150 kg Maximum acceleration without load: (135,000/192) < 703 m=sec2 Maximum acceleration with test item and ﬁxture: (135,000/(192 þ 150)) < 395 m=sec2

No load Test item + fixture mass: 150 kg

2

102 p = 0.05 0.1 0.009

101 −4 10

10−3 10−2 Shock Duration (s)

10−1

FIGURE 5.57 Shaker 135 kN, ^12.7 mm — TPS pulse with half-sine symmetrical pre- and postshocks.

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Vibration Monitoring, Testing, and Instrumentation

The limitation can also be due to: *

*

The resonance of the moving element (a few thousands Hertz; although it is kept to the maximum by design, the resonance of this element can be excited in the presence of signals with very short rise time). The strength of the material (very great accelerations can involve a separation of the coil of the moving component).

Mechanical limitations of electrodynamic shakers for shocks: *

*

*

5.9.4.2

Maximum stroke of the coil-table unit: 25.4 to 75 mm peak-to-peak. Maximum acceleration, related to the maximum force: according to the author, # 4 times the maximum force in sine mode, with the proviso of not exceeding 300g on the armature assembly, more than eight times the maximum force in sine mode in certain cases (very short shocks; 0.4 msec, for example). Maximum velocity: 1.5 to 3 m/sec in sine mode.

Electronic Limitations

1. Limitation of the output voltage of the ampliﬁer (Smallwood, 1974), which limits coil velocity. 2. Limitation of the acceptable maximum current in the ampliﬁer, related to the acceptable maximum force (i.e., with acceleration). 3. Limitation of the bandwidth of the ampliﬁer. 4. Limitation in power, which relates to the shock duration (and the maximum displacement) for a given mass. Current transistor ampliﬁers make it possible to increase the low frequency bandwidth but do not handle even short overtensions well, and thus are limited in mode shock (Miller, 1964).

5.9.5

The Use of Electrohydraulic Shakers

Shocks are realizable on the electrohydraulic exciters, but with additional stresses. *

*

Contrary to the case of the electrodynamic shakers, one cannot obtain via these means shocks of amplitude larger than realizable accelerations in the steady mode. The hydraulic vibration machines are in addition strongly nonlinear (Favour, 1974).

However, their long stroke, required for long duration shocks, is an advantage.

5.10 5.10.1

Control by a Shock Response Spectrum Principle

The exciters are actually always controlled by a signal that is a function of time. An acceleration –time signal gives only one SRS. However, there is an inﬁnity of acceleration –time signals with a given spectrum. The general principle thus consists in searching out one of the signals, x€ ðtÞ; having the speciﬁed spectrum. Historically, the simulation of shocks with spectrum control was ﬁrst carried out using analog and then digital methods (Smallwood and Witte, 1973; Smallwood, 1974).

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Time

(a)

Shock response spectrum

5-53

Acceleration

Mechanical Shock

Frequency f0

(b)

FIGURE 5.58 Elementary shock (a) and its SRS (b). (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

From the data of selected points on the shock spectrum to be simulated, the calculator of the control system uses an acceleration signal with a very tight spectrum. For that, the calculation software proceeds as follows (Lalanne, 2002b): *

*

*

At each frequency, f0 , of the reference shock spectrum, the software generates an elementary acceleration signal, for example, a decaying sinusoid. Such a signal has the property of having a SRS presenting a peak of the frequency of the sinusoid whose amplitude is a function of the damping of the sinusoid (Figure 5.58). With an identical shock spectrum, this property makes it possible to realize shocks on the shaker that would be unrealizable with a control carried out by a temporal signal of simple shape. For high frequencies, the spectrum of the sinusoid tends roughly towards the amplitude of the signal. All the elementary signals are added by possibly introducing a given delay (and variable) between each one of them in order to control to a certain extent the total duration of the shock, which is primarily due to the lower frequency components (Figure 5.59). The total signal being thus made up, the software proceeds to processes correcting the amplitudes of each elementary signal so that the spectrum of the total signal converges towards the reference spectrum after some iterations.

Shock response spectrum (m/s2)

60 50 40 30 20 10 0

0

100

200 300 Frequency (Hz)

400

500

FIGURE 5.59 SRS of the components of the required shock. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Vibration Monitoring, Testing, and Instrumentation

The operator must provide to the software with, at each frequency of the reference spectrum: *

*

*

*

The frequency of the spectrum Its amplitude A delay The damping of sinusoids or other parameters characterizing the number of oscillations of the signal

When a satisfactory spectrum time signal has been obtained, it remains to be checked that the maximum velocity and displacement during the shock are within the authorized limits of the test facility (by integration of the acceleration signal). Lastly, after measurement of the transfer function of the facility, one calculates the electric excitation which will make it possible to reproduce on the table the acceleration pulse with the desired spectrum, as in the case of control from a signal according to the time (Powers, 1974).

5.10.2

Principal Shapes of Elementary Signals

5.10.2.1

Decaying Sinusoid

The shocks measured in the ﬁeld environment are very often responses of structures to an excitation applied upstream, and are thus composed of the superposition of several modal responses of a damped sine type (Smallwood and Witte, 1973; Crimi, 1978; Boissin et al., 1981; Smallwood, 1985). Electrodynamic shakers are completely adapted to the reproduction of this type of signal. According to this, one should be able to reconstitute a given SRS from such signals of the form: 9 t $ 0= t , 0;

aðtÞ ¼ A e2hVt sin Vt aðtÞ ¼ 0

ð5:43Þ

where V ¼ 2pf , f ¼ frequency of the sinusoid, and h ¼ damping factor. Velocity and displacement are not zero at the end of the shock with this type of signal. These nonzero values are very awkward for a test on a shaker. Compensation can be carried out in 150 several ways: 100 50 0 m/s2

1. By truncating the total signal until it is realizable on the shaker. This correction can, however, lead to an important degradation of the corresponding spectrum (Smallwood and Witte, 1972). 2. By adding to the total signal (sum of all the elementary signals) a highly damped decaying sinusoid at low frequency, shifted in time, deﬁned to compensate for the velocity and the displacement (Smallwood and Nord, 1974; Smallwood 1975, 1985). 3. By adding to each component two exponential compensation functions, with a phase in the sinusoid (Nelson and Prasthofer, 1974; Smallwood, 1975).

−50

−100 −150 −200 −250 −300 −350

0

20

40 60 Time (ms)

80

100

FIGURE 5.60 Shock pulse generated from decaying sinusoids, compensated by a decaying sinusoid. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

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Mechanical Shock

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0 −50 −100 −150 −200

ZERD Function

The use of a decaying sinusoid with its compensation waveform modiﬁes the response spectrum at the low frequencies and, in certain cases, can harm the quality of simulation. Fisher and Posehn (1977) proposed using a waveform, which they named “ZERD” (Zero Residual Displacement), deﬁned by the expression: 1 aðtÞ ¼ A e2hVt sin Vt 2 t cosðVt þ fÞ V ð5:44Þ where f ¼ arc tanð2h=1 2 h2 Þ: This function resembles a damped sinusoid and has the advantage of leading to zero velocity and displacement at the end of the shock (Figure 5.62).

Example The reference SRS is that of Figure 5.24 (Section 5.5.2.5). Figure 5.63 shows an example of acceleration signal generated from ZERD functions having approximately the same SRS. 5.10.2.3

50

−250 −300

0

10

20

60

70

80

1.5 f = 1 Hz h = 0.05 A=1

1.0 0.5 0.0 −0.5 −1.0 −1.5

0

4

8 12 Time (s)

WAVSIN Waveform

Yang (1970, 1972) and Smallwood (1974, 1975, 1985) proposed (initially for the simulation of the earthquakes) a signal of the form:

30 40 50 Time (ms)

FIGURE 5.61 Acceleration signal generated from decaying sinusoids, compensated by two exponentia functions. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

Acceleration (m/s2)

5.10.2.2

100

m/s2

Example The reference SRS is that of Figure 5.24 (Section 5.5.2.5). Examples of acceleration signals generated from decaying sinusoids and having approximately the same SRS are shown in Figure 5.60 and Figure 5.61 in the cases of a compensation by a decaying sinusoid and by two exponential functions.

16

20

FIGURE 5.62 ZERD waveform of D.K. Fisher and M.R. Posehn (example).

aðtÞ ¼ am sin 2pbt sin 2pft

0#0#t

aðtÞ ¼ 0

elsewhere

) ð5:45Þ

where f ¼ Nb

ð5:46Þ

1 2b

ð5:47Þ

t¼

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Vibration Monitoring, Testing, and Instrumentation

where N is an integer (which must be odd and higher than on 1). The ﬁrst term of aðtÞ is a window of half-sine form of half-period t: The second describes N half-cycles of a sinusoid of greater frequency ð f Þ; modulated by the preceding window (Figure 5.64). This function leads also to zero velocity and displacement at the end of the shock.

200 150 100 50

m/s2

5-56

0 −50

Example

−100

Figure 5.65 shows an example of acceleration signal generated from WAVSIN functions having approximately the same SRS as the reference SRS of Figure 5.24 (Section 5.5.2.5).

−150

The cases treated by Smallwood (1974) seem to show that these three methods give similar results. It is noted, however, in practice, that, according to the shape of the reference spectrum, one or other of these waveforms allows a better convergence. The ZERD waveform very often gives good results.

20

30

40

50

60

70

80

90

FIGURE 5.63 Acceleration signal generated from ZERD functions.

1.5

WAVSIN (N = 5 f = 1 t = 2.5)

1.0 0.5 0.0 −0.5 −1.0 −1.5 0.0

5.10.4 Criticism of Control by a Shock Response Spectrum

0.5

1.0 1.5 Time t (s)

2.0

2.5

FIGURE 5.64 Example of WAVSIN waveform. (Source: Lalanne, Chocs Mecaniques, Hermes Science Publications. With permission.)

150 100 50

m/s2

Whatever the method adopted, simulation on a test facility of shocks measured in the real world requires the calculation of their SRSs and the search for an equivalent shock. If the speciﬁcation must be presented in the form of a time-dependent shock pulse, the test requester must deﬁne the characteristics of shape, duration, and amplitude of the signal, with the already quoted difﬁculties. If the speciﬁcation is given in the form of an SRS, the operator inputs in the control system the given spectrum, but the shaker is always controlled by a signal according to the time calculated and according to procedures described in the preceding sections. It is known that the transformation shock spectrum signal has an inﬁnite number of solutions, and that very different signals can have identical SRSs. This phenomenon is related to the loss of most of the information initially contained in the signal, x€ ðtÞ; during the calculation of the spectrum (Metzgar, 1967).

10

Time (ms)

Acceleration (m/s2)

5.10.3 Comparison of WAVSIN, SHOC Waveforms, and Decaying Sinusoid

0

0 −50 −100 −150

0

20 40 60 80 100 120 140 160 180

Time (ms)

FIGURE 5.65 Acceleration signal generated from WAVSIN functions.

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Mechanical Shock

5-57

The oscillatory shock pulses have a spectrum that presents an important peak to the frequency of the signal. This peak can, according to choice of parameters, exceed by a factor of ﬁve the amplitude of the same spectrum at the high frequencies; that is, ﬁve times the amplitude of the signal itself. Being given a point of the speciﬁed spectrum of amplitude, S; it is thus enough to have a signal vs. time of amplitude S=5 to reproduce the point. For a simple shaped shock, this factor does not exceed two in the most extreme case. All these remarks show that the determination of a signal of the same spectrum can lead to very diverse solutions, the validity of which one can question. If any particular precaution is not taken, the signals created by these methods have, in a general way, one duration much larger and an amplitude much smaller than the shocks that were used to calculate the reference SRS (a factor of about ten in both cases). Figure 5.66 and Figure 5.67 give an example. In the face of such differences between the excitations, one can legitimately wonder whether the SRS is a sufﬁcient condition to guarantee a representative test. It is necessary to remember that this equivalence is based on the behavior of a linear system that one chooses the Q factor a priori. One must be aware of the following. 60 40

Shock A

m/s2

20

Shock B

0 −20 −40 −60

0

0.5

FIGURE 5.66

1

1.5

2

2.5 3 Seconds

3.5

4

4.5 5 × 10E-2

Example of shocks having spectra near the SRS.

90 80

Shock A

70

m/s2

60

Shock B

50 40 30 20 Q = 10

10 0

0

0.2

0.4

FIGURE 5.67

0.6

0.8

1 Hz

1.2

1.4

1.6

1.8 2 × 10E3

SRS of the shocks shown in Figure 5.66.

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5-58

Vibration Monitoring, Testing, and Instrumentation *

*

*

The behavior of the structure is in practice far from linear and that the equivalence of the spectrum does not lead to stresses of the same amplitude. Another effect of these nonlinearities appears sometimes by the inaptitude of the system to correct the drive waveform to take account of the transfer function of the installation. Even if the amplitudes of the peaks of acceleration and the maximum stresses of the resonant parts of the tested structure are identical, the damage by the fatigue generated by accumulation of the stress cycles is rather different when the number of shocks to be applied is signiﬁcant. The tests carried out by various laboratories do not have the same severity.

These questions did not receive a really satisfactory response. By prudence rather than by rigorous reasoning, many agree, however, on the need for placing parapets, while trying to supplement the speciﬁcation deﬁned by a spectrum with complementary data (DV; duration of the shock, require SRS at two different values of damping, etc.; Favour, 1974; Smallwood, 1974, 1975, 1985).

There is an inﬁnity of acceleration –time signals with a given spectrum. Several elementary waveforms can be used to build a signal of acceleration having a given SRS. They give similar results. Without particular precaution, the signals thus obtained generally have one duration much larger and an amplitude much smaller than the shocks which were used to calculate the reference SRS. A complementary parameter (shock duration, velocity change, etc.) is often speciﬁed with the SRS to limit this effect.

5.11

Pyrotechnic Shock Simulation

Many works have been published on the characterization, measurement, and simulation of shocks of pyrotechnic origin, generated by bolt cutters, explosive valves, separation nuts, and so on (Zimmerman, 1993). The test facilities suggested are many, ranging from traditional machines to very exotic means. The tendency today is to consider that the best simulation of shocks measured in near-ﬁeld can be obtained only by subjecting the material to the shock produced by the real device. For shocks in the far-ﬁeld, simulation can be carried out either using the real pyrotechnic source and a particular mechanical assembly, using speciﬁc equipment using explosives, or by impacting metal to metal if the structural response is more important. When the real shock is practically made up only of the response of the structures, a simulation on a shaker is possible (when performances by this means are allowed).

5.11.1

Simulation Using Pyrotechnic Facilities

The simplest solution consists of making functional, real pyrotechnic devices on real structures. The simulation is perfect but (Conway et al., 1976; Luhrs, 1976): *

*

It can be expensive and destructive. One cannot apply an uncertainty factor (Lalanne, 2002d) to cover the variability of this shock without being likely to create unrealistic local damage (a larger load, which requires an often expensive modiﬁcation of the devices and can be much more destructive). To avoid this problem, an expensive solution consists of carrying out several tests in a statistical matter.

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Mechanical Shock

5-59

One often prefers to carry out a simulation on a reusable assembly, the excitation still being pyrotechnic in nature. Several devices have been designed, some examples of which are as follows: *

*

*

A test facility made up of a cylindrical structure (Ikola, 1964), which comprises a “consumable” sleeve cut out for the test by an explosive cord. Preliminary tests are carried out to calibrate the facility while acting on the linear charge of the explosive cord and/or the distance between the FIGURE 5.68 Plate with resonant system subjected equipment to be tested (ﬁxed on the to detonation. (Source: Lalanne, Chocs Mecaniques, structure as in the real case if possible) Hermes Science Publications. With permission.) and the explosive cord. A greater number of small explosive charges near the equipment to be tested on the structure in “ﬂowers pots.” The number of pots to be used on each axis depends on the amplitude of the shock, the size of the equipment, and the local geometry of the structure. The shape of the shock can be modiﬁed within certain limits by use of damping devices, placing the pot closer to or further from the equipment, or by putting suitable padding in the pot (Aerospace Systems pyrotechnic shock data, 1970). A test facility made up of a basic rectangular steel plate (Figure 5.68) suspended horizontally. This plate receives on its lower part, directly or by the intermediary of an “expendable” item, an explosive load (chalk line, explosive in plate or bread).

A second plate supporting the test item rests on the base plate via four elastic supports (Thomas, 1973). The reproducibility of the shocks is better if the explosive charge is not in direct contact with the base plate.

5.11.2

Simulation Using Metal-to-Metal Impact

The shock obtained by a metal-to-metal impact has similar characteristics to those of a pyrotechnical shock in an intermediate ﬁeld: great amplitude; short duration; high frequency content; SRS comparable with a low frequency slope of 12 dB per octave, etc. The simulation is in general satisfactory up to approximately 10 kHz. The shock can be created by the impact of a hammer on the structure itself, a Hopkinson bar or a resonant plate (Bai and Thatcher, 1979; Luhrs, 1981; Davie, 1985; Dokainish and Subbaraj, 1989; Davie and Batemen, 1992). With all these devices, the amplitude of the shock is controlled while acting on the velocity of impact. The frequency components are adjusted by modifying the resonant geometry of system (changing the length of the bar between two points of ﬁxing, adding or removing runners, etc.) or by the addition of a deformable material between the hammer and the anvil.

5.11.3

Simulation Using Electrodynamic Shakers

The limitation relating to the stroke of the electrodynamic shaker is not very constraining for the pyrotechnical shocks since they are at high frequencies. The possibilities are limited especially by the acceptable maximum force and then concern the maximum acceleration of the shock

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5-60

Vibration Monitoring, Testing, and Instrumentation

(Conway et al., 1976; Luhrs, 1976; Powers, 1976; Caruso, 1977). If one agrees to cover only part of the SRS where material has resonant frequencies, then when one makes a possible simulation on the shaker, which gives a better approach to matching the real spectrum. Exciters have the advantage of allowing the realization of any signal shape such as shocks of simple shapes (Dinicola, 1964; Gallagher and Adkins, 1966), but also of reproducing a speciﬁed SRS (direct control from an SRS; see Section 5.10.1). It is possible, in certain cases, to reproduce the real SRS up to 1000 Hz. If one is sufﬁciently far away from the source of the shock, the transient has a lower level of acceleration and the only limitation is the bandwidth of the shaker, which is about 2000 Hz. Certain facilities of this type were modiﬁed to make it possible to simulate the effects of pyrotechnical shocks up to 4000 Hz. One can thus manage to simulate shocks whose spectrum can reach 7000g (Moening, 1986).

5.11.4

Simulation Using Conventional Shock Machines

We saw that, generally, the method of development of a speciﬁcation of a shock consists of replacing the transient of the real environment, whose shape is in general complex, with a simple shape shock, such as half-sine, triangle, trapezoid, and so on, starting from the SRS equivalence criterion, with the application of a given or calculated uncertainty factor (Lalanne, 2002d) to the shock amplitude (Luhrs, 1976). With the examination of the shapes of the response spectra of standard simple shocks, it seems that the signal best adapted is the TPS pulse, whose spectra are also appreciably symmetrical. SRSs of the pyrotechnical shocks with, in general, averages close to zero have a very weak slope at low frequencies. The research of the characteristics of such a triangular pulse (amplitude, duration) having an SRS envelope of that of a pyrotechnical shock led often to a duration of about 1 msec and to an amplitude being able to reach several tens of thousands of msec22. Except in the case of very TABLE 5.12

Advantages and Drawbacks of Various Test Facilities for the Pyroshock Simulation

Shock Facility

Field

Advantages

Real pyrotechnic devices on real structures

Near

Very good simulation

Expensive, generally destructive, no uncertainty factor/test factor

Reusable assembly with pyrotechnic excitation

Near

Good simulation

Necessity of preliminary tests, no uncertainty factor/test factor, use of explosive (speciﬁc conditions to ensure safety), expensive

Metal to metal impact

Near

Good simulation, no explosive charge

Necessity of preliminary tests, limitations in acceleration and frequency (approximately 10 kHz)

Electrodynamic shaker

Far

Easy implementation, control using any time history signal or SRS, possibility of using an uncertainty factor or a test factor

Necessity of one test by axis, maximum frequency up to about 1 to 2 kHz

Conventional shock-test machine

Far

Easy implementation, possibility of using an uncertainty factor or a test factor

Use of a shock pulse with velocity change instead of an oscillatory shock pulse (over test at low frequency), necessity of one test by axis, shock duration higher than 2 msec (0.1 msec using a speciﬁc device for very light test item), limitation in amplitude, useable for very small test items only

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Drawbacks

Mechanical Shock

5-61

small test items, it is in general not possible to carry out such shocks on the usual drop tables due to certain limitations: *

*

*

Limitation in amplitude (acceptable maximum force on the table). Duration limit: the pneumatic shock simulators do not allow it to go below 3 to 4 msec; even with the lead shock simulators, it is difﬁcult to obtain a duration of less than 2 msec and a larger shock duration leads to a signiﬁcant overtest at low frequency. The SRSs of the pyrotechnical shocks are much more sensitive to the choice of damping than simple shocks carried out on shock machines.

A comparison of some pyroshock-test facilities is given in Table 5.12.

References Aerospace Systems pyrotechnic shock data (Ground test and ﬂight), Final Report, Contract NAS 5, 15208, June 1966, March 1970. ASTM D3332, Standard Test Method for Mechanical-Shock Fragility of Products, Using Shock Machines. American Society for Testing and Materials, Philadelphia, PA. Bai, M. and Thatcher, W., High G pyrotechnic shock simulation using metal-to-metal impact, Shock Vib. Bull., 49, Part 1, 97 –100, 1979. Benioff, H., The physical evaluation of seismic destructiveness, Bull. Seismol. Soc. Am., 398 –403, 1934. Biot, M.A. 1932. Transient oscillations in elastic systems, Thesis No. 259, Aeronautics Department, California Institute of Technology, Pasadena. Biot, M.A., A mechanical analyzer for the prediction of earthquake stresses, Bull. Seismol. Soc. Am., 31, 2, 151–171, 1941. Boissin, B., Girard, A., and Imbert, J.F. 1981. Methodology of uniaxial transient vibration test for satellites, In Recent Advances in Space Structure Design-Veriﬁcation Techniques, ESA SP 1036, pp. 35–53. Bort, R.L., Use and misuse of shock spectra, Shock Vib. Bull., 60, Part 3, 79–86, 1989. Bresk, F., Shock programmers, IES Proc., 141 –149, 1967. Bresk, F. and Beal, J., Universal impulse impact shock simulation system with initial peak sawtooth capability, IES Proc., 405 –416, 1966. Brooks, R.O., Shock springs and pulse shaping on impact shock machines, Shock Vib. Bull., 35, Part 6, 23– 40, 1966. Brooks, R.O. and Mathews, F.H., Mechanical shock testing techniques and equipment, IES Tutorial Lect. Ser., 69, 1966. Burgess, G., Effects of fatigue on fragility testing and the damage boundary curve, J. Test. Eval., 24, 6, 419–426, 1996. Burgess, G., Extension and evaluation of fatigue model for product shock fragility used in package design, J. Test. Eval., 28, 2, 116– 120, 2000. Caruso, H., Testing the Viking lander, J. Environ. Sci., March/April, 11 –17, 1977. Clough, R.W., On the importance of higher modes of vibration in the earthquake response of a tall building, Bull. Seismol. Soc. Am., 45, 4, 289–301, 1955. Colvin, V.G. and Morris, T.R., Algorithms for the rapid computation of response of an oscillator with bounded truncation error estimates, Int. J. Mech. Sci., 32, 3, 181– 189, 1990. Conrad, R. W., Characteristics of the navy medium weight high-impact shock machine, NRL Report 3852, September 14, 1951. Conrad, R.W., Characteristics of the light weight high-impact shock machine, NRL Report 3922, January 23, 1952. Conway, J.J., Pugh, D.A., and Sereno, T.J., Pyrotechnic shock simulation, IES Proc., 12 –16, 1976. Coty, A. and Sannier, B. 1966. Essais de chocs sur excitateur de vibrations, Note Technique No. 170/66/ EM, LRBA, France, December 1966.

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5-62

Vibration Monitoring, Testing, and Instrumentation

Cox, F.W., Efﬁcient algorithms for calculating shock spectra on general purpose computers, Shock Vib. Bull., 53, Part 1, 143 –161, 1983. Crimi, P., Analysis of structural shock transmission, J. Spacecraft, 15, 2, 79 –84, 1978. Davie, N.T., Pyrotechnic shock simulation using the controlled response of a resonating bar ﬁxture, IES Proc., 344– 351, 1985. Davie, N.T. and Bateman, V.I. 1992. Pyroshock simulation for satellite components using a tunable resonant ﬁxture — Phase 1, SAND 92-2135, October 1992. DEF STAN 00 35, Environmental Handbook for Defence Materials, Environmental Test Methods, Part 3, Ministry of Defence, Issue 3, May 7, 1999. Dinicola, D.J., A method of producing high-intensity shock with an electro-dynamic exciter, IES Proc., 253 –256, 1964. Dokainish, M.A. and Subbaraj, K., A survey of direct time-integration methods in computational structural dynamics: I. Explicit methods; II. Implicit methods, Comput. Struct., 32, 6, 1371–1389, 1989, see also 1387– 1401. Fandrich, R.T., Optimizing pre and postpulses for shaker shock testing, Shock Vib. Bull., 51, Part 2, 1 –13, 1981. Favour, J.D., Transient waveform control — a review of current techniques, J. Environ. Sci., November/ December, 9–19, 1974. Favour, J.D., Lebrun, J.M., and Young, J.P., Transient waveform control of electromagnetic test equipment, Shock Vib. Bull., 40, Part 2, 157–171, 1969. Fisher, D.K. and Posehn, M.R., Digital control system for a multiple-actuator shaker, Shock Vib. Bull., 47, Part 3, 79 –96, 1977. Fung, Y.C. and Barton, M.V., Some shock spectra characteristics and uses, J. Appl. Mech., 25, 365–372, 1958. Gaberson, H.A., Shock spectrum calculation from acceleration time histories, Civil Engineering Laboratory TN 1590, September 1980. Galef, A.E., Approximate response spectra of decaying sinusoids, Shock Vib. Bull., 43, Part 1, 61–65, 1973. Gallagher, G.A. and Adkins, A.W., Shock testing a spacecraft to shock response spectrum by means of an electrodynamic exciter, Shock Vib. Bull., 35, Part 6, 41– 45, 1966. GAM EG 13, 1e`re Partie, Recueil des Fascicules d’Essais, Ministe`re de la De´fense, De´le´gation Ge´ne´rale pour l’Armement, France, Juin 1986. Gray, R.P., Shock test programming — some recent developments, Test Eng., May, 28– 41, 1966. Grivelet, P., SRS calculation using prony and wavelet transforms, Shock Vib. Bull., 1, 67, 123 –132, 1996. Hale, M.T. and Adhami, R., Time –frequency analysis of shock data with application to shock response spectrum waveform synthesis, Proc. IEEE, 213 –217, 1991, Southeastcon, 91 CH 2998-3, April 1991. Hay, W.A. and Oliva, R.M., An improved method of shock testing on shakers, IES Proc., 241–246, 1963. Hug, G., Me´thodes d’essais de chocs au moyen de vibrateurs e´lectrodynamiques. IMEX, France, 1972. Hughes, M.E., Pyrotechnic shock test and test simulation, Shock Vib. Bull., 53, Part 1, 83 –88, 1983. Hughes, T.J.R. and Belytschko, T., A precis of developments in computational methods for transient analysis, J. Appl. Mech., 50, 1033– 1041, 1983. IEC 60068-2-27, Environmental testing, Part 2: Tests, Test Ea and guidance: Shock, June 1987a. IEC 60068-2-29, Ed. 2.0, Basic Environmental Testing Procedures, Part 2: Tests, Test Eb and guidance: Bump, 1987b. Ikola, A.L., Simulation of the pyrotechnic shock environment, Shock Vib. Bull., 34, Part 3, 267 –274, 1964. IMPAC 6060F — Shock Test Machine — Operating Manual, MRL 335 Monterey Research Laboratory, Inc. Irvine, M. 1986. Structural Dynamics for the Practising Engineer. Unwin Hyman, London, pp. 114–153. Jennings, R.L. 1958. The response of multi-storied structures to strong ground motion, MSc Thesis, University of Illinois, Urbana.

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Mechanical Shock

5-63

Keegan, W.B. 1973. Capabilities of electrodynamic shakers when used for mechanical shock testing, NASA Report N74-19083, July 1973. Lalanne, C. 1975. La simulation des environnements de chocs me´caniques, Rapport CEA-R-4682 (1) et (2), Commissariat a` l’Energie Atomique, France, 1975. Lalanne, C. 2002a. Mechanical Vibration and Shock, Sinusoidal Vibration, Vol. I. Taylor & Francis, New York. Lalanne, C. 2002b. Mechanical Vibration and Shock, Mechanical Shock, Vol. II. Taylor & Francis, New York. Lalanne, C. 2002c. Mechanical Vibration and Shock, Fatigue Damage, Vol. IV. Taylor & Francis, New York. Lalanne, C. 2002d. Mechanical Vibration and Shock, Speciﬁcation Development, Vol. V. Taylor & Francis, New York. La Verne Root and Bohs, C., Slingshot shock testing, Shock Vib. Bull., 39, Part 5, 73–81, 1969. Lax, R., A new method for designing MIL-STD shock tests, Test Eng. Manage., 63, 3, 10 –13, 2001. Lazarus, M., Shock testing, Mach. Des., October 12, 200 –214, 1967. Lonborg, J.O., A slingshot shock tester, IES Proc., 457–460, 1963. Luhrs, H., Equipment sensitivity to pyrotechnic shock, IES Proc., 3 –4, 1976. Luhrs, H.N., Pyrotechnic shock testing — past and future, J. Environ. Sci., Vol. XXIV, 6, 17–20, 1981. Magne, M. 1971. Essais de chocs sur excitateur e´lectrodynamique — me´thode nume´rique, Note CEADAM Z — SDA/EX — DO 0016, Commissariat a` l’Energie Atomique, France, December 1971. Magne, M., and Leguay, P. 1972. Re´alisation d’essais aux chocs par excitateurs e´lectrodynamique, Rapport CEA-R-4282, Commissariat a` l’Energie Atomique, France, 1972. Marshall, S., La Verne Root, and Sackett, L., 10 000 g Slingshot shock tests on a modiﬁed sand-drop machine, Shock Vib. Bull., 35, Part 6, 1965. McClanahan, J.M. and Fagan, J., Shock capabilities of electro-dynamic shakers, IES Proc., 251–256, 1965. McClanahan, J.M. and Fagan, J., Extension of shaker shock capabilities, Shock Vib. Bull., 35, Part 6, 111–118, 1966. McWhirter, M., Shock machines and shock test speciﬁcations, IES Proc., 497–515, 1963. McWhirter, M., Methods of Simulating Shock and Acceleration and Testing Techniques. Sandia Corporation SCDC 2939, 1961. Mercer, C.A. and Lincoln, A.P., Improved evaluation of shock response spectra, Shock Vib. Bull., 62, Part 2, 350–359, 1991. Merchant, H.C. and Hudson, D.E., Mode superposition in multi-DoF systems using earthquake response spectrum data, Bull. Seismol. Soc. Am., 52, 2, 405 –416, 1962. Merritt, R.G., A note on variation in computation of shock response spectra, IES Proc., 2, 330–335, 1993. Metzgar, K.J., A test oriented appraisal of shock spectrum synthesis and analysis, IES Proc., 69 –73, 1967. Miller, W.R., Shaping shock acceleration waveforms for optimum electrodynamic shaker performance, Shock Vib. Bull., 34, Part 3, 345 –354, 1964. MIL STD 810 F, Test Method Standard for Environmental Engineering Considerations and Laboratory Tests, Department of Defence, January 1, 2000. Moening, C., Views of the world of pyrotechnic shock, Shock Vib. Bull., 56, Part 3, 3 –28, 1986. NATO Standardization Agreement, STANAG 4370, Environmental Testing, Allied Environmental Conditions and Test Publications (AECTP) 400, Mechanical Environmental Tests, Method 403, Classical waveform shock, Edition 2, June 2000. Nelson, D.B. and Prasthofer, P.H., A case for damped oscillatory excitation as a natural pyrotechnic shock simulation, Shock Vib. Bull., 44, Part 3, 57–71, 1974. O’Hara, G.J. 1962. A numerical procedure for shock and Fourier analysis, NRL Report 5772, June 5, 1962. Operating manual for the MRL 2680 Universal programmer, MRL 519, Monterey Research Laboratory, Inc. Ostrem, F.E. and Rumerman, M.L. 1965. Final report. Shock and Vibration Transportation Environmental criteria, NASA Report CR 77220. Powers, D.R., Development of a pyrotechnic shock test facility, Shock Vib. Bull., 44, Part 3, 73 –82, 1974.

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Powers, D.R., Simulation of pyrotechnic shock in a test laboratory, IES Proc., 5 –9, 1976. Ridler, K.D. and Blader, F.B., Errors in the use of shock spectra, Environ. Eng., July, 7 –16, 1969. Rubin, S., Response of complex structures from reed-gage data, J. Appl. Mech., 35, 1958. Seipel, W.F., The SRC shock response spectra computer program, Shock Vib. Bull., 62, Part 1, 300–309, 1991. Selected ASTM Standards on Packaging, 4th Ed., ASTM, Philadelphia, 1994. Shell, E.H., Errors inherent in the speciﬁcation of shock motions by their shock spectra, IES Proc., 439 –448, 1966. Sinn, L.A. and Bosin, K.H., Sampling rate detection requirements for digital shock response spectra calculations, IES Proc., 174–180, 1981. Smallwood, D.O., Methods used to match shock spectra using oscillatory transients, IES Proc., 28 April –1 May, 409 –420, 1974. Smallwood, D.O. 1975. Time history synthesis for shock testing on shakers, Sand 75-5368. Smallwood, D.O., An improved recursive formula for calculating shock response spectra, Shock Vib. Bull., 51, Part 2, 211 –217, 1981. Smallwood, D.O. 1985. Shock testing by using digital control, Sand 85-03552 J. Smallwood, D.O., Shock response spectrum at low frequency, Shock Vib. Bull., 56, Part 1, 279–288, 1986. Smallwood, D.O., Calculation of the shock response spectrum (SRS) with a change in sample rate, ESTECH 2002 Proc., April 28–May 1, 2002. Smallwood, D.O. and Nord, A.R., Matching shock spectra with sums of decaying sinusoids compensated for shaker velocity and displacement limitations, Shock Vib. Bull., 44, Part 3, 43 –56, 1974. Smallwood, D.O. and Witte, A.F., The use of shaker optimized periodic transients in watching ﬁeld shock spectra, Shock Vib. Bull., 43, Part 1, 139– 150, 1973, or Sandia Report, SC-DR-710911, May 1972. Smith, J. L. 1984. Shock response spectra variational analysis for pyrotechnic qualiﬁcation testing of ﬂight hardware, NASA Technical Paper 2315, N84-23676, May 1984. Smith, J.L. 1986. Effects of variables upon pyrotechnically induced shock response spectra, NASA Technical Paper 2603. Thomas, C.L., Pyrotechnic shock simulation using the response plate approach, Shock Vib. Bull., 43, Part 1, 119 –126, 1973. Thorne, L.F., The design and the advantages of an air-accelerated impact mechanical shock machine, Shock Vib. Bull., 33, Part 3, 81 –84, 1964. Vibrations et Chocs Me´caniques — Vocabulaire, Norme AFNOR NF E 90-001 (NF ISO 2041), June 1993. Vigness, I., Some characteristics of navy high impact type shock machines, SESA Proc., 5, 1, 101–110, 1947. Vigness, I. 1961a. Shock testing machines. Shock and Vibration Handbook, Vol. 2, C.M. Harris and C.E. Crede, eds., McGraw-Hill, New York, chap. 26. Vigness, I. 1961b. Navy High Impact Shock Machines for High Weight and Medium Weight Equipment. U.S. Naval Research Laboratory, Washington, DC, NRL Report 5618, AD 260-008, June 1961b. Vigness, I. and Clements, E.W. 1963. Sawtooth and Half-sine Shock Impulses from the Navy Shock Machine for Medium Weight Equipment. U.S. Naval Research Laboratory, NRL Report 5943, June 3, 1963. Wells, R.H. and Mauer, R.C., Shock testing with the electrodynamic shaker, Shock Vib. Bull., 29, Part 4, 96 –105, 1961. Wise, J.H., The effects of digitizing rate and phase distortion errors on the shock response spectrum, IES Proc., 36– 43, 1983. Yang, R.C. 1970. Safeguard BMD system-development of a waveform synthesis technique, Document No. SAF-64, The Ralph M. Parsons Company, August 28, 1970. Yang, R.C. and Saffell, H.R., Development of a waveform synthesis technique. A supplement to response spectrum as a deﬁnition of shock environment, Shock Vib. Bull., 42, Part 2, 45 –53, 1972. Yarnold, J.A.L., High velocity shock machines, Environ. Eng., 17, 11 –16, 1965, November. Young, F.W., Shock testing with vibration systems, Shock Vib. Bull., 34, Part 3, 355–364, 1964. Zimmerman, R.M., Pyroshock — bibliography, IES Proc., 471 –479, 1993.

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6

Machine Condition Monitoring and Fault Diagnostics 6.1 6.2

Introduction ........................................................................ Machinery Failure .............................................................. Causes of Failure of Failure

†

Types of Failure

†

Frequency

6-2 6-2

6.3

Basic Maintenance Strategies .............................................

6-4

6.4 6.5

Factors Which Inﬂuence Maintenance Strategy .............. Machine Condition Monitoring .......................................

6-7 6-8

6.6

Transducer Selection .......................................................... 6-10

6.7 6.8

Transducer Location ........................................................... 6-14 Recording and Analysis Instrumentation ......................... 6-14

6.9

Display Formats and Analysis Tools ................................. 6-16

Run-to-Failure (Breakdown) Maintenance † Scheduled (Preventative) Maintenance † Condition-Based (Predictive, Proactive, Reliability Centered, On-Condition) Maintenance

Periodic Monitoring

†

Continuous Monitoring

Noncontact Displacement Transducers † Velocity Transducers † Acceleration Transducers

Vibration Meters † Data Collectors † FrequencyDomain Analyzers † Time-Domain Instruments † Tracking Analyzers Time Domain † Frequency Domain Quefrency Domain

†

Modal Domain

†

6.10 Fault Detection ................................................................... 6-21 Standards Limits

†

Acceptance Limits

†

Frequency-Domain

6.11 Fault Diagnostics ................................................................ 6-25

Chris K. Mechefske Queen’s University

Forcing Functions † Speciﬁc Machine Components † Speciﬁc Machine Types † Advanced Fault Diagnostic Techniques

Summary The focus of this chapter is on the deﬁnition and description of machine condition monitoring and fault diagnosis. Included are the reasons and justiﬁcation behind the adoption of any of the techniques presented. The motivation behind the decision making in regard to various applications is both ﬁnancial and technical. Both of these aspects are discussed, with the emphasis being on the technical side. The chapter deﬁnes machinery failure (causes, types, and frequency), and describes basic maintenance strategies and the factors that should be considered when deciding 6-1 © 2007 by Taylor & Francis Group, LLC

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6-2

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which to apply in a given situation. Topics considered in detail include transducer selection and mounting location, recording and analysis instrumentation, and display formats and analysis tools (speciﬁcally, time domain, frequency domain, modal domain, and quefrency domain-based strategies). The discussion of fault detection is based primarily on standards and acceptance limits in the time and frequency domains. The discussion of fault diagnostics is divided into sections that focus on different forcing functions, speciﬁc machine components, speciﬁc machine types, and advanced diagnostic techniques. Further considerations on this topic are found in Chapter 7 through Chapter 9.

6.1

Introduction

Approximately half of all operating costs in most processing and manufacturing operations can be attributed to maintenance. This is ample motivation for studying any activity that can potentially lower these costs. Machine condition monitoring and fault diagnostics is one of these activities. Machine condition monitoring and fault diagnostics can be deﬁned as the ﬁeld of technical activity in which selected physical parameters, associated with machinery operation, are observed for the purpose of determining machinery integrity. Once the integrity of a machine has been estimated, this information can be used for many different purposes. Loading and maintenance activities are the two main tasks that link directly to the information provided. The ultimate goal in regard to maintenance activities is to schedule only what is needed at a time, which results in optimum use of resources. Having said this, it should also be noted that condition monitoring and fault diagnostic practices are also applied to improve end product quality control and as such can also be considered as process monitoring tools. This deﬁnition implies that, while machine condition monitoring and fault diagnostics is being treated as the focus of this chapter, it must also be considered in the broader context of plant operations. With this in mind, this chapter will begin with a description of what is meant by machinery failure and a brief overview of different maintenance strategies and the various tasks associated with each. A short description of different vibration sensors, their modes of operation, selection criteria, and placement for the purposes of measuring accurate vibration signals will then follow. Data collection and display formats will be discussed with the speciﬁc focus being on standards common in condition monitoring and fault diagnostics. Machine fault detection and diagnostic practices will make up the remainder of the chapter. The progression of information provided will be from general to speciﬁc. The hope is that this will allow a broad range of individuals to make effective use of the information provided.

6.2

Machinery Failure

Most machinery is required to operate within a relatively close set of limits. These limits, or operating conditions, are designed to allow for safe operation of the equipment and to ensure equipment or system design speciﬁcations are not exceeded. They are usually set to optimize product quality and throughput (load) without overstressing the equipment. Generally speaking, this means that the equipment will operate within a particular range of operating speeds. This deﬁnition includes both steady-state operation (constant speed) and variable speed machines, which may move within a broader range of operation but still have ﬁxed limits based on design constraints. Occasionally, machinery is required to operate outside these limits for short times (during start-up, shutdown, and planned overloads). The main reason for employing machine condition monitoring and fault diagnostics is to generate accurate, quantitative information on the present condition of the machinery. This enables more conﬁdent and realistic expectations regarding machine performance. Having at hand this type of reliable information allows for the following questions to be answered with conﬁdence: *

*

Will a machine stand a required overload? Should equipment be removed from service for maintenance now or later?

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Machine Condition Monitoring and Fault Diagnostics *

*

*

6-3

What maintenance activities (if any) are required? What is the expected time to failure? What is the expected failure mode?

Machinery failure can be deﬁned as the inability of a machine to perform its required function. Failure is always machinery speciﬁc. For example, the bearings in a conveyor belt support pulley may be severely damaged or worn, but as long as the bearings are not seized, it has not failed. Other machinery may not tolerate these operating conditions. A computer disk drive may have only a very slight amount of wear or misalignment resulting in noisy operation, which constitutes a failure. There are also other considerations that may dictate that a machine no longer performs adequately. Economic considerations may result in a machine being classiﬁed as obsolete and it may then be scheduled for replacement before it has “worn out.” Safety considerations may also require the replacement of parts in order to ensure the risk of failure is minimized.

6.2.1

Causes of Failure

When we disregard the gradual wear on machinery as a cause of failure, there are still many speciﬁc causes of failure. These are perhaps as numerous as the different types of machines. There are, however, some generic categories that can be listed. Deﬁciencies in the original design, material or processing, improper assembly, inappropriate maintenance, and excessive operational demands may all cause premature failure.

6.2.2

Types of Failure

As with the causes of failure, there are many different types of failure. Here, these types will be subdivided into only two categories. Catastrophic failures are sudden and complete. Incipient failures are partial and usually gradual. In all but a few instances, there is some advanced warning as to the onset of failure; that is, the vast majority of failures pass through a distinct incipient phase. The goal of machine condition monitoring and fault diagnostics is to detect this onset, diagnose the condition, and trend its progression over time. The time until ultimate failure can then hopefully be better estimated, and this will allow plans to be made to avoid undue catastrophic repercussions. This, of course, excludes failures caused by unforeseen and uncontrollable outside forces.

6.2.3

Frequency of Failure

Anecdotal and statistical data describing the Wear In Normal Wear Wear Out frequency of failures can be summarized in what is called a “bathtub curve.” Figure 6.1 shows a Failure typical bathtub curve, which is applicable to an Rate individual machine or population of machines of the same type. The beginning of a machine’s useful life is Time In Service usually characterized by a relatively high rate of failure. These failures are referred to as “wear-in” FIGURE 6.1 Typical bathtub curve. failures. They are typically due to such things as design errors, manufacturing defects, assembly mistakes, installation problems and commissioning errors. As the causes of these failures are found and corrected, the frequency of failure decreases. The machine then passes into a relatively long period of operation, during which the frequency of failures occurring is relatively low. The failures that do occur mainly happen on a random basis. This period of a machine’s life is called the “normal wear” period and usually makes up most of the life of a machine. There should be a relatively low failure rate during the normal wear period when operating within design speciﬁcations.

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Vibration Monitoring, Testing, and Instrumentation

As a machine gradually reaches the end of its designed life, the frequency of failures again increases. These failures are called “wearout” failures. This gradually increasing failure rate at the expected end of a machine’s useful life is primarily due to metal fatigue, wear mechanisms between moving parts, corrosion, and obsolescence. The slope of the wearout part of the bathtub curve is machine-dependent. The rate at which the frequency of failures increases is largely dependent on the design of the machine and its operational history. If the machine design is such that the operational life ends abruptly, the machine is underdesigned to meet the load expected, or the machine has endured a severe operational life (experienced numerous overloads), the slope of the curve in the wearout section will increase sharply with time. If the machinery is overdesigned or experiences a relatively light loading history, the slope of this part of the bathtub curve will increase only gradually with time.

*

*

*

*

*

6.3

Generally (outside of start-up and shutdown) machinery is required to operate at constant speed and load. Machinery failure is deﬁned based on performance, operating condition, and system speciﬁcations. Machinery failure can be deﬁned as the inability of a machine to perform its required function. Causes of machinery failure can be generally deﬁned as being due to deﬁciencies in the original design, material or processing, improper assembly, inappropriate maintenance, or excessive operation demands. The frequency of failure for an individual machine or a population of similar machines can be summarized using a “bathtub curve.”

Basic Maintenance Strategies

Maintenance strategies can be divided into three main types: (1) run-to-failure, (2) scheduled, and (3) condition-based maintenance. Each of these different strategies has distinct advantages and disadvantages, which will be described below. Speciﬁc situations within any large facility may require the application of a different strategy. Therefore, no one strategy should be considered as always superior or inferior to another.

6.3.1

Run-to-Failure (Breakdown) Maintenance

Run-to-failure, or breakdown maintenance, is a strategy where maintenance, in the form of repair work or replacement, is only performed when machinery has failed. In general, run-to-failure maintenance is appropriate when the following situations exist: *

*

*

*

*

*

*

The equipment is redundant. Low cost spares are available. The process is interruptible or there is stockpiled product. All known failure modes are safe. There is a known long mean time to failure (MTTF) or a long mean time between failure (MTBF). There is a low cost associated with secondary damage. Quick repair or replacement is possible.

An example of the application of run-to-failure maintenance can be found when one considers the standard household light bulb. This device satisﬁes all the requirements above and therefore the most cost-effective maintenance strategy is to replace burnt out light bulbs as needed.

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Machine Condition Monitoring and Fault Diagnostics

Machine Capacity (Est.) Estimated Capacity and Load

6-5

Failures

Maintenance Activities

Machine Duty (Load) Time In Service FIGURE 6.2

Time vs. estimated capacity and actual load (run-to-failure maintenance).

Figure 6.2 shows a schematic demonstrating the relationship between a machine’s time in service, the load (or duty) placed on the machine, and the estimated remaining capacity of the machine. Whenever the estimated capacity curve intersects with (or drops below) the load curve, a failure will occur. At these times, repair work must be carried out. If the situation that exists ﬁts within the “rules” outlined above, all related costs (repair work and downtime) will be minimized when using run-to-failure maintenance.

6.3.2

Scheduled (Preventative) Maintenance

When speciﬁc maintenance tasks are performed at set time intervals (or duty cycles) in order to maintain a signiﬁcant margin between machine capacity and actual duty, the type of maintenance is called scheduled or preventative maintenance. Scheduled maintenance is most effective under the following circumstances: *

*

*

*

*

*

*

*

Data describing the statistical failure rate for the machinery is available. The failure distribution is narrow, meaning that the MTBF is accurately predictable. Maintenance restores close to full integrity of the machine. A single, known failure mode dominates. There is low cost associated with regular overhaul/replacement of the equipment. Unexpected interruptions to production are expensive and scheduled interruptions are not so bad. Low cost spares are available. Costly secondary damage from failure is likely to occur.

An example of scheduled maintenance practices can be found under the hood of your car. Oil and oil ﬁlter changes on a regular basis are part of the scheduled maintenance program that most car owners practice. A relatively small investment in time and money on a regular basis acts to reduce (but not eliminate) the likelihood of a major failure taking place. Again, this example shows how when all, or most, of the criteria listed above are satisﬁed, overall maintenance costs are minimized. Figure 6.3 shows a schematic demonstrating the relationship between a machine’s time in-service, the load (or duty) placed on the machine and the estimated remaining capacity of the machine when scheduled maintenance is being practiced. In this case, maintenance activities are scheduled at regular intervals in order to restore machine capacity before a failure occurs. In this way, there is always a margin between the estimated capacity and the actual load on the machine. If this margin is always present, there should theoretically never be an unexpected failure, which is the ultimate goal of scheduled maintenance.

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Vibration Monitoring, Testing, and Instrumentation

Machine Capacity (Est.) Estimated Capacity and Load

Margin Maintenance Activities

Machine Duty (Load)

Time In Service FIGURE 6.3

Time vs. estimated capacity and actual load (scheduled maintenance).

6.3.3 Condition-Based (Predictive, Proactive, Reliability Centered, On-Condition) Maintenance Condition-based maintenance (which is also known by many other names) requires that some means of assessing the actual condition of the machinery is used in order to optimally schedule maintenance, in order to achieve maximum production, and still avoid unexpected catastrophic failures. Conditionbased maintenance should be employed when the following conditions apply: *

*

*

*

*

*

*

*

*

*

Expensive or critical machinery is under consideration. There is a long lead-time for replacement parts (no spares are readily available). The process is uninterruptible (both scheduled and unexpected interruptions are excessively costly). Equipment overhaul is expensive and requires highly trained people. Reduced numbers of highly skilled maintenance people are available. The costs of the monitoring program are acceptable. Failures may be dangerous. The equipment is remote or mobile. Failures are not indicated by degeneration of normal operating response. Secondary damage may be costly.

An example of condition-based maintenance practices can again be found when considering your car, but this time we consider the tires. Regular inspections of the tires (air pressure checks, looking for cracks and scratches, measuring the remaining tread, listening for slippage during cornering) can all be used to make an assessment of the remaining life of the tires and also the risk of catastrophic failure. In order to minimize costs and risk, the tires are replaced before they are worn out completely, but not before they have given up the majority of their useful life. A measure of the actual condition of equipment is used to utilize maintenance resources optimally. Figure 6.4 shows a schematic drawing that demonstrates the relationship between a machine’s time in service, the load (or duty) placed on the machine, and the estimated remaining capacity of the machine when condition-based maintenance is being practiced. Note that the margin between duty and capacity is allowed to become quite small (smaller than in scheduled maintenance), but the two lines never touch (as in run-to-failure maintenance). This results in a longer time between maintenance activities than for scheduled maintenance. Maintenance tasks are scheduled just before a failure is expected to occur, thereby optimizing the use of resources. This requires that there exists a set of accurate measures that can be used to assess the machine integrity. Each of these maintenance strategies has its advantages and disadvantages and situations exist where one or the other is appropriate. It is the maintenance engineer’s role to decide on and justify the use of any one of these procedures for a given machine. There are also instances where a given machine will require more than one maintenance strategy during its operational life, or perhaps even at one

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Machine Condition Monitoring and Fault Diagnostics

6-7

Machine Capacity (Est.) Estimated Capacity and Load

Minimum Margin

Machine Duty (Load)

Maintenance Activity

Time In Service FIGURE 6.4

Time versus estimated capacity and actual load (condition-based maintenance).

time, and situations where more than one strategy is appropriate within a particular plant. Examples of these situations include the need for an increased frequency of monitoring as the age of a machine increases and the likelihood of failure increases, and the scheduling of maximum time between overhauls during the early stages of a machine’s useful life, with monitoring in between looking for unexpected failures.

*

*

*

*

*

6.4

Maintenance strategies can be divided into three main types: (1) run-to-failure, (2) scheduled, and (3) condition-based maintenance. No one strategy should be considered as always superior or inferior to another. Run-to-failure, or breakdown maintenance, is a strategy where maintenance, in the form of repair work or replacement, is only performed when machinery has failed. When speciﬁc maintenance tasks are performed at set time intervals (or duty cycles) in order to maintain a signiﬁcant margin between machine capacity and actual duty, the type of maintenance is called scheduled or preventative maintenance. Condition-based maintenance requires that some means of assessing the actual condition of the machinery is used in order to optimally schedule maintenance, in order to achieve maximum production and still avoid unexpected catastrophic failures.

Factors Which Inﬂuence Maintenance Strategy

While there are some general guidelines for choosing the most appropriate maintenance strategy, each case must be evaluated individually. Principal considerations will always be deﬁned in economic terms. Sometimes, a speciﬁc company policy (such as safety) will outweigh all other considerations. Below is a list of factors (in no particular order) that should be taken into account when deciding which maintenance strategy is most appropriate for a given situation or machine: *

*

*

*

*

*

*

*

Classiﬁcation (size, type) of the machine Critical nature of the machine relative to production Cost of replacement of the entire machine Lead-time for replacement of the entire machine Manufacturers’ recommendations Failure data (history), MTTF, MTBF, failure modes Redundancy Safety (plant personnel, community, environment)

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Vibration Monitoring, Testing, and Instrumentation *

*

*

6.5

Cost and availability of spare parts Personnel costs, administrative costs, monitoring equipment costs Running costs for a monitoring program (if used)

Machine Condition Monitoring

With the understanding that condition-based maintenance may not be appropriate in all situations, let us say that some preliminary analysis has been carried out and a decision made to apply machine condition monitoring and fault diagnostics in a selected part of a plant or on a speciﬁc machine. The following is a list of potential advantages that should be realized: *

*

*

*

*

*

*

*

*

*

*

Increased machine availability and reliability Improved operating efﬁciency Improved risk management (less downtime) Reduced maintenance costs (better planning) Reduced spare parts inventories Improved safety Improved knowledge of the machine condition (safe short-term overloading of machine possible) Extended operational life of the machine Improved customer relations (less planned/unplanned downtime) Elimination of chronic failures (root cause analysis and redesign) Reduction of postoverhaul failures due to improperly performed maintenance or reassembly

There are, of course, also some disadvantages that must be weighed in the decision to use machine condition monitoring and fault diagnostics. These disadvantages are listed below: *

*

*

*

*

*

Monitoring equipment costs (usually signiﬁcant). Operational costs (running the program). Skilled personnel needed. Strong management commitment needed. A signiﬁcant run-in time to collect machine histories and trends is usually needed. Reduced costs are usually harder to sell to management as beneﬁts when compared with increased proﬁts.

The ultimate goal of machine condition monitoring and fault diagnostics is to get useful information on the condition of equipment to the people who need it in a timely manner. The people who need this information include operators, maintenance engineers and technicians, managers, vendors, and suppliers. These groups will need different information at different times. The task of the person or group in charge of condition monitoring and diagnostics must ensure that useful data is collected, that data is changed into information in a form required by and useful to others, and that the information is provided to the people who need it when they need it. Further general reading can be found in these references: Mitchell (1981), Lyon (1987), Mobley (1990), Rao (1996), and Moubray (1997). The focus of this chapter will be on vibration-based data, but there are several different types of data that can be useful for assessing machine condition and these should not be ignored. These include physical parameters related to lubrication analysis (oil/grease quality, contamination), wear particle monitoring and analysis, force, sound, temperature, output (machine performance), product quality, odor, and visual inspections. All of these factors may contribute to a complete picture of machine integrity. The types of information that can be gleaned from the data include existing condition, trends, expected time to failure at a given load, type of fault existing or developing, and the type of fault that caused failure. The speciﬁc tasks which must be carried out to complete a successful machine condition monitoring and fault diagnostics program include detection, diagnosis, prognosis, postmortem, and

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prescription. Detection requires data gathering, comparison to standards, comparison to limits set in-plant for speciﬁc equipment, and trending over time. Diagnosis involves recognizing the types of fault developing (different fault types may be more or less serious and require different action) and determining the severity of given faults once detected and diagnosed. Prognosis, which is a very challenging task, involves estimating (forecasting) the expected time to failure, trending the condition of the equipment being monitored, and planning the appropriate maintenance timing. Postmortem is the investigation of root-cause failure analysis, and usually involves some research-type investigation in the laboratory and/or in the ﬁeld, as well as modeling of the system. Prescription is an activity that is dictated by the information collected and may be applied at any stage of the condition monitoring and diagnostic work. It may involve recommendations for altering the operating conditions, altering the monitoring strategy (frequency, type), or redesigning the process or equipment. The tasks listed above have relatively crisp deﬁnitions, but there is still considerable room for adjustment within any condition monitoring and diagnostic program. There are always questions, concerning such things as how much data to collect and how much time to spend on data analysis, that need to be considered before the ﬁnal program is put in place. As mentioned above, things such as equipment class, size, importance within the process, replacement cost, availability, and safety need to be carefully considered. Different pieces of equipment or processes may require different monitoring strategies.

6.5.1

Periodic Monitoring

Periodic monitoring involves intermittent data gathering and analysis with portable, removable monitoring equipment. On occasion, permanent monitoring hardware may be used for this type of monitoring strategy, but data is only collected at speciﬁc times. This type of monitoring is usually applied to noncritical equipment where failure modes are well known (historically dependable equipment). Trending of condition and severity level checks are the main focus, with problems triggering more rigorous investigations.

6.5.2

Continuous Monitoring

Constant or very frequent data collection and analysis is referred to as continuous monitoring. Permanently installed monitoring systems are typically used, with samples and analysis of data done automatically. This type of monitoring is carried out on critical equipment (expensive to replace, with downtime and lost production also being expensive). Changes in condition trigger more detailed investigation or possibly an automatic shutdown of the equipment.

*

*

*

*

Potential advantages of machine condition monitoring include increased machine availability and reliability, improved efﬁciency, reduced costs, extended operational life, and improved safety. Some of the disadvantages of condition monitoring include monitoring equipment costs, operational costs, and training costs. The ultimate goal of machine condition monitoring and fault diagnostics is to get useful information on the condition of equipment to the people who need it, in a timely manner. The speciﬁc tasks which must be carried out to complete a successful machine condition monitoring and fault diagnostics program include detection, diagnosis, prognosis, postmortem, and prescription.

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6.6

Transducer Selection Relative Amplitude Response

A transducer is a device that senses a physical quantity (vibration in this case, but it can also be Displacement 10 temperature, pressure, etc.) and converts it into an electrical output signal, which is proportional to Velocity 1.0 the measured variable (see Chapter 1). As such, the Acceleration transducer is a vital link in the measurement chain. 0.1 Accurate analysis results depend on an accurate electrical reproduction of the measured par0.1 1.0 10 100 1,000 10,000 ameters. If information is missed or distorted Frequency (Hz) during measurement, it cannot be recovered later. Hence, the selection, placement, and proper use of FIGURE 6.5 Frequency versus response amplitude for the correct transducer are important steps in the various sensor types. implementation of a condition monitoring and fault diagnostics program. Considerable research and development work has gone into the design, testing, and calibration of sensors (transducers) for a wide range of applications. The transducer must be: *

*

*

*

Correct for the task Properly mounted In good working order (properly calibrated) Fully understood in terms of operational characteristics

Transducers usually require ampliﬁcation and conversion electronics to produce a useful output signal. These circuits may be located within the sealed sensor unit or in a separate box. There are advantages and disadvantages to both of these conﬁgurations but they will not be detailed here. Traditional vibration sensors fall into three main classes: *

*

*

Noncontact displacement transducers (also known as proximity probes or eddy current probes) Velocity transducers (electro-mechanical, piezoelectric) Accelerometers (piezoelectric)

Force and frequency considerations dictate the type of measurements and applications that are best suited for each transducer. Recently, laser-based noncontact velocity/displacement transducers have become more commonplace. These are still relatively expensive because of their extreme sensitivity, and hence are still predominantly used in the laboratory setting. Figure 6.5 shows the relationship between the different transducer types in terms of response amplitude and frequency. For constant velocity vibration amplitude across all frequencies, a displacement transducer is more sensitive in the lower frequency range, while an accelerometer is more sensitive at higher frequencies. While it may appear as if the velocity transducer is the best compromise, transducers are selected to optimize sensitivity over the frequency range that is expected to be recorded. The type of motion sensed by displacement transducers is the relative motion between the point of attachment and the observed surface. Velocity transducers and accelerometers measure the absolute motion of the structure to which they are attached.

6.6.1

Noncontact Displacement Transducers

These types of sensors ﬁnd application primarily in ﬂuid ﬁlm (journal) radial or thrust bearings. With the rotor resting on a ﬂuid ﬁlm there is no way to easily attach a sensor. A noncontact approach is then the best alternative. Noncontact measurements indicate shaft motion and position relative to the bearing. Radial shaft displacements and seal clearances can be conveniently measured. Another advantage of using

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Vibration Amplitude

Vibration Direction

Output

(a)

Time

Vibration

Direction

X Output

Y Direction Displacement

Y Output

(b)

X Direction Displacement

FIGURE 6.6 (a) Shaft displacement with one sensor; (b) shaft orbit with two sensors (x direction versus y direction displacement — assumes shaft is circular).

noncontact displacement probes is that when they are used in pairs, set 908 apart, the signals can be used to show shaft dynamic motion (orbit) within the bearing. Figure 6.6 shows a single channel and dual channel measurement result. When the two channels are plotted against one another, they clearly show what are known as shaft orbits. These orbits deﬁne the dynamic motion of the shaft in the bearing, and are valuable fault detection and diagnostic tools. The linearity and sensitivity of the proximity probe depends on the target conductivity and porosity. Calibration of the probe on the speciﬁc material in use is recommended. This type of sensor is capable of both static and dynamic measurements, but temperature and pressure extremes will affect the transducer output. The probe will detect small defects in the shaft (cracks or pits), and these may seem like vibrations in the output signal. Installation of these sensors requires a rigid mounting. Adaptors for quick removal and replacement without machine disassembly can be useful. The minimum tip clearance from all adjacent surfaces should be two times the tip diameter. Probe extensions must be checked to ensure that the resonant frequency of the extension is not excited during data gathering. As with all sensors, care must be taken when handling the cables and the connections must be kept clean.

6.6.2

Velocity Transducers

There are two general types of velocity transducers. They can be distinguished by considering the mode of operation. The two types are electro-mechanical and piezoelectric crystal based. Piezoelectric crystal-based transducers will be discussed in the next section, so the focus here will be on electro-mechanical (see Chapter 1). Electro-mechanical velocity transducers function with a permanent magnet (supported by springs) moving within a coil of wire. As the sensor experiences changes in velocity, as when attached to a vibrating surface, the movement of the magnet within the coil is proportional to force acting on the

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Vibration Monitoring, Testing, and Instrumentation

Amplitude Response

sensor. The current in the coil, induced by the moving magnet, is proportional to velocity, which in turn is proportional to the force. This type of device is known as “self-generating” and produces a low impedance signal; therefore, no additional signal conditioning is generally needed. Electro-mechanical velocity sensors require the spring suspension system to be designed with a relatively low natural frequency. These devices have good sensitivity, typically above 10 Hz, but Natural Frequency ≈10 Hz their high-frequency response is limited (usually Frequency around 1500 Hz) by the inertia of the system. Some devices may obtain a portion, or all, of FIGURE 6.7 Output sensitivity vs. frequency for an their damping electrically. This type must be electro-mechanical velocity transducer. loaded with resistance of a speciﬁc value to meet design constraints. These are usually designed for use with a speciﬁc data collection instrument and must be checked and modiﬁed if they are to be used with other instruments. Figure 6.7 shows a plot of the sensitivity vs. frequency for an electro-mechanical velocity transducer. While electro-mechanical velocity transducers can be designed to have good dynamic range within a speciﬁc frequency range, there are several functional limitations. Because a damping ﬂuid is typically used to provide most of the damping, this type of transducer is limited to a relatively narrow temperature band, below the boiling point of the damping liquid. The mechanical reliability of these sensors is also limited by the moving parts within the transducer, which may become worn or fail over time. This has resulted in this type of transducer being replaced by piezoelectric sensors in machine condition monitoring applications. The orientation of the sensor is also limited to only the vertical or horizontal direction, depending on the type of mounting used. Finally, as a damped system, such as an electro-mechanical velocity transducer, approaches its natural frequency, a shift in phase relationships may occur (below 50 Hz). This phase shift at low frequencies will affect analysis work.

6.6.3

Acceleration Transducers Amplitude Response

By far the most commonly used transducers for measuring vibration are accelerometers (see Chapter 1). These devices contain one or more Natural Frequency piezoelectric crystal elements (natural quartz or 3 man-made ceramics), which produce voltage when stressed in tension, compression or shear. This is the piezoelectric effect. The voltage generated across the crystal pole faces is proportional to the 0.1 0.2 0.4 0.6 0.8 1.0 applied force. Forcing Frequency Accelerometers have a linear response over a Natural Frequency wide frequency range (0.5 Hz to 20 kHz), with specialty sensors linear up to 50 kHz. This wide FIGURE 6.8 Typical accelerometer sensitivity vs. linear frequency range and the broad dynamic frequency. amplitude range make accelerometers extremely versatile sensors. Figure 6.8 shows the sensitivity vs. frequency relationship for a typical accelerometer. In addition, the signal can be electronically integrated to give velocity and displacement measurements. This type of transducer is relatively resistant to temperature changes, reliable (having no moving parts), produces a self-generating output signal meaning no external power supply is needed unless there are onboard electronics, is available in a variety of sizes, is usually relatively insensitive to nonaxial vibration

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Accelerometer Output

Accelerometer Output

Accelerometer Output

(,3% of main axis sensitivity), and can function well in any orientation. Signals from this type of transducer contain signiﬁcantly more vibration components than other types. This means that there is a large amount of information available in the raw vibration signal. Installation of accelerometers requires as rigid ≈28kHz (a) Frequency a mount as possible. Permanent installation with studs or bolts is usually best for high speed machinery where high-frequency measurements are required. The close coupling between the machine and the sensor allows for direct transmission of the vibration to the sensor. Stud mounting requires a ﬂat surface to give the best amplitude linearity and frequency (b) Frequency ≈ 7kHz response. This type of mounting is expensive and may not be practical if large numbers of measurements are being recorded with a portable instrument. Magnetic mounts have the advantage of being easily movable and provide good repeatability in the lower frequency range, but have limited high-frequency sensitivity (4 to 5 kHz). Hand-held measurements are useful (c) ≈2kHz Frequency when conducting general vibration surveys, but usually result in signiﬁcant variation between measurements. The hand-held mount is least FIGURE 6.9 Accelerometer response vs. frequency for expensive but only offers frequency response various types of mounts (a — stud; b — magnet; c — below 1 kHz. Figure 6.9 shows a plot of the hand held). response curves for the same accelerometer with different mounts. For machine condition monitoring and fault diagnostics applications, there will typically be a combination of all three mounting methods used, depending on the equipment being monitored and the monitoring strategy employed. As with the other types of vibration sensor, accelerometers have certain limitations. Because of their sensitivity and wide dynamic range, accelerometers are also sensitive to environmental input not related to the vibration signal of interest. Temperature (ambient and ﬂuctuations) may cause distortion in the recorded signal. General purpose accelerometers are relatively insensitive to temperatures up to 2508C. At higher temperatures, the piezoelectric material may depolarize and the sensitivity may be permanently altered. Temperature transients also affect accelerometer output. Shear-type accelerometers have the lowest temperature transient sensitivity. A heat sink or mica washer between the accelerometer and a hot surface may help reduce the effects of temperature. Piezoelectric crystals are sensitive to changes in humidity. Most accelerometers are epoxy bonded or welded together to provide a humidity barrier. Moisture migration through cables and into connections must be guarded against. Large electro-magnetic ﬁelds can also induce noise into cables that are not double shielded. If an accelerometer is mounted on a surface that is being strained (bent), the output will be altered. This is known as base strain, and thick accelerometer bases will minimize this effect. Shear-type accelerometers are less sensitive because the piezoelectric crystal is mounted to a center post not the base. Accelerometers are designed to remain constant for long periods of time; however, they may need calibrating if damaged by dropping or high temperatures. A known amplitude and frequency source (or another accelerometer that has a known calibration) should be used to check the calibration of accelerometers from time to time.

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6.7

Transducer Location

The placement (location) of a vibration transducer is a critical factor in machine condition monitoring and fault diagnostics. Using several locations and directions when recording vibration information is recommended. As always, it depends on the application and whether or not the expense is warranted. If the vibration component which relates to a given fault condition is not recorded, no amount of analysis will extract it from the signal. When selecting sensor locations consider the following: *

*

*

Mechanical independence The vibration transmission path Locations where natural frequency vibrations may be excited (ﬂexible components or attachments)

*

*

*

*

*

*

*

6.8 6.8.1

A transducer is a device that senses a physical quantity and converts it into an electrical output signal, which is proportional to the measured variable. The selection, placement, and proper use of the correct transducer are important steps in the implementation of a condition monitoring and fault diagnostics program. The transducer must be correct for the task, properly mounted, in good working order (properly calibrated), and fully understood in terms of operational characteristics. Traditional vibration sensors fall into three main classes: noncontact displacement transducers, velocity transducers, and accelerometers. Transducers are selected to optimize sensitivity over the frequency range that is expected to be recorded. Using several locations and directions when recording vibration information is recommended. When selecting sensor locations, one must consider mechanical independence, the vibration transmission path, and locations where natural frequency vibrations may be excited.

Recording and Analysis Instrumentation Vibration Meters

Vibration meters are generally small, hand-held (portable), inexpensive, simple to use, self-contained devices that give an overall vibration level reading (see Chapter 1). They are used for walk-around surveys and measure velocity and/or acceleration. Generally, these devices have no built-in diagnostics capability, but the natural frequency of an accelerometer can be exploited to look for speciﬁc machinery faults. As an example, rolling element bearings generally emit “spike” energy during the early stages of deterioration. These are sharp impacts as rollers strike defects (pits, cracks) in the races. A spike energy meter is an accelerometer that has been tuned to have its resonant frequency excited by these impacts, thus giving a very early warning of deteriorating bearings.

6.8.2

Data Collectors

Most vibration data collectors available today for use in machine condition monitoring and fault diagnostics are microcomputer based. They are used together with vibration sensors to measures vibration, to store and transfer data, and for frequency domain analysis. Considerably more data can be recorded in a digital form, but the cost of these devices can also be considerable. Another

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advantage of most data collectors is the ability to use these devices to conduct on-the-spot diagnostics or balancing. They are usually used with a PC to provide permanent data storage and a platform for more detailed analysis software. Data collectors are usually used on general-purpose equipment.

6.8.3

Frequency-Domain Analyzers

The frequency-domain analyzer is perhaps the key instrument for diagnostic work. Different machine conditions (unbalance, misalignment, looseness, bearing ﬂaws) all generate characteristic patterns that are usually visible in the frequency domain. While data collectors do provide some frequency domain analysis capability, their main purpose is data collection. Frequency-domain analyzers are specialized instruments that emphasize the analysis of vibration signals. As such, they are often treated as a laboratory instrument. Generally, analyzers will have superior frequency resolution, ﬁltering ability (including antialiasing), weighting functions for the elimination of leakage, averaging capabilities (both in the time and frequency domains), envelope detection (demodulation), transient capture, large memory, order tracking, cascade/waterfall display, and zoom features. Dual-channel analysis is also common.

6.8.4

Time-Domain Instruments

Time-domain instruments are generally only able to provide a time domain display of the vibration waveform. Some devices have limited frequency-domain capabilities. While this restriction may seem limiting, the low cost of these devices and the fact that some vibration characteristics and trends show up well in the time domain make them valuable tools. Oscilloscopes are the most common form. Shaft displacements (orbits), transients and synchronous time averaging (and negative averaging) are some of the analysis strategies that can be employed with this type of device.

6.8.5

Tracking Analyzers

Tracking analyzers are typically used to record and analyze data from machines that are changing speed. This usually occurs during run-up and coast-down of large machinery or turbo-machinery. These measurements are typically used to locate machine resonances and unbalance conditions. The tracking rate is dependent on ﬁlter bandwidth, and there is a need for a reference signal to track speed (tachometer input). These devices usually have variable input sensitivity and a large dynamic range.

*

*

*

*

*

Vibration meters are generally small, hand-held (portable), inexpensive, simple to use, selfcontained devices that give an overall vibration level reading. Most vibration data collectors available today for use in machine condition monitoring and fault diagnostics are microcomputer based. They are used together with vibration sensors to measures vibration, to store and transfer data, and for frequency domain analysis. Frequency domain analyzers are specialized instruments that emphasize the analysis of vibration signals, and as such they are perhaps the key instrument for diagnostic work. Time domain instruments are generally only able to provide a time domain display of the vibration waveform. Tracking analyzers are typically used to record and analyze data (locate machine resonances and unbalance conditions) from machines that are changing speed. This usually occurs during run-up and coast-down of large machinery or turbo-machinery.

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6.9

Vibration Monitoring, Testing, and Instrumentation

Display Formats and Analysis Tools

Vibration signals can be displayed in a variety of different formats. Each format has advantages and disadvantages, but generally the more processing that is done on the dynamic signal, the more speciﬁc information is highlighted and the more extraneous information is discarded. The broad display formats that will be discussed here are the time domain, the frequency domain, the modal domain, and the quefrency domain. Within each of these display formats, several different analysis tools (some speciﬁc to that display format) will be described.

6.9.1

Time Domain

The time domain refers to a display or analysis of the vibration data as a function of time. The principal advantage of this format is that little or no data are lost prior to inspection. This allows for a great deal of detailed analysis. However, the disadvantage is that there is often too much data for easy and clear fault diagnosis. Time-domain analysis of vibration signals can be subdivided into the following sections. 6.9.1.1

Time-Waveform Analysis

Time-waveform analysis involves the visual inspection of the time-history of the vibration signal. The general nature of the vibration signal can be clearly seen and distinctions made between sinusoidal, random, repetitive, and transient events. Nonsteady-state conditions, such as run-up and coast-down, are most easily captured and analyzed using time waveforms. High-speed sampling can reveal such defects as broken gear teeth and cracked bearing races, but can also result in extremely large amounts of data being collected — much of which is likely to be redundant and of little use. 6.9.1.2

Time-Waveform Indices

A time-waveform index is a single number calculated in some way based on the raw vibration signal and used for trending and comparisons. These indices signiﬁcantly reduce the amount of data that is presented for inspection, but highlight differences between samples. Examples of time-waveform-based indices include the peak level (maximum vibration amplitude within a given pﬃﬃtime signal), mean level (average vibration amplitude), root-mean-square (RMS) level (peak level= 2; reduces the effect of spurious peaks caused by noise or transient events), and peak-to-peak amplitude (maximum positive to maximum negative signal amplitudes). All of these measures are affected adversely when more than one machinery component contributes to the measured signal. The crest factor is the ratio of the peak level to the RMS level ðpeak level=RMS levelÞ; and indicates the early stages of rolling-element-bearing failure. However, the crest factor decreases with progressive failure because the RMS level generally increases with progressive failure. 6.9.1.3

Time-Synchronous Averaging

Averaging of the vibration signal synchronous with the running speed of the machinery being monitored is called time-synchronous averaging. When taken over many machine cycles, this technique removes background noise and nonsynchronous events (random transients) from the vibration signal. This technique is extremely useful where multiple shafts that are operating at only slightly different speeds and in close proximity to one another are being monitored. A reference signal (usually from a tachometer) is always needed. 6.9.1.4

Negative Averaging

Negative averaging works in the opposite way to time-synchronous averaging. Rather than averaging all the collected data, a baseline signal is recorded and then subtracted from all subsequent signals to reveal changes and transients only. This type of signal processing is useful on equipment or components that are isolated from other sources of vibrations.

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6-17

Orbits

As described above, orbits are plots of the X direction displacement vs. the Y direction displacement (phase shifted by 908). This display format shows journal bearing relative motion (bearing wear, shaft misalignment, shaft unbalance, lubrication instabilities [whirl, whip], and seal rubs) extremely well, and hence is a powerful monitoring and diagnostic tool, especially on relatively low-speed machinery. 6.9.1.6

Probability Density Functions Probability Density(dB)

The probability of ﬁnding the instantaneous Faulty Bearing amplitude value from a vibration signal within a Normal Bearing certain amplitude range can be represented as a probability density function. Typically, the shape of the probability density function in these cases will be similar to a Gaussian (or normal) probability distribution. Fault conditions will have different characteristic shapes. Figure 6.10 shows two probability density functions. One is Normalized Vibration Amplitude characteristic of normal machine operating conditions, and the other represents a fault condition. FIGURE 6.10 Normalized vibration amplitude vs. A high probability at the mean value with a wide probability density (normal and faulty bearings). spread of low probabilities is characteristic of the impulsive time-domain waveforms that are typical for rolling-element-bearing faults. This type of display format can be used for condition trending and fault diagnostics. 6.9.1.7

Probability Density Moments

Probability density moments are single-number indices (descriptors), similar to the time-waveform indices except they are based on the probability density function. Odd moments (ﬁrst and third, mean and skewness) reﬂect the probability density function peak position relative to the mean. Even moments (second and fourth, standard deviation and kurtosis) are proportional to the spread of the distribution. Perhaps the most useful of these indices is the kurtosis, which is sensitive to the impulsiveness in the vibration signal and therefore sensitive to the type of vibration signal generated in the early stages of a rolling-element-bearing fault. Because of this characteristic sensitivity, the kurtosis index is a useful fault detection tool. However, it is not good for trending. As a rolling-element-bearing fault worsens, the vibration signal becomes more random, the impulsiveness disappears, and the noise ﬂoor increases in amplitude. The kurtosis then increases in value during the early stages of a fault, and decreases in value as the fault worsens.

6.9.2

Frequency Domain

The frequency domain refers to a display or analysis of the vibration data as a function of frequency. The time-domain vibration signal is typically processed into the frequency domain by applying a Fourier transform, usually in the form of a fast Fourier transform (FFT) algorithm. The principal advantage of this format is that the repetitive nature of the vibration signal is clearly displayed as peaks in the frequency spectrum at the frequencies where the repetition takes place. This allows for faults, which usually generate speciﬁc characteristic frequency responses, to be detected early, diagnosed accurately, and trended over time as the condition deteriorates. However, the disadvantage of frequency-domain analysis is that a signiﬁcant amount of information (transients, nonrepetitive signal components) may be lost during the transformation process. This information is nonretrievable unless a permanent record of the raw vibration signal has been made.

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6.9.2.1

Vibration Monitoring, Testing, and Instrumentation

Band-Pass Analysis

Band-pass analysis is perhaps the most basic of all frequency-domain analysis techniques, and involves ﬁltering the vibration signal above and/or below speciﬁc frequencies in order to reduce the amount of information presented in the spectrum to a set band of frequencies. These frequencies are typically where fault characteristic responses are anticipated. Changes in the vibration signal outside the frequency band of interest are not displayed. 6.9.2.2

Shock Pulse (Spike Energy)

The shock-pulse index (also known as spike energy; Boto, 1979) is derived when an accelerometer is tuned such that the resonant frequency of the device is close to the characteristic responses frequency caused by a speciﬁc type of machine fault. Typically, accelerometers are designed so that their natural frequency is signiﬁcantly above the expected response signals that will be measured. If higher frequencies are expected, they are ﬁltered out of the vibration signal. High-speed rollingelement bearings that are experiencing the earlier stages of failure (pitting on interacting surfaces) emit vibration energy in a relatively high, but closely deﬁned, frequency band. An accelerometer that is tuned to 32 kHz will be a sensitive detection device. This type of device is simple, effective, and inexpensive tool for fault detection in high-speed rolling-element bearings. The response from this type of device is load-dependent and may be prone to false alarms if measurement conditions are not constant. 6.9.2.3

Enveloped Spectrum

Another powerful analysis tool that is available in the frequency domain and can be effectively applied to detecting and diagnosing rolling-element-bearing faults is the enveloped spectrum (Courrech, 1985). When the vibration signal time waveform is demodulated (high-pass ﬁltered, rectiﬁed, then low-pass ﬁltered) the frequency spectrum that results is said to be enveloped. This process effectively ﬁlters out the impulsive components in signals that have high noise levels and other strong transient signal components, leaving only the components that are related to the bearing characteristic defect frequencies. This method of analysis is useful for detecting bearing damage in complex machinery where the vibration signal may be contaminated by signals from other sources. However, the ﬁltering bands must be chosen with good judgment. Recall also, the impulsive nature of the fault signal at the characteristic defect frequency leaves as the fault deteriorates. 6.9.2.4

Signature Spectrum

The signature spectrum (Braun, 1986) is a baseline frequency spectrum taken from new or recently overhauled machinery. It is then later compared with spectra taken from the same machinery that represent current conditions. The unique nature of each machine and installation is automatically taken into account. Characteristic component and fault frequencies can be clearly seen and comparisons made manually (by eye), using indices, or using automated pattern recognition techniques. 6.9.2.5

Cascades (Waterfall Plots)

Cascade plots (also known as waterfall plots) are successive spectra plotted with respect to time and displayed in a three-dimensional manner. Changing trends can be seen easily, which makes this type of display a useful fault detection and trending tool. This type of display is also used when a transient event, such as a coast-down, is known to be about to occur. Cascade plots can also be linked to the speed of a machine. In this case, the horizontal axis is labeled in multiples of the rotational speed of the machine. Each multiple of the rotational speed is referred to as an “order.” “Order tracking” is the name commonly used to refer to cascade plots that are synchronously linked to the machine rotational speed via a tachometer. As the speed of the machine changes, the responses at speciﬁc frequencies change relative to the speed, but are still tracked in each time-stamped spectra by the changing horizontal axis scale.

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6-19

Masks

Like negative averaging in the time domain, masks are baseline spectra that are used with an allowable tolerance limit to “ﬁlter out,” or block, speciﬁc frequencies. This technique is similar to band-pass analysis and requires a good knowledge of the full range of each machine’s operating limits (varying load or speed). 6.9.2.7

Frequency-Domain Indices

It has been noted that frequency spectra are more sensitive to changes related to machine condition (Mathew, 1987). Because of this sensitivity, several single number indices based on the frequency spectra have been proposed. Like the time-waveform indices, frequency-domain indices reduce the amount of information in frequency spectra to a single number. Because they are based on the frequency spectra, they are generally more sensitive to changes in machine condition than time domain indices. They are used as a means of comparing original spectra or previous spectra to the current spectra. Several frequency domain indices are listed below: *

Arithmetic mean (Grove, 1979):

( 20 log

*

*

*

*

6.9.3

N 1 X A N i¼1 i

!

Ai ¼ amplitude of ith frequency spectrum component N ¼ total number of frequency spectrum components Geometric mean (Grove, 1979): (N 1 X A 20 log pﬃﬃi N i¼1 2

) 10

25

) 25

10

Matched ﬁlter RMS (Mathew and Alfredson, 1984): ( N 1 X Ai 10 log N i¼1 Ai ðref Þ

2

)

Ai ðref Þ ¼ amplitude of ith component in the reference spectrum RMS of spectral difference (Alfredson, 1982): ( )1=2 N 1 X 2 ðL 2 Loi Þ N i¼1 ci Lci ¼ amplitude (dB) of ith component Loi ¼ amplitude (dB) of ith reference component Sum of squares of difference (Mathew and Alfredson, 1984): ( N 1 X ðL þ Loi Þ £ lLci 2 Loi l N i¼1 ci

) 1=2

Modal Domain

Modal analysis is not traditionally listed as a machine condition monitoring and fault diagnostics tool, but is included here because of the ever-increasing complexity of modern machinery. Often, unless the

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natural (free and forced response) frequencies of machinery, their support structure, and the surrounding buildings are fully understood, a complete and accurate assessment of existing machinery condition is not possible. A complete overview of modal analysis will not be provided here, but a speciﬁc approach to modal analysis (operational deﬂection shape [ODS] analysis) will be described. ODS analysis is like other types of modal analysis in that a force input is provided to a structure or machine and then the response is measured. The response at different frequencies deﬁnes the natural frequencies of the structure or machine. Typically, an impact or constant frequency force is used to excite the structure. In the case of ODSs, the regular operation of the machinery provides the excitation input. With vibration sensors placed at critical locations and a reference signal linking together all the recorded signals, a simple animation showing how the machine or structure deﬂects under normal operation can be generated. These animations, along with the frequency information contained in each individual signal, can provide signiﬁcant insights into how a machine or structure deforms under a dynamic load. This information, in turn, can be a useful addition to other data when attempting to diagnose problems.

6.9.4

Quefrency Domain

A quefrency-domain (Randall, 1981, 1987) plot results when a Fourier transform of a frequency spectra (log scale) is generated. As the frequency spectra highlight periodicities in the time waveform, so the quefrency “cepstra” highlights periodicities in a frequency spectra. This analysis procedure is particularly useful when analyzing gearbox vibration signals where modulation components in spectrum (sidebands) are easily detected and diagnosed in the cepstrum.

*

*

*

*

*

*

*

*

Generally, the more processing that is done on the dynamic signal, the more speciﬁc useful information is highlighted and the more extraneous information is discarded. The primary display formats used in machine condition monitoring are the time domain, the frequency domain, the modal domain, and the quefrency domain. The time domain refers to a display or analysis of the vibration data as a function of time, allowing for little or no data to be lost prior to inspection. Time domain analysis includes: waveform analysis, time waveform indices, time synchronous averaging, negative averaging, orbit analysis, probability density functions, and probability density moments. The frequency domain refers to a display or analysis of the vibration data as a function of frequency, where the time domain vibration signal is typically processed into the frequency domain by applying a Fourier transform, usually in the form of a FFT algorithm. The principal advantage of frequency-domain analysis is that the repetitive nature of the vibration signal is clearly displayed as peaks in the frequency spectrum at the frequencies where the repetition takes place. This allows for faults, which usually generate speciﬁc characteristic frequency responses, to be detected early, diagnosed accurately, and trended over time as the condition deteriorates. Frequency-domain analysis includes the use of band pass analysis, shock pulse (spike energy), envelope spectrum, signature spectrum, cascades (waterfall plots), masks, and frequencydomain indices. Quefrency-domain analysis involves a Fourier transform of a frequency spectra (log scale). As the frequency spectra highlight periodicities in the time waveform, so the quefrency “cepstra” highlights periodicities in a frequency spectra.

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Fault Detection

In many discussions of machine condition monitoring and fault diagnostics, the distinction between fault detection and fault diagnosis is not made. Here, they have been divided into separate sections in order to highlight the differences and clarify why they should be treated as separate tasks. Fault detection can be deﬁned as the departure of a measurement parameter from a range that is known to represent normal operation. Such a departure then signals the existence of a faulty condition. Given that measurement parameters are being recorded, what is needed for fault detection is a deﬁnition of an acceptable range for the measurement parameters to fall within. There are two methods for setting suitable ranges: (1) comparison of recorded signals to known standards and (2) comparison of the recorded signals to acceptance limits.

6.10.1

Standards

One of the best known sources of standards is the International Organization for Standardization (ISO). These standards are technology oriented and are set by teams of international experts. ISO Technical Committee 108, Sub-Committee 5 is responsible for standards for condition monitoring and diagnostics of machines. This group is further divided into a number of working groups who review data and draft preliminary standards. Each working group has a particular focus such as terminology, data interpretation, performance monitoring, or tribology-based machine condition monitoring. While ISO is perhaps the most widely known standardization organization, there are several others that are focused on speciﬁc industries. Examples of these include the International Electrical Commission, which is primarily product oriented, and the American National Standards Institute (ANSI), which is a nongovernment agency. There are also different domestic government agencies that vary from country to country. National defense departments also tend to set their own standards. Standards Based on Machinery Type

Because different machines that are designed to perform approximately the same task tend to behave in a similar manner, it is not surprising that many standards are set based on machinery type. Figure 6.11 shows a generic plot separating vibration amplitude vs. rotating speed into different zones. For a speciﬁc type, size, or class of machine, a plot like this can be used to distinguish gross vibration limits relative to the speed of operation. Machines are usually divided into four basic categories:

ZONE D Vibration Amplitude

6.10.1.1

ZONE C

ZONE B

ZONE A

1. Reciprocating machinery: These machines may contain both rotating and reciprocatRotational Speed ing components (e.g., engines, compressors, pumps). FIGURE 6.11 Normalized vibration amplitude vs. 2. Rotating machinery (rigid rotors): These probability density (zone A — new machine; zone machines have rotors that are supported B — acceptable; zone C — monitor closely; zone D — on rolling element bearings (usually). The damage occurring). vibration signal can be measured from the bearing housing because the vibration signal is transmitted well through the bearings to the housing (e.g., electric motors, single-stage pumps, slow-speed pumps).

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3. Rotating machinery (ﬂexible rotors): These machines have rotors that are supported on journal (ﬂuid ﬁlm) bearings. The movement of the rotor must be measured using proximity probes (e.g., large steam turbines, multistage pumps, compressors). These machines are subject to critical speeds (high vibration levels when the speed of rotation excites a natural frequency). Different modes of vibration may occur at different speeds. 4. Rotating machinery (quasi-rigid rotors): These are usually specialty machines in which some vibration gets through the bearings, but it is not always trustworthy data (e.g., low-pressure steam turbines, axial ﬂow compressors, fans). 6.10.1.2

Standards Based on Vibration Severity

It is an oversimpliﬁcation to say that vibration levels must always be kept low. Standards depend on many things, including the speed of the machinery, the type and size of the machine, the service (load) expected, the mounting system, and the effect of machinery vibration on the surrounding environment. Standards that are based on vibration severity can be divided into two basic categories: 1. Small-to-medium sized machines: These machines usually operate with shaft speeds of between 600 and 12,000 rpm. The highest broadband RMS value usually occurs in the frequency range of 10 to 1000 Hz. 2. Large machines: These machines usually operate with shaft speeds of 600 to 1200 rpm. If the machine is rigidly supported, the machine’s fundamental resonant frequency will be above the main excitation frequency. If the machine is mounted on a ﬂexible support, the machine’s fundamental resonant frequency will be below the main excitation frequency. While general standards do exist, there are also a large number of standards that have been developed for speciﬁc machines. Figure 6.12 shows a table with generic acceptance limits based on vibration severity.

6.10.2

Acceptance Limits

Standards developed by dedicated organizations are a useful starting point for judging machine condition. They give a good indication of the current condition of a machine and whether or not a fault exists. However, judging the overall condition of machinery is often more involved. Recognizing the changing machinery condition requires the trending of condition indicators over time. The development and use of acceptance limits that are close to the normal operating values for speciﬁc machinery will detect even slight changes in condition. While these acceptance limits must be tight enough to allow even small changes in condition to be detected, they must also tolerate normal operating variations without Vibration Amplitude Increasing

Vibration Severity for Separate Classes of Machines Class I Class II Class III Class IV A B

A B

C

C

D

D

A B C D

A B C D

FIGURE 6.12 Acceptance limits based on vibration severity levels (zone A — new machine; zone B — acceptable; zone C — monitor closely; zone D — damage occurring).

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generating false alarms. There are two types of limits: 1. Absolute limits represent conditions that could result in catastrophic failure. These limits are usually physical constraints such as the allowable movement of a rotating part before contact is made with stationary parts. 2. Change limits are essentially warning levels that provide warning well in advance of the absolute limit. These vibration limits are set based on standards and experience with a particular class of machinery or a particular machine. Change limits are usually based on overall vibration levels. It is important to note that the early discovery of faulty conditions is a key to optimizing maintenance effort by allowing the longest possible lead-time for decision making. As well as the overall vibration levels being monitored, the rates of change are also important. The rate of change of a vibration level will often provide a strong indication of the expected time until absolute limits are exceeded. In general, relatively high but stable vibration levels are of less concern than relatively low but rapidly increasing levels. An example of how acceptance limits may be used to detect faults and trend condition is provided when the gradual deterioration of rolling-element bearings is considered. Rolling-element bearings generate distinctive defect characteristic frequencies in the frequency spectrum during a slow, progressive failure. Vibration levels can be monitored to achieve maximum useful life and failure avoidance. Typically, the vibration levels increase as a fault is initiated in the early stages of deterioration, but then decrease in the later stages as the deterioration becomes more advanced. Appropriately, set acceptance levels will detect the early onset of the fault and allow subsequent monitoring to take place even after the overall vibration level has dropped. However, rapid bearing deterioration may still occur due to a sudden loss of lubrication, lubrication contamination, or a sudden overload. The possibility of these situations emphasizes the need for carefully selected acceptance limits. It should also be noted that changes in operating conditions, such as speed or load changes, could invalidate time trends. Comparisons must take this into consideration. 6.10.2.1

Statistical Limits

Statistical acceptance limits are set using statistical information calculated from the vibration signals measured from the equipment that the limits will ultimately be used with. As many vibration signals as possible are recorded, and the average of the overall vibration level is calculated. An alert or warning level can then be set at 2.5 standard deviations above or below the average reading (Mechefske, 1998). This level has been found to provide optimum sensitivity to small changes in machine condition and maximum immunity to false alarms. A distinct advantage to using this method to set alarm levels is the fact that the settings are based on actual conditions being experienced by the machine that is being monitored. This process accommodates normal variations that exist between machines and takes into account the initial condition of the machine.

6.10.3

Frequency-Domain Limits

Judging vibration characteristics within the frequency spectra is sometimes a more accurate method of detecting and trending fault conditions. It can also provide earlier detection of speciﬁc faults because, as mentioned previously, the frequency domain is generally more sensitive to changes in the vibration signal that result from changes in machine condition. The different speciﬁc methods are listed and described below. 6.10.3.1

Limited Band Monitoring

In limited band monitoring, the frequency spectrum is divided into frequency bands. The total energy or highest amplitude frequency is then trended within each band. Each band has its own limits based on experience. Generally, ten or fewer bands are used. Small changes in component-speciﬁc frequency

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ranges are more clearly shown using this strategy. Bandwidths and limits must be speciﬁc to the machine, sensor type, and location. Narrowband monitoring is the same as limited band monitoring, except it has ﬁner deﬁnition of the bands. 6.10.3.2

Constant Bandwidth Limits

When limited band monitoring is practiced and the bands have same width at high and low frequencies, the procedure is called constant bandwidth monitoring (see Figure 6.13). This technique is useful for constant speed machines where the frequency peaks in the spectra remain relatively ﬁxed. 6.10.3.3

Constant Percentage Bandwidth Limits

Amplitude

Constant percentage bandwidth monitoring involves using bandwidths that remain a constant percentage of the frequency being monitored (see Figure 6.14). This results in the higher frequency bands being proportionally wider than the lower frequency bands. This allows for small variations

Frequency Constant bandwidth acceptance limits.

Amplitude

FIGURE 6.13

Frequency FIGURE 6.14

Constant percentage bandwidth acceptance limits.

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in speed without the frequency peaks moving between bands, which may have different acceptance limits.

*

*

*

*

*

*

*

6.11

Fault detection can be deﬁned as the departure of a measurement parameter from a range that is known to represent normal operation. Such a departure signals the existence of a faulty condition. ISO Technical Committee 108, Sub-Committee 5 is responsible for standards for Condition Monitoring and Diagnostics of Machines. Standards are based on machinery type or vibration severity. The development and use of acceptance limits that are close to normal operating values for speciﬁc machinery will detect even slight changes in condition. Statistical acceptance limits are set using statistical information calculated from the vibration signals measured from the equipment that the limits will ultimately be used with. Judging vibration characteristics within the frequency spectra is sometimes a more accurate method of the early detecting and trending of fault conditions because the frequency domain is generally more sensitive to changes in the vibration signal that result from changes in machine condition. Frequency domain limits include limited band monitoring, constant bandwidth limits, and constant percentage bandwidth limits.

Fault Diagnostics

Depending on the type of equipment being monitored and the maintenance strategy being followed, once a faulty condition has been detected and the severity of the fault assessed, repair work or replacement will be scheduled. However, in many situations, the maintenance strategy involves further analysis of the vibration signal to determine the actual type of fault present. This information then allows for a more accurate estimation of the remaining life, the replacement parts that are needed, and the maintenance tools, personnel, and time required to repair the machinery. For these reasons, and many more, it is often advantageous to have some idea of the fault type that exists before decisions regarding maintenance actions are made. There are obviously a large number of potential different fault types. The description of these faults can be systemized somewhat by considering the type of characteristic defect frequencies generated (synchronous to rotating speed, subsynchronous, harmonics related to rotating speed, nonsynchronous harmonics, etc.). Such a systemization requires a focus on frequency-domain analysis tools (primarily frequency spectra). While this organization strategy is effective, it inherently leaves out potentially valuable information from other display formats. For this reason, the various faults that usually develop in machinery are listed here in terms of the forcing functions that cause them and speciﬁc machine types. In this way, a diagnostic template can be developed for the different types of faults that are common in a given facility or plant. Further reading on machinery diagnostics can be found in these references: Wowk (1991), Taylor (1994), Eisenmann and Eisenmann (1998), Goldman (1999), and Reeves (1999).

6.11.1

Forcing Functions

Listed and described below are a variety of forcing functions that can result in accelerated deterioration of machinery or are the result of damaged or worn mechanical components. The list is not meant to be exhaustive and is in no particular order.

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Vibration Monitoring, Testing, and Instrumentation

Unbalance

Unbalance (also referred to as imbalance) exists when the center of mass of a rotating component is not coincident with the center of rotation. It is practically impossible to fabricate a component that is perfectly balanced; hence, unbalance is a relatively common condition in a rotor or other rotating component (ﬂywheel, fan, gear, etc.). The degree to which an unbalance affects the operation of machinery dictates whether or not it is a problem. The causes of unbalance include excess mass on one side of the rotor, low tolerances during fabrication (casting, machining, assembly), variation within materials (voids, porosity, inclusions, etc.), nonsymmetry of design, aerodynamic forces, and temperature changes. The vector sum of all the different sources of unbalance can be combined into a single vector. This vector then represents an imaginary heavy spot on the rotor. If this heavy spot can be located and the unbalance force quantiﬁed, then placing an appropriate weight 1808 from the heavy spot will counteract the original unbalance. If left uncorrected, unbalance can result in excessive bearing wear, fatigue in support structures, decreased product quality, power losses, and disturbed adjacent machinery. Unbalance results in a periodic vibration signal with the same amplitude each shaft rotation (3608). A strong radial vibration at the fundamental frequency, 1X, (1 £ rotational speed) is the characteristic diagnostic symptom. If the rotor is overhung, there will also be a strong axial vibration at 1X. The amplitude of the response is related to the square of the rotational speed, making unbalance a dangerous condition in machinery that runs at high rotational speeds. In variable speed machines (or machines that must be run-up to speed gradually), the effects of unbalance will vary with the shaft rotational speed. At low speeds, the high spot (location of maximum displacement of the shaft) will be at the same location as the unbalance. At increased speeds, the high spot will lag behind the unbalance location. At the shaft ﬁrst critical speed (the ﬁrst resonance), the lag reaches 908, and at the second critical and above, the lag reaches 1808. A special form of unbalance is caused by a bent shaft or bowed rotor. These two conditions are essentially the same; only the location distinguishes them. A bent shaft is located outside the machine housing, while a bowed rotor is inside the machine housing. This condition is seen on large machines (with heavy rotors) that have been allowed to sit idle for a long time. Gravity and time cause the natural sag in the rotor to become permanent. The vibration spectrum from a machine with a bent shaft or bowed rotor is identical to unbalance, largely because it is an unbalanced condition. Bent shafts and bowed rotors are difﬁcult to correct (straighten), so they need to be balanced by adding counterweights as described above. The best way to avoid this condition is to keep the shaft/rotor rotating slowly when the machine is not in use. 6.11.1.2

Misalignment

While misalignment can occur in several different places (between shafts and bearings, between gears, etc.), the most common form is when two machines are coupled together. In this case, there are two main categories of misalignment: (1) parallel misalignment (also known as offset) and (2) angular misalignment. Parallel misalignment occurs when shaft centerlines are parallel but offset from one another in the horizontal or vertical direction, or a combination of both. Angular misalignment occurs when the shaft centerlines meet at an angle. The intersection may be at the driver or driven end, between the coupled units or behind one of the coupled units. Most misalignment is a combination of these two types. Misalignment is another major cause of excessive machinery vibration. It is usually caused by improper machine installation. Flexible couplings can tolerate some shaft misalignment, but misalignment should always be minimized. The vibration caused by misalignment results in excessive radial loads on bearings, which in turn causes premature bearing failure. Elevated 1X vibrations with harmonics (usually up to the third, but sometimes up to the sixth) in the frequency spectrum are the usual diagnostic signatures. The harmonics

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TABLE 6.1 Characteristics that Can Help Distinguish between Unbalance and Misalignment Unbalance High 1X response in frequency spectra Low axial vibration levels Measurements at different locations are in phase Vibration levels are independent of temperature Vibration level at 1X increases with rotational speed. Centrifugal force increases as the square of the shaft rotational speed

Misalignment High harmonics of 1X relative to 1X High axial vibration levels Measurements at different locations are 1808 out of phase Vibration levels are dependent on temperature (change during warm-up) Vibration level does not change with rotational speed. Forces due to misalignment remain relatively constant with changes in shaft rotational speed

allow misalignment to be distinguished from unbalance. High horizontal relative to vertical vibration amplitude ratios (greater than 3:1) may also indicate misalignment. One ﬁnal note regarding misalignment is that the heat of operation causes metal to expand resulting in thermal growth. Vibration readings should be taken when the equipment is cold and again after normal operating temperature has been reached. The changes in alignment due to thermal growth may be minimal, but should always be measured since they can lead to signiﬁcant vibration levels. Because unbalance and misalignment are perhaps the two most common causes of excessive machinery vibrations and they have similar characteristic indicators, Table 6.1 has been included to help distinguish between them. 6.11.1.3

Mechanical Looseness

While there are many ways in which mechanical looseness may appear, there are two main types: (1) a bearing loose on a shaft and (2) a bearing loose in a housing. A bearing that is loose on a shaft will display a modulated time signal with many harmonics. The time period of modulation will vary and the time signal will also be truncated (clipped). A bearing that is loose in its housing will display a strong fourth harmonic, which can sometimes be mistaken for the blade-pass frequency on a four-blade fan. These faults may also look like rolling-element-bearing characteristic defect frequencies, but always contain a signiﬁcant amount of wideband noise. Another way to diagnose mechanical looseness is by tracking the changes in the vibrations signal as the condition worsens. In the early stages, mechanical looseness generates a strong 1X response in the frequency spectrum along with some harmonics. At this stage, the condition could be mistaken for unbalance. As the looseness worsens, the amplitude of the harmonics will increase relative to the 1X response (which may actually decrease). The overall RMS value of the time waveform may also decrease. Further deterioration of the condition results in fractional harmonics 12 ; 13 ; 1 12 ; 2 12 increasing in amplitude. These harmonics are most visible in signals taken when the machine is only lightly loaded. These harmonics show up because of the clipping described above. 6.11.1.4

Soft Foot

Another condition that is in fact a type of mechanical looseness, but often masquerades as misalignment, unbalance, or a bent shaft, is soft foot. Soft foot occurs when one of a machine’s hold-down bolts is not tight enough to resist the dynamic forces exerted by the machine. That part of the machine will lift off and set back down as a function of the cyclical forces acting on it. All the diagnostic signs associated with mechanical looseness will be present in the vibration signal. If the foundation (hold-down points) of a machine does not form a plane, then tightening the holddown bolts will cause the casing and/or rotor to be distorted. This distortion is what leads to the misalignment, unbalance, and bent shaft vibration signatures. In order to check for a soft foot,

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the vibration level must be monitored while each hold-down bolt is loosened and then retightened. The appearance and/or disappearance of the diagnostic indicators mentioned above will determine if soft foot is the problem. When a machine’s vibration levels cannot be reduced by realignment or balancing, soft foot could well be the cause. 6.11.1.5

Rubs

Rubs are caused by excessive mechanical looseness or oil whirl. The result is that moving parts come into contact with stationary parts. The vibration signal generated may be similar to that of looseness, but is usually clouded with high levels of wideband noise. This noise is due to the impacts. If the impacts are repetitive, such as occurring each time a fan blade passes, there may be strong spectral responses at the striking frequency. In many cases, rubs are the result of a rotor pressing too hard against a seal. In these cases, the rotor will heat up unsymmetrically and develop a bowed shape. Subsequently, a vibration signal will be generated that shows unbalance. To diagnose this condition, it will be noted that the unbalance is absent until the machine comes up to normal operating temperature. 6.11.1.6

Resonances

The analysis of resonance problems is beyond the scope of this chapter. However, some basic description is provided here because of the high likelihood that at some time a resonance will be excited by repetitive or cyclic forces acting on or nearby a machine. A resonance is the so-called “natural frequency” at which all things tend to vibrate. A machine’s natural resonant frequency is dictated by the relationship vn ¼ ðk=mÞ1=2 ; where vn is the natural frequency, k is the spring stiffness, and m is the mass. Most systems will have more than one resonance frequency. These resonances (also called modes) can be excited by any forcing function that is at or close to that frequency. The response amplitude can be 10 to 100 times that of the forcing function. The term “critical speed” is also used to refer to resonances when the machine rotating speed equals the natural frequency. The amount of response ampliﬁcation depends on the damping in the system. A highly damped system will not show signs of resonance excitation, while a lightly damped system will be prone to resonance excitations. Resonances can be diagnosed by monitoring the vibration level while the speed of rotation of the machine is changed. A resonance will cause a dramatic increase in the 1X vibration levels as the speed is slowly changed. Most machines are designed to operate well away from known resonance frequencies, but changes to the machine (support structure, piping connections, etc.) and proximity to other machines may excite a resonance. 6.11.1.7

Oil Whirl

Oil whirl occurs when the ﬂuid in a lightly loaded journal bearing does not exert a constant force on the shaft that is being supported and a stable operating position is not maintained. In most journal bearing designs, this situation is prevented by using pressure dams or tilt pads to insure that the shaft rides on an oil pressure gradient that is sufﬁcient to support it. During oil whirl, the shaft pushes a wedge of oil in front of itself and the shaft then migrates in a circular fashion within the bearing clearance at just less than one half the shaft rotational speed. The rotor is actually revolving around inside the bearing in the opposite direction from shaft rotation. Because of the inherent instability of oil whirl, in many situations where oil whirl occurs, the time waveform will show intermittent whirl events. The shaft makes a few revolutions while whirl is present and then a few revolutions where the whirl is not present. This “beating” effect is often evident in the time waveform and can be used as a diagnostic indicator. Persistent oil whirl usually requires a replacement of the bearing. However, temporary measures to mitigate the detrimental effects include changing the oil viscosity (changing the operating temperature or the oil), running the machine in a more heavily loaded manner, or introducing a misalignment that will load the bearing asymmetrically. This last course of action is of course not recommended for more than relatively short-term relief.

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Oil Whip

Oil whip occurs when a subsynchronous instability (oil whirl) excites a critical speed (resonance), which then remains at a constant frequency regardless of speed changes. Oil whip often occurs at two times the critical speed because, at that speed, oil whirl matches the critical speed. Figure 6.15 shows a waterfall (cascade) plot of a mass unbalance that excites oil whirl and oil whip. Note how the oil whip “locks on” to the critical speed resonance. 6.11.1.9

Structural Vibrations

Structural vibrations can range dramatically in amplitude and frequency. Large-amplitude, lowfrequency vibrations can be excited in multistory buildings during an earthquake or by the wind. These vibrations are usually the result of a building resonance being excited. While these sources of structural vibration are important, the source that we are concerned with here is that of machinery operating as part of a building’s utility system, as part of the production plant, or construction equipment close-by. Fans, blowers, compressors, piping systems, elevators, and other building service machines all produce vibrations in a building and, if they are not properly isolated they can cause disruption and/or damage to other machines or processes operating close-by. The same is true of heavy machinery operating within a plant (stamping machines, presses, forges, etc.) and construction equipment. High-impact and repetitive vibrations can excite resonances large distances from the source of the excitation. 6.11.1.10

Foundation Problems

Machine foundations provide rigidity and inertia so that the machine stays in alignment. The energy generated by a machine in the form of vibrations is transmitted, reﬂected, or absorbed by the foundation. Especially on larger machines, the foundation is paramount to successful dynamic behavior. Maximum energy is transmitted through the foundation to the earth when the mechanical impedance of the foundation is well matched to that of the source of vibration. That is, the source of vibration and the foundation should have the same natural frequency. If this is the case, all frequencies of vibration below Rotor Speed Oil Whirl

OIL WHIP

Critical Speed FIGURE 6.15

Mass Unbalance

Frequency

Waterfall (cascade) plot of a mass unbalance that excites oil whirl and oil whip.

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the natural frequency will be transmitted by the foundation to earth. A poor match will mean that more energy is reﬂected or absorbed by the foundation, which could effect the operation of the machine attached. Changing foundations can grossly affect amplitude and phase measurements, which means that vibration measurements can be used to easily detect a changing foundation or hold-down system.

6.11.2

Speciﬁc Machine Components

6.11.2.1

Damaged or Worn Rolling-Element Bearings

Rolling-element bearings produce very little vibration (low level random signal) when they are fault free, and have very distinctive characteristic defect frequency responses (see Eschman, 1985, for the equations for calculation of defect frequencies) when faults develop. This, and the fact that most damage in rollingelement bearings occurs and worsens gradually, makes fault detection and diagnosis on this component relatively straightforward. Faults due to normal use usually begin as a single defect caused by metal fatigue in one of the raceways or on a rolling element. The vibration signature of a damaged bearing is dominated by impulsive events at the ball or roller passing frequency. Figure 6.16 shows the characteristic time waveform and frequency spectra at various stages of damage. As the damage worsens, there is a gradual increase in the characteristic defect frequencies followed by a drop in these amplitudes and an increase in the broadband noise. In machines where there is little other vibration that would contaminate or mask the bearing vibration signal, the gradual deterioration of rolling-element bearings can be monitored by using the crest factor or the kurtosis measure (see above for deﬁnitions). A key factor in being able to accurately detect and diagnose rolling-element-bearing defects is the placement of the vibration sensor. Because of the relatively high frequencies involved, accelerometers should be used and placed on the bearing housing as close as possible to, or within, the load zone of the stationary outer race. Speciﬁc applications can also pose signiﬁcant challenges to fault diagnosis. Very low-speed machines have bearings that generate low energy signals and require special processing to extract useful bearing condition indications (Mechefske and Mathew, 1992a). Machines that operate at varying speeds also pose a problem because the characteristic defect frequencies are continuously changing (Mechefske and Liu, 2001). Bearings located close to, or within, gearboxes are also difﬁcult to monitor because the high energy at the gear meshing frequencies masks the bearing defect frequencies (Randall, 2001).

Amplitude

Amplitude

Time Domain

Frequency Domain FIGURE 6.16 bearing.

Characteristic time waveform and frequency spectra at various stages of damage in a rolling-element

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6-31

Damaged or Worn Gears

Amplitude

Amplitude

Because gears transmit power from one rotating Sidebands Harmonics shaft to another, signiﬁcant forces are present within the mating teeth. While gears are designed for robustness, the teeth do deﬂect under load and then rebound when unloaded. The local stresses are high at the tooth interface and root, which leads to fatigue damage. Proper design and perfect fabrication of gears (with perfect form and no defects) would result in relatively low vibration levels and a Frequency long life. However, the presence of nonperfect gears Rahmonics gives rise to excessive vibration (Smith, 1983). The time waveform, the frequency spectral, and the cepstral patterns generated by gear vibrations all contain critical information needed to diagnose defects (see Figure 6.17). In relatively simple gearboxes, the time waveform can be used to distinguish impacts due to cracked, chipped, or Quefrency missing teeth (McFadden and Smith, 1984, 1985). The frequency spectra and cepstra are powerful FIGURE 6.17 Frequency and quefrency plots tools when the gearbox contains several sets of (damaged gear). mating gears, which is most often the case. Even a signiﬁcant defect on one tooth (or even a missing tooth) often does not produce an abnormally strong frequency spectral response at 1X. However, the defect will modulate the gear mesh frequency (number of teeth times the shaft rotational speed) and appear as 1X sidebands of the gear mesh frequency. That is, smaller spectral responses that appear a distance of 1X (and multiples of 1X for more severe gear faults) above and below the gear mesh frequency. Because these sidebands occur at multiples of 1X and a spectral plot can become quite cluttered with response lines, cepstral analysis is well suited to distinguish the frequency components that are strong fault indicators. Often, a change in the response at two times the gear mesh frequency is a good indicator of developing gear problems. The amplitude of the gear mesh frequency, and its multiples, vary with load. This makes it important to sample the vibration signal at the same load conditions. When unloaded, excessive gear backlash may also cause an increase in the amplitude of the gear mesh frequency. Because each gear tooth meshes with an impact, structural resonances may be excited in the gears, shafts, and housing. Proper design of a gearbox will minimize this effect, but resonances in gearboxes cause accelerated gear wear and should be monitored. Gears provide an excellent example of how machines must wear-in during early use. New gears will have defects that are quickly worn away in the machine’s early life. Vibration levels will become steady and only increase gradually later in the machines life as the gears wearout. These gradual increases in vibration level are normal. Sudden changes in vibration levels (at gear mesh frequency, two times gear mesh frequency, or sidebands), especially decreases, are very signiﬁcant. A drop in the vibration level usually means a decrease in stiffness, and that more of the transmission forces are being absorbed due to bending of the gear teeth. Catastrophic failure is imminent. Premature gear failures are usually a symptom of other problems such as unbalance, misalignment, bent shaft, looseness, improper lubrication, or contaminated lubrication.

6.11.3

Speciﬁc Machine Types

6.11.3.1

Pumps

There are two principal types of pumps: (1) centrifugal pumps and (2) reciprocating pumps. Reciprocating pumps will be discussed in a later section. The sources of vibration in pumps are widely

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varied. In addition to the standard mechanical problems (unbalance, misalignment, worn bearings, etc.), problems that are particular to pumps include vane-pass frequency generating conditions (starvation, impeller loose on the shaft, impeller hitting something) and cavitation. Starvation occurs when not enough liquid is not present to ﬁll each vane on the impeller every revolution of the shaft. Pump starvation can be confused with unbalance. However, it can be distinguished by the varying amplitude 1X vibration at constant speed and the reduced load on the driving motor. When the vanes on the impeller are striking something, the vane-pass frequency (the number of vanes times the rotational speed) is excited. Because the striking causes a force on the shaft, an unbalance is also present. The frequency spectrum will show a response at 1X and vane-pass frequency. The time waveform will show a high-frequency response (vane pass) riding on a frequency response at 1X. The vane-pass frequency is in phase with the shaft speed. If the impeller is loose on the shaft, the vane-pass frequency will be modulated by the shaft speed. Cavitation occurs when there is sufﬁcient negative pressure (suction) acting on the liquid in the system that it becomes a gas (it boils). This usually takes place in localized parts of the system. Cavitation usually occurs in a pump when the suction intake is restricted and the liquid vaporizes when coming off the impeller. As the ﬂuid moves past the low pressure region, the gas bubble collapses. If the collapsing bubble is close to a solid surface, it will aggressively erode the surface. Cavitation may be caused by a local decrease in atmospheric pressure, an increase in ﬂuid temperature, an increase in ﬂuid velocity, a pipe obstruction, or abrupt change in direction. The vibration signal that results will have signiﬁcant vibration levels at 1X with harmonics and strong spectral responses at vane-pass frequency. High-frequency broadband noise is also common. An increase in the system pressure can reduce cavitation. Hydraulic unbalance will result if there has been poor design of suction piping (elbow close to inlet) or poor impeller design (unsymmetrical). The vibration signal will contain high 1X axial vibration components. Impeller unbalance is a speciﬁc form of mechanical unbalance as discussed above. High 1X vibration levels will result. Pipe stresses result from inadequate pipe support and cause stress on the pump casing. This may also cause misalignment. Pipe resonances can also be excited by vane-pass frequency pressure pulsations. Diagnosis of pump problems can be improved by installing a pressure transducer in the discharge line of the pump. The measured pressure ﬂuctuations can be processed in the same way as vibration signals. The frequencies measured represent the pressure ﬂuctuations and the amplitude is the zero-to-peak pressure change. 6.11.3.2

Fans

Fans account for a signiﬁcant number of ﬁeld vibration problems due to their function and construction. Fans move air or exhaust gases that are often laden with grease, dust, sand, ash, and other corrosive and erosive particles. Under these conditions, fans blades gain and lose material resulting in the need to regularly rebalance. The level of balance must also be relatively ﬁne because fans often have large fanblade diameters and operate at relatively high speeds. Fans are usually mounted on spring/damper systems to help isolate vibrations, but they are also constructed in a relatively ﬂexible manner, which adds to the demands for ﬁne balancing. Along with ﬁne balancing requirements, typical problems include looseness, misalignment, bent shaft, and defective bearings. Fans also generate a strong response at blade-pass frequency (number of blades times the shaft rotational speed). This frequency response is present during normal operation, but it can become elevated if the blades are hitting something, the fan housing is excessively ﬂexible, or an acoustical resonance is present. Acoustical resonances are relatively common where large volumes of air are being moved through large ﬂexible ducts and/or the fan blades are of an air-foil design. 6.11.3.3

Electric Motors

Electric motors can be divided into two groups: (1) induction motors and (2) synchronous motors. A full description will not be given here as to the differences. Like any machine, electric motors are subject to a

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full range of mechanical problems, and vibrations signals can be used to detect and diagnose these problems. Apart from the conditions described elsewhere in this section, there are some problems that occur only in electric motors. For the sake of brevity, these problems and the vibration signals that typically accompany them are summarized in Table 6.2. 6.11.3.4

Steam and Gas Turbines

Steam and gas turbines (and high-speed compressors) require special mention because of the high speeds and temperatures involved. Problems on steam turbines are usually limited to looseness, unbalance, misalignment, soft foot, resonance, and rubs. As discussed above, each of these conditions has a set of characteristic vibration responses that allow for relatively straightforward diagnosis. However, because of the high speeds, this type of machinery is usually designed to be lighter and less rigid than other rotating machines. Excessive vibration can therefore lead to catastrophic failure very quickly. Because of this, high-speed turbines and compressors are designed to closer tolerances than other types of machines, and extra care is taken when balancing rotors. These machines also frequently operate above their ﬁrst critical speed and sometimes between their second and third critical speeds. At these speeds, the rotor becomes quite ﬂexible and the support bearings become very important in that they must provide the appropriate amount of damping. Because steam and gas turbines are supported on journal bearings, most monitoring and diagnostics work will be based solely on proximity probe signals. While this is not a problem in and of itself, accelerometer signals should also be taken in order to cover the higher frequencies, which are excited by conditions such as looseness and rubs. 6.11.3.5

Compressors

Compressors act in much the same way as pumps, except that they are compressing some type of gas. They come in many different sizes, but only two principal types: (1) screw-type and (2) reciprocating compressors. Reciprocating compressors will be discussed in a later section. Screw-type compressors have a given number of lobes or vanes on a rotor and generate a vane-passing frequency. Screw compressors with multiple rotors can also generate strong 1X and harmonics up to vane-pass frequency. The close tolerances involved result in relatively high vibration levels, even when the machine is in good condition. As with pumps, signals taken from pressure transducers in the discharge line can be useful for diagnostics. 6.11.3.6

Reciprocating Machines

Reciprocating machines (gas and diesel engines, steam engines, compressors, and pumps) all have one thing in common — a piston that moves in a reciprocating manner. These machines generally have high overall vibration levels and particularly strong responses at 1X and harmonics, even when in good condition. The vibrations are caused by compressed gas pressure forces and unbalance. Vibrations at 12 X may be present in four-stroke engines because the camshaft rotates at one half the crankshaft speed. TABLE 6.2

Mechanical Problems Particular to Electric Motors Condition

Motor out of magnetic center Motors with broken rotor bars Motor with turn-to-turn shorts in the windings Motor out of magnetic center with broken rotor bars or turn-to-turn shorts in the windings

Vibration Indicator High spectral response at 60 Hz High spectral response at motor running speed and/or second harmonic Motor runs at a slower than expected speed (high slit frequency) Side bands of slip frequency times the number of poles centered around the motor running speed and harmonics of the running speed

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Many engines operate at variable speeds, which will allow the strong forcing functions to excite resonances of the components and the mounting structure, if it is not designed in a robust manner. Excessive vibrations in reciprocating machines usually occur due to operational problems such as misﬁring, piston slap, compression leaks, and faulty fuel injection. These problems result in elevated 12 X vibrations, if only one cylinder is affected, and a decrease in efﬁciency and power output. Gear and bearing problems may also occur in reciprocating machines, but the characteristic defect frequencies for these faults are signiﬁcantly higher.

6.11.4

Advanced Fault Diagnostic Techniques

Much of the discussion in the previous sections has highlighted the fact that many machine defects generate distinctive vibration signals. This fact has been exploited recently with the development of a variety of different automatic fault diagnostics techniques (Mechefske and Mathew, 1992b; Mechefske, 1995). The details of these systems will not be provided here, but the goal of automatic diagnostics is to augment and assist, rather than replace, the vibration signal analyst. If characteristic defect indicators can be detected and extracted from a vibration signal without the intervention of a signal analyst, the analyst will have more time for other duties and will also have access to information that may not have been uncovered through normal signal processing and analysis. There are, however, still many situations where machine defects do not generate distinctive vibration signals or when the vibration signals are masked by large amounts of noise or vibrations from other machinery. In such cases, advanced diagnostic algorithms incorporating new signal processing techniques are currently being developed and implemented. Artiﬁcial neural networks (Timusk and Mechefske, 2002) have been found to provide an excellent basis for detecting and diagnosing faults. Wavelet analysis (Lin et al., 2004) and short-time Fourier transforms (STFTs) have also been shown to effectively allow both time domain and frequency domain information to be displayed on the same plot. This provides an opportunity to clearly see short duration transient events as well as detect faults in machinery that is operating in nonsteady-state conditions.

*

*

*

*

*

Analysis of the vibration signal to determine the actual type of fault present will allow for more accurate estimation of the remaining life, the replacement parts that are needed, and the maintenance tools, personnel, and time required to repair the machinery. A diagnostic template can be developed for the different types of faults that are common in a given facility or plant by listing various faults that usually develop in machinery in terms of the forcing functions that cause them and speciﬁc machine types. Common forcing functions include unbalance, misalignment, mechanical looseness, soft foot, rubs, resonances, oil whirl, oil whip, structural vibrations, and foundation problems. Speciﬁc machine components that need to be monitored include damaged or worn rollingelement bearings and gears. Speciﬁc machine types that can be treated as common groups include pumps, fans, electric motors, steam and gas turbines, compressors, and reciprocating machines.

References Alfredson, R.J. 1982. A computer based system for condition monitoring, Symposium on Reliability of Large Machines, The Institute of Engineers Australia, Sydney, pp. 39–46. Boto, P.A., Detection of bearing damage by shock pulse measurement, Ball Bearing J., 5, 167–176, 1979. Braun, S. 1986. Mechanical Signature Analysis: Theory and Applications, Academic Press, London.

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Courrech, J. 1985. New techniques for fault diagnostics in rolling element bearings, In Proceedings of the 40th Meeting of the Mechanical Failure Prevention Group, National Bureau of Standards, Gaithersburg, MD, pp. 47 –54. Eisenmann, R.C. Sr. and Eisenmann, R.C. Jr. 1998. Machinery Malfunction: Diagnosis and Correction, Prentice Hall, Newark. Eschman, P. 1985. Ball and Roller Bearings, 2nd ed., Wiley, New York. Goldman, S. 1999. Vibration Spectrum Analysis, Industrial Press, New York. Grove, R.C. 1979. An investigation of advanced prognostic analysis techniques, Paper USARTL-TR-79-10. Northrup Research and Technology Center, Palos Verdes, CA. Lin, J., Zuo, M.J., and Fyfe, K.R., Mechanical fault detection based on the wavelet de-noising technique, ASME J. Vib. Acoust., 126, 9 –16, 2004. Lyon, R.H. 1987. Machinery Noise and Diagnostics, Butterworth, Boston. Mathew, J., Machine condition monitoring using vibration analysis, J. Aust. Acoust. Soc., 15, 7 –21, 1987. Mathew, J. and Alfredson, R.J., The condition monitoring of rolling element bearings using vibration analysis, Trans. ASME J. Vib. Acoust. Stress Reliab. Des., 106, 447–457, 1984. McFadden, P.D. and Smith, J.D., A model for the vibration produced by a single point defect in a rolling element bearing, J. Sound Vib., 96, 69 –77, 1984. McFadden, P.D. and Smith, J.D., A signal processing technique for detecting local defects in a gear from the average of the vibration, Proc. IMechE, 199, 99 –112, 1985. Mechefske, C.K., Fault detection and diagnosis in low speed rolling element bearings using inductive inference classiﬁcation, Mech. Syst. Signal Process., 9, 275–282, 1995. Mechefske, C.K., Objective machinery fault diagnosis using fuzzy logic, Mech. Syst. Signal Process., 12, 855–864, 1998. Mechefske, C.K. and Liu, L., Fault detection and diagnosis in variable speed machines, Int. J. Condition Monit. Diagn. Eng. Manage., 5, 29 –42, 2001. Mechefske, C.K. and Mathew, J., Fault detection in low speed rolling element bearings, part I: the use of parametric spectra, Mech. Syst. Signal Process., 6, 297–308, 1992a. Mechefske, C.K. and Mathew, J., Fault detection in low speed rolling element bearings, part II: the use of nearest neighbour classiﬁcation, Mech. Syst. Signal Process., 6, 309 –317, 1992b. Mitchell, J.S. 1981. An Introduction to Machinery Analysis and Monitoring, Penwell, Los Angeles. Mobley, R.K. 1990. An Introduction to Predictive Maintenance, Van Nostrand Reinhold, New York. Moubray, J. 1997. Reliability Centered Maintenance, Industrial Press, New York. Randall, R.B. 1981. A new method of modeling gear faults, ASME Annual Congress. Paper No. 81-Set-10. Randall, R.B. 1987. Frequency Analysis, Bruel & Kjaer, Copenhagen. Randall, R.B. 2001. Bearing diagnostics in helicopter gearboxes, Condition Monitoring and Diagnostic Engineering Management Conference, Manchester, U.K., August 2001. Rao, B.K.N. 1996. The Handbook of Condition Monitoring, Elsevier, Oxford. Reeves, C.W. 1999. The Vibration Monitoring Handbook, Coxmoor, Oxford. Smith, J.D. 1983. Gears and Their Vibrations, Marcel Dekker, New York. Taylor, J.I. 1994. The Vibration Analysis Handbook, Vibration Consultants, Tampa. Timusk, M. and Mechefske, C.K. 2002. Applying neural network based novelty detection to industrial machinery, 6th International Conference on Knowledge-Based Intelligent Information Engineering Systems, Crema, Italy, November 2002. Wowk, V. 1991. Machinery Vibration: Measurements and Analysis, McGraw-Hill, New York.

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7

Vibration-Based Tool Condition Monitoring Systems 7.1 7.2 7.3

7.4

C. Scheffer University of Stellenbosch

P.S. Heyns

University of Pretoria

Introduction ............................................................................ 7-1 Mechanics of Turning ............................................................. 7-2

General Terms

†

Chatter Vibrations

Signal Processing for Sensor-Based Tool Condition Monitoring ............................................................7-11 †

Feature Selection

Wear Model/Decision-Making for Sensor-Based Tool Condition Monitoring ...................................................7-15

Trending, Threshold Other Methods

7.6

Tool Wear

Direct and Indirect Systems † Sensor Requirements for Tool Wear Monitoring † Force Measurement † Acceleration Measurement † Acoustic Emission Measurement † Sensor Comparisons

Feature Extraction

7.5

†

Vibration Signal Recording .................................................... 7-7

†

Neural Networks

†

Fuzzy Logic

†

Conclusion ...............................................................................7-20

Summary Despite the high level of technology built into every aspect of modern metal cutting operations, the phenomenon of tool wear still hampers the reliability and complete automation of machining processes. Tool wear is the loss of material on the edge of the cutting tool. This chapter concerns sensor-based tool condition monitoring (TCM), and speciﬁcally those methods that are based on vibration related properties such as force, acceleration, and acoustic emission (AE). References are made to systems proposed in the literature and also to commercially available hardware. The chapter focuses on turning operations. The mechanics of turning are brieﬂy discussed. Various methods of obtaining vibration signals from turning operations are described. The vibration signal has to be processed in order to estimate the level of wear in the cutting edge of the tool, and several state-of-the-art approaches are discussed. Effective methods of constructing a model relating sensor data and the tool wear, using processed vibration signals, are described. The chapter concludes by indicating some important points that should be considered when using vibration-based systems for TCM, and some interesting topics for future research in this ﬁeld of study. Chapters 6, Chapter 8, and Chapter 9 present further information on the present subject.

7.1

Introduction

Millions of products are manufactured daily by a variety of processes. A basic method to form bulk metal into a desired ﬁnal shape is through the process of metal cutting, also referred to as machining. Metal cutting is essentially the removal of excess material from a workpiece by moving a working tool over the 7-1 © 2007 by Taylor & Francis Group, LLC

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surface of the workpiece at a certain depth, speed, and feed rate. Conventional machining operations are turning, milling, and drilling. Despite the high level of technology built into every aspect of modern metal cutting operations, the phenomenon of tool wear still hampers the reliability and complete automation of machining processes. Tool wear is the loss of material on the edge of the cutting tool. Although tool wear can be minimized, it cannot be eliminated. Unfortunately, excessive or even a small quantity of tool wear may cause a defect in a machined component, and therefore it is always necessary to be aware of the extent of the current tool wear before machining can commence. Economic losses due to tool wear occur as a result of the scrapping of expensive parts and the nonoptimal use of tool inserts. A conservative approach is often taken, and the insert is recycled long before it should have been. Furthermore, secondary damage due to tool wear can be extreme and even catastrophic. For this reason, many approaches to tool condition monitoring (TCM) have been proposed through the years. There exist sensorless and sensor-based TCM approaches. Sensorless approaches are generally tool-life equations and not monitoring methods. Thus, sensorless approaches attempt to determine the optimal tool life under certain machining conditions. These are often extended versions of the famous Taylor equation, which is described by vT n ¼ C ð7:1Þ where v is the cutting speed, T is the tool life, and n and C are constants that must be determined experimentally for a given tool and workpiece combination. This chapter is focused on sensor-based TCM, and speciﬁcally those methods that are based on vibration related properties such as force, acceleration, and acoustic emission (AE). These sensor types are known to be most effective for TCM. Furthermore, discussions will be focused on the application of TCM in turning operations, though reference will be made to other machining operations as well. Besides vibration-based approaches, other sensor based TCM methods are: *

*

*

*

*

*

*

*

Use of noncontact capacitive sensors Vision systems Measurement of the motor current Surface roughness monitoring Ultrasonic monitoring Temperature monitoring Laser scatter methods Audible emission monitoring

The reader is also referred to other excellent overviews of sensor-assisted TCM, published by Dan and Mathew (1990), Byrne et al. (1995), Scheffer and Heyns (2001a), and Dimla (2000). A TCM database was also published by the CIRP, supervised by Teti (1995), which includes more than 500 research papers focusing only on TCM.

7.2 7.2.1

Mechanics of Turning General Terms

A typical turning operation is schematically shown in Figure 7.1. The cutting tool moves parallel to the workpiece and spindle, and hence reduces the diameter of the shaft. The most important machining parameters are: *

*

*

Cutting speed (usually expressed in m/min) Feed rate (usually expressed in mm/rev) Depth of cut (usually expressed in mm)

The force response on the tool tip due to the turning operation consists of three components: Fx, Fy, and Fz. These forces consist of a static and a dynamic part, as shown in Figure 7.2. The static forces are governed by the static pressure between the tool and workpiece, and are a function of the machining

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FIGURE 7.1

7-3

Turning operation.

parameters. The dynamic forces are governed by forced and free vibrations due to excitation from the cutting operation. Analytical models exist that can describe the static forces for basic machining operations (Merchant, 1945). The dynamic behavior is more difﬁcult to model theoretically, although there is also continuous research in this area (Kapoor et al., 1998). One of the main difﬁculties of monitoring tool wear with vibration is to identify the frequency range that is inﬂuenced by tool wear, since machining processes entail various mechanisms that produce vibrations that are not related to tool wear. The frequency range of vibrations produced during ordinary machining operations usually falls between 0 and 10 kHz. From the literature, it can be concluded that the frequency range sensitive to tool wear depends entirely on the type of machining operation, and must be determined experimentally for each individual case. There are two important vibration frequencies present during cutting: *

*

The natural frequencies of the tool holder and its components The frequency of chip formation

force [N]

Dynamic tests should be conducted to identify the dynamic properties of tool holders (Scheffer and Heyns, 2002a). However, the interaction of the working tool engaged into the rotating workpiece complicates the situation, and as a result the dynamic behavior during cutting could be differdynamic cutting ent from the expected behavior obtained from offforce line tests. Scheffer and Heyns (2004) compared continuous cantilever models with modal hammer tests for different tool holder overhang lengths. static cutting force The natural frequency of the ﬁrst mode as a function of overhang length is plotted in Figure 7.3 (for a speciﬁc tool holder). It can be seen that a 0 time [s] continuous ﬁxed-free cantilever beam model corresponds well with the results obtained with hammer tests. FIGURE 7.2 Static and dynamic forces.

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The chip formation frequency can be calculated with simple equations that take the machining conditions into account (Lee et al., 1989). The tool holder natural frequencies and chip formation frequency are independent. Generally, tool wear has a larger effect on the vibration amplitudes at the tool holder natural frequencies but can inﬂuence chip formation frequencies as well.

7.2.2

Chatter Vibrations

7 1st mode frequency [kHz]

7-4

6 5 4 3 2 1

fixed-free cantilever beam hammer tests

Another phenomenon important to machining 0 operations is tool chatter vibrations. These are 40 45 50 55 60 overhang length [mm] self-excited vibrations resulting from the generation of different chip thicknesses during machining. Initially, cutting forces excite a structural FIGURE 7.3 Frequency of ﬁrst tool holder mode. mode of the machine –workpiece system. This (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal leaves a wavy surface ﬁnish on the workpiece. Process, Elsevier, 2004. With permission.) During the next revolution, another wavy surface is produced in the same way. Depending on the phase shift between these two waves, the maximum chip thickness can grow and oscillate at a particular frequency that is close to that of a structural mode. This is called the regenerative chatter frequency. Chatter cause a poor surface ﬁnish and can also lead to tool breakage. The analysis and prediction of chatter has been the subject of research for many years. Morimoto et al. (2000) developed a piezoelectric shaker/actuator to regenerate the vibrations of the cutting process. In this way, unwanted vibrations such as chatter can be attenuated. The system is also helpful to determine the dynamic properties of the machine tool. Koizumi et al. (2000) used a very interesting approach called the correlation integral in the time domain to identify chatter onset. Lago et al. (2002) designed a sensor and actuator integrated tool for turning and boring to control chatter. The tool holder shank vibrations are sent to the actuator via a digital controller. An adaptive feedback control system is used to perform broadband vibration attenuation up to 40 dB at different frequencies simultaneously.

7.2.3

Tool Wear

7.2.3.1

Tool Failure Mechanisms

Tool wear is caused by mechanical loads, thermal loads, chemical reactions, and abrasive loads. The load conditions are in turn inﬂuenced by the cutting conditions and materials. The different loads can cause certain wear mechanisms that may occur in combination. These mechanisms have either a physical or chemical characteristic that causes loss or deformation of tool material. Tool wear mechanisms can be classiﬁed into several types, summarized as follows (Du, 1999): *

*

*

*

Abrasive wear resulting from hard particles cutting action Adhesive wear associated with shear plane deformation Diffusion wear occurring at high temperatures Fracture wear due to fatigue

Other wear mechanisms are plastic deformation and oxidation, which are not very common in industry. It is estimated that 50% of all tool wear is caused by abrasion, 20% by adhesion, 10% by chemical reactions and the remaining 20% by the other mechanisms (Kopac, 1998). Abrasion is basically the grinding of the cutting tool material. The volume of abrasive wear increases linearly with the cutting forces. Higher hardness of the tool material can reduce the amount of abrasive wear. During adhesion, the high pressures and temperatures on the roughness peaks on the tool and the workpiece cause

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welding. These welding points are broken many times every second due to the workpiece movement and as a result cause removal of the tool material (Kopac, 1998). Diffusion wear occurs at even higher cutting speeds, where very high temperatures are present (especially when using hard metal tools). 7.2.3.2

Tool Failure Modes

Tool wear will generally occur as a combination of a number of wear modes, with one mode predominant. The dominant mode will depend on the dominant wear mechanism, which in its turn is inﬂuenced by the machining conditions and the choice of tool and workpiece material. For a given tool and workpiece combination, the dominant wear mode can be determined at different cutting speeds using the product of the cutting speed and the undeformed chip thickness (Dimla, 2000). The common wear modes are: *

*

*

*

*

*

*

*

Nose wear Flank wear Crater wear Notch wear Chipping Cracking Breakage Plastic deformation

Figure 7.4 is a graphical representation of the different tool failure modes. The consequences of tool wear are deviations in shape and roughness of the machined part that cause the part to be discarded because it is out of tolerance. Most wear modes cause an increase in cutting forces, although this is not always the case for all tool and workpiece combinations. The most widely researched tool failure modes for turning with single point tools are ﬂank wear, breakage (fracture), and crater wear. Flank and crater wear are accepted as normal tool failure modes, because the other failure modes can be avoided by selecting the proper machining parameters. The growth of ﬂank and crater wear is directly related to the total cutting time, unlike some of the other failure modes, which can occur unexpectedly even with a new tool.

FIGURE 7.4 Tool failure modes. (Source: Scheffer, C. and Heyns, P.S. 2001a. COMA ’01, University of Stellenbosch. With permission.)

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7.2.3.3

Tool Wear Measurement

Wear measurements of tool inserts are done through the implementation of an appropriate international standard, ISO 3685. Flank wear is quantiﬁed in terms of VB, which is the mean of the wear height on the tool ﬂank. The length of ﬂank wear is also measured in terms of b. Crater wear is quantiﬁed in terms of the crater depth, K. The parameters are depicted in Figure 7.5, which is a scanning electron microscope (SEM) picture of a worn turning insert. 7.2.3.4

Tool Wear Stages

It is assumed by most authors that tool wear consists of an initial, a regular, and a fast wear stage (Zhou et al., 1995). Some authors divide tool wear into ﬁve distinct stages (Bonifacio and Diniz, 1994):

It has been established by various researchers that the initial and fast (before tool breakage) stages occur more rapidly than the regular stage. Bonifacio and Diniz (1994) explain that, during the fast wear stage with coated carbide tools, the tool loses its coating and the tool substrate (which has less resistance) begins to perform the cut and wears faster. During the initial stage, the tool edge loses its sharp edge rapidly, after this the process stabilizes for a given time. Flank wear in relation to total cutting time will typically appear as depicted in Figure 7.6. The geometrical growth and rate of wear is unique for every tool insert, even those used with the same machining parameters. Wear measurements conducted on the shop ﬂoor of a piston manufacturer by Scheffer and Heyns (2004) are shown in Figure 7.7. It was found that the tools last between 1000 and 6000 components, which makes the optimal use of the tool extremely problematic if the wear is not monitored on-line. The reason for this behavior is mainly attributed to ﬂuctuating conditions on the shop ﬂoor, for example, the rate at which components are manufactured. If the time allowed for the tool to cool down between workpieces is not constant, large variations in the tool life can be expected.

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flank wear

Initial stage of wear Regular stage of wear Microbreakage stage of wear Fast wear stage Tool breakage

Tool wear parameters.

initial

regular

1

2

fast 3

5

4

cutting time

FIGURE 7.6

Flank wear in relation to cutting time.

0.25 flank wear VB [mm]

1. 2. 3. 4. 5.

FIGURE 7.5

0.2 0.15 0.1 0.05 0

2000

4000

6000

number of workpieces

FIGURE 7.7 Typical variations in tool life. (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)

Vibration-Based Tool Condition Monitoring Systems

7.3

7-7

Vibration Signal Recording

The information from vibration sensors can be treated in numerous ways. The overall aim of a tool condition monitoring system (TCMS) is to utilize the best processing techniques to extract the relevant information from sensor signals. Generally, a TCMS consists of the steps depicted in Figure 7.8. Various methods that could be used in each step will be discussed in more detail.

7.3.1

Direct and Indirect Systems

TCMSs can be divided in two categories, namely, direct and indirect. Direct methods are concerned with a measurement of volumetric loss at the tool tip, while indirect methods use a pattern in sensor data to detect a failure mode (Byrne et al., 1995). Direct methods do not utilize vibration and will not be discussed here. In general, direct methods are sensitive to dirt and cutting chips, and consequently they are not commonly accepted in industry. Indirect methods have found more acceptance in industry due to the fact that most indirect methods are easily interpreted, cost-effective, and often more reliable than direct methods. Also, for some applications, it might not be possible to use a direct monitoring method due to the nature of the process.

7.3.2

Sensor Requirements for Tool Wear Monitoring

Machine tools represent very hostile environments for sensors. Sensors used for TCM (also see Chapter 1) must meet certain requirements, such as (Byrne et al., 1995) the following: *

*

*

*

*

*

*

Must measure as close as possible to the point of metal removal Must not cause a reduction in the stiffness of the machine tool Must not cause a restriction of the working space of the machine Should be wear and maintenance free, easy to replace, and of low cost Must have resistance to dirt, chips, and electromagnetic and thermal inﬂuences Should function independent of tool and workpiece Must provide reliable signal transmission, e.g., from rotating to ﬁxed machine components

7.3.3

Force Measurement

Worn tools cause an increase in the cutting force components. It is also known that both the dynamic and static components generally increase with tool wear due to frictional effects. The three components of the cutting force each responds uniquely to varying machining parameters and the different wear modes. Depending on the type of process that is investigated and the speciﬁc experimental setup, results among researchers vary. This can be attributed to dynamic effects of the machine tool and the measurement equipment. There are a number of different sensor conﬁgurations to collect forces from machining operations and these are described below. 7.3.3.1

Direct Measurement Dynamometers

Tool holder dynamometers are by far the most popular method for collecting cutting forces. These sensors utilize the piezoelectric effect and can measure quasistatic and dynamic cutting forces very accurately. However, dynamometers are very expensive and bulky instruments and are not practical for a sensor selection and deployment

signal recording and conditioning

generate signal features

FIGURE 7.8

select wear sensitive features

TCMS steps.

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model features and wear relationship

7-8

Vibration Monitoring, Testing and Instrumentation

typical shop ﬂoor. Furthermore, their usable frequency range is limited to approximately 1 kHz. An example of a tool holder dynamometer is shown in Figure 7.9. Tarmal and Opavsky (2000) investigated the dynamics of a conventional force dynamometer for machining operations. It was found that the dynamometer has signiﬁcant amplitude distortion in the frequency range that is quoted as the operating range by the manufacturer. The authors suggest that the dynamic characteristics of the dynamometer (while clamped as it would be during measurements) be identiﬁed with a modal test and the effect of dynamometer dynamics be compensated for after measurements are made to obtain the true cutting force. 7.3.3.2

Indirect Force Sensors

There are numerous small force sensors available for the purpose of force measurement on machine tools. These measure forces in load-carrying components of the machine tool and are thus not direct force measurement devices. The advantages of these sensors are their size, low cost, and signiﬁcantly higher operational frequency range. A disadvantage is that a suitable position for the sensor can only be determined experimentally. These sensors are suitable for tool breakage monitoring in rough machining or detection of other catastrophic events such as collisions. An example of a three-component force sensor is shown in Figure 7.10. 7.3.3.3

FIGURE 7.9 KISTLER force dynamometer type 9121. (Source: KISTLER Brochure 2002. Courtesy of Kistler Instrumente AG.)

FIGURE 7.10 KISTLER three-component force sensor type 9251A. (Source: KISTLER Brochure 2003. Courtesy of Kistler Instrumente AG.)

Piezoelectric Strain Sensors

The use of piezoelectric strain sensors for wear monitoring of synthetic diamond tool inserts was reported by Scheffer and Heyns (2000a). These sensors are ultrasensitive to changes in cutting forces if they are installed in an appropriate location. The best location for the sensor must once again be determined experimentally, but generally it should be installed on a load-carrying FIGURE 7.11 KISTLER strain sensor type 9232A. component of the machine as close as possible to (Source: KISTLER Brochure 2004. Courtesy of Kistler the tool tip, for example, on the tool holder itself Instrumente AG.) (Scheffer and Heyns, 2001b). An example of a piezoelectric strain sensor that can be used on machine tools is shown in Figure 7.11. 7.3.3.4

Resistance Strain Gauges

A quite simple method to estimate both the static and dynamic components of cutting forces without any distortion is to use resistance strain gauges (see Chapter 1). These comply with most of the requirements for TCM sensors, and they can accurately follow the static and dynamic response of a system up to

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FIGURE 7.12 Application of resistance strain gauges. (Source: Scheffer, C. and Heyns, P. S., Mech. Syst. Signal Process., Elsevier, 2004. With permission.)

50 kHz. Scheffer and Heyns (2002a) developed a sensor-integrated tool holder using strain gauges. It was shown that the system is robust, cost-effective, and ﬁt for an industrial TCMS. The physical layout of the strain gauges on a boring bar is shown in Figure 7.12. The system was calibrated with a special device to directly obtain the three cutting forces from the strain gauge signals. 7.3.3.5

Customized Force Sensors

There are a number of customized force sensors available that can be used with speciﬁc machining operations. These are: *

*

*

7.3.4

Force measuring plates, pins, and bearings Special force measuring bolts Force and torque measuring rings that ﬁt on spindles

Acceleration Measurement

Piezoelectric accelerometers can measure the machine vibration caused by oscillations of cutting forces. It is well known that high-frequency vibrations (higher than 1 kHz) yield large acceleration levels, giving accelerometers an advantage over force-based monitoring. Accelerometers fulﬁll the environmental requirements for tool wear monitoring because they are resistant to the aggressive media present during machining. Accelerometers are also less expensive than force dynamometers and can measure vibration levels within a very wide frequency range, typically 5 Hz to 10 kHz. Various authors have shown that acceleration levels change with tool wear. Li et al. (1997) found that the coherence function of two crossed accelerations can be used as an easy and effective way to identify tool wear and chatter. They found that with progressive tool wear, the autospectra of the two accelerations and their coherence function increase gradually in magnitude around the ﬁrst natural frequencies of the cross-bending vibration of the tool shank. As the tool approaches a severe wear stage, the peaks of the coherence function increase to values close to unity. Scheffer et al. (2003) reported on the use of an accelerometer for wear monitoring during hard turning. It was found that certain frequencies show repeatable amplitude increase with increasing tool wear. These frequencies corresponded to the tool holder natural frequencies. Some authors, for example, Bonifacio and Diniz (1994), also found that a wear sensitive frequency will increase with increasing tool wear and then suddenly decrease near the end of tool life. This can be attributed to an increased damping effect due to plastic deformation and microbreakage of the cutting edge.

7.3.5

Acoustic Emission Measurement

Cutting processes produce elastic stress waves that propagate through the machine structure. Different sources in the cutting process generate these stress waves known as acoustic emission (AE). Sources of AE

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Vibration Monitoring, Testing and Instrumentation

in metal cutting are: *

*

*

*

*

Friction on the tool face and ﬂank Plastic deformation in the shear zone Crack formation and propagation Impact of the chip at the workpiece Chip breakage

A typical AE sensor for use on machine tools is shown in Figure 7.13. FIGURE 7.13 Kistler AE sensor type 8152B. (Source: The fact that crack formation generates AE PCB Website 2002. Courtesy of Kistler Instrumente AG.) makes AE ideal for tool breakage detection. Collection of the AE requires special hardware that can bandpass ﬁlter the signals to the AE range (between approximately 50 and 250 kHz). Furthermore, ampliﬁcation is required and an analogue root-mean-square (RMS) circuit with a short time constant is generally also included to collect the AERMS. The different steps required to collect AE are depicted in Figure 7.14 (adapted from Jemielniak, 2000). Araujo et al. (2000) investigated sliding friction as a possible source of AE during metal cutting. The AERMS values in different frequency ranges were collected for different widths of cut and also with the tool rubbing against the workpiece without cutting. It was found that the level of AE remains almost constant for all width of cut conditions, and hence it was concluded that the main mechanism for AE during metal cutting is the sliding friction between the tool and workpiece. Consequently, an increase or decrease of AE can be expected with tool wear depending on the effect on the sliding friction due to that tool wear. Furthermore, it is believed that the cutting temperatures will affect the AE due to thermal expansion effects. Chiou and Liang (2000) investigated AE with tool wear and chatter effects in turning. A model is presented that can predict the chatter AERMS amplitude with certain levels of ﬂank wear. Good correlation was found between the model and the experimental results. Kim et al. (1999) reported on the use of AE to monitor the tool life during a gear shaping process. The AERMS is collected and used in a software program to predict the remaining tool life. Li (2002) presented an overview of using AE for TCM in turning operations. It is stated the AE is heavily dependent on cutting conditions and, as a result, methods should be employed to handle this problem effectively. Some methods are proposed that include advanced signal processing, sensor fusion and modeling techniques. Many other AE-based tool wear and breakage monitoring systems have been implemented successfully in research. One problem still lies with an appropriate interpretation of the AE frequency spectrum. In most studies, an explanation for the choice of certain frequencies and their advantages are not given or not investigated. In fact, Jemielniak (2000) found that using the average value of AE (or AERMS) is the most suitable. A similar conclusion was made by Scheffer et al. (2003), who compared different processing methods of the AE signal during hard turning.

FIGURE 7.14

Steps for collecting AE during turning. (Adapted from Jemielniak, K., Ultrasonics, 2000.)

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7.3.6

7-11

Sensor Comparisons

Future research should be directed towards directly comparing different sensors for tool wear monitoring. Choi et al. (1999) developed a single sensor for parallel measurement of force and AE. A ﬁnite element analysis was carried out to determine the optimal position for the sensor away from the tool holder because the sensor obstructed the working space of the machine. The approach was successful for breakage detection but no wear estimations are reported. Barrios et al. (1993) compared AE, vibration, and spindle current for TCM during milling. It was found that the spindle current is the most sensitive sensor for detecting tool wear, with AE the least sensitive. However, contradictory results are reported in other publications, and hence more research would be required to determine ultimately which sensor is the best for which machining operation. Govekar et al. (2000) compared force and AE methods for TCM, and concluded that the best result is achieved when sensory information is combined. Dimla and Lister (2000a) compared the use of force and vibration signals for TCM and also combined the information in a single decision-making technique (Dimla and Lister, 2000b). Similar comparative studies were reported by Scheffer et al. (2003).

7.4 Signal Processing for Sensor-Based Tool Condition Monitoring Using the sensor information from the different sensor systems described in the previous section, a decision must be made with respect to the tool condition. This decision is generally referred to as the data classiﬁcation. It is often better to combine sensory information to solve a complex problem such as TCM. Such a combined approach is referred to as sensor fusion. Sick (2002) proposed a generic sensor fusion architecture for TCM, which summarizes the various sensor fusion levels of a TCMS. These are: *

*

*

*

*

Analogue preprocessing Digital preprocessing Feature extraction Wear model Decision making

Fusion of sensor information can occur at any of these levels. Analogue and digital preprocessing are activities such as signal ampliﬁcation, conditioning, ﬁltering, calibration, and temperature compensation. The feature extraction step is probably the most important step, because here the sensor signals must be condensed and reduced to only a few appropriate wear sensitive values. Many different methods are available to achieve this. The wear model level establishes a relationship between the chosen features and the tool condition. In many cases, neural networks (NNs) are used in this step, and sensor fusion takes place within the NN. A decision level can also be included where a ﬁnal decision is made with respect to the tool condition, for instance a “competing experts” formulation if a TCMS is used in conjunction with a tool-life equation. Discussions on the various techniques follow.

7.4.1

Feature Extraction

Most decision-making techniques for process monitoring are based on signal features. Through appropriate signal processing, features can be extracted from signals that show consistent trends with respect to tool wear. Features are mainly derived through processing in the time, frequency, or joint timefrequency domain or statistical analysis. 7.4.1.1

Time Domain

Features extracted from the time domain are usually fundamental values such as the signal mean or RMS. Other techniques include the shape of enveloping signals, threshold crossings, ratios between

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time-domain signals, peak values, and polynomial approximations of time-domain signals. Examples of time-domain features from an interrupted cutting operation are shown in Figure 7.15. It has been found that some of the time-domain features show good correlation with tool wear and are easy to implement (Scheffer, 1999). Bayramoglu and Du¨ngel (1998) investigated the use of several different force ratios (calculated from the static cutting forces). It was found that certain force ratios can be used to monitor tool wear under a wide range of cutting conditions. Most commercial TCMS rely on time-domain information. However, time-domain features are known to be sensitive to disturbances and should be complemented with features from another domain. 7.4.1.2

FIGURE 7.15

Simple time-domain features.

Frequency Domain

The power or energy of certain frequency bands in the fast Fourier transform (FFT) is often suggested as a feature for TCM. It is very challenging to identify spectral bands that are sensitive to tool wear. It is even more difﬁcult to determine exactly why these frequencies are inﬂuenced by tool wear. Power in certain bands will often increase due to higher excitation forces because of the increase in friction when the tool starts to wear. Sometimes a peak in the FFT will also shift due to changing process dynamics as a result of tool wear. An early frequency-domain approach is reported by Jiang et al. (1987), in which frequency-band energy is determined from FIGURE 7.16 Frequency-domain features. the power spectral density (PSD) function as a feature for tool wear. Some authors suggest that two frequency ranges be identiﬁed from the original signal (Bonifacio and Diniz, 1994). The one range must be sensitive to tool wear, the other must be insensitive. For instance, if the measurement was made from 0 to 8 kHz, it must be split (using appropriate ﬁlters) into a 0 to 4 kHz signal and a 4 to 8 kHz signal. If the lower range is more sensitive to tool wear, a ratio between the two ranges can be calculated. If this ratio exceeds a certain pre established value, it can be concluded that the end of the tool life has been reached. This can also apply for a ratio between the signal recorded from a fresh tool to that compared with a worn tool. Examples of frequency-domain features from cutting forces are shown in Figure 7.16. One difﬁculty with frequency-domain approaches is that the dynamics of the operation and measurement hardware is not always fully understood. The fact that measurement hardware dynamics instead of process dynamics are often measured was also recently identiﬁed by Warnecke and Siems (2002). The response of a force dynamometer is inﬂuenced by its clamping condition, which may cause it to experience nonlinearities at relatively low frequencies. There are also some uncertainties when using these instruments, relating to their calibration and other varying parameters. A model for expressing the uncertainties when collecting cutting forces with a dynamometer was proposed by Axinte et al. (2001). These uncertainties might be responsible for the scatter of force components often reported in the

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literature. An interesting study is also reported by Ba¨hre et al. (1997), concerning determination of the natural frequencies of the machine tool components using the ﬁnite element method (FEM). These are taken into account for interpretation of the vibration/AE signal. 7.4.1.3

Statistical Processing

In the case of statistical features, signals are assumed to have a probabilistic distribution, and consequently, useful information can be extracted from the statistics of the distribution. Hence, the signal is regarded as a random process. Generally, machining processes are nonstationary but are assumed to be stationary for the short periods during which features are calculated. Several statistical features have been investigated for TCM and can be applied to several machining operations. The main features are those that describe the probability distribution of a random process (variance, standard deviation, skewness, kurtosis, etc.) and coefﬁcients of time-series models. There are also miscellaneous other statistical features, such as cross-correlations, the coherence function, and the harmonic mean. One useful approach is the use of autoregressive (AR) and autoregressive moving average (ARMA) coefﬁcients. AR coefﬁcients computed for a signal represent its characteristic behavior. When the signal changes during the cutting operation due to tool wear, the model coefﬁcients also change and can then be utilized to monitor the progressive tool wear. Baek et al. (2000) report on the use of an eighth order AR model for tool breakage detection during end milling. It was found that the AR approach is somewhat more accurate than the frequency band energy method. Yao et al. (1990) used the ARMA method to decompose the dynamic cutting force signals, and wear-sensitive frequencies were identiﬁed. This assisted in identifying the importance of certain vibration modes with respect to TCM. The use of statistical process control (SPC) methods is also reported by some authors. Jun and Suh (1999) considered the X-bar and exponentially weighted moving average (EWMA) for tool breakage detection in milling. Jennings and Drake (1997) used statistical quality control charts for TCM. Different statistical parameters are calculated and examples of one-, two- and three-variable control charts are given. 7.4.1.4

Time–Frequency Domain

The most common time –frequency domain processing method in TCM applications is wavelet analysis. A comprehensive discussion on the advantages and disadvantages of wavelet analysis for TCM is described by Sick (2002). It is often stated that wavelets are used because they provide information about the localization of an event in the time as well as in the frequency domain. However, locating discrete frequency-related events in the time domain is rarely of importance with respect to tool wear (which is a gradually increasing phenomenon). In contrast, tool breakage will have a large localized effect in the time domain, but this can be monitored more effectively using time-domain techniques. Furthermore, wavelets are time variant and the exact contribution of a particular frequency at any given time can never be determined accurately due to Heisenberg’s uncertainty principle. Despite the above arguments, the use of wavelet analysis for TCM is reported in several publications. Lee and Tarng (1999) use the discrete wavelet transform for cutter breakage detection in milling and ﬁnd that the technique is reliable even under changing machining conditions. Luo et al. (2002) published results of a TCMS using wavelet analysis of vibration signals. In this case, the wavelet is used as a ﬁlter to enhance wear-sensitive features in the signals. However, the results are not compared with conventional digital ﬁltering. A comparative study between wavelets and digital ﬁltering for tool wear monitoring was carried out by Scheffer (2002). It was found that, although the wavelet packets act as automated ﬁlters, a very similar (if not better) result could be achieved with appropriate digital ﬁltering. The use of wavelets increase the complexity of the TCMS, which is a disadvantage for shop-ﬂoor implementations. Furthermore, the results from digital ﬁltering can be physically related to the machining operation and tool wear, whereas the behavior of wavelet packets is more difﬁcult to interpret. Another method of time –frequency analysis that can be applied for TCM is spectograms (e.g., the Gabor distribution). Spectograms are very useful to identify stationarity in dynamic signals, and for detection of disturbances that may be time-localized in signals. The use of the Choi –Williams time –frequency distribution for TCM during multimilling is described by James and Tzeng (2000).

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Wear-sensitive regions on the time –frequency distribution are calculated and used as inputs to a NN for wear classiﬁcation. An example of a change in the dominant chip curl frequency during hard turning is shown in Figure 7.17. It is obvious that, due to some disturbance (perhaps tool wear), the dominant dynamic force frequency “jumps” from 27 to 9 Hz.

7.4.2

Feature Selection

Various authors attempt to generate features that are sensitive to tool wear but insensitive to changing machining parameters. For most operations, the machining parameters can be included in the wear model and hence the sensitivity of the features is not such an important issue. There are FIGURE 7.17 Time-frequency distribution of cutting also other techniques to normalize sensor data force signal. (Source: Scheffer, et al., Int. J. Mach. Tools Manuf., Elsevier, 2003. With permission.) with respect to machining parameters, for instance the use of a theoretical model (Sick, 1998). This is very useful if the machining conditions change so often that not enough data can be collected for training or calibrating a model. Numerous techniques exist to select the most wear-sensitive features or to reduce the input feature matrix to a lower dimension. The main techniques for feature selection and reduction are listed below: *

*

*

*

*

*

*

*

Principal component analysis (PCA) Statistical overlap factor (SOF) Genetic algorithm (GA) Partial least squares (PLS) Automatic relevance determination (ARD) Analysis of variance (ANOVA) Correlation coefﬁcient Simulation error calculations

Al-Habaibeh et al. (2000) presented a TCMS for a parallel kinematics machine tool for high-speed milling of titanium. An interesting approach to feature selection is employed, called self-learning automated sensors and signal processing selection (ASPS). This approach is based on an on-line selflearning methodology, whereby a certain feature will be selected automatically based on a correlation with tool wear. A linear regression is performed on each feature in the sensory feature matrix to detect the sensitivity of each feature with respect to tool wear. A very interesting cost analysis is then preformed to determine if the installation of a sensor justiﬁes its costs. Ruiz et al. (1993) proposed the use of a discrimination power for feature selection in a TCM application. The method is similar to that of the SOF. An automated version is proposed that also checks for linear correlation between features. It is difﬁcult to assess the success rate of the automated procedure because the experiments/simulations are not described in enough detail. Lee et al. (1998) describe the use of ANOVA to determine the best force ratio for TCM statistically. Several ratios between the three main cutting forces are computed and the inﬂuence of controllable parameters (e.g., machining conditions) on these ratios are investigated by means of ANOVA. Du (1999) describes the use of a blackboard system, which is a knowledge-based approach for feature selection and decision-making. An advantage is the fact that a physical interpretation of a feature can be linked to phenomena in the machining operation. The method is also ﬂexible, but suffers from the disadvantage of requiring a large quantity of data and expertise to establish the knowledge-based rules.

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FIGURE 7.18 Comparison of correlation coefﬁcient and SOF for feature selection. (Source: Scheffer, C. and Heyns, P.S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)

Mdlazi et al. (2003) compared the performance of ARD and PCA for feature selection for two damage detection case studies. It was found that the performance of the methods is similar, but one might perform better on a particular data set. Generally speaking, the PCA yields better results for damage detection problems. Scheffer and Heyns (2002b) compared several feature selection methods for TCM, such as SOF, PCA, GA, ANOVA, and the linear correlation coefﬁcient. It was found that the correlation coefﬁcient approach and the SOF should be preferred for TCM applications. PCA could also be of assistance, but the feasibility of PCA for on-line applications is still questionable. The correlation coefﬁcient and SOF is expressed as percentages in Figure 7.18 (from Scheffer and Heyns, 2004) for 30 different wear monitoring features in a turning tool wear case study. Ideally, a feature with a high level of correlation and SOF should be selected. As a last step, engineering judgment is required for proper feature selection because automated methods will often select features that are dependant on one another, thus not achieving the goals of sensor fusion. The following rules can be used as a guideline for selecting features for TCM: *

*

*

*

*

Select features from the static and dynamic parts of force signals. Select features measured in different directions. Use time- and frequency-domain features. Features based on simple signal processing methods are preferred. There should be a reasonable physical explanation for the behavior of a feature with respect to tool wear.

7.5 Wear Model/Decision-Making for Sensor-Based Tool Condition Monitoring Trending, Threshold

A very simple decision-making technique is to trend features and to establish threshold values. When a certain feature or set of features crosses a threshold value, an estimation of the tool condition can be made. Unfortunately, these threshold values can only be determined experimentally. The difﬁculty with this method is to determine the correct threshold value, especially under diverse cutting conditions. Furthermore, the method is extremely sensitive to disturbances. The trend of the mean feed force with increasing ﬂank wear is shown in Figure 7.19, and two thresholds are shown as examples. It is clear that this technique is not very reliable due to the large variance in the trend.

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1

feature value [normalised]

7.5.1

0.5 0

threshold-replace threshold-warning

−0.5 −1 0.04

FIGURE 7.19

0.06 0.08 flank wear VB [mm]

Example of trend and thresholds.

7-16

7.5.2

Vibration Monitoring, Testing and Instrumentation

Neural Networks

The use of NNs as a secondary, more sophisticated signal processing and decision-making technique is often found in TCM applications. The simultaneous utilization of many features and the robustness towards distorted sensor signals are two of the most attractive properties of NNs. Neural networks also assist in the fusion of sensor information for TCM. In other words, combining features from acceleration, AE, and force signals in a NN can result in a method that can predict the tool condition with increased accuracy (Silva et al., 1998). The successful implementation of NNs is dependent on the proper selection of the network structure, as well as the use of the correct training and testing methods. It is important to make a distinction between supervised and unsupervised NN paradigms. Unsupervised NNs are trained with input data only, and are usually used for discrete classiﬁcation of different stages of tool wear. Supervised NNs are trained with input and output data, and these are used for continuous estimations of tool wear. Furthermore, a distinction should be made between dynamic and static NNs. In the case of dynamic NNs, temporal (time) information is included in the network with the aim to model a time series. This can be done explicitly by using a time-based feature as an input to the network, or implicitly by using recurrent networks or networks with tapped delay lines (TDLs). Dynamic networks are preferred for TCM because tool wear is time-dependent (tool wear is a monotonically increasing parameter that is partly a function of machining time). 7.5.2.1

Unsupervised Networks

There are two basic network paradigms for unsupervised classiﬁcations, namely adaptive resonance theory (ART) and the self-organizing map (SOM). ART is based on competitive learning, addressing the stability –plasticity dilemma of NNs. The main advantage is its ability to adapt to changing conditions. ART networks also have self-stability and self-organization capabilities. The SOM is actually a data-mining method used to cluster multidimensional data automatically. A highdimensional feature matrix can be displayed on a FIGURE 7.20 Schematic representation of the SOM. two-dimensional grid of neurons that are arranged in clusters with similar feature values. Clusters for new and worn tools can be formed, and these are used for automatic classiﬁcation of the tool condition. A SOM is depicted schematically in Figure 7.20. There are many practical advantages for using unsupervised networks. One is the fact that the machining operation is not interrupted for tool wear measurements during the training phase. There is also the advantage of practical implementation if machining conditions change very often and appropriate training samples for supervised learning cannot be collected. Furthermore, the numerous different combinations of tool and workpiece materials and geometries can make supervised learning impossible. Normally, unsupervised NNs are used to identify discrete wear classes and cannot be used for a continuous estimation of tool wear. Silva et al. (2000) investigated the adaptability of the SOM and ART for tool wear monitoring during turning with changing machining conditions. It was found that, with appropriate training, the methods have enough adaptive capabilities to be employed in industrial applications. Govekar and Grabec (1994) use the SOM for drill wear classiﬁcation, where the SOM is used as a kind of empirical modeler. It was found that the adaptability of the SOM and its ability to handle noisy data makes the technique viable for on-line TCM. Scheffer and Heyns (2000b, 2001b) showed how a TCMS can be adaptable using SOMs. Different network sizes were compared with deﬁne discrete classes of new and worn tools. Larger networks yielded more continuous results. The TCMS using SOMs was applied to monitoring synthetic diamond tools for an industrial turning operation. It was found that the SOM can be used for industrial applications, especially if tool wear measurements are not available.

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FIGURE 7.21 Unsupervised approach to wear monitoring with the SOM. (Source: Scheffer, C. and Heyns, P.S., South African Inst. Tribol. 2002. With permission.)

Different NN paradigms were compared on a wear monitoring application for aluminum turning by Scheffer and Heyns (2002b). It was shown that the SOM is useful to identify discrete wear classes, as shown in Figure 7.21. If an exact value of the tool wear is required, supervised networks will yield better results but will require proper training samples. 7.5.2.2

Supervised Networks

Common supervised NNs used for TCM are the multilayer perceptron (MLP), multilayer feedforward (FF) network, recurrent neural network (RNN), supervised neuro-fuzzy system (NFS-S), time delay neural network (TDNN), single layer perceptron (SLP) and the radial basis function (RBF) network. The use of an SLP for TCM is described by Dimla et al. (1996), using the perceptron learning rule for training. The SLP is useful to identify discrete classes of the tool condition. FF networks are usually trained with the backpropagation algorithm. However, backpropagation should not always be the preferred choice because other methods are known that outperform this technique in terms of training time and generalization. The size of the hidden layers in multilayer networks should be optimized for performance. Many contradictory statements about the use of MLP networks can be found in the literature. One of the main problems is the selection of the number of input features, size of the network, and the number of training examples that should be used. A multilayer feedforward (FF) network is shown schematically in Figure 7.22. Normally, a nonlinear activation function should be used in the ﬁrst layer, and linear neurons in the subsequent layers. In the case of the FF networks, the backpropagation algorithm is often used for training. Backpropagation is an optimization algorithm based on steepest gradient descent. The use of FF networks with the backpropagation training rule is reported by authors such as Zhou et al. (1995), Das et al. (1996), and Zawada-Tomkiewicz (2001). Cutting conditions can also be included in such networks. Lou and Lin (1997) describe the use of a FF network using a Kalman ﬁlter to avoid the training problems encountered with backpropagation for a TCM application. The proposed method is less sensitive to the network initializations that often cause convergence problems with backpropagation. Monitoring a dynamic system such as a cutting process should be done with a dynamic modeling technique such as dynamic NN paradigms, for example, recurrent networks, TDNNs, or explicit FIGURE 7.22 Multilayer FF network.

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FIGURE 7.23 Recurrent networks: feedback connection (left) and Elman network (right). (Source: Scheffer, C. and Heyns, P.S., South African Inst. Tribol. 2002. With permission.)

inclusion of temporal information in static networks. Recurrent NNs have feedback connections from their output to their input. There are various types of recurrent NNs that are useful for speciﬁc applications. Elman networks are quite interesting. Generally, they are two-layer networks with feedbacks from the ﬁrst layer output to the ﬁrst layer input. This type of network can be used to learn and model temporal patterns. A recurrent network and an Elman network are shown schematically in Figure 7.23. Liu and Altintas (1999) report on the use of a FF network using a combination of TDLs and recurrent connections. Machining conditions are also included. It is stated that the system was integrated into an industrial TCMS, but was never put to use due to lack of “… robust, practical cutting force sensors …” (Liu and Altintas 1999). Scheffer and Heyns (2002b) report on the use of an Elman NN for TCM. It was found that the Elman network has a very smooth response and yielded better results than static NN paradigms. It should be mentioned that the Elman network requires more time for training, but because this is done off-line, training time should not be a criterion for evaluating NNs. Neuro-fuzzy systems (NFS-S) attempts to combine the learning ability of NNs with the interpretation ability of fuzzy logic. A TCMS using an NFS-S can be generated almost automatically because the fuzzy rules can be learned by the NN. A combination of supervised and unsupervised training is used for NFS-S. An in-process NFS-S system to monitor tool breakage was designed and implemented successfully by Chen and Black (1997), concentrating on end milling operations. Xiaoli et al. (1997) as well as Chungchoo and Saini (2002) also propose some of the advantages of using an NFS-S for TCM. RBF networks are often preferred because of the convergence properties of the training algorithm. In essence, convergence can be guaranteed and is often achieved much faster than in MLPs. The accuracy of RBFs depends on the choice of the centers for the basis functions, and should be treated with care. Pai et al. (2001) reported on the use of a resource allocation network (RAN) for TCM. The RAN is a RBF network utilizing sequential learning. The RAN is compared with the MLP for wear estimation during face milling. It was found that the RAN has faster learning ability but the MLP is more robust. TDNNs have delay elements in the feedforward connections, called TDLs. One advantage of TDNNs over RNNs is that stability problems are avoided. An investigation towards the inclusion of one and two phase delays for a TCM application was reported by Venkatesh et al. (1997). Different network sizes were also investigated, and it was found that the NNs with temporal memory generally perform better than those without memory. It is also stated that new algorithms should be investigated for training. Sick and Sicheneder (1997) also describe the use of TDNNs for TCM in turning. The TDNN is compared with the MLP and a signiﬁcant improvement was found when using

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TDNNs. In another instance, Sick et al. (1998) compare the SOM, NFS-S, and MLP networks for wear estimation. The following critical questions are used to evaluate the different NN paradigms (Sick et al., 1998): *

*

*

Are the generalization capabilities of the NN sufﬁcient (tested on previously unseen data)? What rate of correct classiﬁcation can be achieved for different wear stages? Are the results repeatable (e.g., with a new initialization)?

In the case study presented by Sick et al. (1998), the best results were found with MLPs. It is stated, however, that the results can be improved when using TDNNs, and such results are reported in Sick (1998). A novel combined approach is suggested by Sick (1998) to handle the effect of machining parameters. An empirical model is used to normalize the data with respect to machining parameters before the data are entered into the NN. Thus, machining parameters are not included in the NN itself. This approach solves the extrapolation limitations encountered when an NN is tested with data recorded with machining parameters it was not trained with. Although many authors test their NNs’ paradigms in such a way, NNs cannot be expected to extrapolate. NNs should instead be tested with previously unseen data recorded with same machining parameters it was trained with (hence an interpolation effect). This is a problem because training and testing patterns for each condition must be supplied. However, if data can be normalized with respect to machining parameters, training is only required for the normalized condition. This was in effect achieved by Sick (1998). A difﬁculty still lies with establishing an appropriate model, and in many cases it will also require a large number of experimental tests. A possible solution lies in the incorporation of numerical models, for example, ﬁnite element models. Scheffer (2002) presented another approach to tool wear monitoring of turning operations, using a combination of static and dynamic NNs. Static networks are trained off-line to model selected features from cutting forces. A dynamic NN that uses explicit temporal information is then trained on-line with the particle swarming optimization algorithm (PSOA). The training goal of the dynamic NN is to minimize the errors between the outputs of the static NNs and the on-line measurements. The method was tested on various turning operations and was also tested on an industrial shop ﬂoor. It was found that the method is more accurate and reliable than other NN paradigms and can be used with cost-effective hardware (Scheffer and Heyns 2002a, 2004). The method is depicted schematically in Figure 7.24.

FIGURE 7.24 Combined static and dynamic NN approach for turning. (Source: Scheffer, C. and Heyns, P. S., Mech. Syst. Signal Process, Elsevier, 2004. With permission.)

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7.5.3

Fuzzy Logic

Many authors have investigated the use of fuzzy logic to classify tool wear. It has been shown that fuzzy logic systems demonstrate great potential for use in intelligent manufacturing applications. While NN models cannot directly encode structured knowledge, it is often stated that fuzzy systems can directly encode structured knowledge in a numerical framework. Additionally, fuzzy systems are capable of estimating functions of a system with only a partial description of the system’s behavior. Du et al. (2002) propose a very interesting method called transition fuzzy probability, which was applied to a boring operation. This formulation can deal with the uncertainty of process conditions. The method performs well because TCM has two uncertainties: that of occurrence and that of appearance. The transition fuzzy probability solves this issue through the use of temporal information, similar to dynamic NNs. The method was shown to outperform a backpropagation NN, although very few details are given. It would be interesting to compare this method with dynamic NNs such as TDNNs. Fu et al. (1997) combined force, vibration, and AE in a fuzzy classiﬁer for TCM during milling. Timeand frequency-domain features were used, and it was found that combining the sensory information achieved the best result. This is done within the fuzzy classiﬁer. Li and Elbestawi (1996) and Kuo and Cohen (1998) combine fuzzy modeling steps with NNs at different levels for TCM. The latter combined force, vibration and AE in a multisensor approach with satisfactory results.

7.5.4

Other Methods

There are also a number of other decision-making and modeling methods that have been applied to TCM, and these include: *

*

*

*

Knowledge-based expert systems (Du, 1999) Pattern recognition algorithms (Kumar et al., 1997) Dempster–Shafer theory of evidence (Beynon et al., 2000) Hidden Markov models (Ertunc and Loparo, 2001; Ertunc et al., 2001)

Of these four approaches, only hidden Markov models have the potential possibly to outperform NNs and fuzzy systems. However, not enough comparable research has been conducted in this area, and is it certainly a worthwhile topic for future research.

7.6

Conclusion

Techniques for achieving TCM with vibration-based properties were presented in this chapter. The sensing methods that have proved to be effective for TCM are force, acceleration, and AE. The sensors employed must comply with certain requirements such as robustness and cost-effectiveness. Sensors must be installed as close as possible to the point of metal removal in order to avoid signal-to-noise ratio problems. Various techniques exist to condition and process the signals in analogue and digital formats. The aim of signal processing is to generate wear sensitive features from the vibration signals. This could be done by time, frequency, joint time –frequency, and statistical analysis. Feature selection can be automated with a variety of procedures, but care must be taken when using these to avoid selection of linearly dependent data. The selected features can be used to establish a model of tool wear. Numerous research papers have shown that NNs should be used due to the many advantages of NN modeling. The training and testing procedures of NNs are of utmost importance if the system is considered for industrial implementation. Care must be taken not to overtrain the networks because they will lose their ability to generalize. Furthermore, NNs cannot be expected to perform well if they are tested with previously unseen machining parameters. They should also be trained with the minimum and maximum tool wear that is

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expected. Future work should be directed towards incorporating numerical machining models into the wear monitoring system to normalize the data with respect to machining parameters. If this can be achieved, the amount of training data required for an effective TCMS will be reduced, which in turn will provide a better solution to TCM for the manufacturing industry.

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8

Fault Diagnosis of Helicopter Gearboxes 8.1 8.2 8.3 8.4 8.5

Kourosh Danai University of Massachusetts

8.6

Introduction ............................................................................ 8-1 Abnormality Scaling ............................................................... 8-5 The Structure-Based Connectionist Network ...................... 8-8 Sensor Location Selection ......................................................8-11

Coverage Index Effectiveness

†

Overlap Index

†

Monitoring

A Case Study ...........................................................................8-14

Structural and Featural Inﬂuences † Evaluation of Inﬂuences Fault Detection Results † Fault Diagnostic Results † Sensor Location Evaluation † Sensor Location Validation

†

Conclusion ..............................................................................8-23

Summary An overview of fault detection and diagnosis of helicopter gearboxes is presented in this chapter. Between oil analysis and vibration monitoring, the two predominant methods of gearbox monitoring, vibration monitoring is much more studied and relied upon because of its ability to reﬂect a wider variety of faults. In order to cover the concepts used in vibration monitoring, a method of diagnosis is discussed that relies on the knowledge of the gearbox structure and characteristics of the features of vibration for component fault isolation. Since a necessary part of vibration monitoring is selection of accelerometer locations on the housing, a method is also described whereby the suitability of accelerometer locations is quantiﬁed based on the structure of the gearbox and the monitoring effectiveness of the locations. Finally, a case study is included to illustrate the application of the concepts presented in the chapter to fault detection and diagnosis, as well as the ranking of suites of accelerometer locations. Some topics related to this chapter are covered in Chapter 1, Chapter 6, Chapter 7, and Chapter 9.

8.1

Introduction

Present helicopter power trains are signiﬁcant contributors to both ﬂight safety incidents and maintenance costs. For example, for large and medium civil transport helicopters in the period 1956 to 1986, gearboxes were the principal cause of 22% of the accidents with potential loss of life and aircraft (Astridge, 1989). To prevent such incidents, routine maintenance is scheduled at a signiﬁcant ratio of the total maintenance cost for the helicopter. Rapid and reliable detection and diagnosis (isolation) of faults1 in helicopter gearboxes is therefore necessary to prevent major breakdowns due to progression of undetected faults, and for enhancing personnel safety by 1 Determining whether the overall machinery is healthy is referred to as fault detection, whereas diagnosis is analogous to isolating the source of failure.

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8-2

Vibration Monitoring, Testing and Instrumentation

preventing catastrophic failures. Fault detection and diagnosis is also necessary for reducing maintenance costs by eliminating the need for routine disassembly of the gearbox, and for saving time during inspection. Detection and diagnosis of helicopter gearbox faults, like most rotating machinery, is based on oil analysis and vibration monitoring. Oil analysis, which is used to detect the presence of metallic debris, is performed by: (1) magnetic plugs (chip detectors), (2) oil ﬁlters, and (3) spectrometric oil analysis (SOA) (Lurton, 1994). Among these, magnetic plugs are the most popular because of their in-ﬂight utility and their ability to quantify the severity of wear by measuring the rate of detected debris. However, magnetic plugs can only collect ferromagnetic particles, and their capture efﬁciency may be poor. Oil ﬁlters used in helicopter gearboxes range between 3 and 150 mm, with ﬁner meshes of 3 mm more common in the recent years. SOA, which is a ground-based technique, is useful for detecting ﬁne debris, typically below 10 mm in size, caused by wear conditions such as rubbing, cutting, and corrosion wear, and ﬁne surface fatigue such as micropitting. Therefore, development of a reliable method of identifying common forms of fatigue such as spalling or wear that generate particles greater than 10 mm in size is of particular interest. The effectiveness of SOA is also affected by the level of ﬁltering performed on the used oil, which often leaves the oil free of particles (Lurton, 1994). The need for improving wear detection has motivated development and use of other analysis methods such as image processing of oil ﬁlters, ferrography, thermography, and ultrasonic analysis (Thornton, 1994). The other method of gearbox fault detection and diagnosis, and by far the more popular one, is based on vibration monitoring. The basic principle behind vibration monitoring is that, under normal operating conditions, each component in the gearbox produces vibrations at speciﬁc frequencies related to the component’s rotational frequency. In the case of a component fault, the vibration generated by the faulty component is different from the normal vibration, and will be reﬂected at the component’s rotational frequency and its harmonics. As such, monitoring the changes of vibration should theoretically give an indication of the fault. In practice, however, changes in the measured vibration as a Sun Planet Gear(SG) Bearing(PB)

Ring Gear(RG)

Gear Roller Bearing(GRB)

Mast Ball Bearing(MBB)

Planet Gear(PG)

Spiral Bevel Gear(SBG) Spiral Bevel Pinion(SBP) Triplex Bearing(TB)

Duplex Bearing(DB)

FIGURE 8.1

Mast Roller Bearing(MRB)

Pinion Roller Bearing(PRB)

Layout of various components in a typical helicopter (OH-58A) gearbox.

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Fault Diagnosis of Helicopter Gearboxes

8-3

result of component faults are not always distinct due to the attenuation of vibration by the housing and other components it travels through, as well as the noise in the signal. To provide a more tangible framework for the concepts in vibration monitoring, the layout of a typical helicopter gearbox is shown in Figure 8.1, with the location of the accelerometers on its test stand shown in Figure 8.2. In order to enhance identiﬁcation of vibration changes by component faults, the raw vibration is generally processed to obtain “features” that characterize the vibration at the frequencies associated with the gearbox components. Accordingly, the main focus of research in vibration monitoring has been the identiﬁcation of individual features that consistently reﬂect speciﬁc gearbox faults (Dyer and Stewart, 1978; McFadden and Smith, 1986; Mertaugh, 1986; Zakrajsek et al., 1995). A typical set of vibration features obtained from each vibration measurement is shown in Figure 8.3 (Stewart Hughes Ltd., 1986). Among them, envelope band and tone energies, cepstra, and synchronous-time averaged signals, are associ- FIGURE 8.2 Location and orientation of the accelated with various bearing frequencies and gear erometers on the OH-58A test stand. mesh tones. Envelope band and tone energies, which are the sum of the harmonics of the Tape Recorder bearings’ fundamental rolling-element frequencies Digitization / within a ﬁltered bandwidth, could be used for early Processing detection of failure in rolling-element bearings (Barkov and Barkova, 1995). Similarly, a cepstrum, STAT BBPS BRGA SGAV which is the power spectrum of the logarithm of a power spectrum, is often used because of its (16) BE For each of the 5 Gears (5) RMS (1) Skewness (6) WHT (17) BKV (1) MF1 (2) Kurtosis insensitivity to transmission path effects for (18) EB (2) FM4a (3) Crest Factor (7) RFR identiﬁcation of families of uniformly spaced (3) FM4B (4) Peak-to-Peak (8) TEO-G (19) ET (9) TEO-P (4) FM4-B sidebands in gearbox vibration spectra (Randall, (10) TM1-G (5) ACH 1982). Synchronous-time averaging is a signal (11) TM1-P (6) WCH (12) CEP(1911) (7) SCH processing technique that isolates the fundamental (13) CEP(572) (14) TON(1911) and harmonics of the gear meshing frequency, and (15) TON(572) is a primary analysis technique for detection of gear and shaft faults (McFadden and Smith, 1986). FIGURE 8.3 A typical set of vibration features The traditional approach to fault diagnosis has extracted from each accelerometer. relied on human expertise to relate vibration features to faults. In this approach, a diagnostician ﬁrst identiﬁes abnormalities in vibration features, then relates them to component faults, considering both the proximity of the accelerometer producing the feature to various components and the information about the type of fault characterized by the feature. Using this information, the diagnostician hypothesizes faults in speciﬁc components and then veriﬁes or discards the hypothesis by examining features from other accelerometers in the proximity of the suspect component. The disadvantages of this approach arise from: (1) the difﬁculty in identifying abnormality in features which are contaminated with noise, and (2) the tediousness of examining the numerous features obtained from all of the accelerometers. Owing to the large number of features

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8-4

Vibration Monitoring, Testing and Instrumentation

Raw Vibration

associated with a gearbox2, the diagnostician Fault Detection cannot often pay equal attention to all the features and is likely to ignore information that contradicts Signal Abnormality Diagnostic the hypothesis. Processing Scaling features The most efﬁcient method for integration of features is pattern classiﬁcation. To this end, FIGURE 8.4 Generic structure of an unsupervised artiﬁcial neural networks have been widely inves- diagnostic system. tigated and shown to provide excellent results (Solorzano et al., 1991; Chin et al., 1993; Kazlas et al., 1993; Chin et al., 1995). Neural networks offer the following advantages in diagnosis: (1) as pattern classiﬁers, they can efﬁciently cope with noise in features, (2) through training, they can form the signatures of individual faults in the multidimensional space of features, and (3) they can process vibration features in parallel, so diagnosis is not hindered by the enormity of feature space. The main disadvantage of supervised neural networks, however, is their need for prior training, which requires a comprehensive set of features during normal operation and at various fault instances. While training data can be obtained experimentally through accelerated fatigue tests (Lewicki et al., 1992) or seeded fault studies (Naval Command, Control, and Ocean Surveillance Center, 1995), their cost is considered too high, limiting the utility of supervised neural networks in fault diagnosis. The lack of training data poses a similar restriction for statistical pattern classiﬁers which need a priori statistics of the features. In the absence of training data required by supervised neural networks, the monitoring system can be formatted as in Figure 8.4 to take advantage of unsupervised pattern classiﬁcation. The strategy depicted in Figure 8.4 considers fault detection independent of abnormality scaling of features, so it can be performed before or in parallel to this stage, based on the overall deviation of all of the features from their normal state. An example of abnormality scaling based on unsupervised pattern classiﬁcation is described in the Section 8.2. For fault diagnosis, the proposed system should have detailed information about the relation between the individual features and component faults. One format for providing that information is expert diagnostic software, which can be developed at two different levels. At one level, “shallow expert systems” can be developed to compile a human diagnostician’s knowledge relating measurements to faults, often as “if …then” rules, similar to those already developed for simpler rotating machinery (Liddle and Reilly, 1993). The main obstacle in such a development will be the design of a robust inference engine that can resolve conﬂicting conclusions from the large pool of suspect features. At another level, “deep expert systems” can be developed to represent the diagnostic knowledge derived from the physics of the process. In deep expert systems, measurements need to be related to component faults by a model of the energy ﬂow via the structural connections between components and sensors. A suitable format for deﬁning the relation between components and sensors is the fuzzy set theory, which can address the approximate nature of vibration modeling. An example of a deep expert diagnostic system for helicopter gearboxes is the structure-based connectionist network (SBCN) (Jammu et al., 1998) described in Section 8.3, which takes advantage of the integration capability of neural networks while avoiding the need for supervised training. The salient feature of this connectionist network is that its weights can be determined a priori, based on the proximity of components to various accelerometers and the type of faults characterized by individual features. Although this system has been developed on a very approximate model of vibration ﬂow between the gearbox components and accelerometers, it has produced promising results when used for fault diagnosis of helicopter gearboxes (Jammu et al., 1998). A side beneﬁt of a model of the structural connections between gearbox components and sensors is its use in sensor location selection (Wang et al., 1999). An important issue in helicopter gearbox diagnostics is the determination of the number of accelerometers to be used for monitoring and their location on the gearbox housing. Accelerometers are generally located by experts, based on their 2 The large number of frequencies (tones) associated with the various components of the gearbox necessitates a huge number of features (often in excess of a hundred) to be obtained for vibration readings from each of the several accelerometers.

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Fault Diagnosis of Helicopter Gearboxes

8-5

proximity to gearbox components, orientation, and ease of mounting on the housing. However, this approach often leads to too many accelerometers and too high a demand for on-line monitoring on the on-board computer. Another problem with the extra accelerometers is the cost of extra mountings, cabling, and signal conditioning equipment. Sensor location selection based on a gearbox model is discussed in Section 8.4.

8.2

Abnormality Scaling

Vibration Features

Abnormality-Scaled Features

The abnormality-scaling module described here Weight relies solely on the value of each feature during normal operation. Accordingly, it is referred to as the single category-based classiﬁer (SCBC) to signify its independence from feature values associated Matching with faulty conditions (Jammu et al., 1996). In order to perform abnormality scaling, the SCBC compares features with their normal values, and if they are sufﬁciently different, assigns values between zero and one to characterize their degree of deviation from their normal values. The Abnormality-Scaling schematic of the SCBC, which is implemented by a connectionist network, is shown in Figure 8.5. FIGURE 8.5 Schematic of the SCBC network. The inputs to SCBC are the raw features from signal processing, si ðtÞ; i ¼ 1; …; n; and its outputs are abnormality-scaled features, fi ðtÞ; with values between zero and one. The value of zero indicates normality, and the other extreme of one denotes complete abnormality. The individual weights of the SCBC network, wi ; represent the normal values of the features, which are initially set equal to the ﬁrst corresponding feature value supplied to the SCBC. Classiﬁcation in the SCBC is performed by ﬁrst measuring the Euclidean distance of each feature, si ðtÞ; from its weight value, wi ; and normalizing it into the range ½0; 1 using a matching factor, fi ; deﬁned as (Figure 8.6)

fi ðtÞ ¼ 1 2 exp

2ðsi ðtÞ 2 wi Þ wi

2

ð8:1Þ

A fi value of zero indicates that the feature value matches the weight value precisely, and a value of one denotes that it deviates from it considerably. Note that the exponential function used here is not unique, and that other functions that can map the Euclidean distance into the range ½0; 1 can also be used for the matching factor. Since during normal operation of the gearbox, noise in the features usually causes them Matching

Abnormality-Scaling

1

1 Normal Region FIGURE 8.6

Matching and abnormality-scaling in SCBC.

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fmin

8-6

Vibration Monitoring, Testing and Instrumentation

to deviate from their normal values, a threshold, u, is considered to account for deviations by noise. The threshold, u, is used to hard-limit fi ðtÞ in SCBC as ( 0 ðnormal Þ if fi ðtÞ , u ð8:2Þ fi ðtÞ ¼ otherwise fi ðtÞ ðhard-limited Þ In the above relationship, the threshold u is obtained as n 1 X 2½maxðsi Þ 2 mi 1 2 exp u¼ n i¼1 m2i

2

! ð8:3Þ

where maxðsi Þ denotes the maximum value of the ith feature in a set of k samples of this feature recorded during normal operation, and mi represents its mean, estimated as

mi ¼

k 1 X s ðtÞ k t¼1 i

ð8:4Þ

The matching factor, deﬁned by Equation 8.1, suppresses any positive value in ½0; 1 into the range ½0; 1 : As such, only very large deviations in the feature values will be scaled to the value of one. Since large deviations in feature values are uncommon for gearboxes, the value of matching factor is further scaled to yield abnormality-scaled feature values fi ðtÞ as (Figure 8.6) fi ðtÞ ¼ fmin þ expða p fi ðtÞÞ

ð8:5Þ

where fmin represents the minimum abnormality value assigned to any feature that violates the threshold, u, and a controls the slope of the exponential curve. Since fi ðtÞ is deﬁned to have a value between zero and one, it is set to one when fi ðtÞ in Equation 8.5 exceeds the value of one. After each round of classiﬁcation of the vibration features, the weight values in the SCBC are updated so as to cope with noise and small variations in the operating conditions. Adaptation is carried out in two stages. In the ﬁrst stage, called primary adaptation, a network weight is adapted if the feature associated with it is classiﬁed as normal. In the second stage, referred to as contrast enhancement (CE) (Carpenter and Grossberg, 1987), the rest of the weights are adapted to achieve homogeneity in the abnormalityscaled values, thus increasing the likelihood of all of them being classiﬁed as normal or abnormal. Homogeneity, however, needs to exist only within speciﬁc feature groups, because gearbox faults do not necessarily cause abnormality in all of the features. For example, a gear fault will be reﬂected only by the features related to the gear, and is not expected to cause abnormality in bearing features. In order to preserve the functionality of individual feature groups (i.e., general features, gear features, and bearing features), CE is performed exclusively for each feature group. Adaptation in SCBC is performed as follows. Let wI represent the weight which is presently being updated and wi the remaining weights in the group. In primary adaptation, the weight value wI is modiﬁed according to the relationship wI ¼ wI þ dwI

ð8:6Þ

where ( dwI ¼

h½sI ðtÞ 2 wI

if fI ðtÞ ¼ 0

0

otherwise

ð8:7Þ

with the parameter h denoting the learning rate. For CE, if the majority of features are classiﬁed as normal, then the weight values associated with the features classiﬁed as abnormal will be adjusted such that the likelihood of all of the features being

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Fault Diagnosis of Helicopter Gearboxes

8-7

classiﬁed as normal is increased for the same feature. CE is performed as wi ¼ wi þ dwi

for all i – I

ð8:8Þ

where ( dwi ¼

hL½si ðtÞ 2 wi

if fI ðtÞ ¼ 0

2hL½si ðtÞ 2 wi

otherwise

ð8:9Þ

In CE, the amount by which the weight values are adjusted is controlled by a neighborhood function, L (Kohonen, 1989), which is assigned a value between zero and one. A value of zero is used for inputs with no noise, and a value at the other extreme of one is used for unreliable features with large amounts of noise. Usually, in practice, the value of L is set less than 0.5. For each round of primary adaptation (Equation 8.6 and Equation 8.7), I is varied to include all the features in the group. If the jth group of features contains mj features, then primary adaptation is applied by varying I from one to mj ; to cover all the weight values wI in the jth feature group. For each I, the remaining weight values, wi ; in the group (i ¼ 1 to mj and i – I) are adapted using CE according to Equation 8.8 and Equation 8.9. The adaptation algorithm presented in Equation 8.6 to Equation 8.9 is biased towards the most recent feature vector if only this vector were used for adaptation, whereas adaptation should be ideally performed using all of the feature vectors that pertain to the current operating conditions. However, as the number of available feature vectors for the operating condition progressively increases, adaptation based on all of the features becomes computationally demanding. As a compromise, in SCBC only the b most recent feature vectors are utilized for each adaptation sweep, such that adaptation is performed iteratively over the b most recent feature vectors. The learning rate, h, is progressively reduced for each adaptation iteration (Equation 8.7 and Equation 8.9). Abnormality scaling formulae are summarized in Table 8.1. TABLE 8.1

Summary of the Abnormality-Scaling Formulae

Classiﬁcation in the single category-based classiﬁer is performed by ﬁrst measuring the Euclidean distance of each vibration feature, si ðtÞ; from its weight value wi ; and normalizing it into the range ½0; 1 using a matching factor, fi ; deﬁned as

fi ðtÞ ¼ 1 2 exp

2ðsi ðtÞ 2 wi Þ wi

2

The matching factor is then hard-limited by a threshold, u, as (

fi ðtÞ ¼

0

if fi ðtÞ , u

ðnormalÞ

fi ðtÞ ðhard-limitedÞ otherwise

In the above relationship, the threshold, u, is obtained as

u¼

n 1 X 2½maxðsi Þ 2 mi 1 2 exp m2i n i¼1

2

!

where maxðsi Þ denotes the maximum value of the ith feature in a set of k samples of this feature, recorded during normal operation, and mi represents its mean, estimated as

mi ¼

k 1 X s ðtÞ k t¼1 i

The vibration feature is then abnormality-scaled as fi ðtÞ ¼ fmin þ expða p fi ðtÞÞ where fmin represents the minimum abnormality value assigned to any feature that violates the threshold, u, and a controls the slope of the exponential curve.

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8-8

8.3

Vibration Monitoring, Testing and Instrumentation

The Structure-Based Connectionist Network

The SBCN (Jammu et al., 1998) is developed to take advantage of the integration capability of neural networks, but to avoid the need for supervised training. It deﬁnes the weights of the network according to the structural knowledge of the gearbox and the type of fault represented by various vibration features. In the SBCN, the structural inﬂuences, which represent the proximity effect of faults on accelerometers, are determined based on the root-mean-square (RMS) value of the frequency response of a simpliﬁed lumped-mass model of the gearbox. Fault diagnosis in SBCN is performed by propagating the abnormality-scaled vibration features from SCBC to produce outputs representing the fault possibility values for each gearbox component as (Figure 8.7) n X pk ðtÞ ¼ fi ðtÞvik ð8:10Þ

Abnormality-Scaled Features

Fuzzy Influence Weights

lik

uik

Faulty Components FIGURE 8.7 The structure-based connectionist network (SBCN) with its fuzzy weights for isolating faulty components.

i¼1

In the above equation, pk ðtÞ represents the fault possibility value for the kth component of the gearbox, fi ðtÞ denotes the abnormality-scaled value of a feature and vik represents the weighting factor determined based on the lower and upper bounds of fuzzy inﬂuence weights ðlik ; uik Þ as vik ¼ lik þ ðuik 2 lik Þfi ðtÞ

ð8:11Þ

Note that Equation 8.10 represents propagation of inputs through the weights of the SBCN, similar to a regular connectionist network with vik as weights (Hertz et al., 1991). The difference is that the weights of the SBCN vary within the range ðlik ; uik Þ according to the magnitude of the corresponding input, fi ðtÞ: According to Equation 8.11, a higher abnormality-scaled feature value produces a higher weight value, vik ; emulating the reasoning by the human expert who pays more attention to the features that exhibit higher abnormality values. In SBCN, in order to make uniform the interpretation of the fault possibility values pk ðtÞ; they are normalized to have values between zero and one as pk ðtÞ ck ðtÞ ¼ X n uik

ð8:12Þ

i¼1

Accordingly, a ck ðtÞ equal to one denotes a deﬁnite fault, whereas a value of zero represents normality. In this system, fault diagnosis is performed hierarchically. First, the faulty subsystem within the gearbox is identiﬁed by using the structural inﬂuences as vik : Then, the faulty components within the suspect subsystem(s) are isolated using the product of structural and featural inﬂuences as vik : The connection weights of the SBCN are deﬁned based on the inﬂuences between the component faults and vibration features. Ideally, the structural inﬂuences should represent the strength of component vibration at the particular frequency (frequencies) represented by the feature. For this, the attenuation property of the “travel path” between each component and accelerometer needs to be modeled as a function of the moment of inertia, stiffness, and damping of the components in the path (Badgley and Hartman, 1974; Smith, 1983; Lyon, 1995). In order to appreciate the difﬁculties associated with vibration modeling of gearboxes, the vibration model of a simple gearbox is considered

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Fault Diagnosis of Helicopter Gearboxes

8-9

(Choy and Qian, 1993): _ v ½X_ þ ½GA ½X þ ½Cb ½X_ 2 X_ c þ ½Kb ½X 2 Xc þ ½Ks ½X 2 Xr ¼ ½FðtÞ þ ½FG ðtÞ ½M ½X€ þ ½G ð8:13Þ where X represents the generalized displacement vectors in the lateral x, y, and z, and rotational ux ; uy ; and uz directions, M denotes the inertia matrix, Gv represents gyroscopic forces, GA denotes the rotor angular acceleration, Cb and Kb represent the damping and stiffness matrices of bearings, respectively, ½X 2 Xc denotes the casing vibration, ½X 2 Xr represents the shaft residual bow, and Ks denotes the shaft bow stiffness matrix. The excitation force [F(t)] is due to mass imbalance, and [FG(t)] represents the nonlinear gear mesh force, which in the x direction has the form (Choy and Qian, 1993) FGxk ¼

n X i¼1;i–k

Ktki ½2Rci uci 2 Rck uck þ ðXci 2 Xck Þ cosðaki Þ

þ ðYci 2 Yck Þ sinðaki Þ ½cosðaki Þ þ SIGNðmÞ sinðaki Þ

ð8:14Þ

where FGxk denotes the gear mesh force in the x direction on the (n 2 1)th gear due to its mesh with (n 2 1) other gears, Ktki represents the nonlinear gear mesh stiffness between the kth gear and ith gear, Rci denotes the radius of the ith gear, aki represents the orientation angle between the kth and ith gears, and m denotes the coefﬁcient of friction. Mathematical relationships similar to Equation 8.14 can be deﬁned to represent the force in the y direction as well as torsional gear mesh forces. Furthermore, the vibration of the casing due to the vibration of the gear–shaft system needs to be represented by a separate set of coupled equations of motion similar to Equation 8.13. The above equations need to be integrated numerically in order to estimate the vibration signal recorded on the housing, but the following difﬁculties exist. (1) The values of stiffness and damping coefﬁcients for the components are not readily available. (2) The cross-coupling terms in the stiffness matrices in the x, y, and z directions cannot be easily deﬁned (Mitchell and Davis, 1985; Choy and Qian, 1993). (3) It is difﬁcult to take into account the multitude of travel paths and the associated models of attenuation for the many component– accelerometer pairs in the gearbox. For example, Ktki, the gear mesh stiffness, which is obtained by considering the gear tooth as a nonuniform cantilever beam (Lin et al., 1988; Boyd and Pike, 1989), is a function of the cross section of the tooth at the point of loading as well as load variation due to changes in the direction of load application (Mark, 1987; Lin et al., 1988; Choy and Qian, 1993), friction between the meshing teeth (Rebbechi et al., 1991), contact ratio (Cornell and Westervelt, 1978), the type of gears (spur, helical, etc.) (Mark, 1987; Lin et al., 1988; Boyd and Pike, 1989), and gear errors such as proﬁle, transmission, and manufacturing errors (Smith, 1983; Mark, 1987). Similarly, the stiffness of bearings is a time-varying, nonlinear function of bearing displacement and the number of rolling elements in the load zone, as well as the bearing type (roller, ball, etc.), axial preload, clearance, and race waviness (Harris, 1966; While, 1979; Walford and Stone, 1983). All these factors make it very difﬁcult to obtain an accurate and computationally inexpensive vibration attenuation model for gearboxes. In order to avoid the difﬁculties associated with accurate modeling of vibration transfer, a simpliﬁed method is described here that accounts separately for the two main aspects of vibration change by faulty components: (1) the proximity effect of the faulty component on the accelerometer generating the feature (structural inﬂuence), and (2) the frequency related information represented by the feature (featural inﬂuence). The main simpliﬁcation in this method is in the deﬁnition of structural inﬂuences as the average strength of the vibration signal across all frequencies measured by an accelerometer due to a component fault. To compute this average vibration, several simpliﬁcations have been adopted: 1. A lumped-mass model of the gearbox is used to model vibration. 2. Only the average static values for the stiffness coefﬁcients are used, to cope with the absence of accurate values for stiffness coefﬁcients. 3. Damping ratios of bearings and shafts are neglected (Lin et al., 1988). 4. A damping ratio of 0.1 is used for all the gears as an approximation to actual gear ratios estimated between 0.03 and 0.17 (Kasuba and Evans, 1981).

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8-10

Vibration Monitoring, Testing and Instrumentation

5. The cross-coupling terms in the stiffness matrix are neglected. 6. Only the shortest vibration travel path between each component –accelerometer pair is considered. Using the above simpliﬁcations, the average vibration registered by an accelerometer due to a faulty ith component can be simulated by applying an excitation source y at the ith component in the lumped-mass model (Figure 8.8). In order to represent all frequencies in the excitation source, y can be selected to consist of unit amplitude sine waves of all frequencies. The displacement of various components in the travel path due to an excitation exerted at the ith component can be obtained for a typical N-mass path as (James et al., 1994) 2 32 3 2 3 a11 a12 0 ··· 0 x1 y 6 76 7 6 7 6a 7 6 7 ··· 0 76 x2 7 6 07 6 21 a22 a23 7 6 76 7 6 6 7 ¼ ð8:15Þ 6 . 7 6 7 .. 7 .. .. .. 76 .. 7 6 6 . 6 . 76 . 7 6 .7 . . · · · . 4 5 4 54 5 0 xN 0 · · · 0 aN N21 aNN where ½x1 ; x2 ; · · ·; xN T represent the displacements of the N components in the path, y denotes the magnitude of excitation at the ﬁrst component, and the coefﬁcients aij are deﬁned as ann21 ¼ 2jvcn21 2 kn21

ann ¼ 2mn v2 þ jvðcn21 þ cn Þ þ ðkn21 þ kn Þ

annþ1 ¼ 2jvcn 2 kn

ð8:16Þ

In the above equation, mn denotes the mass of the nth component, kn and cn represent the stiffness and damping coefﬁcients between the nth and (n þ 1)th components, respectively, and v denotes frequency. In SBCN, the average vibration representing the overall vibration transferred from the component to the accelerometer is characterized by the RMS value of vibration across all frequencies. RMS values of vibration are readily obtained from Equation 8.15 by numerical integration of the square of displacements across all frequencies. In these calculations, to avoid unnecessary numerical problems at the natural frequencies of the components with negligible damping, the integration is carried out excluding the natural frequencies. For the purpose of assigning structural inﬂuences, the RMS values are scaled so that the component directly adjacent to the accelerometer has the highest inﬂuence. Different functions can be used for deﬁning the inﬂuences. For example, Ii can be deﬁned as Ii ¼

logðri Þ logðrN Þ

ð8:17Þ

where ri represents the RMS value of vibration with the excitation source at the ith component, and rN denotes the RMS value when the excitation is at the Nth component. In both cases, the accelerometer is considered at the Nth component. The inﬂuence for the other components is obtained in a similar fashion, by moving the excitation source to them in the travel path. The RMS values, ri, are only approximate estimates due to the simpliﬁcations made for their computation. Such approximate RMS values would, in turn, result in approximate inﬂuences. To yi k0

k1 m2

m1 c0 FIGURE 8.8

k2

c1

Sensor k3

kN-1

mi c2

mN c3

cN-1

Illustration of the lumped-mass model of a vibration travel path consisting of N masses.

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characterize the approximate nature of inﬂuences, they can be deﬁned as fuzzy variables (Zadeh, 1975) by mapping the inﬂuences into the range associated with fuzzy variables, such as nil: (0, 0.1), low: (0.1, 0.4), medium: (0.4, 0.6), high: (0.6, 0.9) and deﬁnite: (0.9, 1). Another body of knowledge commonly used by diagnosticians is that of the type of fault represented by a feature. To incorporate this knowledge, fuzzy featural inﬂuences can be deﬁned according to the relation between the frequency content of each feature and the rotational frequencies of various components (McFadden and Smith, 1986; Stewart Hughes Ltd., 1986). For example, a feature such as envelope band energy (BE), which represents the energy at the bearings rotational frequencies and harmonics, is assigned a featural inﬂuence of “high” in relation to bearing faults. The SBCN is designed to provide fault possibility values for gearbox components without any prior training. However, its design does not preclude the possibility of training when confronted with misclassiﬁcations, which are in the form of undetected faults, false alarms, and misdiagnoses. Among these, undetected faults are safety hazards that should be avoided at all costs, and false alarms and misdiagnoses, although not as crucial as undetected faults, should be minimized so as to improve the reliability of the diagnostic system. One of the features of the SBCN is its ability to beneﬁt from connectionist learning (Hertz et al., 1991) to improve diagnostic performance after each misdiagnosis. For this purpose, an error-minimizing adaptation algorithm can be considered for adapting the fuzzy inﬂuence weights of SBCN so as to avoid reoccurrence of misdiagnosis. This algorithm reduces the error between the outputs of the SBCN ck(t) and the binary target Tk(t) obtained after inspection. The binary target takes the value of zero for all the normal components and one for the faulty components. Sequential update rules for adapting the fuzzy inﬂuences in SBCN have the form ( lik þ h3 ðTk ðtÞ 2 ck ðtÞÞð1 2 fi ðtÞÞfi ðtÞ if 0 , lik , 1 lik ¼ ð8:18Þ lik otherwise ( uik ¼

uik þ h3 ðTk ðtÞ 2 ck ðtÞfi ðtÞÞ2

if 0 , uik , 1

uik

otherwise

ð8:19Þ

where h3 represents the learning rate. In this method, in order to allow uniform interpretation of the trained fuzzy inﬂuences with respect to their original values, adaptation is stopped when the weight values reach the bound of zero or one.

8.4

Sensor Location Selection

Suitable mounting locations for accelerometers on the housing are generally those that provide both close proximity to gearbox components and ease of mounting (also see Chapter 1). However, considering that only a limited number of accelerometers can be used, the candidate locations need to be selected so as to provide a comprehensive coverage of the components. The method presented here relies on indices to quantify the suitability of various mounting locations. It uses a coverage index to deﬁne the reach of each accelerometer, and an overlap index to represent the overlap of the locations in coverage of the gearbox components. The monitoring effectiveness of various combinations of accelerometer locations can then be estimated as a function of the coverage and overlap indices of the accelerometer locations (Wang et al., 1999).

8.4.1

Coverage Index

The coverage index denotes the reach of each accelerometer in monitoring various components of the gearbox. It can be computed based on an inﬂuence matrix such as that deﬁned in the SBCN. It can be deﬁned, for example, as the sum of the corresponding row of the inﬂuence matrix, as Cj ¼

N uij þ lij 1 X N i¼1 2

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ð8:20Þ

8-12

Vibration Monitoring, Testing and Instrumentation

where uij and lij represent the upper and lower limits, respectively, of the fuzzy inﬂuence coefﬁcients between the gearbox component, i, and accelerometer, j, and N denotes the total number of gearbox components. According to this index, an accelerometer location with a higher coverage index should be considered a better candidate.

8.4.2

Overlap Index

The coverage index can be used to rank accelerometers individually. However, for a complex gearbox where more than one accelerometer is needed for monitoring, the sum of the coverage indices associated with all of the accelerometers in the suite will not provide a correct representation of their overall effectiveness because it will ignore the overlap among them. Although the overlap between accelerometer pairs may be deﬁned according to the inﬂuence coefﬁcients, that deﬁnition will only consider the distance of the accelerometers from the components, and will ignore factors such as accelerometer orientation and gearbox size. In order to consider these other factors, the overlap index may be deﬁned independent of the inﬂuence coefﬁcients, based on an overlap matrix that represents the level of cocoverage between all of the accelerometer pairs. The individual components Ojk in this matrix can be deﬁned as Ojk ¼

Ejk ð1 þ Sjk Þ eaDjk

ð8:21Þ

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of orientation between the two accelerometers, a denotes a constant to account for the size of the gearbox, Djk accounts for the physical distance between the two accelerometers, and Sjk represents a symmetry factor between the two accelerometers. The formulation of Equation 8.21 is based on the following understanding. The overlap in coverage of two accelerometers depends mainly on their orientation and location, that is, the more identical their orientation is and the closer they are mounted to each other, the higher is their expected level of overlap. In the above equation, the orientation factor, Ejk [ ½0; 1 ; is set to one when the accelerometers, j and k, have identical orientation. The distance factor, Djk [ ½0; 1 ; is deﬁned as the normalized shortest geometrical distance between the two accelerometers along the casing, where the term eaDjk approximates the attenuation of vibration due to distance (Lindsay, 1960) and the constant a, with values ranging between zero and two, accounts for the effect of gearbox size on vibration attenuation. Although the main factors in the coverage overlap of two accelerometers are orientation and distance, the accelerometers that are symmetrically positioned are believed to have a larger overlap with each other. In order to account for this factor, a symmetry factor, Sjk [ ½0; 1 ; is also included in Equation 8.21, which is set to one when two accelerometers are perfectly symmetrical with respect to the housing. Based on Equation 8.21, the values of Ojk would ordinarily fall in the range ½0; 1 ; where the value of one indicates that the two accelerometers, j and k, have 100% overlap with each other and that one of them can be removed. Similarly, when Ojk is zero then the two accelerometers are assumed to be covering completely different components. It should be noted, of course, that some overlap between accelerometers is considered useful and is often factored in when selecting accelerometer locations. The overlap coefﬁcients only represent the level of overlap between pairs of accelerometers and not the speciﬁc components covered jointly by each accelerometer pair. As such, deﬁning the level of overlap between several accelerometers is not as straightforward. At one extreme, one can assume that the coverage overlap of each accelerometer pair in the suite does not coincide with the coverage overlap of the other accelerometer pairs, so the total overlap, Os, for the accelerometer suite can be computed as 0 1 m m X X Cj þ Ck Os ¼ min@ Ojk ; Cj A ð8:22Þ 2 j¼1 k¼jþ1 where m denotes the number of accelerometers in the suite, and the min function ensures that the computed overlap will not exceed the total coverage of accelerometer j. At the other extreme, it can

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be assumed that the overlaps of all of the accelerometer pairs coincide, and that the total overlap of the accelerometer suite can be deﬁned as Os ¼

m X j¼1

max Ojk

Cj þ Ck for all k . j 2

ð8:23Þ

The latter formulation only considers the largest overlap between accelerometer j and the other accelerometers in the suite. It is also possible to use the average of the above two extremes as a compromise: 0 1 20 13 m m X X Cj þ Ck Cj þ Ck 1 4@ Os ¼ ; Cj A þ max Ojk for all k . j A5 ð8:24Þ min@ Ojk 2 2 2 j¼1 k¼jþ1 It should be noted that, unlike the coverage index, the overlap index is suite related because the overlap of an accelerometer depends on the other accelerometers in the suite.

8.4.3

Monitoring Effectiveness

As mentioned earlier, the coverage index can be used to evaluate the effectiveness of individual accelerometers in monitoring, but its sum cannot be a sole measure of effectiveness of suites of accelerometers. For instance, a suite consisting of the top three accelerometer locations in terms of coverage may be inferior to another suite of three accelerometer locations with a lower total level of coverage but less overlap among the accelerometers. The best set of accelerometer locations is, therefore, one which provides the highest coverage of the gearbox components and the least overlap among the accelerometers. As such, the monitoring effectiveness (ME) of a suite can be expressed as 0 1 m X MEs ¼ @ Cj A 2 Os ð8:25Þ j¼1

where m represents the number of accelerometers in the suite, Ci denotes the coverage of individual accelerometers in the suite, and Os represents the total overlap among the accelerometers in the suite (Equation 8.24), estimated from Equation 8.22, Equation 8.23, or Equation 8.24. The indices used in sensor location are summarized in Table 8.2. TABLE 8.2

Summary of Indices Used for Sensor Location Selection

Sensor location is performed according to the following indices: Coverage index Cj ¼

N uij þ lij 1 X N i¼1 2

where uij and lij represent the upper and lower limits of the fuzzy inﬂuence coefﬁcients between the gearbox component, I, and accelerometer, j, and N denotes the total number of gearbox components. Overlap index Ojk ¼

Ejk ð1 þ Sjk Þ eaDjk

where Ojk denotes the overlap between the accelerometers, j and k, Ejk represents the similarity of orientation between the two accelerometers, a denotes a constant to account for the size of the gearbox, Djk accounts for the physical distance between the two accelerometers, and Sjk represents a symmetry factor between the two accelerometers. According to the above indices, the monitoring effectiveness of an accelerometer suite is deﬁned as 0 1 m X MEs ¼ @ Cj A 2 Os j¼1

where m represents the number of accelerometers in the suite, Cj denotes the coverage of individual accelerometers in the suite and Os represents the total overlap among the accelerometers in the suite.

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8.5

Vibration Monitoring, Testing and Instrumentation

A Case Study

This case study illustrates the application of the SBCN to an OH-58A main rotor gearbox (Figure 8.1; Jammu et al., 1996). Experimental vibration data for the OH-58A gearbox were collected at the NASA Lewis Research Center as part of a joint NASA/Navy/Army advanced lubricants program (Lewicki et al., 1992). Various component failures in the OH-58A transmission were produced during accelerated fatigue tests. The vibration signals were recorded by eight piezoelectric accelerometers (frequency range of up to 10 kHz) using an FM tape recorder. The signals were recorded once every hour, for about 1 to 2 min per recording (using a bandwidth of 20 kHz). Two magnetic chip detectors were also used to detect the debris caused by component failures. The location and orientation of the accelerometers are shown in Figure 8.2. The OH-58A gearbox was run under a constant load, and was disassembled and inspected periodically or when one of the chip detectors indicated a failure. A total of eleven failures occurred during these tests. They consisted of three cases of planet bearing pitting fatigue, three cases of sun gear pitting fatigue, two cases of top housing cover cracking, and one case each of spiral bevel pinion (SBP) pitting fatigue, mast bearing micropitting, and planet gear pitting fatigue. In order to extract the vibration features, the vibration signals from the gearbox were digitized and processed by a commercially available signal analyzer (Stewart Hughes Ltd., 1986). Overall, 54 vibration features were extracted from each accelerometer for the OH-58A gearbox. Out of these features, 35 features were gear-related features (7 for each of the ﬁve gears). The remaining 19 features were indicators of either general faults (e.g., wear and out-of-balance), or general gear and bearing faults.

8.5.1

Structural and Featural Inﬂuences

The structural inﬂuences for the OH-58A gearbox were obtained through ﬁve primary vibration travel paths: (1) duplex bearing-spiral bevel mesh-triplex bearing, (2) duplex bearing-sun-planet mesh-ring gear, (3) mast roller bearing-main shaft-mast ball bearing, (4) ring gear-planet bearing-mast ball bearing, and (5) duplex bearing-sun planet mesh-mast ball bearing. The ﬁrst travel path was in connection to accelerometers 4, 5, and 6, whereas all the other paths were connected to accelerometers 1, 2, 3, 6, 7, and 8. The RMS values of vibration were then computed using the lumped-mass model of these paths with excitation sources at each of the gearbox components. These RMS values were then used as the basis for deﬁning the fuzzy structural inﬂuences between each component–accelerometer pair (Table 8.3). The structural inﬂuences in Table 8.3 indicate that all of the components in the gearbox are covered by the accelerometers, and that some accelerometers have identical inﬂuences with respect to the components. Although these inﬂuences are not completely accurate due to their neglect of the orientation of accelerometers and various approximations (Jammu et al., 1998), they can still be used for TABLE 8.3

Structural Inﬂuences between the Components of the OH-58A Gearbox and the Eight Accelerometers

Component/Accelerator Triplex bearing Spiral bevel pinion Pinion roller bearing Spiral bevel gear Duplex bearing Gear roller bearing Mast roller bearing Main shaft Mast ball bearing Sun gear Planet bearing Planet gear Ring gear

1

2

3

4

5

6

7

8

— — — — M M M M M H H H H

— — — — M M M M M H H H H

— — — — M M M M M H H H H

H H H H H H — — — L L L L

H H H H H H — — — M M M M

— — — — M M M M M H H H H

— — — — M M M M M H H H H

— — — — M M M M M H H H H

The inﬂuences shown are: “—” nil, “L” low, “M” medium, and “H” high.

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an overall assessment of the effectiveness of various accelerometers and their redundancy. For example, the inﬂuences in Table 8.3 are identical for accelerometers 1, 2, 3, 6, 7, and 8. This would indicate that one or more of these accelerometers can be discarded without any drastic effect on fault diagnostic effectiveness. However, it should be noted that the strategy used in SBCN relies on the averaging effect of accelerometers, and that it will not function without a certain level of overlap in accelerometer coverage. The featural inﬂuences were deﬁned for individual vibration features according to the type of fault they are supposed to represent. For example, bearing-related features such as envelope band and tone energy were assigned high in association with gearbox bearings. Similarly, the signal averaged features for the ﬁve gears in the OH-58A gearbox were assigned high with respect to these gears. Features that are indicators of faults in all rotating elements in the gearbox were assigned an inﬂuence of medium for gears as well as bearings. The structural and featural inﬂuences constitute the basis of the SBCN connection weights. As discussed in Jammu et al. (1998), diagnosis in SBCN is performed hierarchically, ﬁrst isolating the faulty subsystems and then faulty components. As such, only the average of the structural inﬂuences associated with each subsystem is used as the connection weights, vik, of SBCN to yield the fault possibility values, pk(t), associated with each subsystem, according to Jammu et al. (1998): pk ðtÞ ¼

n X i¼1

fi ðtÞvik

ð8:26Þ

In subsystem diagnosis, fi(t) denotes the average abnormality-scaled value of general features. In the second stage of fault diagnosis, at the component level, the combination of structural and featural inﬂuences are used as vik, and fi(t) consist of individual abnormality-scaled features.

8.5.2

Evaluation of Inﬂuences

As explained in Jammu et al. (1998), the structural inﬂuences representing the proximity effect of component faults on accelerometer readings were obtained from the RMS value of the frequency response of the lumped-mass model of the gearbox. In this case, however, the actual RMS values of the vibration were available at several fault instances for the OH-58A gearbox, which could also be used to yield a set of experimentally obtained structural inﬂuences. In order to evaluate the modeled inﬂuences, a comparison between these two sets of structural inﬂuences was conducted. The experimental inﬂuences were obtained by normalizing the experimental RMS values when a faulty component had been detected and using them as the basis for assigning the level of fuzzy inﬂuences (Table 8.4). The results in Table 8.4 indicate mixed agreement between experimental and structural inﬂuences. For example, the structural inﬂuences of sun gear, planet bearing, and planet gear on accelerometers 1, 2, and 3 are close to the experimentally obtained inﬂuences, but the inﬂuence of mast ball bearing on accelerometers 2 and 3 TABLE 8.4 Inﬂuences between Accelerometers and Gearbox Components Obtained from Experimental RMS Values of Vibration for the OH-58A Gearbox Inﬂuences from RMS Values of Vibration Accelerator/Parts

Spiral Bevel Pinion

Mast Ball Bearing

Sun Gear

Planet Bearing

Planet Gear

1 2 3 4 5 6 7 8

— ( –) M ( –) M ( –) L (H) H (H) L (– ) — ( –) — ( –)

— ( –) — (H) — (H) — ( –) — ( –) — ( –) — ( –) — ( –)

H (H) M (H) M (H) H (L) H (M) L (H) H (H) M (H)

M (H) M (H) H (H) — ( –) H (M) M (H) M (H) M (H)

M (H) M (M) M (M) M (–) — (M) L (H) L (H) M (H)

For comparison, the inﬂuences from the lumped-mass model are shown inside parentheses.

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Vibration Monitoring, Testing and Instrumentation TABLE 8.5 Normalized Weight Values of the Supervised Connectionist Network Accelerator

1 2 3 4 5 6 7 8

Subsystem 1

2

3

p (–) p (–) p (–) H (H) M (H) M (M) M (–) p (–)

p (M) H (M) p (M) p ( –) p ( –) p (M) L (M) H (M)

p (H) H (H) L (H) p (L) H (M) H (H) H (H) M (H)

For comparison, the subsystem inﬂuences of the SBCN are included inside parentheses. A “p” indicates a negative weight value.

do not match. In this case, the mismatch between the two sets of inﬂuences may be due to (1) the limitation of the RMS value in reﬂecting the change in vibration as a result of various component faults (e.g., mast ball bearing micropitting), (2) variation in the level of change of the RMS values as a function of the type and size of the fault, and (3) lack of faults in every component of the OH-58A gearbox, which limits the ability to determine the inﬂuences for every gearbox component. Deﬁning inﬂuences based on an approximate model of the gearbox was motivated by the need to avoid supervised training of the SBCN. However, given that experimental data were available for the OH-58A gearbox, an evaluation of the structural inﬂuences could be performed by comparing them with the weights of a connectionist network structurally similar to SBCN, but trained by supervised learning. For this purpose, the OH-58A gearbox was divided into three subsystems (Figure 8.1), and the supervised network, having three output units for the three subsystems and eight input units for the eight accelerometers, was trained using least-mean-square (LMS) learning. The weights of this network were trained until the number of false alarms and misdiagnoses were reduced to zero. The trained weights were then normalized and converted into fuzzy variables for comparison with the structural inﬂuences obtained from the lumped-mass model of the gearbox. Table 8.5 includes the inﬂuences of the two networks, where the modeled inﬂuences (inside parentheses) represent the average of component inﬂuences within each subsystem (Table 8.3). The results in Table 8.3 indicate general agreement between the trained weights and modeled inﬂuences. Some of the trained weights had negative values (indicated by “ p ”), which is inevitable due to the use of LMS learning. However, all of these negative weights were quite small in magnitude, which makes them consistent with their modeled nil counterparts (denoted by “ –”). As in the case of inﬂuences from the RMS values (Table 8.4), some of the mismatches in Table 8.5 are expected to be due to the limited number of faults represented in the experimental data. For example, the considerably different inﬂuence of subsystem 2 on accelerometer 1, or subsystem 3 on accelerometer 1, is attributed to the lack of speciﬁc faults in these subsystems that would lead to a more accurate inﬂuence on accelerometer 1. The comparison between the modeled structural inﬂuences and those obtained from the supervised neural network indicates that the modeled inﬂuences are in good agreement with the trained inﬂuences, and that the lumped-mass modeling used here provides an acceptable set of structural inﬂuences for the SBCN.

8.5.3

Fault Detection Results

The fault detection network (FDN) in the proposed diagnostic system (Figure 8.4) is used ﬁrst to identify the presence of faults in the gearbox. Fault diagnosis is then performed when the presence of a fault is prompted. A total of eight FDNs, one for each of the eight OH-58A accelerometers, were used. The inputs to each FDN were the 19 general features not speciﬁc to any particular gear or bearing. The initial weight

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Fault Diagnosis of Helicopter Gearboxes TABLE 8.6

Fault Detection Results for the OH-58A Gearbox

Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

8-17

Fault Detection Network: Predicted and Actual Failures Test 1

Test 2

Test 3

Test 4

Test 5

— ( –) — ( –) — ( –) — ( –) 1 (1) — (1) 1 (1) 1 (1) — (1p)

— — — — — — — — —

— — 1 1 — — — — — — 1 1 —

— — — — — — — — — 1 1 1 1 1 1

— ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — (1) 1 (1) — (1) — (1p)

(–) (–) (–) (–) (–) (–) (–) (–) (–)

(–) (–) (1) (1p) (–) (–) (–) (–) (1p) (–) (1) (1) (1p)

(–) (–) (–) (–) (–) (–) (–) (–) (–) (1) (1) (1p) (–) (1) (1p)

A “—” indicates normality and a “1” represents the presence of a fault. For reference, the expected faults determined by an expert are included inside the parentheses, with “ p ” indicating actual observation of the fault.

values of the FDNs were set as the values of the ﬁrst set of features for each of the ﬁve tests, and were subsequently adapted using 50 adaptation sweeps for each training batch. The occurrence of a fault was prompted when any of the FDNs indicated a fault. The detection results for individual test sets obtained from the FDNs are shown in Table 8.6. A “—” in this table indicates normal conditions, whereas a “1” indicates the presence of a fault. The expected detection results are indicated inside parentheses. The results for test 1 indicate that the presence of faults was detected on days 5, 7, and 8, while faults were expected to be present from days 5 to 9. Of course, it should be noted that the gearbox was not inspected on a daily basis, so the actual condition of the gearbox is unknown for each day of the tests. In test 1, which was run for nine days, a fault was actually observed only on day 9 during a routine inspection of the gearbox (indicated by 1p). However, based on an inspection of the vibration features, it was estimated that the fault could have been present as early as day 5. For the other tests, the days when the faults were actually observed are also indicated by 1p. While it is discouraging to note that day 6 of test 1 was classiﬁed as normal though a fault was present on day 5, the results are in agreement with observations by experts who believe that sometimes increased noise levels immediately after the occurrence of faults mask the effect of faults on vibration features. For the other tests, the results indicate that except for an undetected fault in test 3 and a false alarm in test 4, excellent fault detection was obtained. It should also be noted that the fault on day 9 of test 3 was a hairline crack, which was perhaps undetectable through vibration monitoring. The quality of fault detection in the proposed system is particularly important to the overall diagnostic results, since it is only after a fault is detected that SBCN is engaged in diagnosis. In summary, the detection results obtained (Table 8.6) indicate that the occurrences of most of the faults were identiﬁed. This provides assurance that the later stage of diagnostics would not be hampered by the detection phase.

8.5.4

Fault Diagnostic Results

In the proposed system, fault diagnosis is performed by the SBCN only after the presence of a fault is detected. In this system, fault diagnosis is performed in two hierarchies so as to take full advantage of the separation of the structural and featural inﬂuences. In the ﬁrst hierarchy the gearbox is divided into subsystems (Figure 8.1) and the faults in individual subsystems are isolated by the SBCN based on the structural inﬂuences alone. For each subsystem, the weights of the SBCN are set equal to the average of the structural inﬂuences of the components within that subsystem (Table 8.5). The inputs to the SBCN

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for this stage of the diagnosis consist of the averaged values of abnormality-scaled features from each accelerometer, and its outputs are the fault possibility values for each subsystem. In the second hierarchy, the faulty components within each subsystem are isolated. The inputs to the SBCN for this stage of the diagnosis are the abnormality-scaled vibration features, and the weights are the product of featural inﬂuences and structural inﬂuences of the subsystem containing each component. The averaging of fuzzy inﬂuences here was done by taking the average of upper bounds and lower bounds of individual inﬂuences separately, and then deﬁning the fuzzy variable that would match the range. The product of fuzzy inﬂuences was determined by multiplying the upper bounds and lower bounds of fuzzy variables separately, and then deﬁning the fuzzy variable for the resultant range. 8.5.4.1

Faulty Subsystem Isolation

The fault possibility values for the three subsystems of the OH-58A gearbox are shown in Table 8.7. The results in this table represent the hard-limited fault possibility values (threshold of 0.5) and include, for comparison, the actual condition of the gearbox reported from routine inspection inside parentheses. As before, a “ p ” indicates actual observation of the fault during inspection of the gearbox. The results in Table 8.7 indicate that in test 1, faults in subsystems 1 and 3 were correctly identiﬁed on days 5, 7, and 8. In test 3, the faults in subsystem 3 on days 3 and 4 were correctly identiﬁed, along with a possible fault in subsystem 1. The housing crack on day 9 of this test was left unidentiﬁed because it was never prompted during the detection phase. In any case, this particular fault (a housing crack) could not be isolated by the current SBCN due to absence of features that reﬂect this fault. Also for this test, faults in subsystems 2 and 3 were correctly identiﬁed on days 11 and 12. In Test 4, the fault in subsystem 3 was correctly diagnosed on days 10, 11, 12, 14, and 15. Moreover, on day 13 of test 4, even though the gearbox was supposed to be normal, the SBCN indicated faults in subsystem 3. This was due to the replacement of the three-planet assembly with a four-planet assembly, which changed the vibration characteristic of subsystem 3. In test 5, the fault in subsystem 3 was correctly identiﬁed on day 9. There was also a misdiagnosis in subsystem 1. In summary, the diagnostic results from the gearbox subsystems indicate that all of the eight subsystem faults were correctly identiﬁed in the OH-58A gearbox and that four faults were misdiagnosed. Considering that these results were obtained by using structural inﬂuences alone as the connection weights of the SBCN, the results validate the utility of these inﬂuences and of lumped-mass modeling as a means of representing the vibration travel path of gearboxes for their model-based fault diagnosis. TABLE 8.7

Faulty Subsystem Isolation Results for the OH-58A Gearbox

Day

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Faulty Subsystems Isolation for OH-58A Test 1

Test 2

Test 3

Test 4

Test 5

— ( –) — ( –) — ( –) — ( –) 1, 3 (1, 3) — (1, 3) 1, 3 (1, 3) 1, 3 (1, 3) — (1p, 3p)

— ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –)

— — 1, 3 1, 3 — — — — — — 2, 3 2, 3 —

— ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) 3 (3) 3 (3) 3 (3p) 3 ( –) 3 (3) 3 (3p)

— ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — ( –) — (3) 1, 3 (3) — (3) — (3p)

(–) (–) (3) (3p) (–) (–) (–) (–) (3) (–) (2, 3) (2, 3) (2p, 3p)

The three subsystems in the table are the input subsystem (1), the output subsystem (2), and the transmission subsystem (3). For comparison, the actual faults are included inside parentheses with “ p ” indicating the observed faults.

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Fault Diagnosis of Helicopter Gearboxes

8.5.4.2

8-19

Faulty Component Isolation

Fault possibility values associated with the components of the OH-58A gearbox obtained from the SBCN are included in Table 8.8. The results indicate that the diagnostics associated with individual components are not as accurate as those obtained for the subsystems. Brieﬂy, for test 1, the SBP fault in subsystem 1 and the sun gear (SG) fault in subsystem 3 were correctly identiﬁed only on days 5 and 8, respectively, while other components were assigned higher fault possibility values on other days. For test 3, the three bearing faults in subsystems 2 and 3 (BRG2 and BRG3, respectively) were correctly identiﬁed on days 3, 4, and 12, but other components were also given high fault possibility values. In test 4, the bearing fault in subsystem 3 (BRG3) was correctly identiﬁed only on day 10, while the SG fault remained misdiagnosed. In test 5, the SG fault was correctly identiﬁed on day 9, while the planet gear (PG) fault was misdiagnosed. In view of the promising results obtained at the subsystem level, which conﬁrm the validity of the structural inﬂuences, the cause of diagnostic inaccuracies at the component level should be attributed mainly to the deﬁciency of gear and bearing speciﬁc features used in this study. The strong cross-coupling TABLE 8.8

Faulty Component Isolation by SBCN for the OH-58A Gearbox

Days

Faulty Component Isolation for OH-58A SS1

SS2

SBP

SBG

BRG1

BRG2

— 0.90p —p 0.68p 0.65p —

— 0.62 — 0.43 0.74 —

— 0.89 — 0.79 0.18 —

—

—

— 0.43 0.38 — — — —

SS3 SG

PG

RG

BRG3

— — — — — —

— 0.52p —p 0.67p 0.98p —

— 0.73 — 1.00 0.70 —

— 0.12 — 0.23 0.70 —

— 0.86 — 0.72 0.33 —

—

—

—

—

—

—

— 0.77 0.60 — — — —

— 0.80 0.78 — — — —

— — — — — 0.74p —p

— 0.65 0.56 — 0.67 0.67 —

— 0.56 0.47 — 0.79 0.71 —

— 0.71 0.04 — 0.52 0.55 —

— 0.72p 0.79p — —p 1.00p —p

— — — — — — —

— — — — — — —

— — — — — — —

— — — — — — —

— 0.34 0.54 0.59 0.72 0.81p 0.79p

— 0.41 0.53 0.50 0.85 0.90 0.90

— 0.75 0.79 0.91 0.83 0.88 0.93

— 0.79p —p 0.64p 1.00 0.68 0.48

— 0.58 —

— 0.24 —

— 0.68 —

— — —

— 0.60p —p

— 0.54p —p

— 0.50 —

— 0.58 —

Test 1 1–4 5 6 7 8 9 Test 2 1–9 Test 3 1 and 2 3 4 5–10 11 12 13 Test 4 1–9 10 11 12 13 14 15 Test 5 1–8 9 10 and 11

The components listed are SBP: spiral bevel pinion; SBG: spiral bevel gear; BRG1: bearings in subsystem (SS) 1; BRG2: bearings in SS2; SG: sun gear; PG: planet gear; RG: ring gear; and BRG3: bearings in SS3. As before, “ p ” indicates observation of the faulty component.

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8-20

Vibration Monitoring, Testing and Instrumentation TABLE 8.9

Maximum Values of Correlation Coefﬁcients between Features and Faults Fault-Feature Correlation Coefﬁcients

Features/Faults

Gear Faults

Bearing Faults

Gear features Bearing features

0.49 (1) 0.38 (0)

0.57 (0) 0.44 (1)

The values inside parentheses are the expected ideal values.

between these features is illustrated by the maximum values of the correlation coefﬁcients between the gear and bearing features and gear and bearing faults in Table 8.9. The results indicate reasonable correlation values of 0.49 between the gear features and gear faults, and 0.44 between the bearing features and bearing faults. These numbers, however, are not as impressive when they are compared with the cross-correlation values of 0.57 between gear features and bearing faults, and 0.38 between bearing features and gear faults. The high cross-correlation values in Table 8.9 indicate that the gear and bearing features do not provide the resolution necessary for faulty component isolation. The manifestation of the resolution problem caused by the coupling between gear and bearing features is observed in the similar fault possibility values of 0.9 and 0.89 on day 5 of test 1, in Table 8.8, for the SBP and bearings in subsystem 1 (BRG1). Based on the results in Table 8.8, one can conclude that the unsupervised pattern classiﬁcation scheme incorporated in this research cannot be a substitute for well-deﬁned features, and that a more effective set of features with smaller cross-correlation values are needed for diagnosis at the component level.

8.5.5

Sensor Location Evaluation

The eight candidate accelerometer locations (see Figure 8.2 for their locations and orientations), which were actually used for vibration measurement during accelerated fatigue tests of the gearbox, were analyzed and ranked for their signiﬁcance in monitoring. In order to obtain the overlap coefﬁcients for the OH-58A gearbox, the values of the orientation factors, Ejk, distance factors, Djk, and symmetry factors, Sjk, for the eight accelerometer locations were deﬁned as 2

1

6 6 0:5 6 6 6 0:3 6 6 6 6 1 6 Ejk ¼ 6 6 0:4 6 6 6 6 0:8 6 6 6 1 4

0:5 0:3 1 0:3 0:5 0:4 0:7 0:5

1

0:4 0:8

1

0:8

3

2

0

0

0

7 6 6 0 0:3 0:5 0:4 0:7 0:5 0:7 7 0 0 7 6 7 6 7 6 1 0:4 1 0:3 0:3 0:3 7 0 0 6 0 7 6 7 6 6 1:5 1:5 1:5 0:4 1 0:4 0:5 1 0:8 7 7 6 7; aDjk ¼ 6 7 6 1:5 1:5 1:5 1 0:4 1 0:4 0:4 0:4 7 6 7 6 7 6 6 0:5 0:5 0:5 0:3 0:5 0:4 1 0:8 0:6 7 7 6 7 6 7 6 2:0 2:0 2:0 0:3 1 0:4 0:8 1 0:8 5 4

0:8 0:7 0:3 0:8 0:4 0:6 0:8 2 3 0 0 0 0 0 0 1 0 6 7 60 0 0 0 0 0 1 0 7 6 7 6 7 60 0 0 0 0 0 1 0 7 6 7 6 7 6 7 60 0 0 0 0 0 1 7 0 6 7 Sjk ¼ 6 7 60 0 0 0 0 0 0 7 0 6 7 6 7 6 7 60 0 0 0 0 0 0 7 1 6 7 6 7 6 1 1 1 1 0 0 0 0:5 7 4 5 0 0 0 0 0 1 0:5

1

1:5 1:5 0:5 2:0 0:5

7 1:5 1:5 0:5 2:0 0:5 7 7 7 1:5 1:5 0:5 2:0 0:5 7 7 7 7 0 0:8 1:0 0:5 2:0 7 7 7 0:8 0 1:0 0:5 2:0 7 7 7 7 1:0 1:0 0 1:5 1:0 7 7 7 1:5 1:5 1:5 0 1:5 7 5

0:5 0:5 0:5 2:0 2:0 1:0 1:5

0

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3

0

Fault Diagnosis of Helicopter Gearboxes TABLE 8.10

8-21

Best and Worst Accelerometer Suites within Each Suite Size

Suite Size 1 accelerometer 2 accelerometers 3 accelerometers 4 accelerometers 5 accelerometers 6 accelerometers 7 accelerometers

Best Suites 5 (5, (2, (3, (2, (1, (1,

Worst Suites 3 (4, 7); (1, 6) (1, 6, 8); (1, 2, 6); (1, 2, 8) (1, 2, 7, 8); (1, 4, 7, 8); (1, 6, 7, 8) (1, 4, 6, 7, 8); (1, 2, 6, 7, 8); (1, 2, 4, 7, 8) (1, 2, 4, 6, 7, 8); (1, 3, 4, 6, 7, 8) (1, 2, 3, 4, 6, 7, 8)

8); (2, 5) 5, 7); (4, 5, 8); (2, 4, 5) 4, 5, 8); (2, 4, 5, 8); (2, 4, 5, 7) 3, 4, 5, 8); (2, 3, 5, 7, 8); (2, 3, 5, 6, 7) 2, 3, 4, 5, 7); (2, 3, 4, 5, 6, 8) 2, 3, 4, 5, 6, 8)

Using the above values, the overlap coefﬁcients for the eight locations were determined from Equation 8.21 as 2

1 6 6 0:500 6 6 6 0:300 6 6 6 6 0:223 6 Ojk ¼ 6 6 0:089 6 6 6 6 0:485 6 6 6 0:271 4

0:50

0:3

1

0:3

0:300

1

0:112 0:089 0:089 0:223 0:425 0:182 0:135 0:081

0:2

0:089 0:485 0:271 0:485

3

7 0:112 0:089 0:425 0:135 0:425 7 7 7 0:089 0:223 0:182 0:081 0:182 7 7 7 7 1 0:180 0:184 0:446 0:108 7 7 7 0:180 1 0:147 0:089 0:054 7 7 7 7 0:184 0:147 1 0:179 0:441 7 7 7 0:446 0:089 0:179 1 0:268 7 5

0:485 0:425 0:182 0:108 0:054 0:441 0:268

ð8:27Þ

1

Maximum monitoring effectiveness

Given the above values of coverage and overlap, the ME values were determined from Equation 8.24 for various suites from the eight candidate accelerometer locations of the OH-58A main rotor gearbox. The ME values can be used, for example, to select the best set of accelerometer locations given a number of accelerometers to be used. A sample set of accelerometer locations with the highest and lowest rankings (from the 254 possible suites of the eight OH-58A locations) are shown in Table 8.10. Note that, for some suite sizes, several combinations of accelerometer locations are selected as the “best” and “worst” suites, because in these cases the ME values were too close to render a suite better or worse than the others. The ME values can also be used to assess the beneﬁt of additional accelerometers, leading to larger suite sizes. To demonstrate this utility of monitoring effectiveness, the maximum ME value for each suite size was obtained (Figure 8.9). While additional accelerometers are expected to add to monitoring effectiveness, the added coverage they provide will 1 diminish for larger suite sizes due to the increase in 0.9 overlap between the accelerometers. This is clearly 0.8 reﬂected in Figure 8.9 where the maximum ME 0.7 value of the eight-strong accelerometer suite is the 0.6 same as that of the seven-strong accelerometer 0.5 suite, and the suite with ﬁve accelerometers has a 0.4 0.3 maximum ME value of 0.90, only 10% less than 0.2 that of the suite with eight accelerometers.

8.5.6

Sensor Location Validation

The validity of the ME values was evaluated by comparing the rankings they provided for various accelerometer suites to the rankings obtained

0.1

0

1

2 5 6 3 4 7 Number of accelerometers in the suite

8

FIGURE 8.9 Maximum values of monitoring effectiveness for different accelerometer suite sizes.

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8-22

Vibration Monitoring, Testing and Instrumentation

empirically from a series of diagnostic tests. The suites were ranked into four evenly spaced categories. The suites with the highest monitoring effectiveness were ranked as one, and those with the lowest values were ranked into the fourth category. The empirical rankings were obtained from the SBCN according to the fault possibility values, pi [ ð0; 1Þ; of the individual gearbox subsystems (Jammu et al., 1998). For quantiﬁcation purposes, a performance index, ps, was obtained for each suite of accelerometers to represent the accuracy of the diagnostic results, as X ps ¼ pi ð8:28Þ faults

where the summation was carried out over all the available faults that had occurred during the OH-58A experiments. Under ideal conditions and with a perfect accelerometer suite, the values of pi should be all equal to one (i.e., the value of ps should be equal to the number of faults). However, in practice, the fault possibility values are smaller than one due to imperfect signal conditioning, presence of noise, and so on. The understanding used here is that, everything else being the same, the value of ps is only affected by the quality of the accelerometer suite when the vibration features from different suites are used as inputs to the diagnostic system. The value of ps was computed for various accelerometer suites, and then normalized against the largest value of ps obtained for the same suite size. These normalized ps values were then used as the basis for ranking the suites. As with the monitoring effectiveness values, the suite with the highest normalized ps value was assigned to the ﬁrst category. It should also be noted that, in both cases, the rankings provide only a relative measure of effectiveness among suites of the same size, and that they should not be perceived as global measures. The normalized ps values and the associated rankings for suites of seven accelerometers are included in Table 8.11, along with the ME values and their corresponding rankings. The results indicate remarkably close agreement between the estimated and empirical rankings and that, except for one mismatch, the rankings are identical. Similar analyses were performed for suites of other sizes. A summary of matches and mismatches for all the suites is given in Table 8.12. The results indicate that, out of the 254 possible suites, the estimated rankings of 174 suites match exactly the empirical rankings, and 103 mismatch by only one rank. The results summarized in Table 8.12 indicate that the proposed selection method is effective in assessing the monitoring effectiveness of suites of accelerometers. The experimental data set, although one of the most complete sets available in the industry, is not comprehensive enough to render a complete evaluation of the method. The main limitation is the absence of faults in all of the components of the gearbox. This could result in an overestimation of the signiﬁcance of accelerometers that cover faulty components during the experiments. Similarly, it could lead to devaluation of accelerometers which cover healthy components during the experiments. For example, there was only a single fault in subsystem 2 (mast bearing micropitting), therefore, accelerometer locations that covered this subsystem were given a lower empirical ranking than they actually deserved. TABLE 8.11 Rankings Obtained from the Monitoring Effectiveness Values and from the Diagnostic Results for Suites of Seven Accelerometers Accelerometers Included

2, 3, 4, 5, 6, 7, 1, 3, 4, 5, 6, 7, 1, 2, 4, 5, 6, 7, 1, 2, 3, 5, 6, 7, 1, 2, 3, 4, 6, 7, 1, 2, 3, 4, 5, 7, 1, 2, 3, 4, 5, 6, 1, 2, 3, 4, 5, 6,

8 8 8 8 8 8 8 7

Monitoring Effectiveness

Empirical

MEs

Rank

ps

Rank

0.983 0.791 0.705 0.908 0.200 0.990 1.000 0.940

1 2 2p 1 4 1 1 1

0.888 0.702 0.844 0.926 0.200 1.000 0.938 0.843

1 2 1 1 4 1 1 1

“ p ” indicates a mismatch between the rankings.

© 2007 by Taylor & Francis Group, LLC

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Fault Diagnosis of Helicopter Gearboxes

8-23

TABLE 8.12 Summary Comparison of Rankings Obtained from the Monitoring Effectiveness Values and from the Diagnostic Results for Suites of All sizes Suites of

7 6 5 4 3 2 1 Total

8.6

Match Exactly

7 17 39 48 38 17 8 174

Mismatch by 1

2

3

1 11 17 22 18 11 0 80

0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0

Conclusion

Commercial fault detection systems developed for helicopter gearboxes, generally as part of a Health and Usage Monitoring System (HUMS)3, follow the traditional approach to fault detection and diagnosis. In these systems, the individual vibration signals obtained from each accelerometer are processed ﬁrst to extract various features, and are then evaluated statistically or in comparison with established thresholds to identify possible abnormalities. The abnormal features are then scaled and weighted to provide a weighted average “ﬁgure of merit” that can be used for fault detection. Once a gearbox is identiﬁed as abnormal, diagnosis is performed by evaluating the synchronized time averaged features associated with individual shafts and gears — the signal average of a particular shaft contains the mounted gear meshing characteristics which can be used to determine the gear condition. In HUMS, statistical analysis is usually based on the mean and standard deviation of individual features, and thresholds are established based either on experience or according to base-lines determined from initial data points at the beginning of aircraft operation. It should be noted that the simplistic approach used by major HUMS producers for abnormality identiﬁcation and fault detection and diagnosis is often due to the stringent restrictions imposed by the certiﬁcation process4; otherwise, much development work has been conducted in the recent years that can be included in commercial systems. A truly integrated mechanical diagnostic system should have the following characteristics: (1) be able to automatically correlate the information from various sensor suites including accelerometers, oil debris sensors, acoustic emission (AE) transducers, and so on, and to consolidate the results into a reliable solution; (2) be able to “predict” (prognose) a degrading mechanical condition based on trending of the “features” or condition indices; (3) be compatible with the maintenance policies and operational situations, (4) account for the fatigue life of the monitored components; (5) account for ﬂight conditions (such as transmission torque) that would affect the diagnostic results; (6) be based on open systems architecture (both hardware and software) for future technology insertion; and (7) be lightweight (including cabling and sensors) such that no signiﬁcant extra weight will be added to the aircraft. Oil analysis will undoubtedly play a more critical role in fault detection and diagnosis. One logical step will be development of more reliable in-ﬂight methods of chip detection based on optical technology and image processing (Lukas and Yurko, 1996). The chip detectors of the future are expected to beneﬁt from the micro-electro-mechanical systems (MEMS) technology, making available microdetectors that can be 3 A HUMS generally consists of structural usage monitoring, gearbox, engine, and rotor system fault detection and diagnosis, rotor track and balance, and the associated data processing and maintenance logistics. 4 No HUMS has been certiﬁed by the FAA yet because of the relative newness of the technology and lack of data to verify it. Some HUM systems have been ﬂying in the North Sea area, mostly manufactured by Teledyne-Stewart Hughes and GECBristows, but they need to be developed further before they can be totally reliable.

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8-24

Vibration Monitoring, Testing and Instrumentation

located practically anywhere within the gearbox. Continued emphasis is also expected on postﬂight oil analysis such as ferrography, thermography, ultrasonic analysis, and passive electric current analysis (Saba 1996). As for vibration monitoring, there will be continued need for algorithmic development to produce new features for reliable identiﬁcation of component faults. In this domain, there will be more focus on acoustic monitoring and stress wave analysis. Accelerometers normally have a frequency response range of a few Hz to over 50 kHz. In acoustic monitoring, a much higher bandwidth is considered — microphones are currently available that have a ﬂat response to over 100 kHz. This provides the possibility of measuring a cleaner signal in higher frequency ranges, which may prove useful in detecting certain bearing faults. Another advantage of microphones is that they do not need to be mounted on the surface of the housing, therefore, unlike accelerometers, they are not affected by mounting resonance. For stress wave monitoring, AE sensors are available that detect the stress waves generated by strain (elastic) energy spontaneously released by materials when they undergo deformation. This type of energy, which has a very broad bandwidth (DC to several MHz) due to the impulse nature of the strain release, propagates away from a crack tip and throughout the structure body. For crack growth monitoring, analysis is usually focused on the high frequency region (. 100 kHz) of the AE signal so that the effect of low frequency noise caused by airframe, gearbox, and/or drive shaft vibration is minimized. Stress wave monitoring through AE signals may, therefore, reveal the presence of cracks, as well as their location and severity (Teller and Kwun, 1994). Another important area of research and development in fault diagnosis is sensor technology. A generic restriction in vibration monitoring is the limited number of mounting locations for accelerometers on the housing. This restriction, however, is expected to be alleviated with the advancement of the MEMS technology which will eventually make available miniaturized sensors that can be mounted practically anywhere on a housing, and perhaps inside it. These sensors are expected to have the added capability of processing the vibration signal and producing the features locally, so that the central processor can be dedicated solely to integration of features for fault detection and diagnosis.

References Astridge, D.G., Helicopter transmission — design for safety and reliability, Proc. Inst. Mech. Engrs., 203, 123 –138, 1989. Badgley, R.H. and Hartman, R.M., Gearbox noise reduction: prediction and measurement of meshfrequency vibrations within an operating helicopter rotor-drive gearbox, J. Engng Ind., May, 567 –577, 1974. Barkov, A. and Barkova, N., Condition assessment and life prediction of rolling element bearings — parts 1 and 2, Sound Vibr., June, 14–20, 1995. Boyd, L.S. and Pike, J., Epicyclic gear dynamics, AIAA J., 27, 5, 603 –609, 1989. Carpenter, G.A. and Grossberg, S., ART2: self-organization of stable category recognition codes for analog input patterns, Appl. Optics, 1987. Chin, H., Danai, K., and Lewicki, D.G., Pattern classiﬁer for fault diagnosis of helicopter gearboxes, IFAC J. Control Eng. Practice, 1, 5, 771–778, 1993. Chin, H., Danai, K., and Lewicki, D.G., Fault detection of helicopter gearboxes using the multivalued inﬂuence matrix method, ASME J. Mech. Des., 117, 2, 248– 253, 1995. Choy, F.K. and Qian, W. 1993. Global Dynamic Modeling of a Transmission System. NASA Contractor Report 191117, ARL-CR-11. Lewis Research Center, Cleveland, OH. Cornell, R.W. and Westervelt, W.W., Dynamic tooth loads and stressing for high contact ratio spur gears, J. Mech. Des., 100, 69 –76, 1978. Dyer, D. and Stewart, R.M., Detection of rolling element bearing damage by statistical vibration analysis, ASME J. Mech. Des., 100, 229–235, 1978. Harris, T.A. 1966. Rolling Bearing Analysis, Wiley, New York.

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Fault Diagnosis of Helicopter Gearboxes

8-25

Hertz, J., Krogh, A., Palmer, R.G., and Palmer, R.G. 1991. Introduction to the Theory of Neural Computation, Addison-Wesley, Redwood City, CA. James, M.L., Smith, G.M., Wolford, J.C., and Whaley, P.W. 1994. Vibration of mechanical and structural systems, Harper Collins College Publishers, New York. Jammu, V.B., Danai, K., and Lewicki, D.G., Unsupervised pattern classiﬁer for abnormality scaling of vibration features for helicopter gearbox diagnosis, Machine Vibr., 5, 3, 154– 162, 1996. Jammu, V.B., Danai, K., and Lewicki, D.G., Structure-based connectionist network for fault diagnosis of helicopter gearboxes, ASME J. Mech. Des., 120, 100– 112, 1998. Kasuba, R. and Evans, J.W., An extended model for determining dynamic loads in spur gearing, J. Mech. Des., 103, 398 –409, 1981. Kazlas, P.T., Monsen, P.T., and LeBlanc, M.J. 1993. Neural Network-Based Helicopter Gearbox Health Monitoring System, presented at IEEE-SP Workshop on Neural Networks for Signal Processing, Linthicum, MD, September. Kohonen, T. 1989. Self-Organization and Associative Memory, Springer, Berlin. Lewicki, D.G., Decker, H.J., and Shimski, J.T. 1992. Full-Scale Transmission Testing to Evaluate Advanced Lubricants. Technical Report, NASA TM-105668, AVSCOM TR-91-C-035. Liddle, I. and Reilly, S., Automatic analysis of rotating machinery using an expert system, Sound Vibr., February, 7–11, 1993. Lin, H.-H., Huston, R.L., and Coy, J.J., On dynamic loads in parallel shaft trans-mission: Part I — modelling and analysis, J. Mech. Transm. Automation Des., 110, 221 –229, 1988. Lindsay, R.B. 1960. Mechanical Radiation, McGraw-Hill, New York. Lukas, M. and Yurko, R.J. 1996. Current technology in oil analysis spectrometers and what we may expect in the future, pp. 161 –171. Proceedings of Integrated Monitoring Diagnostics and Failure Prevention, Mobile, AL. Lurton, E.H. 1994. Navy oil analysis program overview, pp. 4 –6. In Proc. JOAP International Condition Monitoring Conference. Lyon, R.H., Structural diagnostics using vibration transfer functions, Sound Vibr., January, 28–31, 1995. Mark, W.D., Use of the generalized transmission error in equations of motion of gear systems, J. Mech. Transm. Automation Des., 109, 283 –291, 1987. McFadden, P.D. and Smith, J.D., Effect of transmission path on measured gear vibration, J. Vibr. Acoust. Stress Reliabil. Des., 108, 377– 378, 1986. Mertaugh, L.J. 1986. Evaluation of vibration analysis techniques for the detection of gear and bearing faults in helicopter gearboxes, pp. 28– 30. In Proc. of the Mechanical Failure Prevention Group 41th Meeting, October, 1986. Mitchell, L.D. and Davis, J.W., Proposed solution methodology for the dynamically coupled, nonlinear geared rotor mechanics equations, J. Vibr. Acoust. Stress Reliabil. Des., 107, 112 –116, 1985. Naval Command, Control, and Ocean Surveillance Center. 1995. CH-53E Aft Main Transmission Test Stand Data Set, San Diego, CA. Randall, R.B., A new method of modeling gear faults, ASME J. Mech. Des., 104, 259–267, 1982. Rebbechi, B., Oswald, F.B., and Townsend, D.P. 1991. Dynamic measurements of gear tooth friction and load, NASA Tech. Memorandum 103281, TR-90-C-0023. Lewis Research Center, Cleveland, OH. Saba, C.S. 1996. Alternate techniques for wear metal analysis, pp. 151-160. In Proceedings of Integrated Monitoring Diagnostics and Failure Prevention, Mobile, AL. Smith, J.D. 1983. Gears and Their Vibrations, Marcel Dekker/The Macmillan Press Ltd, Netowork, New York. Solorzano, M.R., Ishii, D.K., Nickolaisen, N.R., and Huang, W.Y. 1991. Detection and Classiﬁcation of Faults from Helicopter Vibration Data Using Recently Developed Signal Processing and Neural Network Techniques, Code 535. Advanced Technology Development Branch, Naval Ocean Systems Center, San Diego, CA. Stewart Hughes Ltd. 1986. MSDA User’s Guide, Technical Report, Southamton, U.K.

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Teller, C.M. and Kwun, H. 1994. Magnetostrictive sensors for structural health monitoring systems, pp. 297– 305. In Proceedings of Mechanical Failure Prevention Group 48th Meeting, April. Thornton, M.G. 1994. Filter debris analysis, a viable alternative to existing spectrometric oil analysis techniques, pp. 108 –119. In Proceedings of JOAP International Condition Monitoring Conference, 1994. Walford, T.L.H. and Stone, B.J. 1983. The sources of damping in rolling element bearings under oscillating conditions, pp. 225 –232. In Proceedings of the Institute of Mechanical Engineers, Vol. 197c, December. Wang, K., Yang, D., Danai, K., and Lewicki, D.G., Model-based selection of accelerometer locations for helicopter gearbox monitoring, J. Am. Helicopter Soc., 44, 4, 269–275, 1999. While, M.F., Rolling element bearing vibration transfer characteristics: effect of stiffness, J. Appl. Mech., 46, 677 –684, 1979. Zadeh, L.A., The concept of a linguistic variable and its application to approximate reasoning-1, Inf. Sci., 8, 199 –249, 1975. Zakrajsek, J.J., Hanschuh, R.F., Lewicki, D.G., and Decker, H.J. 1995. Detecting gear tooth fracture in a high contact ratio face gear mesh, pp. 91 –102. In Proceedings of 49th Meeting of the Society of Machinery Failure Prevention Technology, April.

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9

Vibration Suppression and Monitoring in Precision Motion Systems 9.1 9.2

K.K. Tan, T.H. Lee, K.Z. Tang, and S. Huang National University of Singapore

S.Y. Lim, W. Lin, and Y.P. Leow Singapore Institute of Manufacturing Technology

9.3 9.4 9.5 9.6

Introduction ............................................................................ 9-1 Mechanical Design to Minimize Vibration .......................... 9-2

Stability and Static Determinacy of Machine Structures † Two-Dimensional Structures † Three-Dimensional Structures

Adaptive Notch Filter .............................................................9-10

Fast Fourier Transform

†

Simulation and Experiments

Real-Time Vibration Analyzer ...............................................9-17

Learning Mode † Monitoring Mode Mode † Experiments

†

Diagnostic

Practical Insights and Case Study ..........................................9-29 Conclusions .............................................................................9-35

Summary Much research and development effort is going on to develop precision motion systems that are used in most production equipment. Market demands for better products (products with much higher performance, higher reliability, longer life, lower cost, and increasing miniaturization) are some of the main driving forces behind these efforts. There are several challenges ahead in order to meet these stringent requirements for precision motion systems. One of the main challenges is to suppress the mechanical vibrations in these motion systems. This chapter provides several possible approaches to this objective. The ﬁrst approach will focus on a proper mechanical design, based on the determinacy of machine structure, to reduce the mechanical vibration in the motion systems to a minimum. In addition to a good design, a monitoring and suppression mechanism is necessary to cope with additional and usually unpredictable sources of vibration seeping in during the course of operations. An approach, utilizing an adaptive notch ﬁlter in the control system, is presented in the chapter, to continuously identify resonant frequencies present and suppress signal transmission into the system at these frequencies. Finally, the development of a low-cost real-time vibration analyzer for precision motion systems is presented. A case study is provided at the end of the chapter to illustrate the effectiveness of a remote vibration monitoring and control system for precision motion systems.

9.1

Introduction

Industrialization, propelled by advances in science and technology in the last millennium, has led to the development of mass-production equipment to satisfy man’s relentless pursuit for products such as 9-1 © 2007 by Taylor & Francis Group, LLC

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9-2

Vibration Monitoring, Testing, and Instrumentation

automobiles, air-conditioners, audio and video products, and so on. Today, apart from the consumer’s demand for more functionality in new product lines, a miniaturization phenomenon is also clearly evident where the physical size of the product can be smaller than its predecessor, even when it has expanded performance. In addition, the ever shortening time-to-market of products and productivity cost factors in today’s highly competitive world together pose tough and challenging requirements on the manufacturing and automation systems that produce them. Thus, while high accuracy and precision in production systems are necessary to produce highly delicate parts and products, a high production speed must still be maintained even though speed and precision adversely affect each other. It is not difﬁcult to appreciate this dilemma since a high production speed can lead to excessive mechanical vibration (Fertis, 1995; Uicker et al., 2003), which means inevitably a loss in precision (Tan et al., 2001). This chapter will seek to address the vibration issues in precision motion systems. Different and numerous sources of vibration can be present in an industrial operating environment (Adams, 2000; Kelly, 2000). Vibration may be generated by rotating or reciprocating machineries such as: engines, motors, and compressors; impact processes such as drilling and PCB printing; the ﬂow of ﬂuid; and many others (Raichel, 2000). The vibration level can reach an excessive level when abnormal operating conditions occur due to unbalanced inertia, bearing failures in rotating systems such as turbines, motors, generators, pumps, drives, and turbofans, component failure and operation outside prescribed load ratings, and poor kinematical design (resulting in a nonrigid and nonisolating support structure). Vibration, whether naturally occurring or induced under abnormal conditions, is undesirable as it usually leads to dynamic stresses that, in turn, causes fatigue and deterioration of the machinery. Other adverse consequences include unnecessary energy losses, deterioration in performance, and an unduly high level of noise produced. These effects can be even more severe for high-precision motion systems with stringent requirements on precision and accuracy, since vibration will lead directly to poor repeatability properties, impeding any effort for systematic error compensation. Thus, it is even more essential that the vibration level in such systems be suppressed as far as possible with an efﬁcient mechanical design and that functionality be included in the control to monitor and possibly adaptively reduce excessive vibration when it occurs. This chapter provides several possible approaches to this objective. The ﬁrst approach will focus on a proper mechanical design, based on the determinacy of the machine structure, to reduce the mechanical vibration to a minimum. While the system design approach is certainly a ﬁrst and key step to minimizing vibration in mechanical systems, a parallel monitoring and suppression mechanism is necessary to cope with additional and usually unpredictable sources of vibration seeping in during the course of operations. An approach, utilizing an adaptive notch ﬁlter (narrow bandstop ﬁlter) in the control system, will be presented as a means of continuously identifying resonant frequencies present and suppress signal transmission into the system at these frequencies. Finally, the development of a low-cost real-time vibration analyzer will be presented. This analyzer can be implemented independently of the control system, and as such can be applied to existing equipment without much retroﬁtting being necessary. A vibration signature is derived from the vibration signal acquired using an accelerometer that is attached to the machine under normal operating conditions. A pattern recognition template is used to compare real-time vibration signals against the normal-condition signature and an alarm can be activated when the difference deviates beyond an acceptable threshold. Actions of rectiﬁcation can then be invoked before damage is done to the machine. A case study will be provided at the end of the chapter to illustrate the effectiveness of a remote vibration monitoring and control system for precision motion systems.

9.2

Mechanical Design to Minimize Vibration

In the development of high-speed and high-precision motion systems, the notion of determinism is a key consideration (Evan, 1989), which implies that a physical system complies with the law of cause and effect, and this behavior allows the physical system to be modeled mathematically.

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Vibration Suppression and Monitoring in Precision Motion Systems

9-3

The governing equations describing the model can then be used to predict the behavior of the system and thus allow for the compensation for possible errors to meet the demand of a tight error budget. A mechatronic approach, in which the structural design and the control design are to be seamlessly integrated (Rankers, 1997), is one of the possible approaches for machine design. This approach has been adopted efﬁciently by many scientists and engineers, and the beneﬁts are clearly evident in the end products, such as the wafer scanner and stepper. In this section, the key issues to address in a sound mechanical design to keep mechanical vibration to a minimum will be highlighted. The issue of mechanical design represents a very large area in precision motion systems. In this section, only key points will be highlighted in general, to enable designers to design “rigid” structures during the initial phase, even before the physical modeling stage. Design, being an iterative process, always requires the designer to revisit the drawing board frequently, until an optimum design is achieved. The section will give qualitative ideas with abundant ﬁgures to illustrate key ideas, rather than using a purely quantitative approach. The reason for this is that, during the initial phase of a design, intensive quantiﬁcation is normally not necessary for decision making. Iteration and optimization, which are normally mathematically intensive, should be addressed during the next stage of the design process.

9.2.1

Stability and Static Determinacy of Machine Structures

A structure can be a supporting framework that houses all the subassemblies that make up a machine, or it can be a collection of many smaller structures, or even a single component. The reaction forces of the high-speed moving parts will excite the structural dynamics resulting in mechanical vibrations. These vibrations can be attenuated by reducing either the excitation or the response of the structure to that excitation (Beards, 1983). The ﬁrst factor can be overcome by relocating the source within the structure or by isolating it from the structure so that the generated vibration is not transmitted to the structure via the supports. As for the second factor, changing the mass, the stiffness, or the damping can alter the structural response. In order to understand the dynamic responses of the structure, the real structure can be transformed into a physical model, which is usually a simpliﬁed model representative of the real structure. For example, a real machine can be modeled as a number of coupled spring–mass systems. A derived physical model of the real system can be translated to a mathematical model which can be solved via software or by hand, thereby allowing engineers from different disciplines to communicate and reﬁne their portion of the design. Every designer working on the structure for a machine needs to answer a very important question. Is the designed structure rigid and stable? A structure is rigid if its shape cannot be changed without deforming the members in the structure (Fleming, 1997), and a structure is stable if rigid-body translation or rotation cannot occur. A good way to tell whether a structure is stable or not is the degree of indeterminacy. A structure is considered statically determinate if all the support reactions and internal forces in the members can be determined solely by the equations of static equilibrium. Otherwise, the structure is considered statically indeterminate. Statically indeterminate structures arise owing to the presence of extra supports, members, reaction forces, or reaction moments. For a structure to be statically determinate, it must ﬁrst be constructed correctly and then supported correctly.

9.2.2

Two-Dimensional Structures

Most machine structures, in practice, are three-dimensional. However, it is useful to look at a twodimensional problem ﬁrst, before extending the problem to a three-dimensional one. Generally, machine structures are stationary. Therefore, the sum of the forces and moments acting on it must be zero, which is in accordance with Newton’s Second Law. In mathematical form, X

Fx ¼ 0

© 2007 by Taylor & Francis Group, LLC

53191—5/2/2007—12:26—VELU—246502— CRC – pp. 1–35

ð9:1Þ

e[O

\)!\gd gd\g\dQ8 A\dQ8 !d: dbAd!\gd

<

<

h

Se7

h

Se7

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ß

Þ ±â euj

\Qd 2gdAd\gd Jg p_!dA 2Au

] h b |

SeO7

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>®

V®

rÎ sÏ Ós´Ï -Ì>ÌV>Þ iÌiÀ>Ìi

r{ s| Ó´Ï 1ÃÌ>Li

L®

`®

r{ sy Ós´Ï -Ì>ÌV>Þ iÌiÀ>Ìi

r{ sÉ Ó´Ï -Ì>ÌV>Þ `iÌiÀ>Ìi

eu _!dA 2A7 r!s !d: r)s !\2!__ :AAb\d!A 2A r2s d!)_A 2A r:s !\2!__ \d:AAb\d!A 2Au

© 2007 by Taylor & Francis Group, LLC

Vibration Suppression and Monitoring in Precision Motion Systems

extra member

new members new joint

(a)

FIGURE 9.3

9-5

(b)

ground

(a) Unstable structure; (b) extra member.

dimension of the ﬁfth bar is not exact. Assembly is probably possible using brute force, but internal stresses will be built into the structure even without any external loading. When a structure is statically determinate, it will be stress-free when it is not loaded externally other than by its own weight. In the event of thermal expansion of its member, owing to an increase in temperature, statically determinate structures allow expansion of their members, without inducing any stress resulting from an overconstrained condition due to the redundant members. The triangle is the basic shape for a plane structure as shown in Figure 9.2a. Statically determinate plane structure can be expanded from this basic structure, simply by linking two new members to two different existing joints for every new joint added, as shown in Figure 9.2b. However, the axis of the two new members must not form a line; in other words, the three joints must not be on the same line, as shown in Figure 9.3a. It is also noteworthy that the ground constitutes one member as well, and all joints are pin-joints, as shown in Figure 9.3b. The second part of structure design lies in its supports. In this aspect, the whole structure can be treated as a rigid body. A plane structure has three DoF; that is, the plane structure is capable of motion in the x and y directions, and rotation about the z-axis. Therefore, three members are needed providing three reactive forces to exactly constrain the plane structure in the plane.

Stable

(a)

Stable

(c) FIGURE 9.4

(b)

(d)

Stable

Unstable

(a), (b), and (c) Stable and exactly constraint supports; (d) unstable support.

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9-6

Vibration Monitoring, Testing, and Instrumentation

Figure 9.4a –c show some possible support for plane structure, while Figure 9.4d shows an unstable support scenario. It should be highlighted that the condition of having two support members at the same location can be replaced by a single pin-joint, as shown in Figure 9.5. It is apparent that, in both cases, they constitute two reaction forces and do not constrain FIGURE 9.5 Equivalent of a two-member support. the angular motion about the z-axis present in the plane structure. The correct number of members in a structure, as well as the correct number of supports, must be in place for a stable and statically determinate structure. At this juncture, the issue of where the loads are to be applied on the structure must be addressed. To this end, it is necessary to examine the members that make up the structure. The stiffness of a bar member is affected by the way the load is applied with respect to its axial axis, its cross-sectional geometry (e.g., the diameter, for a round bar) and the modulus of elasticity, E, of its material. In most cases, the bar is either loaded in tension, compression, or bending, as shown in Table 9.1. It is apparent from examining Table 9.1 that the stiffness of a bar is much better in axial loading than in bending loading. For a value of d ¼ 0:05 m and L ¼ 1:2 m, the ratio of kt =kb is 192. That is, a bar is 192 times stiffer when loaded axially than in bending. Therefore, when designing a rigid and stiff structure, the members must be loaded in tension or compression, never in bending. At times, redesigning the way an external load is applied on a structure can greatly improve the stiffness of the structure. Various conﬁgurations are shown in Table 9.2. As a general rule to observe, the loading point should be located at the joints.

≈

9.2.3

Three-Dimensional Structures

Next, space structures or three-dimensional structures will be considered. These are structures that are of interest in most applications. In a very general sense, space structures can be perceived as a combination of many plane structures, arranged in a manner that all the planes are not coplanar. Therefore, for a space structure to be rigid, every plane structure that makes up the space structure

TABLE 9.1

Comparison of Stiffness for Axial Loading versus Bending Loading Conﬁgurations

(a)

F

Stiffness (N/m) 2

Normalized Stiffness

Tension

kt ¼ 0:25pEd =L

1

Compression

kc ¼ 0:25pEd2 =L

1

Bending

kb ¼ 0:75pEd4 =L3

3ðd=LÞ2

F L

(b)

F

F 0.5L (c)

Loading Condition

L

F

Units: E (N/m2), d (m), L (m), and L .. d:

© 2007 by Taylor & Francis Group, LLC

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Vibration Suppression and Monitoring in Precision Motion Systems TABLE 9.2

Comparison of Stiffness for Various Loading Conﬁgurations Conﬁgurations

(a)

9-7

L

F

Stiffness (N/m)

F

Normalized Stiffness

Compare

kt ¼ 0:25pEd2 =L

1

1

kb ¼ 0:75pEd4 =L3

3ðd=LÞ2

1/192

k^ ¼ 0:5pEd2 sin2 b=L

sin2 b

1/2

kcl ¼ 0:047pEd4 =L3

0:1875ðd=LÞ2

1/3072

k. ¼ 0:25pEd2 =L

1

1

0.5L F

F (c)

L b

b F

(d)

L

L (e)

b

b

F

L

(f)

b ¼ 458; d ¼ 0:05 m, L ¼ 1:2 m.

must be rigid in its own right. This is one reason to have a good understanding of plane structural rigidity. Since machine structures are stationary, the sum of the forces and moments acting on the machine must be zero, which is in accordance with Newton’s Second Law. Mathematically, this implies X F¼0 ð9:5Þ X M¼0

ð9:6Þ

where F and M are three-dimensional force and moment vectors, respectively. The sign conventions as depicted in Figure 9.1 will be used. As before, each structural conﬁguration can be tested to verify if the plane structure satisﬁes the equation: 3j ¼ m þ 6

© 2007 by Taylor & Francis Group, LLC

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ð9:7Þ

9-8

Vibration Monitoring, Testing, and Instrumentation

where j denotes the number of joints and m denotes the number of members. There are three possible cases, as follows: 1. If 3j ¼ m þ 6; then the structure is statically determinate. 2. If 3j . m þ 6; then the structure is unstable. 3. If 3j , m þ 6; then the structure is statically indeterminate. In the plane structure, the triangle is the basic shape, and is rigid and statically determinate. In a space structure, the basic form for being rigid and statically determinant is the tetrahedron, which is depicted in Figure 9.6. Adding a new noncoplanar joint to the three existing joints of a triangular plane structure derives the tetrahedron structure. This new joint is connected to the existing joints with three new members. By following this procedure, a rigid and statically determinate space structure can be derived. Other space structures are shown in Figure 9.7 and Figure 9.8. It is also noteworthy that the members are connected with ball joints. Thus far, the approach to obtain the tetrahedron space structure from the triangle plane structure, the pyramid from the tetrahedron, and the box from the tetrahedron has been illustrated. The next aspect of the design is to combine some of these structures. The structures can be treated as being coupled together as rigid bodies, and a rigid body in space has six DoF, i.e., the structure is capable of translations in the x, y, and z directions, and rotation about the x, y, and z axes. Therefore, six members are needed, providing six reactive forces to exactly constrain the structure in space. Figure 9.9 shows a typical gantry conﬁguration, which is used extensively in many coordinate-measuring machines (CMM). However, one of the members is bearing a bending load, which has been shown earlier to be very detrimental to the stiffness of the structure. There are alternative structure conﬁgurations as shown in Figure 9.10 and Figure 9.11, although some redesign may be needed if such a conﬁguration is to be utilized.

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FIGURE 9.6 structure.

Basic space structure — the tetrahedron

FIGURE 9.7 Pyramid structure derived from a tetrahedron structure.

FIGURE 9.8 structure.

Box structure derived from a tetrahedron

Vibration Suppression and Monitoring in Precision Motion Systems

9-9

If the ground is perceived as another rigid body to which the space structure is to be coupled, then the design of the supports for a space structure is similar to those of coupling two space structures together; that is, six reactive forces are needed to exactly constrain the space structure. Some ways to arrange the six supporting members constraining a space structure are suggested in Figure 9.12. Examples of physical supports offering one, two, or three reactive forces are shown in Figure 9.13. This method of design, known as kinematical design, requires the use of point contact at the interfaces. Unfortunately, this method has some disadvantages: *

*

*

Load carrying limitation. Stiffness may be too low for application. Low damping.

There are, however, ways to overcome the disadvantages, which are via the semikinematical approach. This approach is a modiﬁcation of the kinematical approach, and it aims to overcome the limitations of a pure kinematical design. The direct way is to replace all point contact with a small area, as shown in Figure 9.14; doing so decreases the contact stress, but increases the stiffness and load carrying capacity. However, the area contact should be kept reasonably small. This section has only illustrated some fundamental concepts in designing rigid and statically determinate machine structures. Interested readers may refer to Blanding (1999) for more details on designing a machine using the exact constraints principles.

FIGURE 9.9

Gantry space structure.

FIGURE 9.10 Coupling of a tetrahedron structure to a box structure with six members.

SUMMARY The approach taken to reduce the mechanical vibration in precision motion systems is to focus on a proper mechanical design, based on the determinacy of machine structures in this chapter. The aim of this approach is to design systems with stable and statistically determinate structures. Twodimensional structures and three-dimensional structures are considered.

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9-10

9.3

Vibration Monitoring, Testing, and Instrumentation

Adaptive Notch Filter

The task of eliminating or suppressing undesirable narrowband frequencies can be efﬁciently accomplished using a notch ﬁlter (also known as a narrowband-stop ﬁlter), if the frequencies are known. The ﬁlter highly attenuates a particular frequency component and leaves the rest of the spectrum relatively unaffected. An ideal notch ﬁlter has a unity gain at all frequencies except in the so-called null frequency band, where the gain is zero. A single-notch ﬁlter is effective in removing single-frequency or narrowband interference; a multiple-notch ﬁlter is useful for the removal of multiple narrowbands, which is necessary in applications requiring the cancellation of harmonics. Digital notch ﬁlters are widely used to retrieve sinusoids from noisy signals, eliminate sinusoidal disturbances, and track and enhance FIGURE 9.11 Coupling of a triangular plane strutime-varying narrowband signals with wideband cture to a tetrahedron space structure with six members. noise. They have found extensive use in the areas of radar, signal processing, communications, biomedical engineering, and control and instrumentation systems. To create a null band in the frequency response of a digital ﬁlter at a normalized frequency, b0 ; a pair of complex-conjugate zeros can be introduced on the unit circle at phase angles ^ b0, respectively. The zeros are deﬁned as z1;2 ¼ e^jb0 ¼ cos b0 ^ j sin b0

ð9:8Þ

where the normalized null frequency, b0, is deﬁned as

b0 ¼ 2p

f0 fs

ð9:9Þ

FIGURE 9.12 Examples for applying six constraints to a rigid body: (a) three sets of twin reactive forces; (b) 3– 2– 1 reactive forces.

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Vibration Suppression and Monitoring in Precision Motion Systems

FIGURE 9.13

9-11

Examples of support with (a) one reactive force; (b) two reactive forces; and (c) three reactive forces.

Note that fs is the sampling frequency in Hz (or rad/sec) and f0 is the notch frequency in Hz (or rad/sec). This yields a ﬁnite impulse response (FIR) ﬁlter given by the following transfer function: HðzÞ ¼ 1 2 2 cos b0 z 21 þ z 22

ð9:10Þ

A FIR notch ﬁlter has a relatively large notch bandwidth, which means that the frequency components in the neighborhood of the desired null frequency are also severely attenuated as a consequence. The frequency response can be improved by introducing a pair of complex-conjugate poles. The poles are placed inside the circle with a radius of a at phase angles ^ b0. The poles are deﬁned as p1;2 ¼ a e^jb0 ¼ aðcos b0 ^ j sin b0 Þ ð9:11Þ where a # 1 for ﬁlter stability, and ð1 2 aÞ is the distance between the poles and the zeros. The poles introduce a resonance in the vicinity of the null frequency, thus reducing the bandwidth of the notch. The transfer function of the ﬁlter is given by HðzÞ ¼

ðz 2 z1 Þðz 2 z2 Þ ðz 2 p1 Þðz 2 p2 Þ

ð9:12Þ

Substituting the expression for zi and pi ; and dividing throughout by z 2 ; the resulting ﬁlter has

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FIGURE 9.14 Kinematical vs. semikinematical design: (a) ideal condition — point contact; (b) line contact; (c) area contact.

9-12

Vibration Monitoring, Testing, and Instrumentation

the following transfer function: a0 þ a1 z 21 þ a2 z 22 1 þ b1 z 21 þ b2 z 22

ð9:13Þ

1 2 2 cos b0 z 21 þ z 22 1 2 2a cos b0 z 21 þ a2 z 22

ð9:14Þ

HðzÞ ¼ HðzÞ ¼

Digitally, the ﬁltered signal, y, is thus obtained from the raw signal, u, via the recursive formula in the discrete time domain as follows: yðnÞ ¼ a0 uðnÞ þ a1 uðn 2 1Þ þ a2 uðn 2 2Þ 2 b1 yðn 2 1Þ 2 b2 yðn 2 2Þ

ð9:15Þ

where the coefﬁcients ai and bi are the same as those in Equation 9.13 because z 21 corresponds to the time-shift (delay through sampling period) operator. The bandwidth and the Q-factor of the notch ﬁlter are, respectively pﬃﬃ 2 2ð1 2 a2 Þ BW ¼ ð9:16Þ ½16 2 2að1 þ aÞ2 1=2 Q ¼ w0

½16 2 2að1 þ aÞ2 pﬃﬃ 2 2ð1 2 a2 Þ

1=2

ð9:17Þ

The ﬁlter transfer function, HðzÞ; has its zeros on the unit circle. This implies a zero transmission gain at the normalized null frequency, b0. It is interesting to note that the ﬁlter structure, Equation 9.14, allows independent tuning of the null frequency and the 3-dB attenuation bandwidth by adjusting b0 and a, respectively. The performance of the notch ﬁlter depends on the choice of the constant, a, which controls the bandwidth, BW, according to Equation 9.16. The bandwidth, which is a function of the distance of the poles and zeros ð1 2 aÞ; narrows when a approaches unity. Clearly, when a is close to 1, the corresponding transfer function behaves virtually like an ideal notch ﬁlter. Complete narrowband disturbance suppression requires an exact adjustment of the ﬁlter parameters to align the notches with the resonant frequencies. If the true frequency of the narrowband interference that is to be rejected is stable and known a priori, a notch ﬁlter with ﬁxed null frequency and ﬁxed bandwidth can be used. However, if no information is available a priori, or when the resonant frequencies drift with time, the ﬁxed notch may not coincide exactly with the desired null frequency, particularly if the bandwidth is too narrow (i.e., a < 1). In this case, a tunable or adaptive notch ﬁlter is highly recommended. In Ahlstrom and Tompkins (1985) and Glover (1987), it is proposed to adapt the null bandwidth of the ﬁlter to accommodate the drift in frequency. In Bertran and Montoro (1998), it is suggested that an active compensator be used to suppress the vibration signals. Kwan and Martin (1989) adapt the null frequency, b0, while keeping the pole radii, a, constant. In other words, the parameters ai and bi of Equation 9.13 are adjusted such that the notch will center at the unwanted frequency while retaining the null bandwidth of the notch ﬁlter.

9.3.1

Fast Fourier Transform

The discrete Fourier transform (DFT) is a tool that links the discrete-time domain to the discretefrequency domain. It is a popular off-line approach, widely used to obtain the information about the frequency distribution required for the ﬁlter design. However, the direct computation of the DFT is prohibitively expensive in terms of required computation effort. Fortunately, the fast Fourier transform (FFT) is mathematically equivalent to the DFT, but it is a more efﬁcient alternative for implementation purposes (with a computational speed that is exponentially faster) and can be used when the number of samples, n, is a power of two (which is not a serious constraint). For vibration signals where the concerned frequencies drift with time, the FFT can be continuously applied to the latest n samples to update the signal spectrum. Based on the updated spectrum, the ﬁlter characteristics can be continuously

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Vibration Suppression and Monitoring in Precision Motion Systems

. .. xd, xd, xd

e

Controller

Control signal

Adaptive Notch Filter

9-13

PMLM

x

Adjusting mechanism based on FFT

FIGURE 9.15 Block diagram of the adaptive notch ﬁlter with adjusting mechanism. (Source: Tan, K.K., Tang, K.Z., de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

adjusted for notch alignment. The block diagram of the adaptive notch ﬁlter that has been developed in the present work, with its adjusting mechanism, is shown in Figure 9.15.

9.3.2

Simulation and Experiments

A simulation study is carried out to explore the application of the adaptive notch ﬁlter in suppressing undesirable frequency transmission in the control system for a precision positioning system that uses permanent magnet linear motors (PMLM). In the simulation, a sinusoidal trajectory proﬁle is to be closely followed and an undesirable vibration signal is simulated that drifts from a frequency of 500 Hz in the ﬁrst cycle to a frequency of 1 to 5 Hz in the second cycle of the trajectory. Figure 9.16 shows

FIGURE 9.16 Simulation results without a notch ﬁlter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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FIGURE 9.17 Simulation results using a ﬁxednotch ﬁlter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 9.18 Simulation results using an adaptive notch ﬁlter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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the tracking performance of the precision machine without a notch ﬁlter. Figure 9.17 shows the performance when a ﬁxed notch ﬁlter is used, and Figure 9.18 shows the performance with an adaptive notch ﬁlter. It is clearly evident that a time-invariant narrowband vibration signal can be effectively eliminated using just a ﬁxed notch ﬁlter. However, when the vibration frequencies drift, an adaptive notch ﬁlter is able to detect the drift and FIGURE 9.19 Hardware setup for the experimental align the notch to remove the undesirable study of the notch ﬁlter. (Source: Tan, K.K., Tang, K.Z., frequencies, with only a short transient period. de Silva, C.S., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy The notch ﬁlter is subsequently implemented in Syst., 2001, IOS Press. With permission.) the control system of a linear drive tubular linear motor (LD3810) equipped with a Renishaw optical encoder having an effective resolution of 1 mm. The hardware setup for this experimental study is shown in Figure 9.19 whereas Figure 9.20 shows the linear motor in more detail. The components of the linear motor consist of the thrust rod, the thrust block, the motor cable, and the optical encoder. The thrust rod is made of a thin-walled stainless steel tube housing highenergy permanent magnets. To enable the smooth FIGURE 9.20 Experimental platform (LD 3810) — translation of the thrust block along the length of linear drive tubular motor with optical encoder the thrust rod, the thrust block is made from an attached. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., aluminum housing that contains cylindrical coils Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS arranged in a three-phase star pattern. An Press. With permission.) electromagnetic ﬁeld is produced by energizing these coils. The interactions between the permanent magnetic ﬁeld of the thrust rod and the changing magnetic ﬁeld of the thrust block provide the induced force for the translation of the block. Usually, a sinusoidal or trapezoidal motor commutation is utilized to smoothen the translation of the block. The popular proportional –integral –derivative (PID) control is used in this experimental study. The dSPACE DS1102 (dSPACE User’s Guide, 1996) digital signal processing (DSP) board is used as the data acquisition and control card. It is a single-board system, which is speciﬁcally designed for the development of highspeed multivariable digital controllers and real-time simulations in various ﬁelds. The DS1102 is based on the Texas Instruments TMS320C31 third-generation ﬂoating-point DSP, which builds the main processing unit, providing fast instruction cycle time for numeric intensive algorithms. It contains 128K words memory that is fast enough to allow zero wait-state operation. Besides these, the DS1102 DSP board supports a total memory space of 16M 32-bit words, including program, data, and I/O space. All off-chip memory and I/O can be accessed by the host, even while the host is running, thus allowing easy system setup and monitoring. The TMS320C31 is object-code compatible to the TMS320C30. The DSP is fully supplemented by a set of on-board peripherals, frequently used in digital control systems. Analogto-digital and digital-to-analog converters, a DSP-microcontroller-based digital-input/output (I/O) subsystem, and incremental sensor interfaces make the DS1102 an ideal single-board solution for a broad range of digital control tasks. The DS1102 DSP board is well supported by popular software design and simulation tools, including MATLABw and SIMULINK, which offer a rich set of standard and modular design functions for both classical and modern control algorithms. The SIMULINK model developed for the system (Figure 9.15), with the notch ﬁlter, is shown in Figure 9.21. This model is then downloaded to the DS1102 DSP board for real-time implementation using one of the options available from the pull-down menu.

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FIGURE 9.21 SIMULINK model created for the system with the notch ﬁlter incorporated. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

FIGURE 9.22 Experimental results without a notch ﬁlter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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FIGURE 9.23 Experimental results using a notch ﬁlter: (a) error (mm); (b) desired trajectory (mm); (c) control signal (V). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

It is now in order to present the experimental results utilizing the notch ﬁlter in the system. Figure 9.22 shows the performance of the PMLM when no ﬁlter is used. Figure 9.23 shows the improvement in the control performance when the notch ﬁlter is incorporated into the control system.

SUMMARY An approach utilizing an adaptive notch ﬁlter in the control system is elaborated upon in this section. The notch ﬁlter is able to adaptively identify resonant frequencies present in the motion system and suppress vibration signal transmission into the system at these frequencies. The FFT is used to obtain the frequency distribution of the vibration signals. Experimental and simulation results are provided to illustrate the effectiveness of the adaptive notch ﬁlter.

9.4

Real-Time Vibration Analyzer

The development of an alternative, low-cost approach towards real-time monitoring and analysis of machine vibration (Vierck, 1979; de Silva, 2006) is described in this section. The main idea behind this approach is to construct a vibration signature based on pattern recognition of “acceptable” or “healthy” vibration patterns. The vibration analyzer can operate in three modes: learning, monitoring, or diagnostic. The learning mode, to be initiated ﬁrst, will yield a set of vibration signatures based on which the monitoring and diagnostic modes will operate. In the monitoring mode, with the machine

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Machine

Signal conditioning

Accelerometer

Vibration signals

Vibration monitoring program

Activate alarm or corrective action

DSP module

FIGURE 9.24 Schematic diagram of the real-time vibration analyzer. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

under normal closed-loop control, the analyzer only uses a naturally occurring vibration signal to deduce the condition of the machine. No test excitation is deliberately added to the input signal of the machine. More than one criterion may be used in the evaluation of the condition of the machine, in which case a fusion approach will generate a combined output (machine condition) based on the multiple inputs. In the diagnostic mode, explicit input signals are applied to the machine and the output signal (vibration) is logged for analysis with respect to the associated vibration signature. In what FIGURE 9.25 Hardware setup for the standalone follows, the details of the various components DSP module. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, and functions of the analyzer will be described IOS Press. With permission.) systematically. The block diagram of the real-time vibration analyzer system that has been developed is shown in Figure 9.24. It consists of an accelerometer, which is mounted on the machine to be monitored. The accelerometer measures a multifrequency vibration signal and transmits it to an intelligent DSP module after performing appropriate signal conditioning. This module can be a standalone device (Figure 9.25), or one integrated to a personal computer (PC) host. The vibration analysis algorithm is downloaded to this DSP module. With this algorithm, it can be established as to whether the condition of the machine is within a predetermined acceptable threshold. If the condition is determined to be poor, the DSP module will trigger an alarm to the operator who would enable a corrective action, or automatically activate a corrective action (e.g., change the operating conditions of the machine, modify the parameters of the controller or shutdown the machine). The construction of the real-time vibration analyzer is inexpensive and requires only commercially available, low-cost components. The installation can be hassle free, as the accelerometer is able to gather vibration signals independent of the machine’s own control system. Thus, there is no need to disrupt the operation of the machine. In the prototype reported here, a DSP emulator board TMS320C24x model (TMSS320C24x DSP Controllers Evaluation Module Technical References, 1997), from Texas Instruments, is used as the standalone DSP module (Figure 9.25). This C24x series emulator board is built around the F240 DSP controller, operating at

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20MIPS with an instruction cycle time of 50 ns. It is optimized for digital motor control and conversion applications. Other key components supported on this DSP module are analog-to-digital converters (ADCs), dual access RAM (DARAM), on-chip ﬂash memory, and an RS-232 compatible serial port. This DSP module and the accelerometer unit (with signal conditioning) constitute the only hardware requirements of the real-time vibration analyzer. The vibration-analysis algorithm, which is described in the sequel, will be downloaded to the ﬂash ROM on the DSP module after satisfactory evaluation and tests on the PC. The algorithm is programmed using a Visual Cþþ -based compatible environment, that is, a Code Composer integrated environment (IDE). The C-based source code is then compiled into assembly code, using the built-in compiler available in the IDE.

9.4.1

Learning Mode

In the learning mode, with the machine operating under normal conditions, the vibration signals are acquired by the accelerometer and stored in the DSP module. A suitable vibration signature (Ramirez, 1985) is then extracted from the vibration signals. There are many types of vibration signatures that are adequate for the purpose of machine monitoring. For example, one form of vibration signature may be based on the amplitude of the vibration; another form may use a timeseries analysis of the vibration; yet another form may employ the spectrum of the vibration, which can be efﬁciently obtained using the FFT algorithm. Regardless of the type, these vibration signatures are dependent on the nature of the input signals driving the machine. For example, a square wave input will produce a vibration spectrum that can be quite different from that resulting from an input of a chirp signal (i.e., a repeating sine wave of increasing frequency) or a pure sinusoid. Thus, a particular input signal will produce a unique spectrum based on which a unique vibration signature can be derived. Multiple vibration signatures corresponding to the natural vibrations of the machine (useful for the monitoring mode) or corresponding to different input signals (useful for the diagnostic mode) can thus be captured for subsequent diagnosis and monitoring of the machine.

9.4.2

Monitoring Mode

In the monitoring mode, the vibration signals are sampled periodically from the machine to monitor the condition of the machine. No deliberate or additional input signal is required, so the machine operation is not disrupted. The updated spectra are analyzed against the relevant vibration signatures. The analysis and comparison may be done in terms of the shift in the frequency or the amplitude of the spectrum, or a combination of the two. For example, one evaluation criterion (EV) may be based on the mean-square (ms) value of the error between the current real-time vibration spectrum and the vibration signature: N X

EV1 ¼

q¼1

ðSq 2 Spq Þ2 M

ð9:18Þ

where Sq is the discretized current real-time vibration spectrum, the superscript p represents the vibration signature of the “healthy” machine, subscript q is the index for the data points, N is the total number of frequency points, and M is the total number of data points. Another EV may be formulated based on the difference in the amplitude of the current time series vibration pattern and its corresponding vibration signature: EV2 ¼

maxðTq Þ 2 maxðTqp Þ M

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ð9:19Þ

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where maxðTq Þ represents the highest amplitude of the current time series vibration pattern Tq ; and maxðTqp Þ is the highest amplitude of its corresponding vibration signature. More than one evaluation criterion may be used in the determination of machine condition. In this case, a fusion technique is necessary. The key idea of fusion is to associate the machine with a HEALTH attribute, which is computed from multiple evaluation criteria. These criteria are expected to inﬂuence, to a varying degree, the HEALTH of the machine. The HEALTH attribute is thus an appropriate function, I; of the various criteria (EVis); that is HEALTH ¼ IðEV1 ; EV2 ; …; EVn Þ

ð9:20Þ

where n refers to the number of criteria being evaluated. A fuzzy weighted approach may be used to realize the I function as follows. The HEALTH attribute is treated as a fuzzy variable (i.e., HEALTH [ ½0; 1 ). HEALTH ¼ 0 will represent absolute machine failure, while HEALTH ¼ 1 represents a perfectly normal machine condition. This attribute may be computed from a fuzzy operation on a combination of the evaluation criteria (EVis) obtained via an analysis of the vibration signals against their signatures. The ﬁnal decision on the condition of the machine will be derived from the HEALTH attribute. Zadeh (1973) provides a comprehensive review on fuzzy logic. A Takagi and Sugeno (1985) type of fuzzy inference is used in this chapter. Consider the following p rules governing the computation of an attribute: IF EVi1 IS F1i ^ · · · ^EVin IS Fni ; THEN ui ¼ ai p X

ai ¼ 1

ð9:21Þ ð9:22Þ

i¼1

where ui [ ½0; 1 is a crisp variable output representing the extent to which the ith evaluation rule affects the ﬁnal outcome. Thus, ai represents the weight of the ith rule, Fji represents the fuzzy sets in which the input linguistic variables (EVis) are evaluated, and ^ is a fuzzy operator that combines the antecedents into premises. The value of the attribute is then evaluated as a weighted average of the ui s: p X

HEALTH ¼

w i ui

i¼1 p X

ð9:23Þ wi

i¼1

where the weight wi implies the overall truth value of the premise of rule, i, for the input. It is computed as wi ¼

n Y j¼1

mFji ðEVij Þ

mHIGH(MAX _ ERR)

ð9:24Þ

where mFji ðEVij Þ is the membership function for the fuzzy set, Fji ; related to the input linguistic variable, EVij ; (for the ith rule). For example, in this application, EVj may be the maximum error, MAX_ERR, and Fji may be the fuzzy set HIGH. The membership function is represented as mHIGH ðMAX_ERRÞ: It may have the characteristic shown in Figure 9.26. The decision as to whether any corrective action might be

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MAX _ ERR

FIGURE 9.26 Membership function for the fuzzy input MAX_ERR. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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necessary can then be based on a simple IF–THEN –ELSE formulation as follows: IF HEALTH # g, THEN STRATEGY ¼ TRIGGER ALARM ELSE STRATEGY ¼ CONTINUE TO MONITOR Here, g is interpreted as a threshold value. Suitable values for g may be in the range 0.6 # g # 0.9. Here, STRATEGY is stated to trigger an alarm to the operator who will enable a corrective action, or automatically activate a corrective action (e.g., change the operating conditions of the machine, modify the parameters of the controller, or shutdown the machine). Under this framework, it is relatively easy to include additional criteria for analysis and decision making to the system. The procedure will involve setting up the membership functions for the criterion, formulating the additional fuzzy rules required, and adjusting the scaling parameters (the a terms in Equation 9.21) to reﬂect the relative weight of the new criterion as compared with the existing ones. In this manner, in the monitoring mode, foreboding trends can often be spotted long before the vibration condition reaches a level that is seriously detrimental to the machine.

9.4.3

Diagnostic Mode

In the diagnostic mode, the current vibration signal corresponding to each input signal (with standardized amplitude and frequency) is analyzed against the associated signature obtained earlier in the learning mode, depending on the type of machine (also see Chapter 6). Similar to the monitoring mode, there can be multiple evaluation criteria used in the diagnostic mode, so that the fusion technique described earlier is also applicable. The input signals applied to the machine must be designed carefully so as to yield as much information of the machine condition as possible in the operational regime of interest. Two important considerations are the choices of amplitude and frequency. Machines may have constraints in relation to the amount of travel that is possible. Too large an amplitude for the input signal may be not be viable for the machine owing to the limit of travel, or may even damage the machine. Also, the frequency range of the input should be chosen so that it has most of its energy in the frequency bands that are important for the system. Where input signals cannot be applied to the system in the open loop, the setpoint signal will serve as the input for the closed-loop system since it may not be possible to directly access the system under closedloop control. Careful considerations of the mentioned issues will ensure that signiﬁcant information can be obtained from the machine. The input signals considered here are square wave input (Figure 9.27), chirp input (Figure 9.29), and sine wave input (Figure 9.31), standardized in amplitude to 1 V and in frequency to 5 Hz. The corresponding vibration signatures are shown in Figure 9.28, Figure 9.30, and Figure 9.32, respectively.

9.4.4

Experiments

A shaker table (Figure 9.33) is used as the test platform for the experiments presented here (also see Chapter 1). The shaker table can be used to simulate machine vibrations and evaluate the performance, for example, of active inertial dampers. The shaker table is driven by a high-torque direct-drive motor (which has a maximum torque of 1.11 N m, a maximum design load of 11 kg and generates a maximum force of 175 N). The maximum linear travel of the table is ^ 2 cm. The learning mode is ﬁrst initiated to obtain the vibration signals with the shaker table operating under normal conditions. It is assumed in the experiments that the normal condition corresponding to the input is a square wave signal (with a standardized amplitude of 1 V and frequency of 5 Hz). For the purpose of implementing the diagnostic mode, the vibration signals are also obtained for the input signals of the sinusoidal and chirp type, with standardized amplitudes of 1 V and

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FIGURE 9.27 Square wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

frequencies of 5 Hz (see Figure 9.27 to Figure 9.32 for the inputs and their corresponding vibration signatures). 9.4.4.1

Input Variables — the Evaluation Criteria

Different types of EV can be used as input variables for the determination of the machine condition. For the present vibration-analysis application, the input variables chosen for the computation of the HEALTH attribute are given below. 9.4.4.1.1

Monitoring Mode N X

EV1 ¼ EV2 ¼

q¼1

ðSsq;q 2 Spsq;q Þ2 M

ðmaxðTsq;q Þ 2 maxðTsq;q Þp Þ2 M M X p Þ2 ðTsq;q 2 Tsq;q EV3 ¼

q¼1

M

ð9:25Þ ð9:26Þ

ð9:27Þ

where Ssq;q represents the vibration spectrum for a square wave input (driving the machine) at the qth frequency point and Tsq;q represents the timedomain signal of a square wave input at the qth time instant. N refers to the total number of points in the FFT spectrum and M is the number

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FIGURE 9.28 Vibration signature of the square wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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Time domain signal

1 0.8 0.6 0.4

Voltage (V)

0.2 0 –0.2 –0.4 –0.6 –0.8 –1

0

0.5

1

1.5

2

2.5 3 Time (secs)

3.5

4

4.5

5

FIGURE 9.29 Chirp wave input, with a standardized amplitude of 1 V and initial frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

of time-series data points over an operational cycle. Hence, EV1 refers to the ms deviation between the vibration spectrum and its signature; EV2 refers to the square of the difference between the amplitude of the vibration signal over one operational cycle compared with its signature; EV3 refers to the ms deviation between the vibration signal and its signature (time domain) over one operational cycle. The superscript p represents the signature of the healthy machine. 9.4.4.1.2

Diagnostic Mode N X

EV4 ¼

q¼1

M N X

EV5 ¼

q¼1

q¼1

ð9:28Þ

ðScp;q 2 Spcp;q Þ2 M

N X

EV6 ¼

ðSsq;q 2 Spsq;q Þ2

ð9:29Þ

ðSsn;q 2 Spsn;q Þ2 M

ð9:30Þ

Here, cp denotes a chirp input signal and sn denotes a sine input signal. For the monitoring mode, the input attributes are related only to the square input owing to the assumption that the input signal, under normal operating conditions, is the square wave signal (with a standardized amplitude of 1 V and frequency of 5 Hz).

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FIGURE 9.30 Vibration signature of the chirp wave input, with a standardized amplitude of 1 V and starting frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

9.4.4.2

Evaluation Rules

The three rules for the computation of the HEALTH attribute are given below. 9.4.4.2.1

Monitoring Mode

IF EV1 IS LOW, THEN u ¼ a1 IF EV2 IS SHORT, THEN u ¼ a2 IF EV3 IS LOW, THEN u ¼ a3 The values of the scaling parameters, that is, a terms in Equation 9.21, reﬂect the relative importance of the fuzzy rules in the determination of the HEALTH of the machine. The scaling values of a1 ; a2 ; and a3 are set at 0.7, 0.2, and 0.1, respectively. The respective membership functions are

mi ðEVi Þ ¼ e2nðEVi Þ

b

ð9:31Þ

where n and b are scaling factors for normalization of EVi. Here, they are selected to be n ¼ 10 and b ¼ 0:5: 9.4.4.2.2 Diagnostic Mode The three evaluation rules for the computation of the HEALTH attribute in the diagnostic mode are IF EV4 IS LOW, THEN u ¼ a4 IF EV5 IS LOW, THEN u ¼ a5 IF EV6 IS LOW, THEN u ¼ a6

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FIGURE 9.31 Sine wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

The scaling values of a4 ; a5 ; and a6 are set at 0.4, 0.2, and 0.4, respectively. Similar membership functions as for the monitoring mode are used here. The machine condition attribute HEALTH is then computed as in Equation 9.23. 9.4.4.3

Tests

9.4.4.3.1 Monitoring Mode In the monitoring mode, the normal input signal (i.e., the square wave with standardized amplitude of 1 V and frequency of 5 Hz) is applied to the shaker-table system. At t ¼ 5 sec, a sinusoidal signal (with amplitude 0.4 V and frequency f ¼ 5 Hz) is also applied to the system to simulate a fault arising in the machine. The time-domain signal of the machine (corresponding to the square input) is shown in Figure 9.34. The spectra of the machine before and after t ¼ 5 sec are shown in Figure 9.35. The vibration-analysis algorithm is able to detect the fault in the machine. Before the introduction of the fault, the HEALTH attribute of the shaker table is found to be 0.98. After the introduction of the fault, the HEALTH attribute falls to 0.63, which is below the threshold value, set at 0.7. As a result, the alarm is triggered.

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FIGURE 9.32 Vibration signature of the sine wave input, with a standardized amplitude of 1 V and frequency of 5 Hz. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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FIGURE 9.33 Test platform: the shaker table. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

9.4.4.3.2 Diagnostic Mode In the diagnostic mode, three input signals (i.e., sine, square, and chirp wave with standardized amplitude and frequency) are selected to be applied to the shaker table system in turn. To simulate a fault arising at t ¼ 5 sec, the input gain is increased by a factor of 1.4 times at t ¼ 5 sec. The time-domain

FIGURE 9.34 Time-domain vibration signal corresponding to the square input, with a standardized amplitude of 1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated). (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

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FIGURE 9.35 (a) Vibration signature corresponding to the square input, with a standardized amplitude of 1 V and frequency of 5 Hz; (b) spectrum of the machine corresponding to the square input after fault occurs. (Source: Tan, K.K., Tang, K.Z., de Silva, C.W., Lee, T.H., and Chin, S.J., J. Intell. Fuzzy Syst., 2001, IOS Press. With permission.)

vibration signal of the machine (corresponding to the chirp signal, with a standardized amplitude of 1 V and starting frequency of 5 Hz) is shown in Figure 9.36. The spectra (corresponding to the chirp signal) of the machine before and after t ¼ 5 sec are shown in Figure 9.37. The time-domain vibration signal of the machine (corresponding to the sinusoidal wave input, with a standardized amplitude of 1 V and

FIGURE 9.36 Time-domain vibration signal corresponding to the chirp input, with a standardized amplitude of 1 V and starting frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

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FIGURE 9.37 (a) Vibration signature corresponding to the chirp input, with a standardized amplitude of 1 V and starting frequency of 5 Hz; (b) spectrum of the machine corresponding to the chirp input after a fault occurs.

FIGURE 9.38 Time-domain vibration signal corresponding to the sinusoidal input, with a standardized amplitude of 1 V and frequency of 5 Hz (at t ¼ 5 sec, a fault is simulated).

frequency of 5 Hz) is shown in Figure 9.38. The spectra (corresponding to the sinusoidal input) of the machine before and after t ¼ 5 sec are shown in Figure 9.39. The vibration-analysis algorithm is able to detect the fault in the machine. Before the introduction of the fault, the HEALTH attribute of the shaker table is found to be about 0.97. After the introduction of the fault, the HEALTH attribute falls to 0.58, which is below the threshold value, set at 0.7. The alarm is triggered as a result.

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FIGURE 9.39 (a) Vibration signature corresponding to the sinusoidal input, with a standardized amplitude of 1 V and frequency of 5 Hz; (b) spectrum of the machine corresponding to the sinusoidal input after fault occurs.

SUMMARY In this section, the development of a low-cost approach towards real-time monitoring and analysis of machine vibration is described. A real-time analyzer, based on a fuzzy fusion technique, is used to monitor continuously and compare the actual vibration pattern against a vibration signature. This intelligent knowledge-based real-time analyzer is able to detect excessive vibration conditions much sooner than a resulting fault could be detected by an operator. Subsequently, appropriate actions can be taken. This approach may be implemented independently of the control system and as such can be applied to existing equipment without modiﬁcation of the normal mode of operation. Experimental and simulation results are provided to illustrate the effectiveness of this real-time vibration analyzer.

9.5

Practical Insights and Case Study

There are currently many companies in the industry that deal with machinery vibrations. Broadly, these companies can be classiﬁed into those that produce vibration-related testing products (e.g., FFT analyzers [Steinberg, 2000], vibration data collectors, and end-of-line production test equipment), and those that provide solutions in resolving noise- and vibration-related problems for different industries (e.g., automotive, aerospace, manufacturing, and engineering companies). For instance, if one of the machines in operation on the shop-ﬂoor is experiencing unexplained machinery noise or a high level of product failure, vibration analysis (Farrar et al., 2001) may provide some answers. The case study explores the feasibility of remote vibration monitoring and control of a number of machines or mechanical systems located at different sites. For the sake of generality, “machines” is used as a general term to represent vibrating machines, mechanical systems, and other systems that exhibit vibration behavior. There are strong motivations for such setups. Since the middle of the 1960s, the concept of the distributed system has been widely adopted in the process industries, such

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Vibration Monitoring, Testing, and Instrumentation

as the chemical industry, manufacturing industry, food processing industry, primary industry, and many others. These distributed systems are also characterized by smart devices. As the costs of manufacturing, that is, labor costs, the cost of raw materials, ﬁxed costs, rental costs, and other production costs, are a major concern for the location of plants, many processes are currently widely distributed geometrically. The layout of an entire plant can be rather extensive, spreading across continents in certain cases. The extensive distribution of an entire plant requires close coordination and synchronization of the distributed operations, as well as an efﬁcient remote monitoring and control facility in place. The utilization of the Internet for this monitoring and control purpose perfectly complements the trend of distributed intelligence in the ﬁeld controllers and devices on the shop-ﬂoor. As a result, superior control decisions can be made with these readily available resources and information, that is, historical data, knowledge databases, and so on. Internet working has become essential for plant automation. A case study which implements remote vibration monitoring and control of various machines will be illustrated in this section. This vibration monitoring and control system possesses the structure as shown in Figure 9.40. It is able to monitor continuously the real-time health conditions of multiple machines at different sites, connected via the Internet. The system operates in two modes: the learning mode and the monitoring mode, as mentioned earlier. In the learning mode, vibration signatures, representative of the health of the machines to be monitored, are stored in the supervisory controller. Accelerometers mounted on the machine directly provide measurements of the vibration signals. In the monitoring mode, real-time vibration signals are streamed from the front-end controllers to the supervisory controller. Based on these real-time vibration signals from the various machines, the supervisory controller is able to generate decisions on the health condition of the machines, taking into consideration various criteria based on a fuzzy fusion technique, as mentioned earlier. Alarms will be activated when the health condition of the machines falls below an acceptable threshold level. Subsequently, rectiﬁcation actions, to provide a warning or automatic corrective action (e.g., changing the operating conditions of the machine,

FIGURE 9.40

Remote vibration monitoring and control system.

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9-31

modifying the parameters of the controller, or shutting down the machine), may be invoked before extensive damage is done to the machines. Referring to the architecture of the vibration monitoring and control system as shown in Figure 9.40, a data transfer and communication protocol needs to be adopted for such an application. There are several ways of connecting the supervisory controller and the remote machines. One of the possible ways is to apply DataSocket technology; yet another is to use low-level protocols such as TCP/IP and User Datagram Protocol. As can be seen, this vibration monitoring and control system may be applied to existing setups at the shop-ﬂoor level, without much modiﬁcation necessary. The front-end ﬁeld devices (i.e., actuators, sensors, and other I/O modules) are linked up via the ﬁeld-level data highway. Two popular types of ﬁeld-level data highway are ﬁeldbus and twisted pair. Access security is a main issue of such remote applications. There is a need to prevent unauthorized users from accessing and modifying the system so as to maintain the integrity and proper functioning of the system. This can be achieved by

FIGURE 9.41

Learning mode — snapshot of the control panel at the supervisory controller end.

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imposing some access authentication scheme. One commonly used basic access authentication scheme is the challenge–response authentication mechanism whereby the operator or user must provide some user logon information (e.g., user identiﬁcation and a password) before accessing the supervisory controller terminal. The control panel at the supervisory controller end (Figure 9.40) is as shown in Figure 9.41. At the supervisory controller end, the operator is able to monitor and control the health condition of various machines at different remote sites. As shown here, two machines (i.e., machine A and machine B) are being monitored. The learning mode is ﬁrst initiated to obtain the vibration signatures of the two machines in the normal operational condition. The vibration signature of machine A resembles a chirplike signal input whereas that of machine B resembles a sinusoidal-like signal input. At the end of the learning mode, a message is shown in the message box. Figure 9.42 shows a snapshot of the control panel in the monitoring mode. Here, the two machines are in good health, as the health indices

FIGURE 9.42 Monitoring mode — snapshot of the control panel at the supervisory controller end before any fault is invoked on machine A and machine B.

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(i.e., current values are at 0.9) of the two machines are above the threshold value of 0.6. To simulate a fault arising in machine A, the input signal to machine A is changed to a sinusoidal input. This causes a simultaneous change in the vibration pattern of Machine A. A snapshot of the control panel at the supervisory controller end is as shown in Figure 9.43 after the fault is invoked to machine A. As can be clearly seen, the vibration pattern of machine A has changed. The message box displays that the health index of machine A (current value is at 0.49) has fallen below the threshold value of 0.6. To signal the deterioration of the health condition of machine A, the alarm LED is lit up. The health condition of machine B remains in the satisfactory region. To simulate a fault arising in machine B, the amplitude of the input signal to Machine B is increased slightly, by 20%. Consequently, this will also cause simultaneous change in the vibration pattern of machine B. A snapshot of the control panel at the supervisory controller end, after the fault is invoked to machine B, is as shown in Figure 9.44.

FIGURE 9.43 Monitoring mode — snapshot of the control panel at the supervisory controller end when a fault is invoked on machine A (at t ¼ 3 sec).

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 9.44 Monitoring mode — snapshot of the control panel at the supervisory controller end when a fault is invoked on machine B (at t ¼ 5 sec).

The vibration pattern of machine B has changed. The message box displays the health index of machine B (current value is at 0.53) has fallen below the threshold value of 0.6. In the same way, the deterioration of the health condition of machine B is highlighted by activating the alarm LED.

SUMMARY This section gives the reader practical insights into vibration monitoring and control applications in the industry. A case study is provided to explore the feasibility of remote vibration monitoring and control of a number of machines or mechanical systems located at different sites.

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9.6

9-35

Conclusions

Approaches to address undue vibration arising in mechanical systems have been presented in the chapter. The ﬁrst approach presented is concerned with a proper mechanical design with a view to reduce the sources of vibration. The second approach is the possibility of including functionalities in the control system to continuously monitor and possibly suppress vibration by terminating transmission at the resonant frequencies. The third approach uses a low-cost monitoring device, which can trigger alarm when excessive vibration is detected to enable corrective actions. Precision motion systems, where the effects due to vibration will directly affect performance, will usually adopt a number of such strategies to keep vibration level to a minimum and acceptable level.

References Adams, M.L. Jr. 2000. Rotating Machinery Vibration: From Analysis to Troubleshooting, Marcel Dekker, New York. Ahlstrom, M.L. and Tompkins, W.J., Digital ﬁlters for real-time ECG signal processing using microprocessors, IEEE Trans. Biomed. Eng., 32, 708 –713, 1985. Beards, C.F. 1983. Structural Vibration Analysis: Modelling, Analysis and Damping of Vibrating Structures, Ellis Horwood, Chichester, U.K. Bertran, E. and Montoro, G. 1998. Adaptive suppression of narrow-band vibrations, 5th International Workshop on Advanced Motion Control, Coimbra, Portugal, June 1998, pp. 288 –292. Blanding, D.L. 1999. Exact Constraint: Machine Design Using Kinematic Principles, ASME Press, New York. de Silva, C.W. 2006. Vibration: Fundamentals and Practice, 2nd ed., Taylor & Francis, CRC Press, Boca Raton, FL. dSPACE DS1102 User’s Guide, Document Version 3, dSPACE GmbH, Paderborn, Germany, 1996. Evan, C. 1989. Precision Engineering: An Evolutionary View, Cranﬁeld Press, Bedford, U.K. Farrar, C.R., Doebling, S.W. and Duffey, T.A. 2001. Vibration-based Damage Detection, Structural Dynamics @ 2000: Current Status and Future Directions, Research Studies Press, Baldock, U.K., pp. 145–174. Fertis, D.G. 1995. Mechanical and Structural Vibrations, Wiley, New York. Fleming, J.F. 1997. Analysis of Structural Systems, Prentice Hall, Englewood Cliffs, NJ. Glover, J.R. Jr., Comments on digital ﬁlters for real-time ECG signal processing using microprocessors, IEEE Trans. Biomed. Eng., 34, 1, 962 –963, 1987. Kelly, S.G. 2000. Fundamentals of Mechanical Vibrations, 2nd ed., McGraw-Hill, Boston, MA. Kwan, T. and Martin, K., Adaptive detection and enhancement of multiple sinusoids using a cascade IIR ﬁlter, IEEE Trans. Circuits Syst., 36, 7, 937 –947, 1989. Raichel, D.R. 2000. The Science and Applications of Acoustics, Springer-Verlag, New York. Ramirez, R.W. 1985. The FFT Fundamentals and Concepts, Prentice Hall, Englewood Cliffs, NJ. Rankers, A.M. 1997. Machine Dynamics in Mechatronic Systems — An Engineering Approach, In-house Report, Philips, Amsterdam. Steinberg, D.S. 2000. Vibration Analysis for Electronic Equipment, 3rd ed., Wiley, New York. Takagi, T. and Sugeno, M., Fuzzy identiﬁcation of systems and its applications to modelling and control, IEEE Trans. Syst. Man Cybern., 15, 116–132, 1985. Tan, K.K., Lee, T.H., Dou, H.F., Huang, S. 2001. Precision Motion Control: Design and Implementation, Springer-Verlag, London. TMSS320C24x DSP Controllers Evaluation Module Technical References, Tarrant Dallas Printing, Dallas, TX, 1997. Uicker, J.J., Pennock, G.R. and Shigley, J.E. 2003. Theory of Machines and Mechanisms, 3rd ed., Oxford University Press, New York. Vierck, R.K. 1979. Vibration Analysis, Crowell, New York. Zadeh, L.A., Outline of a new approach to the analysis of complex systems and decision process, IEEE Trans. Syst. Man Cybern., 3, 28 –44, 1973.

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10

Vibration and Shock Problems of Civil Engineering Structures

Priyan Mendis University of Melbourne

Tuan Ngo

University of Melbourne

10.1 10.2

Introduction ....................................................................... 10-2 Earthquake-Induced Vibration of Structures .................. 10-3

10.3

Dynamic Effects of Wind Loading on Structures ........... 10-22

10.4

Vibrations Due to Fluid– Structure Interaction .............. 10-33

Seismicity and Ground Motions † Inﬂuence of Local Site Conditions † Response of Structures to Ground Motions † Dynamic Analysis † Earthquake Response Spectra † Design Philosophy and the Code Approach † Analysis Options for Earthquake Effects † Soil –Structure Interaction † Active and Passive Control Systems † Worked Examples

Introduction † Wind Speed † Design Structures for Wind Loading † Along and Across-Wind Loading † Wind Tunnel Tests † Comfort Criteria: Human Response to Building Motion † Dampers † Comparison with Earthquake Loading Added Mass and Inertial Coupling Vibration of Structure

†

Wave-Induced

10.5

Blast Loading and Blast Effects on Structures ................. 10-34

10.6

Impact loading ................................................................... 10-47

10.7

Floor Vibration ................................................................... 10-51

Explosions and Blast Phenomenon † Explosive Air-Blast Loading † Gas Explosion Loading and Effect of Internal Explosions † Structural Response to Blast Loading † Material Behaviors at High Strain Rate † Failure Modes of Blast-Loaded Structures † Blast Wave–Structure Interaction † Effect of Ground Shocks † Technical Design Manuals for Blast-Resistant Design † Computer Programs for Blast and Shock Effects Structural Impact between Two Bodies — Hard Impact and Soft Impact † Example — Aircraft Impact

Introduction † Types of Vibration † Natural Frequency of Vibration † Vibration Caused by Walking † Design for Rhythmic Excitation † Example — Vibration Analysis of a Reinforced Concrete Floor

Summary This chapter provides a concise guide to vibration theory, sources of dynamic loading and effects on structures, options for dynamic analyses, and methods of vibration control. Section 10.1 gives an introduction to 10-1 © 2007 by Taylor & Francis Group, LLC

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different types of dynamic loads. Section 10.2 covers the basic theory underlying earthquake engineering and seismic design. In this section, seismic codes and standards are reviewed including American, British, and European practices. Active and passive control systems for seismic mitigation are also discussed. This section contains analytical and design examples on seismic analysis and building response to earthquakes. Section 10.3 introduces the nature of wind loading, dynamic effects, and the basic principles of wind design. This section includes formulae, charts, graphs, and tables on both static and dynamic approaches for designing structures to resist wind loads. Types of dampers to reduce vibration in tall buildings under wind loads are also introduced. Section 10.4 gives a brief overview of vibration due to ﬂuid– structure interaction. Section 10.5 extensively covers the effects of explosion on structures. An explanation of the nature of explosions and the mechanism of blast waves in free air is given. This section also introduces different methods to estimate blast loads and structural response. Section 10.6 deals with the impact loading. An analytical example of aircraft impact on a building is given. Section 10.7 looks in detail at the problems of ﬂoor vibration. Charts and tables are given for designing ﬂoor slabs to avoid excessive vibrations. A comprehensive list of references is provided.

10.1

Introduction

The different types of dynamic loading considered in this chapter include: earthquakes, wind, ﬂoor vibrations, blast effects, and impact- and wave-induced vibration. The effects of these loadings on different engineering structures are also discussed. It is standard practice to use equivalent static horizontal forces when designing buildings for earthquake and wind resistance. This is the simplest way of obtaining the dimensions of structural members. Dynamic calculations may follow to check, and perhaps modify, the design. However, vibrations caused by extreme loads such as blast and impact must be assessed by methods of dynamical analysis or by experiment. Some examples of dynamic loading are shown in Figure 10.1. The ﬁrst (a) is a record of ﬂuctuating wind velocity. Corresponding ﬂuctuating pressures will be applied to the structure. The random nature of the loading is evident, and it is clear that statistical methods are required for establishing an appropriate design loading. The next ﬁgure (b) shows a typical earthquake accelerogram. As shown, the maximum ground acceleration of the El-Centro earthquake was about 0.33g. The third ﬁgure (c) shows the characteristic shape of the air pressure impulse caused by a Wind speed (v) − V

Dynamic Static component Time (t)

(a)

El Centro ground acceleration

5 a/g 0 (b)

−5

0

10 20 Time (sec)

30

P(t) Pressure

Pso

(c)

Shock velocity

t

Po

Distance from explosion Positive phase

Negative phase

FIGURE 10.1 Examples of dynamic loading. (a) Fluctuating wind velocity; (b) earthquake accelerogram; (c) pressure time history for bomb blasts.

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bomb blast. The shapes of air-blast curves are usually quite similar, having an initial peak followed by an almost linear decay and often followed by some suction. The duration of the impulsive loads and their amplitudes depend on many factors, for example, distance from blast and charge weight. Vibration of structures is undesirable for a number of reasons, as follows: 1. 2. 3. 4. 5.

Overstressing and collapse of structures Cracking and other damage requiring repair Damage to safety-related equipment Impaired performance of equipment or delicate apparatus Adverse human response

With modern forms of construction, it is feasible to design structures to resist the forces arising from dynamic loadings such as major earthquakes. The essential requirement is to prevent total collapse and consequent loss of life. For economic reasons, however, it is the accepted practice to absorb the earthquake energy by ductile deformation, therefore accepting that repair might be required. Some forms of loading are quite well deﬁned and may be quantiﬁed by observation or experiment. Many forms of loading are not at all well deﬁned and require judgment on the part of the engineer. London’s Millennium Bridge, which is a 350-m pedestrian bridge, opened in June 2000. However, local authorities shut it down after two days due to vibration problems. Engineers found that the “synchronous lateral excitation” caused the problem and ﬁtted 91 dampers to reduce the excessive movement. In January 2001, a 2000-strong crowd marched across the bridge to check the performance of the structure before it was reopened to the public. Data on certain types of dynamic loading, such as earthquakes and wind, are readily available in many design codes. Other types of loading are less well covered, though much data may be available in published research papers. One of the aims of this chapter is to discuss the nature of the most important types of dynamic loading and to direct the reader to relevant literature for further information.

10.2 10.2.1

Earthquake-Induced Vibration of Structures Seismicity and Ground Motions

The most common cause of earthquakes is thought to be the violent slipping of rock masses along major geological fault lines in the Earth’s crust, or lithosphere. These fault lines divide the global crust into about 12 tectonic plates, which are rigid, relatively cool slabs about 100 km thick. Tectonic plates ﬂoat on the molten mantle of the Earth and move relative to one another at the rate of 10 to 100 mm/year. The basic mechanism causing earthquakes in the plate boundary regions appears to be that the continuing deformation of the crustal structure eventually leads to stresses/strains which exceed the material strength. A rupture will then initiate at some critical point along the fault line and will propagate rapidly through the highly stressed material at the plate boundary. In some cases, the plate margins are moving away from one another. In those cases, molten rock appears from deep in the Earth to ﬁll the gap, often manifesting itself as volcanoes. If the plates are pushing together, one plate tends to dive under the other and, depending on the density of the material, it may resurface in the form of volcanoes. In both these scenarios, there may be volcanoes and earthquakes at the plate boundaries, both being caused by the same mechanism of movement in the Earth’s crust. Another possibility is that the plate boundaries will slide sideways past each other, essentially retaining the local surface area of the plate. It is believed that approximately three quarters of the world’s earthquakes are accounted for by this rubbing –sticking – slipping mechanism, with ruptures occurring on faults on boundaries between tectonic plates.

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Earthquake occurrence maps tend to outline the plate boundaries. Such earthquakes are referred to as interplate earthquakes. Earthquakes do occur at locations away from the plate boundaries. Such events are known as intraplate earthquakes and they are much less frequent than interplate earthquakes. They are also much less predictable than events at the plate margins and they have been observed to be far more severe. For example, the Eastern United States, which is located well away from the tectonic plate boundaries of California, has recorded the largest earthquakes in the history of European settlement in the country. These major intraplate earthquakes occurred in the middle of last century in South Carolina on the East Coast and Missouri in the interior. However, because of the low population density at the time, the damage caused was minimal. It is signiﬁcant to note, however, that these intraplate earthquakes, although very infrequent, were considerably larger than the moderately sized interplate earthquakes that frequently occur along the plate boundaries in California. (It is thought that, because tectonic plates are not homogeneous or isotropic, areas of local high stress are developed as the plate attempts to move as a rigid body. Accordingly, rupture within the plate, and the consequent release of energy, are believed to give rise to these intraplate events.) The point in the Earth’s crustal system where an earthquake is initiated (the point of rupture) is called the hypocenter or focus of the earthquake. The point on the Earth’s surface directly above the focus is called the epicenter and the depth of the focus is the focal depth. Earthquake-occurrence maps usually indicate the location of various epicenters of past earthquakes and these epicenters are located by seismological analysis of the effect of earthquake waves on strategically located receiving instruments called seismometers. When an earthquake occurs, several types of seismic wave are radiated from the rupture. The most important of these are the body waves (primary (P) and secondary (S) waves). P waves are essentially sound waves traveling through the Earth, causing particles to move in the direction of wave propagation with alternate expansions and compressions. They tend to travel through the Earth with velocities of up to 8000 m/sec (up to 30 times faster than sound waves through air). S waves are shear waves with particle motion transverse to the direction of propagation. S waves tend to travel at about 60% of the velocity of P waves, so they always arrive at seismometers after the P waves. The time lag between arrivals often provides seismologists with useful information about the distance of the epicenter from the recorder. The total strain energy released during an earthquake is known as the magnitude of the earthquake and it is measured on the Richter scale. It is deﬁned quite simply as the amplitude of the recorded vibrations on a particular kind of seismometer located at a particular distance from the epicenter. The magnitude of an earthquake by itself, which reﬂects the size of an earthquake at its source, is not sufﬁcient to indicate whether structural damage can be expected at a particular site. The distance of the structure from the source has an equally important effect on the response of a structure, as do the local ground conditions. The local intensity of a particular earthquake is measured on the subjective Modiﬁed Mercalli scale (Table 10.1) which ranges from 1 (barely felt) to 12 (total destruction). The Modiﬁed Mercalli scale is essentially a means by which damage may be assessed after an earthquake. In a given location, where there has been some experience of the damaging effects of earthquakes, albeit only subjective and qualitative, regions of varying seismic risk may be identiﬁed. The Modiﬁed Mercalli scale is sometimes used to assist in the delineation of these regions. A particular earthquake will be associated with a range of local intensities, which generally diminish with distance from the source, although anomalies due to local soil and geological conditions are quite common. Modem seismometers (or seismographs) are sophisticated instruments utilizing, in part, electromagnetic principles. These instruments can provide digitized or graphical records of earthquake-induced accelerations in both the horizontal and vertical directions at a particular site. Accelerometers provide records of earthquake accelerations and the records may be appropriately integrated to provide velocity records and displacement records. Peak accelerations, velocities, and displacements are all in turn signiﬁcant for structures of differing stiffness (Figure 10.2).

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TABLE 10.1

10-5

Modiﬁed Mercalli Intensity Scale

I. Not felt except by a very few under especially favorable circumstances II. Felt only by a few persons at rest, especially on upper ﬂoors of buildings. Delicately suspended objects may swing III. Felt quite noticeably indoors, especially on upper ﬂoors of buildings, but many people do not recognize it as an earthquake. Standing motor cars may rock slightly. Vibration like passing truck. Duration estimated IV. During the day felt indoors by many, outdoors by few. At night some awakened. Dishes, windows, and doors disturbed; walls make creaking sound. Sensation like heavy truck striking building. Standing motorcars rock noticeably V. Felt by nearly everyone; many awakened. Some dishes, windows, etc., broken; a few instances of cracked plaster; unstable objects overturned. Disturbance of trees, poles, and other tall objects sometimes noticed. Pendulum clocks may stop VI. Felt by all; many frightened and run outdoors. Some heavy furniture moved; a few instances of fallen plaster or damaged chimneys. Damage slight VII. Everybody runs outdoors. Damage negligible in buildings of good design and construction, slight to moderate in well-built ordinary structures; considerable in poorly built or badly designed structures. Some chimneys broken. Noticed by persons driving motor cars VIII. Damage slight in specially designed structures; considerable in ordinary substantial buildings, with partial collapse; great in poorly built structures. Panel walls thrown out of frame structures. Fall of chimneys, factory stacks, columns, monuments, walls. Heavy furniture overturned. Sand and mud ejected in small amounts. Changes in well water. Persons driving motorcars disturbed IX. Damage considerable in specially designed structures; well-designed frame structures thrown out of plumb; great in substantial buildings, with partial collapse. Buildings shifted off foundations. Ground cracked conspicuously. Underground pipes broken X. Some well-built wooden structures destroyed; most masonry and frame structures destroyed with foundations; ground badly cracked. Rails bent. Landslides considerable from riverbanks and steep slopes. Shifted sand and mud. Water splashed over banks XI. Few, if any (masonry), structures remain standing. Bridges destroyed. Broad ﬁssures in ground. Underground pipelines completely out of service. Earth slumps and land slips in soft ground. Rails bent greatly XII. Damage total. Waves seen on ground surfaces. Lines of sight and level distorted. Objects thrown upward into the air Source: Data from Wood, H.O. and Neumann, Fr., Bull. Seis. Soc. Am., 21, 277– 283, 1931.

10.2.2

Inﬂuence of Local Site Conditions

Local geological and soil conditions may have a signiﬁcant inﬂuence on the amplitude and frequency content of ground motions. These conditions affect the earthquake motions experienced (and hence the structural response) in one, or more, of the following ways: *

*

*

Interaction between the bedrock earthquake motion and the soil column will modify the actual ground accelerations input to the structure. This manifests itself by an increase in the amplitude of the ground motion over and above that at the bedrock, and a ﬁltering of the motion so that the range of frequencies present becomes narrow with the high-frequency components being eliminated. This condition particularly arises in areas where soft sediments and alluvial soil overly bedrock. The degree of ampliﬁcation is dependent on the strength of shaking at the bedrock. Because of nonlinear effects in the soil, the ampliﬁcation ratio is less in strong shaking than under base motions of lower amplitude. The soil properties in the proximity of the structure contribute signiﬁcantly to the effective stiffness of the structural foundation. This may be a signiﬁcant parameter in determining the overall structural response, especially for structures that would be characterized as stiff under other environmental loadings. The strength (and response) of the local soil under earthquake shaking may be critical to the overall stability of the structure.

It is also important that information on relevant geological features, such as faulting, be assessed. Geological information on suspected active faults near the site can assist in providing a basis for

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30 20 10 0 −10 −20 −30 −40

cm

cm/sec

a/g

4 3 2 1 0 −1 0 −2 −3 −4

20 15 10 5 0 −5 −10 −15 −20 −25

(A) Ground acceleration

5

10

15

20

25

30 Time (sec)

(B) Ground velocity

0

5

10

15

20

25

30 Time (sec)

(C) Ground displacement

0

5

10

15

20

25

30 Time (sec)

FIGURE 10.2 El-Centro earthquake, north –south component. (A) Record of the ground acceleration; (B) ground velocity, obtained by integration of (A); (C) ground displacement, obtained by integration of (B).

evaluating the intensity of a likely earthquake. It is usual to use this information, together with the regional seismicity data, to determine the likely level of seismic activity.

10.2.3

Response of Structures to Ground Motions

The effect of ground motion on the various categories of structures is dictated almost entirely by the distribution of mass and stiffness in the structure. It is important to appreciate that, in an earthquake, loads are not applied to the structure. Rather, earthquake loading arises because of accelerations generated by the foundation level(s) of the structure intercepting and being inﬂuenced by transient ground motions. Speciﬁcally, the product of the structural mass and the total acceleration produces the inertia loading experienced by the structure. This is an expression of Newton’s Second Law. It is important to appreciate that the total acceleration is the absolute acceleration of the structure, namely, the sum of the ground acceleration and that of the structure relative to the ground. If the structure is stiff there is little, if any, additional acceleration relative to the ground motion and, therefore, the earthquake loading experienced is essentially proportional to the building mass, that is, Feq / M: For structures that are ﬂexible, for example, those in the high-rise or long-span category, the absolute acceleration is low. This occurs because the ground acceleration and the acceleration of the building relative to the ground tend to oppose one another. In this case, the earthquake loading is approximately proportional to the square root of the mass, that is Feq / M 0:5 : For structures in the cantilever category, which are essentially vertical, it is the horizontal accelerations that are signiﬁcant; whereas for structures that are largely horizontal in extent, the effect of the vertical accelerations is dominant. Moreover, if the plan distributions of mass and stiffness are dissimilar in vertical structures, signiﬁcant twisting motions may arise.

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The peak ground acceleration is of importance in the response of stiff structures and peak ground displacements are of importance in the response of ﬂexible structures, with peak ground velocity being of importance for structures of intermediate stiffness. Stiff structures tend to move in unison with the ground while ﬂexible structures, such as high-rise buildings, experience the ground moving beneath them, their upper ﬂoors tending to remain motionless.

10.2.4

Dynamic Analysis

10.2.4.1

Equations of Motion for Linear Single-Degree-of-Freedom Systems

Consider the linear single-degree-of-freedom (single-DoF) system shown in Figure 10.3 subjected to a time varying ground displacement, zðtÞ: Let the relative displacement of the system to the ground be, yðtÞ; y is then the extension of the spring and dashpot. From the equation of motion, it follows that mð€y þ z€Þ ¼ 2ky 2 c_y

ð10:1Þ

Rearranging Equation 10.1, and replacing m; k; and c by the system’s radial frequency v and damping ratio j; gives y€ þ 2jvy_ þ v2 y ¼ 2€z

ð10:2Þ

Viscous damper, c

Stiffness, k

Mass, m

m

Displacement y (t)

Force f (t)

Given a description of the input motion, zðtÞ; (for FIGURE 10.3 Single-DoF system. example, from an accelerograph recording), the solution of Equation 10.2 provides a complete time history of the response of a structure with a given natural period and damping ratio, and can also be used to derive maximum responses for constructing a response spectrum (Figure 10.6). Owing to the random nature of earthquake ground motion, numerical solution techniques are needed for Equation 10.2, as described by Clough and Penzien (1993). 10.2.4.2

Equations of Motion for Linear Multiple-Degree-of-Freedom Systems

The dynamic response of many linear multipledegree-of-freedom (multi-DoF) systems can be split into decoupled natural modes of vibration (Figure 10.4), each mode effectively representing a single-DoF system. A modiﬁed form of Equation 10.2 then applies to each mode, which for mode i becomes L Y€ i þ 2ji vi Y_ i þ v2i Yi ¼ i z€ ð10:3Þ Mi Here, Yi is the generalized modal response in the ith mode. ðLi =Mi Þ is a participation factor, which depends on the mode shape and mass distribution, and describes the participation of the mode in overall response to a particular direction of ground FIGURE 10.4 Typical modes for multistory buildings. motion. For a two-dimensional (2D) structure with n lumped masses, responding in one horizontal direction n X fij mj Li j¼1 ¼ X ð10:4Þ n Mi 2 fij mj j¼1

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In Equation 10.4, fij ; describes the modal displacement of the jth mass in the ith mode. The higher modes often have very low values of ðLi =Mi Þ; and their contribution can then be omitted. In this way, the computational effort is greatly reduced. In cases where only the ﬁrst mode in each direction is signiﬁcant (often the case for low- to medium-rise building structures), equivalent static analysis may be sufﬁcient, as described later.

10.2.5

Earthquake Response Spectra

10.2.5.1

Elastic Response Spectra

For design purposes, it is generally sufﬁcient to know only the maximum value of the response due to an earthquake. A plot of the maximum value of a response quantity as a function of the natural vibration frequency of the structure, or as a function of a quantity which is related to the frequency such as natural period, constitutes the response spectrum for that quantity (see Chapter 3 and Chapter 13). The peak relative displacement is usually called Sd and the peak strain energy of the oscillator is 1 2 KS 2 d 1 K SE ¼ M S2d 2 M SE ¼

or SE ¼

1 MS2v 2

Hence, the pseudo-relative velocity and acceleration spectra are deﬁned as Sv z ¼ v Sd

ð10:5Þ

Saz ¼ v2 Sd

ð10:6Þ

Figure 10.5 shows that a record of peak relative displacement response of an single-DoF oscillator can be plotted for a given earthquake, given damping, and a range of periods, typical of structures. The structure’s natural period (T or 1=n) is conventionally taken as the abscissa, and curves are drawn for various levels of damping (Figure 10.6). It should be noted that the response spectrum gives no information about the duration of response (and hence the number of damaging cycles) that the structure experiences, which can have a very signiﬁcant inﬂuence on the damage sustained. 10.2.5.2

Smoothed Design Spectra

Owing to the highly random nature of earthquake ground motions, the response spectrum for a real earthquake record contains many sharp peaks and troughs, especially for low levels of damping. The peaks and troughs are determined by a number of uncertain factors, such as the precise location of the earthquake source, which are unlikely to be known precisely in advance. Therefore, spectra for design purposes are usually smoothed envelopes of spectra for a range of different earthquakes; indeed, one of the advantages of response spectrum analysis over time history analysis is that it can represent the envelope response to a number of different possible earthquake sources from a single analysis, and is not dependent on the precise characteristic of one particular ground motion record. Codes of practice such as UBC (2000) and Eurocode 8 (ENV 1998, 1994-8) provide smoothed spectra for design purposes. 10.2.5.3

Ductility-Modiﬁed Response Spectrum Analysis

In a ductile structure, or subassemblage, the resistance, R; may be sustained at displacements that are several times those at ﬁrst yield, Dy ; as represented in Figure 10.7. For yielding single-DoF systems, ductility-modiﬁed acceleration response spectra can be drawn, representing the maximum acceleration response of a system as a function of its initial (elastic) period, T; damping ratio, j; and displacement ductility ratio, m (m is the ratio of maximum displacement,

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4

10-9

El Centro ground acceleration

3 2 1 a/g

0 0 −1

5

10

15

20

time (sec)

−2 −3 −4

T = 0.5s x = 2%

Deformation (m)

0.5 0.3

Max deformation = 0.043

0.1 −0.1 0

10

−0.3

20

30

40

50

Natural period (sec)

T = 1s x = 2%

Deformation (m)

−0.5 0.5 0.3

Max deformation = 0.074

0.1 −0.1 0

10

−0.3

20

30

40

50

Natural period (sec)

−0.5

T = 2s x = 2%

Deformation (m)

0.5 0.3

Max deformation = 0.281

0.1 −0.1 0 −0.3

10

20

30

40

50

Natural period (sec)

−0.5 0.35

Sd (m)

0.3 0.25 0.2 0.15 0.1 0.05 0

0

1

2

3

Period (sec) FIGURE 10.5

Compilation of (relative) displacement response spectra.

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30

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Mass, m Stiffness, k

12

Peak ground acceleration

Spectral acceleration

10

Peak spectral acceleration Peak ground acceleration

0%

8

2%

6

Viscous damper, c

3% 10%

4

20%

2

Damping ratio x = 2 km m Undamped natural period T =2π k

0 0

0.5

1 1.5 2 2.5 3 3.5 4 Undamped natural period of structure, s

FIGURE 10.6

4.5

5

Acceleration response spectrum for El-Centro 1940 earthquake.

Resistance

Yield

Peak deflection

R Unloading

∆y FIGURE 10.7

Deflection

∆ max

A simple bilinear elasto-plastic curve of response, representative of ductile performance.

Dmax ; to yield displacement, Dy ). The reduction in acceleration response of the yielding system compared with the elastic one is period dependent; for structural periods greater than the predominant earthquake periods, the reduction is approximately 1=m; for very stiff systems there is no reduction, while at intermediate periods a reduction factor between 1=m and 1 applies. To derive peak accelerations and internal forces, the system can be treated as linear elastic and the ductility-modiﬁed spectrum used exactly like a normal elastic spectrum. However, deﬂections derived from this treatment must be multiplied by m to allow for the plastic deformation. It is now standard practice to analyze multi-DoF systems in the same way. That is, a yielding multiDoF system is treated as elastic, and an appropriate ductility-modiﬁed spectrum is substituted for an elastic one. Acceleration and force responses are derived directly and deﬂections are multiplied by m: However, this procedure is not (contrary to the case for single-DoF systems) rigorously correct. Although it gives satisfactory answers for regular structures, it can be seriously in error for structures (such as those with weak stories) where the plasticity demand is not evenly distributed. Nevertheless, most codes of practice allow the use of ductility-modiﬁed spectra for design, and give appropriate values for the reduction factors (called q; or behavior factors in Eurocode 8 and R factors in UBC) to apply to elastic response spectra.

10.2.6

Design Philosophy and the Code Approach

In areas of the world recognized as being prone to major earthquakes, the engineer is faced with the dilemma of being required to design for an event, the magnitude of which has only a small chance of

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occurring during the life of the facility. If the designer adopts conservative performance criteria for the facility, the client (often society) is faced with costs which may be out of proportion to the risks involved. On the other hand, to ignore the possibility of a major earthquake could be construed as negligent in these circumstances. To overcome this problem, a dual design philosophy has been developed, by which procedure: 1. A moderate earthquake, such as may reasonably be expected at the site, is used as a basis for the seismic design. The facility should be proportioned to resist such an earthquake without signiﬁcant damage. This “damageability” limit state should ensure safety, limited nonstructural damage, and the continued performance of facilities and services, particularly in those with important postearthquake functions. The list includes hospitals, police, ﬁre and civil defense facilities, water supply, telecommunications, electricity generation and distribution systems, and so on. Almost as important is the maintenance of road and rail communications, particularly for food distribution (including warehouses and their contents). Similarly, the protection of industrial complexes, in their own right, as well as the protection of individual items of equipment in other buildings and facilities, is a necessary consequence of adoption of this limit state. 2. The most severe, credible earthquake that may be expected to occur at the site is used to test safety. In this ultimate limit state, signiﬁcant structural and nonstructural damage is expected but neither collapse nor loss of life should occur. The main strategy for preventing collapse has traditionally been provision of ductility. This is the opposite quality to brittleness, and may be deﬁned as the ability to sustain repeated excursions beyond the elastic limit without fracture. Owing to the cyclic, imposed displacement nature of earthquake loading, a ductile structure can absorb very large amounts of energy without collapse; the designer must think in terms of designing for maximum imposed displacements, rather than imposed loads. Achieving ductility is partly a matter of choosing the right structural system, and partly a matter of detailing. In the former category comes the important concept of “capacity design,” as described by Paulay (1993). This involves ensuring a hierarchy of strengths within a structure to ensure that yielding occurs in ductile modes (such as ﬂexure) rather than brittle modes (such as buckling or, for reinforced concrete, shear). There are other aspects of structural form which are important, particularly regularity in elevation (to avoid “soft” or weak stories) and regularity in plan (to minimize torsional response). These aspects are described in many textbooks and are quantiﬁed in some codes of practice (Park and Paulay, 1975). Detailing of the structure is also important to ensure ductility. For concrete structures, this primarily involves reinforcement detailing and in steel structures connection detailing. The latter aspect has been particularly recognized following the failure of H-welded connections in the Northridge earthquake of 1994 (Burdekin, 1996). The primary reliance is on empirical solutions to these problems, as described in codes of practice, such as Eurocode 8 (ENV 1998, 1994-8) Part 1.3 and UBC/IBC (2000). Textbooks discussing these issues for concrete include Paulay and Priestley (1992), Booth (1994), Penelis and Kappos (1996), FEMA 273/274, FEMA 356/357 “NEHRP guidelines for seismic rehabilitation of buildings,” FEMA 368/369 “NEHRP recommended provisions for seismic regulations for new buildings and other structures,” and FEMA 306/307/308 “Evaluation and repair of earthquake damaged concrete and masonry buildings.” Textbooks covering failures of steel structures in recent earthquakes include Burdekin (1996) and FEMA 350-354 (2000). Another important aspect of detailing is to allow for the maximum inelastic deﬂections caused by the design earthquake. Nonseismic-resisting elements of a structure such as cladding and inﬁll walls must be able to accommodate these deﬂections safely, as must (crucially) the gravity load-bearing structure, which still suffers the seismic displacements even when not contributing to seismic resistance. In addition, adequate separation between adjacent structures must be provided. Codes of practice (e.g., Eurocode 8 and UBC) give guidance on suitable limits.

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10.2.6.1

Performance-Based Design

In recent years, seismic design codes throughout the world have been shifting toward the adoption of performance-based design philosophy. The goal of a performance-based design procedure is to produce structures that have predictable seismic performance under multiple levels of earthquake intensity. In order to do so, it is important that the behavior of the structures is targeted in advance, both in the elastic as well as the inelastic ranges of deformation. The four important parameters in seismic design, strength, stiffness, ductility, and deformation, become the primary elements of a performance-based design procedure and have to be designed rationally. The next generation of codes is expected to be based on performance-based principles such as Asian Model Concrete Code (ACMC, 2001).

10.2.7

Analysis Options for Earthquake Effects

Analysis is only one part of the design process; conceptual design, detailing, and proper construction are the other vital components for ensuring good seismic performance. This section provides a brief theoretical review of the basis for seismic analysis, describing the main analytical techniques currently used by designers. More details are given by Clough and Penzien (1993), a standard general text for dynamic analysis, and Chopra (2001), which deals speciﬁcally with concerns for earthquake engineers. Essentially, an earthquake engineer is faced with four possible methods of analysis/design for earthquake loading on a structure. In all methods, the dual damage criteria discussed above may be applied. The four analysis/design options available are: *

*

*

*

Dynamic time history analysis Response spectrum analysis Equivalent static approach (or force-based approach) Displacement-based approach

Equivalent static methods are usually adequate for conventional, regular building structures under about 75 m in height. A response spectrum analysis is required for taller buildings, because higher mode effects may become important, and also for buildings with plan or elevational eccentricities because torsional effects or nonstandard mode shapes may be signiﬁcant. Codes of practice such as Eurocode 8 and UBC (2000) specify the degree of eccentricity at which such analysis is required. Unusual or very important structures may require nonlinear time history analysis, and this may also be required where the inaccuracies implicit in the use of ductility-modiﬁed response spectrum analysis become unacceptable. Displacement-based approach is a new method for seismic design, which is gaining popularity. The above four analysis options are discussed next. 10.2.7.1

Dynamic Time History Analysis

The most rigorous form of dynamic analysis involves stepping a nonlinear model of the structure through a complete time history of earthquake ground motions. The advantage of the method is that it can give direct information on nonlinear response, the duration of response (and hence the number of loading cycles), and the relative phasing of response between various parts. The method involves subjecting an appropriate ﬁnite element computer model of the building, or structural system, to a given, previously recorded, earthquake record and examining its response in real time. Response peaks are generally of most interest. The analysis must be performed for a number of different earthquake time histories to reduce dependence on the random characteristics of a particular record. There are certain special circumstances where this procedure is useful but, for general seismic design, it is of little value as the actual earthquake that the structure may have to resist cannot be guaranteed to have sufﬁciently similar characteristics to the design earthquake. In particular, the intensity, duration, and frequency content of the earthquake may be unsuitable especially if, as often happens, the record comes from another country or continent. Moreover, the method is expensive and time-consuming, so that only for special structures can its use be justiﬁed. If, in addition, inelastic response calculations are involved, another level of complexity (and uncertainty) is introduced. Response then becomes

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dependent, often heavily so, on the nonlinear models chosen and this is in addition to that inherent in choosing to use one particular record. 10.2.7.2

Response Spectrum Analysis

10.2.7.2.1 Response Spectrum Analysis of Single-Degree-of-Freedom Systems With a knowledge of the natural period and damping of an single-DoF system, its peak (i.e., spectral) acceleration, Sa ; can be determined directly from an appropriate response spectrum (see Chapter 3). In undamped systems, this peak response occurs when the equivalent spring is at its maximum extension point, so that the maximum force in the spring is given by F ¼ mSa

ð10:7Þ

From Equation 10.7, the peak (i.e., spectral) displacement, Sx ; of the spring is given by Sx ¼

F T2 S T2 ¼ mSa 2 ¼ a 2 k 4p m 4p

ð10:8Þ

For structures with relatively small viscous damping, the same relationships are still approximately true, because the maximum acceleration occurs when the velocity is low and hence the damping force (which is velocity proportional) is also low. Therefore, Equation 10.7 and hence also Equation 10.8 are still very good approximations for lightly damped systems. Thus the two most important parameters of structural response — maximum force and displacement — can be determined for a linear single-DoF system directly from the acceleration response spectrum, provided only that the mass, natural period, and damping are known. It is important to realize that the spectral acceleration, Sa ; is an absolute value (the true acceleration of the structure in space) whereas the spectral displacement, Sx ; is a relative value, measured in relation to the ground, which itself is moving in the earthquake. This at ﬁrst sight may seem confusing, until it is remembered that the absolute acceleration of the mass is determined by the force on it (Equation 10.7), which itself is determined by the relative compression of the spring with respect to the ground (Equation 10.8). 10.2.7.2.2 Response Spectrum Analysis of Multi-Degree-of-Freedom Systems By considering the response of each mode separately, a response spectrum analysis is also possible for an multi-DoF system, if generalized modal quantities are used (compare Equation 10.2 and Equation 10.3). For example, for a 2D structure with n lumped masses, responding in one horizontal direction, Equation 10.4 is modiﬁed to give the maximum base shear in the ith mode as 0 12 n X @ fij mj A Fi ¼

j¼1 n X j¼1

f2ij mj

Sa ¼ meff ;i Sa

ð10:9Þ

where Sa is the spectral acceleration corresponding to the damping and frequency of mode i: Higher modes with low effective masses, meff;i ; may contribute little to response and can usually be neglected. Since the sum of effective masses, meff ;i ; of all modes equals the total mass, a good test of whether the ﬁrst r modes are sufﬁcient to capture response adequately is r X i¼1

meff;i $ 0:9

n X i¼1

mi ¼ 0:9 ðtotal massÞ

ð10:10Þ

A response spectrum analysis gives the maximum response of the structure for each mode of vibration considered. Although it is rigorously correct to add the response in each mode at any time to obtain the total response, the maximum responses in each mode, calculated from response spectrum analysis, do

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not occur simultaneously, and hence simple addition produces an overestimate of response. A common and usually adequate approximation is the square root of the sum of the squares (SRSS) rule, where the maximum total response is estimated as the SRSS combination of the individual modal responses. However, this may not be conservative enough for closely spaced or high-frequency modes, and other methods, such as the complete quadratic combination (CQC) method, are available (Gupta, 1990). There are many commercially available computer programs which can perform response spectrum analysis, and it is now regarded as a standard rather than a specialist technique. 10.2.7.3

Equivalent Static Analysis (Force-Based Approach)

This is the type of analysis presented in most contemporary codes of practice, and it is conditional for its accuracy upon response being dominated by one mode of vibration in each direction. In the case of buildings, a quantity usually referred to as the “total base shear” is calculated from the product of the weight of the building and a coefﬁcient. This coefﬁcient takes into account the location and importance of the structure, its ductility or energy absorption capacity, its dynamic characteristics, and the local soil conditions and their effect on structural responses. Once the total base shear has been calculated, it is distributed up the structure as a series of horizontal loads at each ﬂoor level and the structure is analyzed with these equivalent horizontal loads applied. The maximum lateral base shear is ﬁrst calculated. Equation 10.11 gives the relevant formulae in UBC (2000). Other current codes follow similar formats V¼

Cv IW 2:5Ca I but V # W and $ 0:11Ca IW RT R

ð10:11Þ

In addition, V $ ð0:8ZNv I=RÞW (high seismicity, Zone 4 only), where: V ¼ ultimate seismic base shear (force units, e.g., kN) Cv ; Ca ¼ seismic coefﬁcients, depending on the zone factor Z as given in UBC I ¼ importance factor ¼ 1 to 1.25 in UBC R ¼ reduction coefﬁcient depending on the ductility of structure ¼ 2.8 to 8.5 in UBC T ¼ ﬁrst mode period of the building (sec) W ¼ building weight (force units, e.g., kN) Z ¼ zone factor expressed as the peak ground acceleration on rock (in gravity units) for a 475-year return period ¼ 0.075 to 0.4 in UBC Nv ¼ factor allowing for proximity to active faults ¼ 1.0 to 2.0 in UBC ðV=WÞ represents the shape of a standard design response spectrum with a peak ampliﬁcation on ground acceleration for 5% damping of 2.5, and a minimum value at long period to allow for the uncertainty in long-period motions and for proximity to active faults. The base shear calculated by these methods is then applied to the structure as a set of horizontal forces, with a vertical distribution based on the ﬁrst mode shape of regular vertical cantilever structures. Horizontal distribution follows the mass distribution, with some additional allowance for torsional effects. 10.2.7.4

Displacement-Based Approach

In the development of performance-based earthquake engineering, which stresses the inelastic behavior of structural system under severe earthquake ground motions (high seismic region), displacement rather than force has been recognized as the most suitable and direct performance or damage indicator. Deformation-controlled design can be achieved either by using the traditional force/strength-based design procedure together with a check on the displacement/drift limit, or by employing a direct displacement-based procedure. The idea of displacement-based design was introduced by Gulkan and Sozen (1974). They developed the concept of substitute structure to estimate the nonlinear structural response through an equivalent elastic model, assuming a linear behavior and a viscous

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damping equivalent to the nonlinear response. This idea has been adopted recently by Priestley and Kowalsky (2000) for a direct displacement design of single-DoF and multi-DoF reinforced concrete structures. Another direct displacement-based design approach was proposed by Fajfar (2000) based on the capacity spectrum method (Chopra and Goel, 1999). In all the above references, seismic demand is speciﬁed as either a displacement response spectrum (D-T format) or an acceleration displacement response spectrum (ADRS format). For a general-purpose spectrum, nonlinear elastic behavior of a structural system can be accounted for by either an equivalent elastic response spectrum or an inelastic response spectrum. The former is associated with effective viscous damping jeff and the latter is directly constructed based on the relation between reduction factors and ductility. Although the elastic acceleration design spectrum is available from codes, it is not appropriate to be a basis for the determination of the elastic displacement design spectrum because the displacement increases with period even at longer periods.

10.2.8

Soil– Structure Interaction

Structural analyses usually assume that ground motions are applied via a rigid base, thus neglecting the effect of ground compliance on response. Although this rigid base assumption may lead to an underestimate of deﬂections, it is usually conservative as far as forces are concerned, because ground compliance reduces stiffness and usually moves structural periods farther from resonance with the ground motion. However, this conservatism may not always apply, and Eurocode 8 Part 5 lists the following cases where soil –structure interaction (SSI) should be investigated: 1. Structures where P-Delta (second order) effects need to be considered 2. Structures with massive or deep-seated foundations, such as bridge piers, offshore caissons, and silos 3. Slender, tall structures such as towers and chimneys 4. Structures supported on very soft soils with an average shear wave velocity less than 100 m/sec Allowance for SSI effects is usually a specialist task. The simplest method is to present the soil ﬂexibility by discrete springs connected to the foundation. These require a knowledge of the shear stiffness of the soil. Further information is given by Pappin (1991) and Wolf (1985, 1994).

10.2.9

Active and Passive Control Systems

Alternative strategies of designing for earthquake resistance involve modiﬁcation of the dynamic characteristics of structure to improve seismic response. The systems can be classiﬁed as either passive or active. The basic role of these systems is to absorb a portion of the input energy, thereby reducing energy dissipation demand on primary structural members and minimizing possible structural damage. The most common type of passive system involves lengthening the structure’s fundamental period of vibration by mounting the superstructure on bearings with a low horizontal stiffness; this is known as base or seismic isolation. Where this increases, the fundamental period above the predominant periods of earthquake excitation, the acceleration (but not necessarily displacement) response is signiﬁcantly reduced. Usually, additional damping is provided in the seismic isolation bearing to control deﬂections. The principle of seismic isolation is illustrated by Figure 10.8. The reduction in response, often of the order of 50, has proved highly effective in recent earthquakes in reducing damage to both building structure and building contents. UBC (2000) provides codiﬁed guidance for seismic isolation of buildings while AASHTO (1991) and Eurocode 8 (ENV 1998, 1994-8) Part 2 treat bridge structures. Seismic isolation has been incorporated in many hundreds of recent structures, particularly in bridges, and also in buildings such as hospitals with contents that must remain functional after an earthquake. It has also been used to improve the seismic resistance of existing structures. Another form of passive

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Spectral acceleration of base shear

10-16

5 4.5 4

Period shift

3.5 3 2.5 2 1.5 1 0.5 0

x= 2%

Isolated period

Reduction in Force

x = 10% x = 20% 0

0.5

1

1.5

2

2.5

Period (sec)

(a)

Spectral displacement (m)

0.45 0.4 Period shift

0.35

Period shift leads to increase of displacement

0.3 0.25 0.2 0.15 0.1

x = 10%

0.05 0

Increased damping leads to decrease in displacement

x = 2%

x = 20% 0

0.5

(b)

1

1.5

2

2.5

Period (sec)

FIGURE 10.8 Effect of seismic isolation on forces and displacements for an earthquake with predominant period around 0.5 sec. (a) Effect of period shift on design forces; (b) Effect of period shift and damping on relative displacement between ground and structure.

system is the provision of additional structural damping in the form of discrete viscous, frictional, or hysteretic dampers. Active systems modify the dynamic characteristics of a structure in real time during an earthquake, by computer-controlled devices such as active mass dampers. Presently, very few buildings are actually constructed in this way, but there has been a recent large international research effort (Casciati, 1996; Kabori, 1996; Soong, 1996). Owing to their adaptability, active systems are less dependent for their effectiveness on the precise nature of the input motion (a concern for passive systems, particularly where they are very close to the earthquake source) but they must have a very high degree of reliability to ensure they function during the crucial few seconds of an earthquake.

10.2.10

Worked Examples

Example 10.1 Seismic Analysis of a 30-Story Frame A 30-story building has the effective stiffness, Ke ¼ 2:5 £ 103 kN/m, together with a mass per unit height of m ¼ 30 tons=m:

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Parabolic mode shape ∆(x,t) = ∆t x h

120 m

2

1st mode shape

42 m FIGURE 10.9

First mode shape of the 30-story frame.

The mass of the building is uniform over its height of 120 m. An appropriate ﬁrst mode shape for the structure is the parabola, as shown in Figure 10.9. For this frame, using the response spectrum for 2% damping (Figure 10.10), ﬁnd: 1. 2. 3. 4. 5.

The peak tip deﬂection The peak base shear The overturning moment The peak interstory drift at the top of the frame The peak acceleration at the top of the frame

The effective mass, Me or Mi ; (see Equation 10.4) Me ¼ f2 m ¼

ðH 0

mðxÞ

x H

4

dx ¼

H

mðxÞ x5 5 H4

0

¼

mH 30 £ 120 ¼ ¼ 720 tons 5 5

This is the effective mass tributary to one frame. The effective earthquake mass, Meq or Li (see Equation 10.4) Meq ¼ fm ¼

ðH 0

mðxÞ

x H

2

dx ¼

mðxÞ x3 3 H2

H 0

¼

mH 30 £ 120 ¼ ¼ 1200 tons 3 3

The natural period is 3.37 sec, a long-period structure, given: Keq ¼ 2:5 £ 103 kN/m T ¼ 2pð720=2500Þ0:5 The participation factor is PF ¼

Meq 1200 ¼ 1:667 ¼ 720 Me

(i) Peak tip deﬂection From Figure 10.9 Sd ¼ 0:32 m ¼ 320 mm ¼ PF £ Sd Dtip ¼ 1:667 £ 320 ¼ 533:44 mm (ii) Peak base shear ¼ PF £ Fmax V0 ¼ PFðDtip Ke Þ ¼ 1:667ð0:53344 £ 2500Þ ¼ 2:22 MN (iii) Peak overturning moment Mot ¼ V0 x

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Sd (m)

10-18

0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0

(a)

0

1

2 Period (sec)

3

4

0

1

2 Period (sec)

3

4

0

1

2 Period (sec)

3

4

1.2 Sv (m/sec)

1 0.8 0.6 0.4 0.2 0

Sa (g)

(b) 1.6 1.4 1.2 1 0.8 0.6 0.4 0.2 0

(c)

FIGURE 10.10 (a) Deformation (or displacement); (b) pseudo-velocity, and (c) pseudo-acceleration response spectra. El-Centro ground motion. Damping ratio j ¼ 2%:

ðH

ðH

x H 0 x ¼ ðH x mðxÞ H 0

x3 dx H2 0 ¼ 2 ðH x2 dx mðxÞ 2 dx H 0

x¼

3H 3 £ 120 ¼ ¼ 90 m 4 4

xmðxÞ

2

dx

mðxÞ

x

Mot ¼ 2223:11 £ 90 ¼ 2:0008 £ 105 kN m Equivalent static loading ¼ mðtÞv2 £ PF £ Sd

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x H

2

Vibration and Shock Problems of Civil Engineering Structures

10-19

55.365 kN/m

100 m

Equivalent static load FIGURE 10.11

V0 = 2.22x103 kN SFD

M0t = 2x105 kNm BMD

Equivalent static load, shear force diagram, and bending moment diagram of the frame.

sﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃ Ke 2500 ¼ ¼ 1:86 rad=sec v¼ Me 720 Hence, equivalent static loading ¼ 30 £ 1.862 £ 1.667 £ 0.32 £ 1 ¼ 55.365 kN/m. The equivalent static load, shear force, and bending moment diagrams of the frame are shown in Figure 10.11. 116 2 (iv) Peak interstory drift ¼ 533:44 2 £ 533:44 ¼ 34:97 mm 120 This will create signiﬁcant constraints on component details such as partitions, windows, and panels at the particular level. To avoid potential problems, the structure might have to be stiffened laterally. f 55:365 (v) Peak acceleration ¼ ¼ ¼ 1:8455 m/sec2 mðxÞ 30 This acceleration is 18.8% of gravity. It is important that the facade attachment, mechanical utilities, or electrical utilities of the structure are appropriately designed according to the peak acceleration.

Example 10.2 Response of Buildings to an Earthquake In the following example, a 52-story ofﬁce/residential building is considered. The structure is founded on a highly soft soil and located in UBC Zone 4 in the USA, which represents a relatively active seismic area. The lateral load resisting system is a concrete core system with concrete moment frames for the perimeter. According to UBC (2000), the structure needs to resist an equivalent horizontal seismic force of 79,113 kN, representing nearly 9.14% of the effective vertical load. Details of story weights, elevation, and interstory height are given in Table 10.2. Sample calculation The base shear value is obtained based on UBC (2000) approach: T ¼ Ct Hn3=4 ¼ 0:03 £ 682:43=4 ¼ 4:01 sec ðheight input must be in feetÞ V¼

Cv IW 2:5Ca I 0:8ZNv I but V # W and V $ W ðsee Equation 10:11Þ RT R R V¼ V$

0:96 £ 1 £ 865:3 £ 103 ¼ 59:25 MN 3:5 £ 4:01

0:8 £ 0:4 £ 1 £ 1 £ 863:5 £ 103 $ 79:113 MN 3:5

Hence, the lower limit applies, V ¼ 79:113 MN.

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10-20

Height (m)

Calculation Details for Structure’s Response to an Earthquake Height (ft)

208.00

682.41

Level

Story Height (m)

Top 52 51 50 49 48 47 46 45 44 43 42 41 40 39 38 37 36 35 34 33 32 31 30 29

0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0

© 2007 by Taylor & Francis Group, LLC

T (sec) 4.01

Ct

Cv

0.03

Height (m)

Story Weight (kN)

212 208.0 204.0 200.0 196.0 192.0 188.0 184.0 180.0 176.0 172.0 168.0 164.0 160.0 156.0 152.0 148.0 144.0 140.0 136.0 132.0 128.0 124.0 120.0 116.0

0 9,300.0 9,300.0 9,300.0 9,300.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0

0.96 Wi Hi 0 1,934,400 1,897,200 1,860,000 1,822,800 3,244,800 3,177,200 3,109,600 3,042,000 2,974,400 2,906,800 2,839,200 2,771,600 2,704,000 2,636,400 2,568,800 2,501,200 2,433,600 2,366,000 2,298,400 2,230,800 2,163,200 2,095,600 2,028,000 1,960,400

Ca 0.36

Nv 1.00

Na 1.00

Force (kN)

Shear (kN)

Moment (kN m)

19,778 1,319 1,294 1,268 1,243 2,213 2,167 2,120 2,074 2,028 1,982 1,936 1,890 1,844 1,798 1,752 1,706 1,660 1,613 1,567 1,521 1,475 1,429 1,383 1,337

19,778 21,097 22,391 23,659 24,902 27,115 29,282 31,402 33,477 35,505 37,487 39,423 41,313 43,157 44,955 46,707 48,412 50,072 51,685 53,252 54,774 56,249 57,678 59,061 60,398

0 79,113 163,503 253,067 347,705 447,315 555,776 672,903 798,511 932,418 1,074,437 1,224,386 1,382,079 1,547,331 1,719,960 1,899,779 2,086,606 2,280,254 2,480,541 2,687,282 2,900,292 3,119,386 3,344,381 3,575,092 3,811,335

R

I

Z

3.50

1.00

0.40

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TABLE 10.2

4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0 4.0

Sum

112.0 108.0 104.0 100.0 96.0 92.0 88.0 84.0 80.0 76.0 72.0 68.0 64.0 60.0 56.0 52.0 48.0 44.0 40.0 36.0 32.0 28.0 24.0 20.0 16.0 12.0 8.0 4.0 0.0

16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0 16,900.0

1,892,800 1,825,200 1,757,600 1,690,000 1,622,400 1,554,800 1,487,200 1,419,600 1,352,000 1,284,400 1,216,800 1,149,200 1,081,600 1,014,000 946,400 878,800 811,200 743,600 676,000 608,400 540,800 473,200 405,600 338,000 270,400 202,800 135,200 67,600 0

865,300.0 kN

87,012,000 kN m

1,291 1,245 1,199 1,152 1,106 1,060 1,014 968 922 876 830 784 738 691 645 599 553 507 461 415 369 323 277 230 184 138 92 46 0 79,113 kN

61,688 62,933 64,131 65,284 66,390 67,450 68,465 69,433 70,355 71,230 72,060 72,844 73,581 74,273 74,918 75,518 76,071 76,578 77,039 77,454 77,822 78,145 78,422 78,652 78,837 78,975 79,067 79,113 79,113

4,052,926 4,299,679 4,551,410 4,807,936 5,069,072 5,334,633 5,604,435 5,878,293 6,156,024 6,437,442 6,722,364 7,010,605 7,301,981 7,596,306 7,893,398 8,193,071 8,495,141 8,799,424 9,105,735 9,413,890 9,723,705 10,034,994 10,347,575 10,661,261 10,975,870 11,291,216 11,607,116 11,923,384 11,923,384

Seismic base shear

Seismic overturning moment

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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 GF

10-21

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10-22

Vibration Monitoring, Testing, and Instrumentation

Distribution of lateral forces UBC (2000) speciﬁes the load at the top to be Ft ¼ 0:07TV ¼ 19:778 MN Extracting calculation for level 48, Wi Hi ¼ 16:9 £ 192 ¼ 3:245 £ 103 MN Fx ¼

ðV 2 Ft ÞðW48 H48 Þ ð79:113 2 19:778Þð3:245 £ 103 Þ ¼ 2:21 MN ¼ 52 X 87:012 £ 103 Wi Hi i

Shear force ¼ Shear49 þ F48 ¼ 24:9 þ 2:21 ¼ 27:1 MN Moment ¼ Moment49 þ ðShear49 £ story heightÞ ¼ 347:7 þ ð24:9 £ 4Þ ¼ 447:3 MN m

10.3 10.3.1

Dynamic Effects of Wind Loading on Structures Introduction

The turbulent nature of the wind is characterized by sudden gusts superimposed upon a mean wind velocity. The wind vector at a point may be regarded as the sum of the mean wind vector (static component) and a dynamic component Vðz; tÞ ¼ VðzÞ þ vðz; tÞ

ð10:12Þ

Wind is a phenomenon of great complexity because of the many ﬂow situations arising from the interaction of wind with structures. Wind is composed of a multitude of eddies of varying sizes and rotational characteristics carried along in a general stream of air moving relative to the Earth’s surface. These eddies give wind its gusty or turbulent character. The gustiness of strong winds in the lower levels of the atmosphere largely arises from interaction with surface features. The average wind speed over a time period of the order of 10 min or more tends to increase with height, while the gustiness tends to decrease with height. A further consequence of turbulence is that dynamic loading on a structure depends on the size of the eddies. Large eddies, whose dimensions are comparable with the structure, give rise to well-correlated pressures as they envelop the structure. On the other hand, small eddies result in pressures at various parts of the structure being practically uncorrelated. Eddies generated around a typical structure are shown in Figure 10.12.

(a) Elevation

FIGURE 10.12

(b) Plan

Generation of eddies. (a) Elevation; (b) plan.

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Vibration and Shock Problems of Civil Engineering Structures

10-23

Some structures, particularly those that are tall or slender, respond dynamically to the wind. The bestknown structural collapse due to wind was the Tacoma Narrows Bridge which occurred in 1940 at a wind speed of only about 19 m/sec. It failed after it had developed a joint torsional and ﬂexural mode of oscillation. There are several different phenomena giving rise to dynamic response of structures in wind. These include buffeting, vortex shedding, galloping, and ﬂutter. Slender structures are likely to be sensitive to dynamic response in line with the wind direction as a consequence of turbulence buffeting. Transverse or crosswind response is more likely to arise from vortex shedding or galloping, but may be excited by turbulence buffeting also. Flutter is a coupled motion, often a combination of bending and torsion, and can result in instability. An important problem associated with the wind-induced motion of buildings is concerned with the human response to vibration. At this point, it will sufﬁce to note that humans are surprisingly sensitive to vibration, to the extent that motions may feel uncomfortable even if they correspond to relatively unimportant stresses. The next few sections give a brief introduction to the dynamic response of structures in wind. More details can be found in wind engineering texts (e.g., Sachs, 1978; Holmes, 2001).

10.3.2

Wind Speed

At great heights above the surface of the Earth, where frictional effects are negligible, air movements are driven by pressure gradients in the atmosphere, which in turn are the thermodynamic consequence of variable solar heating of the Earth. This upper level wind speed is known as the gradient wind velocity. Different terrains can be categorized according to the roughness length. Table 10.3 shows the different categories speciﬁed in the Australian/New Zealand wind code, AS/NZS 1170.2 (2002). Closer to the surface, the wind speed is affected by frictional drag of the air over the terrain. There is a boundary layer within which the wind speed varies from almost zero, at the surface, to the gradient wind speed at a height known as the gradient height. The thickness of this boundary layer, which may vary from 500 to 3000 m, depends on the type of terrain, as depicted in Figure 10.13. As can be seen, the gradient height within a large city center is much higher than it is over the sea where the surface roughness is less. In practice, it has been found useful to start with a reference wind speed based on statistical analysis of wind speed records obtained at meteorological stations throughout the country. The deﬁnition of the reference wind speed varies from one country to another. For example, in Australia/New Zealand, it is the 3-sec gust wind speed at a height of 10 m above the ground assuming terrain category 2. Maps of reference wind speeds applying to various countries are usually available. An engineering wind model for Australia has been developed by Melbourne (1992) from the Deaves and Harris (1978) model. This model is based on extensive full-scale data and on the classic logarithmic law in which the mean velocity proﬁle in strong winds applicable in noncyclonic regions (neutral stability conditions) is given by Equation 10.13 " ! !2 !3 !4 # up z z z z z Vz < loge þ 5:75 2 1:88 2 þ 0:25 ð10:13Þ z0 zg zg zg zg 0:4 The numerical values are based on a mean gradient wind speed of 50 m/sec. TABLE 10.3

Terrain Category and Roughness Length ðz0 Þ Terrain Category

Roughness Length ðz0 Þ

Exposed open terrain with few or no obstructions and water surfaces at serviceability wind speeds Water surfaces, open terrain, grassland with few, well-scattered obstructions having heights generally from 1.5 to 10 m Terrain with numerous closely spaced obstructions 3 to 5 m high such as areas of suburban housing Terrain with numerous large, high (10.0 to 30.0 m high) and closely spaced obstructions such as large city centers and well-developed industrial complexes

0.002 0.02

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0.2 2

10-24

Vibration Monitoring, Testing, and Instrumentation

Elevation

Open sea

Open level country

z 0 = 0.002

z 0 = 0.02 FIGURE 10.13

Woodlands, suburbs z 0 = 0.2

City centre z0 = 2

Mean wind proﬁles for different terrains.

For values of z , 30:0 m the z=zg values become insigniﬁcant and the above equation simpliﬁes to Vz <

up z log 0:4 e z0

ð10:14Þ

where: Vz ¼ the design hourly mean wind speed at height z; in m/sec up ¼ the friction velocity sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ surface friction shear stress p u ¼ atmospheric density z ¼ the distance or height above ground, in m zg ¼ the gradient height in meters (the value ranges from 2700 to 4500 m), see Table 10.4 (derived by the authors) up zg ¼ 6 £ 1024 As given in Table 10.3, there is an interaction between roughness length and terrain category, so it is necessary to deﬁne a terrain category to ﬁnd the design hourly wind speeds and gust wind speeds. The link between hourly mean and gust wind speeds is as follows: V ¼ V 1 þ 3:7

sv Vz

ð10:15Þ

where

sv ¼ 2:63hup 0:538 þ 0:09 loge TABLE 10.4

z z0

h16

ð10:16Þ

Roughness Length, Friction Velocity, and Gradient Height

Terrain Category

z0 (m)

up

zg (m)

1 2 3 4

0.002 0.02 0.2 2

1.662 1.910 2.243 2.708

2769 3184 3738 4514

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Vibration and Shock Problems of Civil Engineering Structures

h ¼ 1:0 2

z zg

10-25

! ð10:17Þ

For design, the basic wind speed is classiﬁed into three different speeds as follows: Vs ¼ V20 yr ¼ serviceability limit state design speed having an estimated probability of exceedance of 5% in any one year, for the serviceability limit states Vp ¼ V50 yr ¼ permissible, or working, stress design wind speed and can be obtained directly from Vu using the relation Vp ¼ Vu =ð1:5Þ0:5 Vu ¼ V1000 yr ¼ ultimate limit state design wind speed having an estimated probability of exceedance of 5% in a lifetime of 50 years, for the ultimate limit states Using rigorous analysis incorporating probability distribution of wind speed and direction, basic design wind speeds for different directions and different return periods can be derived. For example, AS/NZS 1170.2 provides a wind direction multiplier, which varies from 0.80 for wind from the east to 1.0 for wind from the west, and having wind speeds up to a 2000-year return period.

10.3.3

Design Structures for Wind Loading

The characteristics of wind pressures are a function of the characteristics of the approaching wind, the geometry of the structure, and the geometry and proximity of the upwind structures. The pressures are not steady, but highly ﬂuctuating, partly as a result of the gustiness of the wind, but also because of local vortex shedding at the edges of the structures themselves. The ﬂuctuating pressures result in fatigue damage to structures, and in dynamic excitation, if the structure happens to be dynamically wind sensitive. The pressures are also not uniformly distributed over the surface of the structure, but vary with position. The complexities of wind loading should be kept in mind when designing a structure. Because of the many uncertainties involved, the maximum wind loads experienced by a structure during its lifetime may vary widely from those assumed in the design. Thus, the failure or nonfailure of a structure in a windstorm cannot necessarily be taken as an indication of the nonconservativeness, or conservativeness, of the wind-loading standard. The standards do not apply to buildings or structures that are of unusual shape or location. Wind loading governs the design of some types of structures, such as tall buildings and slender towers. Experimental wind tunnel data may be used in place of the coefﬁcients given in the code for these structures. 10.3.3.1

Types of Wind Design

Typically, for wind-sensitive structures, three basic wind effects need to be considered: *

*

Environmental wind studies — to study the wind effects on the surrounding environment caused by erecting the structure (e.g., a tall building). This study is particularly important to assess the impact of wind on pedestrians and motor vehicles and so on, which utilize the public domain within the vicinity of the proposed structure. Wind loads for facade — to assess design wind pressures throughout the surface area of the structure to design the cladding system. Owing to the signiﬁcant cost of typical facade systems in proportion to the overall cost of very tall buildings, engineers cannot afford the luxury of conservatism in assessing design wind loads. With due consideration to the complex building shapes and dynamic characteristics of the wind and building structure, even the most advanced wind codes generally cannot accurately assess design loads. Wind tunnel tests to assess design loads for cladding are now a normal industry practice, with the aim of minimizing initial capital costs, and more signiﬁcantly, to avoid the expensive maintenance costs associated with malfunctions due to leakage and/or structural failure.

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10-26 *

Vibration Monitoring, Testing, and Instrumentation

Wind loads for structure — to determine the design wind load so as to design the lateral loadresisting structural system of a structure and therefore satisfy the various design criteria.

10.3.3.2

Design Criteria

In terms of designing a structure for lateral wind loads, the following design criteria need to be satisﬁed: *

*

*

Stability against overturning, uplift, and/or sliding of the structure as a whole. The strength of the structural components of the building, and stresses that must be withstood without failure during the life of the structure. Serviceability, for example for buildings, where interstory and overall deﬂections are within acceptable limits. The control of deﬂection and drift is imperative for tall buildings in order to limit damage and cracking to nonstructural members such as the facade, internal partitions, and ceilings.

As adopted by most international codes, to satisfy stability and strength limit state requirements, ultimate limit state wind speed is used. In many codes, such a speed has a 5% probability of being exceeded in a 1-year period. An additional criterion that requires careful consideration in wind-sensitive structures such as tall buildings is the control of accelerations when subjected to wind loads under serviceability conditions. Acceptability criteria for vibrations in buildings are frequently expressed in terms of acceleration limits for a 1- or 5-year return period wind speed, and are based on human tolerance to vibration discomfort in the upper levels of buildings. Wind response is relatively sensitive to both mass and stiffness, and response accelerations can be reduced by increasing either or both of these parameters. However, this is in conﬂict with earthquake design optimization where loads are minimized in buildings by reducing both the mass and stiffness. Increasing the damping results in a reduction in both the wind and earthquake responses. The detailed procedure described in wind codes is subdivided into static analysis and dynamic analysis methods. The static approach is based on a quasi-steady assumption. It assumes that the building is a ﬁxed rigid body in the wind. The static method is not appropriate for tall or slender structures or structures susceptible to vibration in the wind. In practice, static analysis is normally appropriate for structures up to 50 m in height. The subsequently described dynamic method is for exceptionally tall, slender, or vibration-prone buildings. The codes not only provide some detailed design guidance with respect to dynamic response, but also state speciﬁcally that a dynamic analysis must be undertaken to determine overall forces on any structure with both a height (or length) to breadth ratio greater than ﬁve, and a ﬁrst mode frequency less than one. Wind-loading codes may give the impression that wind forces are relatively constant with time. In reality, wind forces vary signiﬁcantly over short time intervals, with large amplitude ﬂuctuations at highfrequency intervals. The magnitude and frequency of the ﬂuctuations is dependent on many factors associated with the turbulence of the wind and local gusting effects caused by the structure and surrounding environment. To simplify this complex wind characteristic, most international codes have adopted a simpliﬁed approach by utilizing a quasi-steady assumption. This approach simply uses a single value equivalent, static wind pressure, to represent the maximum peak pressure the structure would experience. 10.3.3.3

Static Analysis

This method assumes the quasi-steady approximation. It approximates the peak pressures on the building surfaces by the product of gust dynamic wind pressure and the mean pressure coefﬁcients. The mean pressure coefﬁcients are measured in a wind-tunnel or full-scale tests and are given by pbar =qzðbarÞ : The implied assumption is that the pressures on the building surface (external and internal) faithfully follow the variations in upwind velocity. Thus, it is assumed that a peak value of wind speed is accompanied by a peak value of pressure or load on the structure. The quasi-steady model has been found to be fairly good for small structures.

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10-27

In static analysis, gust wind speed, Vz ; is used to calculate the forces, pressures, and moments on the structure. The main advantages and disadvantages of the quasi-steady/peak gust format can be summarized as follows: *

Advantages: Simplicity. Continuity with previous practice. Pressure coefﬁcients should need little adjustment for different upwind terrain types. Existing meteorological data on wind gusts are used directly. Disadvantages: The approach is not suitable for very large structures, or for those with signiﬁcant dynamic response. The response characteristics of the gust anemometers and the natural variability of the peak gusts tend to be incorporated into the wind load estimates. The quasi-steady assumption does not work well for cases where the mean pressure coefﬁcient is near zero.

*

*

*

*

*

*

*

*

However, the advantages outweigh the disadvantages — certainly for smaller, stiff structures for which the code is mainly intended. The philosophy used in specifying the peak loads in AS/NZS 1170.2 has been to approximate the real values of the extremes. In many cases, this has required the adjustment of the quasi-steady pressures with factors such as area reduction factors and local pressure factors. The dynamic wind pressure at height z is given by qz ¼ 0:6V2z £ 1023

ð10:18Þ

where Vz ¼ the design gust wind speed at height z; in meters per second ¼ VMðz;catÞ Mz Mt Mi V ¼ the basic wind speed The multiplying factors ðMÞ take into account the type of terrain ðMt Þ; height above ground level ðMz Þ; topography, and the importance of the structure ðMi Þ: The above derivation essentially forms the basis of most international codes. The mean base overturning moment Mbar is determined by summing the moments resulting from the net effect of the mean pressure and leeward sides of the structure given by Fz ¼

X cp;e qz Az

Fd ¼

X cd qz Az

or for structures with discrete elements:

ð10:19Þ

where Fz ¼ the hourly mean net horizontal force acting on a structure at height z Cp;e ¼ the pressure coefﬁcients for both windward and leeward surfaces Az ¼ the area of a structure or a part of a structure, at height z; in square meters Fd ¼ the hourly mean drag force acting on discrete elements Cd ¼ the drag force coefﬁcient for an element of the structure

10.3.4

Along and Across-Wind Loading

Not only is the wind approaching a building a complex phenomenon, but the ﬂow pattern generated around a building is complicated by the distortion of the mean ﬂow, the ﬂow

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Vibration Monitoring, Testing, and Instrumentation

separation, the vortex formation, and the wake development. Large wind pressure ﬂuctuations due to these effects occur on the surface of a building. As a result, large aerodynamic loads are imposed on the structural system and intense localized ﬂuctuating forces act on the facade of such structures. Under the collective inﬂuence of these ﬂuctuating forces, a building vibrates in rectilinear and torsional modes, as illustrated in Figure 10.14. The amplitude of such oscillations is dependant on the nature of aerodynamic forces and the dynamic characteristics of the building. 10.3.4.1

Along-Wind Loading

Torsion

Wind Direction

FIGURE 10.14

Along-wind

Across-wind

Wind response directions.

The along-wind loading or response of a building due to the gusting wind can be assumed to consist of a mean component due to the action of the mean wind speed (e.g., the mean hourly wind speed), and a ﬂuctuating component due to wind speed variations from the mean. The ﬂuctuating wind is a random mixture of gusts or eddies of various sizes, with the larger eddies occurring less often (i.e., with a lower average frequency) than smaller eddies. The natural frequency of vibration of most structures is sufﬁciently higher than the component of the ﬂuctuating load effect imposed by the larger eddies. That is, the average frequency with which large gusts occur is usually much less than any of the structure’s natural frequencies of vibration and so they do not force the structure to respond dynamically. The loading due to those larger gusts (which are sometimes referred to as “background turbulence”) can therefore be treated in similar way to that due to the mean wind speed. The smaller eddies, however, because they occur more often, may induce the structure to vibrate at or near one of the structure’s natural frequencies of vibration. This in turn induces a magniﬁed dynamic load effect in the structure which can be signiﬁcant. The separation of wind loading into mean and ﬂuctuating components is the basis of the so-called “gust factor” approach, which is the basis of many design codes. The mean load component is evaluated from the mean wind speed using pressure and load coefﬁcients. The ﬂuctuating loads are determined separately by a method which makes an allowance for the intensity of turbulence at the site, size reduction effects, and dynamic ampliﬁcation (Davenport, 1967; Vickery, 1971). The dynamic response of buildings in the along-wind direction can be predicted with reasonable accuracy by the gust factor approach, provided the wind ﬂow is not signiﬁcantly affected by the presence of neighboring tall buildings or surrounding terrain. 10.3.4.2

Across-Wind Loading

There are many examples of slender structures that are susceptible to dynamic motion perpendicular to the direction of the wind. Tall chimneys, street lighting standards, towers, and cables frequently exhibit this form of oscillation, which can be very signiﬁcant, especially if the structural damping is small. Crosswind excitation of modern tall buildings and structures can be divided into three mechanisms (AS/NZS 1170.2, 2002). These and higher time derivatives are described as follows: 1. The most common source of crosswind excitation is that associated with “vortex shedding.” Tall buildings are bluff (as opposed to streamlined) bodies that cause the ﬂow to separate from the surface of the structure, rather than follow the body contour (Figure 10.15). For a

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Vibration and Shock Problems of Civil Engineering Structures

10-29

particular structure, the shed vortices have a dominant periodicity that is deﬁned by the Strouhal number. Hence, the structure is subjected to a periodic pressure loading, which results in an alternating crosswind force. If the natural frequency of the structure coincides with the shedding FIGURE 10.15 Vortex formation in the wake of a bluff frequency of the vortices, large amplitude object. displacement response may occur, and this is often referred to as critical velocity effect. The asymmetric pressure distribution created by the vortices around the cross section results in an alternating transverse force as they are shed. If the structure is ﬂexible, oscillation will occur transverse to the wind, and the conditions for resonance would exist if the vortex shedding frequency coincided with the natural frequency of the structure. This situation could give rise to very large oscillations and possibly failure. In practice, vertical structures are exposed to a turbulent wind in which both the wind speed and the turbulence level vary with height, so that excitation due to vortex shedding is effectively broadband. Therefore, the term “wake excitation” is used to include all forms of excitation associated with the wake and not just those associated with the critical wind velocity. 2. The “incident turbulence” mechanism refers to the situation where the turbulence properties of the natural wind give rise to changing wind speeds and directions that directly induce varying lift and drag forces and pitching moments on the structure over a wide band of frequencies. The ability of incident turbulence to produce signiﬁcant contributions to crosswind response depends very much on the ability to generate a crosswind (lift) force on the structure as a function of longitudinal wind speed and angle of attack. In general, this means that sections with a high lift curve slope or pitching moment curve slope, such as a streamlined bridge deck section or a ﬂat deck roof, are possible candidates for this effect. 3. Higher derivatives of crosswind displacement: there are three commonly recognized displacement-dependent excitations (i.e., “galloping,” “ﬂutter,” and “lock-in”), all of which are also dependent on the effects of turbulence (turbulence affects the wake development, and hence, the aerodynamic derivatives). Many formulae are available to calculate these effects (Holmes, 2001). Recently, computational ﬂuid dynamics techniques have also been used (Tamura, 1999) to evaluate these effects.

10.3.5

Wind Tunnel Tests

There are many situations in which analytical methods cannot be used to estimate certain types of wind loads and the associated structural response. For example, when the aerodynamic shape of the building is rather uncommon, or the building is very ﬂexible so that its motion affects the aerodynamic forces acting on the building. In such situations, more accurate estimates of wind effects on buildings are obtained through aeroelastic model tests in a boundary-layer wind tunnel. Wind tunnel tests currently being conducted on buildings and other structures can be divided into two types. The ﬁrst is concerned with the determination of wind-loading effects to enable the design of a wind-resistant structure. The second is concerned with the ﬂow ﬁelds induced around the structure, such as its effects on pedestrian comfort and safety at ground level or air intake concentration levels of exhaust pollutants. Wind tunnel studies involve blowing wind on the subject building model and its surrounding at various angles relative to the building orientation, representing the wind directions. This is typically achieved by placing the complete model on a rotating platform within the wind tunnel. Once testing is

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 10.16

Wind tunnel test.

complete for a select direction, the platform is simply rotated by a chosen increment to represent a new wind direction. A typical wind tunnel model is illustrated in Figure 10.16. The design wind speed is based on meteorological data for the given city or area, which are analyzed to produce the required probability distribution of gust wind speeds. By appropriate integration processes and the application of necessary scaling factors, directional wind speeds for the wind tunnel can be determined. Although wind tunnel testing attempts to duplicate a complex problem, the actual models are quite simple and are based on the premise that the fundamental mode of displacement for a structure such as a tall building can be approximated by a straight line. In general terms, it is not necessary to achieve a correct mass density distribution along the building height as long as the mass moment of inertia about the pivot point is the same as the prototype density distribution. The pivot point is typically chosen to obtain a mode shape which provides the best agreement with the calculated fundamental mode shapes of the prototype. Springs are located near the pivot points to achieve the correct frequencies of vibrations in the two fundamental sway modes corresponding to the orthogonal building axis. An electromagnet or oil dashpot provides the model with a damping corresponding to that of the full scale tower. In addition to the stiffness and damping compatibility, it is essential that structural length scale, timescale, and the inertial force are the same between the model and the full structure. Buildings of similar size located in close proximity to the proposed building can cause large increases in across-wind responses. Fortunately, in wind tunnel studies, surroundings comprising existing and/or future buildings can easily be incorporated with relatively minor costs.

10.3.6

Comfort Criteria: Human Response to Building Motion

There are no generally accepted international standards for comfort criteria. A considerable amount of research has been carried out into the important physiological and psychological parameters that affect human perceptions of motion and vibration in the low-frequency range of 0 to 1 Hz encountered in tall buildings. These parameters include the occupant’s expectations and experience, activity, body posture, and orientation; visual and acoustic cues; and the amplitude, frequency, and acceleration of both the translational and rotational motion to which the occupant is subjected. Table 10.5 gives some guidance on the general human perception levels.

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Vibration and Shock Problems of Civil Engineering Structures TABLE 10.5

Human Perception Levels

Level

Acceleration (m/sec2)

1 2 3

,0.05 0.05 to 0.1 0.1 to 0.25

4 5

0.25 to 0.4 0.4 to 0.5

6 7 8

0.5 to 0.6 0.6 to 0.7 .0.85

Effect Humans cannot perceive motion Sensitive people can perceive motion; hanging objects may move slightly The majority of people will perceive motion; the level of motion may affect desk work; long-term exposure may produce motion sickness Desk work becomes difﬁcult or almost impossible; ambulation still possible People strongly perceive motion; it is difﬁcult to walk naturally; standing people may lose their balance Most people cannot tolerate the motion and are unable to walk naturally People cannot walk or tolerate the motion Objects begin to fall and people may be injured

1

Horizontal acceleration m/s

10-31

, Melbourne s (1988) maximum peak horizontal acceleration criteria based on Irwin (1978) and Chen and Robertson (1972), for T = 600 seconds, and return period R years

RETURN PERIODS

< 0.1

10 YEARS 5 YEARS 1 YEAR Irwin's E2 Curve and ISO 6897 (1984) Curve 1, maximum standard deviation horizontal criteria for 10 minutes in 5 years return period for a building.

0.01 0.01

STET

0.1

1

10

Frequency n 0 (Hz) FIGURE 10.17

Horizontal acceleration criteria for occupancy comfort in buildings.

Acceleration limits are a function of the frequency of the vibration felt. Upper limits have been recommended for corresponding frequencies of vibration with the relationship suggested by Irwin (1978). Peak acceleration limits as suggested by Melbourne (1988) and Chen (1987) have been plotted along with the Irwin E2 curve in Figure 10.17. To obtain the peak acceleration, the root-mean-square (rms) value can be multiplied by a peak factor. The peak factor is generally between 3 and 4.

10.3.7

Dampers

The damping in a mechanical or structural system is a measure of the rate at which the energy of motion of the system is dissipated. All real systems have some damping. An example is friction in a bearing. Another example is the viscous damping created by the oil within an automotive shock absorber. In many systems, damping is not helpful and it has to be overcome by the system input. In the case of windsensitive structures such as tall buildings, however, it is beneﬁcial, as damping reduces motion, making the building feel more stable to its occupants. Controlling vibrations by increasing the effective damping can be a cost-effective solution. Occasionally, it is the only practical and economical solution.

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Vibration Monitoring, Testing, and Instrumentation

The types of damping systems that can be implemented include: *

*

*

*

*

*

*

*

Tuned mass damper (TMD; an example is given in Figure 10.18) Distributed viscous dampers Tuned liquid column dampers (TLCD), also known as liquid column vibration absorbers (LCVA) Tuned sloshing water dampers (TSWD) Impact-type dampers Visco-elastic dampers Semiactive dampers Active dampers

While general design philosophy tends to favor passive damping systems due to their lower capital and maintenance costs, active or semiactive dampers may be the ideal solution for certain vibration problems. More details about passive and active systems to control vibrations are given by Soong and Costantinou (1994).

10.3.8

FIGURE 10.18 One of the TMDs designed for the skybridge legs of the Petronas Towers by RWDI Inc. (12 TMDs were installed, three in each of the four legs).

Comparison with Earthquake Loading

Extremes of wind loading, which may be as much as three or four times the loading associated with the mean result, are possible, and a signiﬁcant contribution to this extreme is often supplied by the resonant component in the turbulence of the wind. Resonance refers to a condition in which the periodicity of forcing is identical to that of the structure, with a consequential ampliﬁcation of response that is limited only by the level of damping of the structure. A typical wind contains a wide range of frequency components in its turbulence, so it is always possible that the peak response has a resonant component. Earthquake ground motions are characterized by a series of rather random spikes, with the range of frequencies present (i.e., the range of intervals between zero crossings on the ground acceleration record) being somewhat narrower than for normal wind turbulence. Structures that are stiff will move essentially in unison with the ground motion. For more ﬂexible structures, response is analogous to that from a series of impulses, with the dominant frequency in the response being that of the structure itself. This frequency, the natural frequency of the structure, is dependent on the mass and stiffness of the system. Wind loading depends on exposed area; earthquake loading depends on the (hidden) mass of the structure. Structures attract wind loadings which increase steadily with the major dimension (height or span, say). The earthquake loading experienced by such structures increases much less rapidly, with the result that, for high-rise structures, wind loading is almost always the dominant lateral loading. This assumes elastic responses for both regimes of loading. Wind loading depends on topography and, in urban areas, on the proximity of other buildings. Earthquake loading, on the other hand, depends to a marked degree on the foundation materials. It is universally observed that buildings founded on soft soils perform much worse than those founded on rock. The most important differences between wind and earthquake loading are summarized in Table 10.6.

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Vibration and Shock Problems of Civil Engineering Structures TABLE 10.6

10-33

Main Differences between Wind and Earthquake Loading

Characteristic

Wind

Earthquake

Source of loading

External force due to wind pressures

Applied base motion from ground vibration

Type and duration of loading

Wind storm of several hours duration. Loads ﬂuctuate, but are predominantly in one direction

Transient cyclic loads of at most a few minutes total duration. Loads change repeatedly in direction

Predictability

Good statistical basis is generally available

Poor

Sensitivity of loading to return period

Moderate; þ15% typical for £ 10 on return period

High; maximum credible earthquake often greatly in excess of “design” values

Inﬂuence of local soil conditions

Little effect on dynamic sensitivity

Soil conditions can be very important

Spectral peak input range

Gust: ,0.1 Hz

Usually 1 to 5 Hz

Main factors affecting building response

External shape of building or structure. Generally only global dynamic properties are important. Dynamic considerations affect only a small fraction of building structures

Response is governed by global dynamic properties (fundamental period, damping, and mass) but plan and vertical regularity of structure also important. All structures are affected dynamically

Normal design basis

Elastic response is required

Inelastic response is usually permitted, but ductility must be provided

Design of nonstructural elements

Applied loading is concentrated on external cladding

Entire building contents are shaken and must be appropriately designed

Source: Data from Maguire, J.R. and Wyatt, T.A., 1999. Dynamics — An Introduction for Civil and Structural Engineers, Thomas Telford, London.

10.4

Vibrations Due to Fluid –Structure Interaction

The principles of vibration due to ﬂuids such as water are very similar to those of wind action presented in the previous section. The ﬂuid ﬂow can signiﬁcantly affect the vibrational characteristics of a structure. The presence of a quiescent ﬂuid decreases the natural frequencies and increases the damping of the structure. Similar to wind, a turbulent ﬂuid ﬂow exerts random pressures on a structure, and these random pressures induce a random response leading to large structural deformations or failure. More details on ﬂuid –structure interaction can be found in text books (e.g., Tiejens and Prandtl, 1957; Milne-Thompson, 1968).

10.4.1

Added Mass and Inertial Coupling

A real ﬂuid is viscous and compressible. In contrast, a perfect ﬂuid is nonviscous and incompressible. Fluid damping is absent in perfect ﬂuids, and therefore, for a structure oscillating in a quiescent perfect ﬂuid, the ﬂuid-force component is associated with the ﬂuid inertia called the added mass. This is of practical importance when the ﬂuid density is comparable to the density of the structure, because then the added mass becomes a signiﬁcant fraction of the total mass in dynamic motion. Added mass and ﬂuid damping associated with single and multiple cylindrical structures are discussed in detail by Chen (1987). For example, the added mass for a circular cylinder of radius a and height h is equal to pa2 h: Added masses for different cross-sectional shapes are presented by Milne-Thompson (1968).

10.4.2

Wave-Induced Vibration of Structure

This effect is similar to wind; however, the density is very much larger than wind, thus making the structural damping less effective. Therefore, it is essential to ensure that resonance does not occur.

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The ﬂuid forces which act in line with the Trough Wave crest D direction of wave propagation (Figure 10.19) can be found using a generalized form of Morrison’s z equation comprising both drag (proportional to h area times velocity squared) and inertia forces U d (proportional to immersed volume times acceleration). More details can be found in Muga and l Wilson (1970). Flexible structures will resonate with the wave if the structural natural period equals the wave period or a harmonic of the wave period. Since FIGURE 10.19 A circular cylindrical structure exposed the wave frequencies of importance are ordinarily to ocean waves. less than 0.2 Hz, such a resonance occurs only for exceptionally ﬂexible structures such as offshore platforms. The amplitude of structural response at resonance is a balance between the wave force and the structural stiffness times the damping. The above discussion considers only the in-line forces. These in-line forces produce an in-line response. However, substantial transverse vibrations also occur for ocean ﬂows around circular cylinders. These vibrations are associated with periodic vortex shedding, which was discussed under wind-induced vibration.

10.5

Blast Loading and Blast Effects on Structures

The use of vehicle bombs to attack city centers has been a feature of campaigns by terrorist organizations around the world. A bomb explosion within or very near a building can have catastrophic effects, destroying or severely damaging portions of the building’s external and internal structural frames, collapsing walls, blowing out large expanses of windows, and shutting down critical life-safety systems, such as ﬁre detection and suppression, ventilation, light, water, sewage, and power systems. Loss of life and injuries to occupants can result from many causes, including direct blast effects, structural collapse, debris impact, ﬁre, and smoke. The indirect effects can combine to inhibit or prevent timely evacuation, thereby contributing to additional casualties. In addition, major catastrophes resulting from gaschemical explosions or nuclear leakage result in large dynamic loads, greater than the original design loads, of many structures. Owing to the threat of such extreme loading conditions, efforts have been made during the past three decades to develop methods of structural analysis and design to resist blast loads. The analysis and design of structures subjected to blast loads requires a detailed understanding of blast phenomena and the dynamic response of various structural elements.

10.5.1

Explosions and Blast Phenomenon

An explosion is deﬁned as a large-scale, rapid, and sudden release of energy. Explosions can be categorized on the basis of their nature as physical, nuclear, or chemical events. In physical explosions, energy may be released from the catastrophic failure of a cylinder of compressed gas, volcanic eruptions, or even the mixing of two liquids at different temperatures. In a nuclear explosion, energy is released from the formation of different atomic nuclei by the redistribution of the protons and neutrons within the interacting nuclei; whereas the rapid oxidation of fuel elements (carbon and hydrogen atoms) is the main source of energy in the case of chemical explosions. Explosive materials can be classiﬁed according to their physical state as solids, liquids, or gases. Solid explosives are mainly high explosives, for which blast effects are best known. They can also be classiﬁed on the basis of their sensitivity to ignition as secondary or primary explosive. The latter is one that can be easily detonated by simple ignition from a spark, ﬂame, or impact. Materials such as mercury fulminate and lead azide are primary explosives. Secondary explosives detonate creating blast (shock) waves, which

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10-35

Pressure

Vibration and Shock Problems of Civil Engineering Structures

Shock velocity Distance from explosion FIGURE 10.20

Blast wave propagation.

P(t) result in damage to the surroundings. Examples include trinitrotoluene (TNT) and ammonium nitrate and fuel oil (ANFO). Pso Positive Specific The detonation of a condensed high explosive Impulse generates hot gases under pressure of up to Negative Specific 300 kbar and a temperature of about 3000 to Impulse 40008C. The hot gas expands, forcing out the t tA tA+td Po volume it occupies. As a consequence, a layer of − Pso compressed air (blast wave) forms in front of this gas volume, containing most of the energy released Positive Negative by the explosion. The blast wave instantaneously duration td− duration t d increases to a value of pressure above the ambient atmospheric pressure. This is referred to as the FIGURE 10.21 Blast wave pressure — time history. side-on overpressure, and decays as the shock wave expands outward from the explosion source. After a short time, the pressure behind the front may drop below the ambient pressure (see Figure 10.20 and Figure 10.21). During such a negative phase, a partial vacuum is created and air is sucked in. This is also accompanied by high suction winds that carry the debris for long distances away from the explosion source.

10.5.2

Explosive Air-Blast Loading

The threat for a conventional bomb is deﬁned by two equally important elements, the bomb size, or charge weight, W; and the standoff distance, R; between the blast source and the target (Figure 10.22). For example, the blast that occurred at the basement of the World Trade Center in 1993 had the charge weight of 816.5 kg TNT. The Oklahoma City bomb in 1995 had a charge weight of 1814 kg at a stand off of 4.5 m (Longinow and Mniszewski, 1996). As terrorist attacks may range from a small letter bomb to a gigantic truck bomb, as experienced in Oklahoma City, the mechanics of a conventional explosion and their effects on the target must be addressed. The observed characteristics of air-blast waves are found to be affected by the physical properties of the explosion source. Figure 10.21 shows a typical blast pressure proﬁle. At an arrival time of tA after the explosion, pressure at that position suddenly increases to a peak value of overpressure, Pso ; over the ambient pressure, P0 : The pressure then decays to the ambient pressure at time td until it reaches a partial 2 vacuum of peak underpressure Pso ; and eventually returns to the ambient pressure at time td þ td2 : The quantity Pso is usually referred to as the peak side-on overpressure, incident peak overpressure, or merely the peak overpressure (TM 5-1300, 1990).

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Vibration Monitoring, Testing, and Instrumentation

Over-pressure

Over-pressure (side-on) Reflected Pressure

Blast wave Stand-off distance FIGURE 10.22

Blast loads on a building.

The incident peak overpressure, Pso ; is ampliﬁed by a reﬂection factor as the shock wave encounters an object or structure in its path. Except for the speciﬁc focusing of high-intensity shock waves at near 458 incidence, these reﬂection factors are typically greatest for normal incidence (a surface adjacent and perpendicular to the source) and diminish with the angle of obliquity or angular position relative to the source. Reﬂection factors depend on the intensity of the shock wave. For large explosions at normal incidence these reﬂection factors may enhance the incident pressures by as much as an order of magnitude. Throughout the pressure –time proﬁle, two main phases can be observed; the portion above ambient pressure is called positive phase of duration td ; while that below ambient is called negative phase of duration, td2 : The negative phase is of a longer duration and a lower intensity than the positive duration. The duration of the positive-phase blast wave increases with range, resulting in a lower amplitude, longer duration shock pulse the further a target structure is situated from the burst. Charges situated extremely close to a target structure impose a highly impulsive, high-intensity pressure load over a localized region of the structure; charges situated further away produce a lower-intensity, longer-duration uniform pressure distribution over the entire structure. Eventually, the entire structure is engulfed in the shock wave, with reﬂection and diffraction effects creating focusing and shadow zones in a complex pattern around the structure. During the negative phase, the weakened structure may be subjected to impact by debris that may cause additional damage. If the exterior building walls are capable of resisting the blast load, the shock front penetrates through window and door openings, subjecting the ﬂoors, ceilings, walls, contents, and people within to sudden pressures and fragments from shattered windows, doors, and other ﬁxtures. Building components not capable of resisting the blast wave will fracture and be further fragmented and moved by the dynamic pressure that immediately follows the shock front. Building contents and people will be displaced and tumbled in the direction of blast wave propagation. In this manner the blast will propagate through the building. 10.5.2.1

Blast Wave Scaling Laws

All blast parameters are primarily dependent on the amount of energy released by a detonation in the form of a blast wave and the distance from the explosion. A universal normalized description of the blast effects can be given by scaling distance relative to ðE=P0 Þ1=3 ; and pressure relative to P0 ; where E is the energy release (kJ) and P0 the ambient pressure (typically 100 kN/m2). For convenience, however, it is general practice to express the basic explosive input or charge weight W as an equivalent mass of TNT. Results are then given as a function of the dimensional distance parameter (scaled distance)

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Z ¼ R=W 1=3 ; where R is the actual effective distance from the explosion. W is generally expressed in kilograms. Scaling laws provide parametric correlations between a particular explosion and a standard charge of the same substance. 10.5.2.2

Prediction of Blast Pressure

Blast wave parameters for conventional high-explosive materials have been the focus of a number of studies during the 1950s and 1960s. Estimations of peak overpressure due to a spherical blast based on the scaled distance Z ¼ R=W 1=3 were introduced by Brode (1955) as Pso ¼

6:7 þ 1 bar ðPso . 10 barÞ Z3

0:975 1:455 5:85 Pso ¼ þ þ 3 2 0:019 bar ð0:1 bar , Pso , 10 barÞ Z Z2 Z

ð10:20Þ

Newmark and Hansen (1961) introduced a relationship to calculate the maximum blast overpressure, Pso ; in bars, for a high-explosive charge detonated at the ground surface as Pso ¼ 6784

W W 3 þ 93 R R3

1 2

ð10:21Þ

Another expression of the peak overpressure in kPa was introduced by Mills (1987), in which W is expressed as the equivalent charge weight in kg of TNT, and Z is the scaled distance Pso ¼

1772 114 108 2 2 þ Z3 Z Z

ð10:22Þ

As the blast wave propagates through the atmosphere, the air behind the shock front is moving outward at a lower velocity. The velocity of the air particles, and hence the wind pressure, depends on the peak overpressure of the blast wave. This later velocity of the air is associated with the dynamic pressure, qðtÞ: The maximum value, qs ; say, is given by 2 qs ¼ 5Pso =2ðPso þ 7P0 Þ

ð10:23Þ

If the blast wave encounters an obstacle perpendicular to the direction of propagation, reﬂection increases the overpressure to a maximum reﬂected pressure Pr as Pr ¼ 2Pso

7P0 þ 4Pso 7P0 þ Pso

ð10:24Þ

A full discussion and extensive charts for predicting blast pressures and blast durations are given by TM 5-1300 (1990) and Mays and Smith (1995). Some representative numerical values of peak-reﬂected overpressure are given in Table 10.7. TABLE 10.7

Peak-Reﬂected Overpressures Pr (in MPa) with Different W – R Combinations

R (m)

1 2.5 5 10 15 20 25 30

W 100 kg TNT

500 kg TNT

1000 kg TNT

165.8 34.2 6.65 0.85 0.27 0.14 0.09 0.06

354.5 89.4 24.8 4.25 1.25 0.54 0.29 0.19

464.5 130.8 39.5 8.15 2.53 1.06 0.55 0.33

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2000 kg TNT 602.9 188.4 60.19 14.7 5.01 2.13 1.08 0.63

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Vibration Monitoring, Testing, and Instrumentation

For design purposes, reﬂected overpressure can be idealized by an equivalent triangular pulse of maximum peak pressure Pr and time duration td ; which yields the reﬂected impulse ir : ir ¼

1 Pt 2 rd

ð10:25Þ

The reﬂection effect dissipates as the perturbation propagates to the edges of the obstacle at a velocity related to the speed of sound ðUs Þ in the compressed and heated air behind the wave front. Denoting the maximum distance from an edge as S (for example, the lesser of the height or half the width of a conventional building), the additional pressure due to reﬂection is considered to reduce from Pr 2 Pso to 0 in time 3S=Us : Conservatively, Us can be taken as the normal speed of sound, about 340 m/sec, and the additional impulse to the structure evaluated on the assumption of a linear decay. After the blast wave has passed the rear corner of a prismatic obstacle, the pressure similarly propagates on to the rear face; linear build-up over duration 5S=Us has been suggested. For skeletal structures, the effective duration of the net overpressure load is thus small, and the drag loading based on the dynamic pressure is then likely to be dominant. Conventional wind-loading pressure coefﬁcients may be used, with the conservative assumption of instantaneous build-up when the wave passes the plane of the relevant face of the building, the loads on the front and rear faces being numerically cumulative for the overall load effect on the structure. Various formulations have been put forward for the rate of decay of the dynamic pressure loading; a parabolic decay (i.e., corresponding to a linear decay of equivalent wind velocity) over a time equal to the total duration of positive overpressure is a practical approximation.

10.5.3

Gas Explosion Loading and Effect of Internal Explosions

In the circumstances of a progressive build-up of fuel in a low-turbulence environment, typical of domestic gas explosions, ﬂame propagation on ignition is slow and the resulting pressure pulse is correspondingly extended. The speciﬁc energy of combustion of a hydrocarbon fuel is very high (46,000 kJ/kg for propane, compared with 4520 kJ/kg for TNT) but widely differing effects are possible according to the conditions at ignition. Internal explosions often produce complex pressure loading proﬁles as a consequence of having two loading phases. The ﬁrst results from the blast overpressure reﬂection and, due to the conﬁnement provided by the structure, re-reﬂection will occur. Depending on the degree of conﬁnement of the structure, the conﬁned effects of the resulting pressures may cause different degrees of damage to the structure. On the basis of the conﬁnement effect, target structures can be described as either vented or unvented. The latter must be stronger to resist a speciﬁc explosion yield than a vented structure where some of the explosion energy would be dissipated by the breaking of window glass or fragile partitions. Generally, venting following the failure of windows (typically at 7 kN/m2) greatly reduces the peak values of internal pressures. A study of this problem at the Building Research Establishment (Ellis and Crowhurst, 1991) showed that an explosion fuelled by a 200 ml aerosol canister in a typical domestic room produced a peak pressure of 9 kN/m2 with a pulse duration over 0.1 sec. This is long by comparison with the natural frequency of wall panels in conventional building construction, and a quasi-static design pressure is commonly advocated. Much higher pressures with a shorter timescale are generated in turbulent conditions. Suitable conditions arise in buildings in multiroom explosions on the passage of the blast through doorways, but can also be created by obstacles closer to the release of the gas. These may be presumed to occur on a release of gas due to a failure of industrial pressure vessels or pipelines.

10.5.4

Structural Response to Blast Loading

Complexity in analyzing the dynamic response of blast-loaded structures involves the effect of high strain-rates, nonlinear inelastic material behavior, uncertainties of blast-load calculations, and timedependent deformations. Therefore, to simplify the analysis, a number of assumptions related to the response of structures and loads have been proposed and widely accepted. To establish the principles of

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10-39

this analysis, the structure is idealized as an single-DoF system and the link between the positive duration of the blast load and the natural period of vibration of the structure is established. This leads to blast-load idealization and simpliﬁes the classiﬁcation of the blast-loading regimes. 10.5.4.1

Elastic Single-Degree-of-Freedom Systems

The simplest discretization of transient problems is by means of the single-DoF approach. The actual structure can be replaced by an equivalent system of one concentrated mass and one weightless spring, representing the resistance of the structure against deformation. Such an idealized system is illustrated in Figure 10.23. The structural mass, M; is under the effect of an external force, FðtÞ; and the structural resistance, R; is expressed in terms of the vertical displacement, y; and the spring constant, K: The blast load can also be idealized as a triangular pulse having a peak force Fm and positive-phase duration td (see Figure 10.23). The forcing function is given as FðtÞ ¼ Fm 1 2

t td

ð10:26Þ

The blast impulse is approximated as the area under the force–time curve, and is given by I¼

1 F t 2 md

ð10:27Þ

The equation of motion of the undamped elastic single-DoF system for a time ranging from 0 to the positive-phase duration, td ; is given by Biggs (1964) as t M y€ þ Ky ¼ Fm 1 2 ð10:28Þ td The general solution can be expressed as Displacement yðtÞ ¼

Fm F ð1 2 cos vtÞ þ m K Ktd

sin vt 2t v

ð10:29Þ

dy F 1 ¼ m v sin vt þ ðcos vt 2 1Þ Velocity y_ ðtÞ ¼ dt K td

in which v is the natural circular frequency of vibration of the structure and T is the natural period of vibration of the structure given as rﬃﬃﬃﬃﬃ 2p K ¼ v¼ ð10:30Þ T M The maximum response is deﬁned by the maximum dynamic deﬂection ym ; which occurs at time tm : The maximum dynamic deﬂection ym can be evaluated by setting dy=dt in Equation 10.29 equal to zero, that is, when the structural velocity is zero. The dynamic load factor (DLF) is deﬁned as the ratio of the maximum dynamic deﬂection ym to the static deﬂection yst which would have resulted from the static F(t)

Force F(t)

Fm

M Stiffness, K

Displacement y(t) (b)

(a) FIGURE 10.23

Time td

(a) Single-DoF system and (b) blast loading.

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application of the peak load Fm ; as follows: DLF ¼

ymax y t ¼ max ¼ cðvtd Þ ¼ C d yst Fm =K T

ð10:31Þ

The structural response to blast loading is signiﬁcantly inﬂuenced by the ratio td =T or vtd ðtd =T ¼ vtd =2pÞ: Three loading regimes are categorized as follows: 1. vtd , 0:4: impulsive loading regime 2. vtd , 0:4: quasi-static loading regime 3. 0:4 , vtd , 40: dynamic loading regime 10.5.4.2

Elasto-Plastic Single-Degree-of-Freedom Systems

Structural elements are expected to undergo Resistance large inelastic deformation under a blast load or Ru high-velocity impact. The exact analysis of dynamic response is then only possible by a step-by-step numerical solution requiring nonlinear dynamic ﬁnite-element software. However, ye ym Deflection the degree of uncertainty in both the determination of the loading, and the interpretation of FIGURE 10.24 Simpliﬁed resistance function of an acceptability of the resulting deformation, is elasto-plastic single-DoF system. such that the solution of a postulated equivalent ideal elasto-plastic single-DoF system (Biggs, 1964) is commonly used. Interpretation is based on the required ductility factor m ¼ ym =ye (Figure 10.24). For example, a uniform simply supported beam has ﬁrst mode shape fðxÞ ¼ sin px=L and the equivalent mass is M ¼ ð1=2ÞmL; where L is the span of the beam and m is mass per unit length. The equivalent force corresponding to a uniformly distributed load of intensity p is F ¼ ð2=pÞpL: The response of the ideal bilinear elasto-plastic system can be evaluated in closed form for the triangular load pulse comprising rapid rise and linear decay, with maximum value Fm and duration td. The result for the maximum displacement is generally presented in chart form (TM 5-1300) as a family of curves for

ym /ye

100

0.1

0.2

0.3 0.4 0.5 0.6 0.7

0.8

50

0.9

10

1.0

5 1.2 1.5 2.0

1 0.5 Numbers next to curves are Ru /Fm 0.1 0.1

FIGURE 10.25

0.5

1

td / T

5

10

20

Maximum response of an elasto-plastic single-DoF system to a triangular load (TM 5-1300).

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selected values of Ru =Fm showing the required ductility m as a function of td =T; in which Ru is the structural resistance of the beam and T is the natural period (Figure 10.25).

10.5.5

Material Behaviors at High Strain Rate

Blast loads typically produce very high strain-rates in the range of 100 to 10,000 sec21. This high straining (loading) rate would alter the dynamic mechanical properties of target structures and, accordingly, the expected damage mechanisms for various structural elements. For reinforced concrete structures subjected to blast effects, the strength of concrete and steel reinforcing bars can increase signiﬁcantly due to the strain-rate effect. Figure 10.26 shows approximate ranges of the expected strain rates for different loading conditions. It can be seen that the ordinary static strain rate is located in the range of 1026 to 1025 sec21, while blast pressures normally yield loads associated with strain rates in the range of 100 to 10,000 sec. 10.5.5.1

Dynamic Properties of Concrete under High Strain Rates

The mechanical properties of concrete under dynamic loading conditions can be quite different from that under static loads. While the dynamic stiffness does not change very much compared with the static stiffness, the stresses that are sustained for a certain period under dynamic conditions may gain values that are remarkably higher than the static compressive strength (Figure 10.27). Strength magniﬁcation factors as high as four in compression and up to six in tension for strain rates in the range of 100 to 1000 sec21 have been reported (Grote et al., 2001). Quasi-static 10−6

10−5

Impact

Earthquake 10−4

10−3

10−2

10−1

100

101

Blast 102

102

103

104

Strain rate (s−1)

Stress (MPa)

FIGURE 10.26

Strain rates associated with different types of loading.

250

. e = 264

200

. e = 233 . e = 97

150 Static 100

. e = 49

50

0 0

0.002

0.004

Strain

0.006

0.008

0.01

FIGURE 10.27 Stress– strain curves of concrete at different strain rates. (Source: Data from Ngo,T. et al., Proc. 18th Australasian Conf. on Mechanics of Structures and Materials, Perth, Australia. With permission.)

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Vibration Monitoring, Testing, and Instrumentation

For the increase in peak compressive stress ðf 0c Þ; a dynamic increase factor (DIF) is introduced in the Comite´ Euro-International du Be´ton (CEBFIP; 1990) model (Figure 10.28) for strain-rate enhancement of concrete as follows: DIF ¼

1_ 1_s

DIF ¼ g

1:026a

1_ 1_s

1=3

for 1_ # 30 sec21

ð10:32Þ

for 1_ . 30 sec21

ð10:33Þ

where

8 Dynamic factor

10-42

6 4 2 0 1.E−04

1.E−02 1.E+00 1.E+02 Strain rate (s−1)

1.E+04

FIGURE 10.28 Dynamic increase factor for peak stress of concrete (CEB-FIP model).

1_ ¼ strain rate 1_s ¼ 30 £ 1026 sec21 (quasi-static strain rate) log g ¼ 6:156a 2 2 a ¼ 1=ð5 þ 9f 0c =fco Þ fco ¼ 10 MPa ¼ 1450 psi 10.5.5.2

Dynamic Properties of Reinforcing Steel under High Strain Rates

Owing to the isotropic properties of metallic materials, their elastic and inelastic response to dynamic loading can easily be monitored and assessed. Norris et al. (1959) tested steel with two different static yield strengths (330 and 278 MPa) under tension at strain rates ranging from 1025 to 0.1 sec21. Strength increases of 9–21% and 10–23% were observed for the two steel types, respectively. Dowling and Harding (1967) conducted tensile experiments using the tensile version of the Split Hopkinton’s Pressure Bar (SHPB) on mild steel using strain rates varying between 1023 and 2000 sec21. It was concluded from this test series that materials of body-centered cubic (BCC) structure (such as mild steel) showed the greatest strain rate sensitivity, the lower yield tensile strength of mild steel was almost doubled, the ultimate tensile stress was increased by about 50%, the upper yield tensile strength considerably increased, and the ultimate tensile strain decreased by different percentages, depending on the strain rate. Malvar (1998) also studied the strength enhancement of steel reinforcing bars under the effect of high strain rates. This was described in terms of the DIF, which can be evaluated for different steel grades and for yield stresses, fy ; ranging from 290 to 710 MPa as DIF ¼

1_ 1024

a

ð10:34Þ

where for calculating yield stress a ¼ afy ;

afy ¼ 0:074 2 0:04 ð fy =414Þ

ð10:35Þ

and for ultimate stress calculation a ¼ afu

afu ¼ 0:019 2 0:009 ð fy =414Þ

10.5.6

ð10:36Þ

Failure Modes of Blast-Loaded Structures

Blast-loading effects on structural members may produce both local and global responses associated with different failure modes. The type of structural response depends mainly on the loading rate, the orientation of the target with respect to the direction of the blast wave propagation, and boundary conditions. The general failure modes associated with blast loading can be ﬂexure, direct shear, or punching shear. Local responses are characterized by localized breaching and spalling, and generally

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result from the close-in effects of explosions, while global responses are typically manifested as ﬂexural failure. 10.5.6.1

Global Structural Behavior

The global response of structural elements is generally a consequence of transverse (out of plane) loads with long exposure time (quasi-static loading), and is usually associated with global membrane (bending) and shear responses. Therefore, the global response of above-ground reinforced concrete structures subjected to blast loading is referred to as membrane/bending failure. The second global failure mode to be considered is shear failure. It has been found that under the effect of both static and dynamic loads, four types of shear failure can be identiﬁed: diagonal tension, diagonal compression, punching shear, and direct (dynamic) shear (Woodson, 1993). The ﬁrst two types are common in reinforced concrete elements under static loads, while punching shear is associated with local shear failure; for example, the familiar case is column punching through ﬂat slabs. These shear response mechanisms have relatively minor structural effect in case of blast loading and can be neglected. The fourth type of shear failure is direct (dynamic) shear. This failure mode is primarily associated with transient short duration dynamic loads that result from blast effects, and it depends mainly on the intensity of the pressure waves. The associated shear force is many times higher than the shear force associated with ﬂexural failure modes. The high shear stresses may lead to a direct global shear failure and it may occur very early (within a few milliseconds of shock wave arrival to the facing structure’s surface) even prior to any signiﬁcant bending deformations. 10.5.6.2

Localized Structural Behavior

The close-in effect of an explosion may cause localized shear or ﬂexural failure in the closest structural elements. This depends mainly on the distance between the explosion center and the target, and the relative strength/ductility of the structural elements. The localized shear failure takes the form of localized punching and spalling, which produces low and high-speed fragments. The punching effect is frequently referred to as breaching, which is well known in high-velocity impact applications and in the case of explosions close to the surface of structural members. Breaching failures are typically accompanied by spalling and scabbing of concrete covers, as well as fragments and debris (Figure 10.29).

FIGURE 10.29 Tuan Ngo).

Breaching failure due to a close-in explosion of 6000 kg TNT equivalent (photograph by

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Impulse is (kPa.sec)

101

100

(I) − Severe damage 10−1 (II) − No damage / minor damage 10−2 100

FIGURE 10.30

10.5.6.3

101 102 Pressure Ps (kPa)

103

Typical pressure– impulse (P– I) diagram.

Pressure –Impulse (P–I ) Diagrams

The P–I diagram is an easy way to mathematically relate a speciﬁc damage level to a combination of blast pressures and the corresponding impulses for a particular structural element. An example P–I diagram is given in Figure 10.30. This ﬁgure shows the levels of damage of a structural member, in which region (I) corresponds to severe structural damage and region (II) refers to no or minor damage. There are other P–I diagrams that are concerned with human responses to blasts, in which three categories of blast-induced injury are identiﬁed as primary, secondary, and tertiary injury (Baker et al., 1983).

10.5.7

Blast Wave– Structure Interaction

The structural behavior of an object or structure exposed to such a wave may be analyzed by dealing with two main issues. Firstly, blast-loading effects, that is, forces that result from the action of the blast pressure; secondly, the structural response, or the expected damage criteria associated with such loading effects. It is important to consider the interaction of the blast waves with target structures. This might be quite complicated in the case of complex structural conﬁgurations. However, it is possible to consider some equivalent simpliﬁed geometry. Accordingly, in analyzing the dynamic response to blast loading, two types of target structures can be considered: diffraction-type and drag-type structures. As these names imply, the former would be affected mainly by diffraction (engulﬁng) loading and the latter by drag loading. It should be emphasized that actual buildings will respond to both types of loading and the distinction is made primarily to simplify the analysis. The structural response will depend upon the size, shape, and weight of the target, how ﬁrmly it is attached to the ground, and also on the existence of openings in each face of the structure.

10.5.8

Effect of Ground Shocks

Above ground or shallow-buried structures can be subjected to ground shock resulting from the detonation of explosive charges that are on, or close to, the ground surface. The energy imparted to the

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ground by the explosion is the main source of ground shock. A part of this energy is directly transmitted through the ground as direct-induced ground shock, while part is transmitted through the air as airinduced ground shock. Air-induced ground shock results when the air-blast wave compresses the ground surface and sends a stress pulse into the ground underlayers. Generally, motion due to air-induced ground shock is maximum at the ground surface and attenuates with depth (TM 5-1300, 1990). The direct-induced shock results from the direct transmission of explosive energy through the ground. For a point of interest on the ground surface, the net experienced ground shock results from a combination of both the air-induced and direct-induced shocks. 10.5.8.1

Loads from Air-Induced Ground Shock

To overcome complications of predicting actual ground motion, one-dimensional wave propagation theory has been employed to quantify the maximum displacement, velocity, and acceleration in terms of the already known blast wave parameters (TM 5-1300). The maximum vertical velocity at the ground surface, Vv ; is expressed in terms of the peak incident overpressure, Pso ; as Vv ¼

Pso r Cp

ð10:37Þ

where r and Cp are, respectively, the mass density and the wave seismic velocity in the soil. By integrating the vertical velocity in Equation 10.37 with time, the maximum vertical displacement at the ground surface, Dv ; can be obtained as is 1000rCp

Dv ¼

ð10:38Þ

Accounting for the depth of soil layers, an empirical formula is given by TM 5-1300 to estimate the vertical displacement in meters so that Dv ¼ 0:09W 1=6 ðH=50Þ0:6 ðPso Þ2=3

ð10:39Þ

9

where W is the explosion yield in 10 kg and H is the depth of the soil layer in meters. 10.5.8.2

Loads from Direct Ground Shock

As a result of the direct transmission of the explosion energy, the ground surface experiences vertical and horizontal motions. Some empirical equations were derived (TM 5-1300) to predict the direct-induced ground motions in three different ground media; dry soil, saturated soil, and rock media. The peak vertical displacement in m/sec at the ground surface for rock, DVrock and dry soil, DVsoil are given as DVrock ¼

0:25R1=3 W 1=3 Z 1=3

ð10:40Þ

DVsoil ¼

0:17R1=3 W 1=3 Z 2:3

ð10:41Þ

The maximum vertical acceleration, Av ; in m/sec2 for all ground media is given by Av ¼

10.5.9

1000 W 1=8 Z 2

ð10:42Þ

Technical Design Manuals for Blast-Resistant Design

This section summarizes applicable military design manuals and computational approaches to predicting blast loads and the responses of structural systems. Although the majority of these design guidelines were focused on military applications, this knowledge is relevant for civil design practice.

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Structures to Resist the Effects of Accidental Explosions, TM 5-1300 (U.S. Departments of the Army, Navy, and Air Force, 1990): This manual appears to be the most widely used publication by both military and civilian organizations for designing structures to prevent the propagation of explosion, and to provide protection for personnel and valuable equipment. It includes step-by-step analysis and design procedures, including information on such items as (1) blast, fragment, and shock-loading; (2) principles of dynamic analysis; (3) reinforced and structural steel design; and (4) a number of special design considerations, including information on tolerances and fragility, as well as shock isolation. Guidance is provided for the selection and design of security windows, doors, utility openings, and other components that must resist blast and forced-entry effects. A Manual for the Prediction of Blast and Fragment Loadings on Structures, DOE/TIC-11268 (U.S. Department of Energy, 1992): This manual provides guidance to the designers of facilities subject to accidental explosions and aids in the assessment of the explosion-resistant capabilities of existing buildings. Protective Construction Design Manual, ESL-TR-87-57 (Air Force Engineering and Services Center, 1989): This manual provides procedures for the analysis and design of protective structures exposed to the effects of conventional (nonnuclear) weapons, and is intended for use by engineers with a basic knowledge of weapons effects, structural dynamics, and hardened protective structures. Fundamentals of Protective Design for Conventional Weapons, TM 5-855-1 (U.S. Department of the Army, 1986): This manual provides procedures for the design and analysis of protective structures subjected to the effects of conventional weapons. It is intended for use by engineers involved in designing hardened facilities. The Design and Analysis of Hardened Structures to Conventional Weapons Effects (DAHS CWE, 1998): This new joint services manual, written by a team of more than 200 experts in conventional weapons and protective structures engineering, supersedes U.S. Department of the Army TM 5-855-1, Fundamentals of Protective Design for Conventional Weapons (1986), and Air Force Engineering and Services Centre ESL-TR-87-57, Protective Construction Design Manual (1989). Structural Design for Physical Security — State of the Practice Report (Conrath et al., 1995): This report is intended to be a comprehensive guide for civilian designers and planners who wish to incorporate physical security considerations into their designs or building retroﬁt efforts.

10.5.10

Computer Programs for Blast and Shock Effects

Computational methods in the area of blast effects mitigation are generally divided into those used for the prediction of blast loads on the structure and those for the calculation of structural responses to the loads. Computational programs for blast prediction and structural response use both ﬁrst-principle and semiempirical methods. Programs using the ﬁrst-principle method can be categorized into uncouple and couple analyses. The uncouple analysis calculates blast loads as if the structure (and its components) were rigid, and then applies these loads to a responding model of the structure. The shortcoming of this procedure is that, when the blast ﬁeld is obtained with a rigid model of the structure, the loads on the structure are often overpredicted, particularly if signiﬁcant motion or the failure of the structure occurs during the loading period. For a coupled analysis, the blast simulation module is linked with the structural response module. In this type of analysis, the computational ﬂuid mechanics (CFD) model for blast-load prediction is solved simultaneously with the computational solid mechanics (CSM) model for structural response. By accounting for the motion of the structure while the blast calculation proceeds, the pressures that arise due to the motion and failure of the structure can be predicted more accurately. Examples of this type of computer software are AUTODYN, DYNA3D, LS-DYNA, and ABAQUS. Table 10.8 provides a listing of computer programs that are currently being used to model blast effects on structures. Prediction of the blast-induced pressure ﬁeld on a structure and its response involves highly nonlinear behavior. Computational methods for blast-response prediction must therefore be validated by comparing calculations to experiments. Considerable skill is required to evaluate the output of the

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10-47

Examples of Computer Programs Used to Simulate Blast Effects and Structural Response

Name

Purpose and Type of Analysis

Author/Vendor

BLASTX CTH FEFLO FOIL SHARC DYNA3D ALE3D LS-DYNA Air3D CONWEP AUTODYN ABAQUS

a

SAIC Sandia National Laboratories SAIC Applied Research Associates, Waterways Experiment Station Applied Research Associates, Inc. Lawrence Livermore National Laboratory (LLNL) Lawrence Livermore National Laboratory (LLNL) Livermore Software Technology Corporation (LSTC) Royal Military Science College, Cranﬁeld University U.S. Army Waterways Experiment Station Century Dynamics ABAQUS Inc.

a

Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Blast prediction, CFD Structural response, CFD (coupled analysis) Coupled analysis Structural response, CFD (coupled analysis) Blast prediction, CFD Blast prediction (empirical) Structural response, CFD (coupled analysis) Structural response, CFD (coupled analysis)

CFD, computational ﬂuid mechanics.

computer software, both as to its correctness and its appropriateness to the situation modeled; without such judgment, it is possible through a combination of modeling errors and poor interpretation to obtain erroneous or meaningless results. Therefore, successful computational modeling of speciﬁc blast scenarios by engineers unfamiliar with these programs is difﬁcult, if not impossible.

10.6

Impact Loading

Impact effects on structures arise over a very broad range of circumstances, from high-velocity missiles or aircraft impact to high-mass ship or vehicle collisions. The requirement may be for the structure to withstand the impact without serious damage, or major inelastic deformation may be permitted.

10.6.1 Structural Impact between Two Bodies — Hard Impact and Soft Impact Impact loads differ from blast loads in duration, and they are applied to a localized area. Blast loads propagate as a wave front, while an impact load is caused by the force resulting from the collision between a moving object and a structure. Impact loading can be classiﬁed as either hard or soft, depending upon the relative characteristics of the impactor and the target structure. Impulsive loading can be considered to be a special case of soft impact. Soft impact occurs when the impactor deforms substantially with respect to a hard structure, and a portion of the impactor’s kinetic energy is absorbed by the impactor’s plastic deformation. For hard impact, the striking object is rigid and the kinetic energy is transmitted to the target and absorbed by deformation and damage in the structure. Impact problems essentially involve all three fundamental conservation laws: conservation of mass, conservation of momentum, and conservation of energy. These three laws are outlined in the following equations (Zukas, 1990), respectively ð r dV ¼ const ð10:43Þ v

where

F ¼ m dv=dt X1 2 X X1 2 X rvi ¼ Ef þ rv þ W Ei þ 2 2 f

r ¼ material density V ¼ volume F ¼ force

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ð10:44Þ ð10:45Þ

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m ¼ mass v ¼ velocity E ¼ stored internal energy W ¼ work i, f ¼ initial and ﬁnal states Upon impact, stresses and strains are induced in the target material. The layers of particles in the target are compressed upon contact, creating compressive stress. When the compression stress between two layers is equal to the applied pressure, compression supports the entire pressure. Through this process, stress waves are developed similar to the shock waves generated by blast loading. The stress waves propagate throughout the material at a speed inherent to that material and reﬂect multiple times as interfaces are reached. Various types of stress waves are developed, depending on the energy imparted into the target. The impact velocity determines the strain rate, mode of response, and the type of impact damage (Zukas et al., 1982). If the impact is below a certain level, only elastic stress waves are generated. Higher velocity impacts create inelastic stress waves. Historically, impact has been considered a localized phenomenon that may cause plastic deformation and/or failure of the target and/or the impactor. During an impact event, some or all of the kinetic energy of the impactor is transferred to the target. This process is a function of the wave propagation in the target, the impactor’s deformation of the target upon contact, and the contact velocity. Because the impact has been considered to be localized, the local behavior deformation and penetration has been the prime consideration. Impact causes elastic and plastic stress waves, and propagation through the structural thickness can cause failure by spalling. Such effects usually occur within microseconds of the impact, and may be referred to as the early time response. The overall dynamic response of the structure usually occurs on a timescale several orders of magnitude longer, and can thus reasonably be decoupled from the early time response and subjected to an initial check against spalling. Impact imparts impulsive loadings to a structure, producing responses within the structure. Three different types of solutions to the impact problem are available: theoretical (analytical), semiempirical, and numerical. Theoretical methods provide closed-form solutions for the governing partial differential

FIGURE 10.31 Transient deformation of a reinforced concrete beam under impact at midspan. (Source: Data from Ngo, T. et al., Proc. of 18th Australasian Conference on the Mechanics of Structures and Materials, Perth, Australia. 2004a. With permission.)

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equations. Semiempirical methods rely on extensive test data to produce a curve-ﬁt solution for a class of similar impact problems. Numerical solutions replace the continuous system with discrete domains and treat the problem as it progresses over time (Figure 10.31).

10.6.2

Example — Aircraft Impact

Design loads resulting from aircraft impacts are governed by the absorption of kinetic energy from the aircraft by the building at its maximum deﬂection. These loads are limited by the yield, buckling, and crushing of the aircraft. Total impact load FðtÞ at the interface of the collapsing aircraft and the building is given by FðtÞ ¼ Fc þ m½mðtÞ VðtÞ

ð10:46Þ

in which mðtÞ is the mass of the aircraft reaching the building per unit time; m is a coefﬁcient for change in momentum (which can be taken conservatively as one); Fc is the crushing load, a constant which can be determined from the design acceleration for failure of the aircraft; and VðtÞ is the velocity of the aircraft. Figure 10.32 compares the impact loads produced by a Boeing 707-320 and a Boeing 767, which hit the World Trade Center. It should be noted that World Trade Center was designed to resist the equivalent impact of a Boeing 707. Figure 10.33 compares the impact loads produced by different aircraft. The peak loads and the duration of loading for different aircraft are given in Table 10.9. These loads were calculated by the method suggested by Kar in 1979 (Mendis and Ngo, 2002). Kinsella and Jowett (1981) suggested a more accurate method in which the crash event is treated as a combination of the separate time-dependent impacts of the aircraft’s frame and engines. The frame is classed as a soft missile which will suffer considerable deformation, and a ﬁnite difference method of calculation is employed to describe its perfectly plastic impact. The engines, which are considered separately, are assumed to constitute a much harder missile which will undergo little deformation. The results obtained by this method for a Phantom F4 aircraft are shown in Figure 10.34. This method gave a maximum load of 233 MN compared with the 145 MN obtained from Kar’s method.

Boeing 767, V = 140 m/s

Impact Load, P (kN × 10 3)

300

200

100

Boeing 707,V = 100 m/s

t (s) 0.1 FIGURE 10.32

0.2

0.3

0.4

Impact load– time history for aircraft impact.

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Load-time history

700

Concorde

Impact load, P (MN)

600 500 400

Boeing 767

300 200

F4

100 0

Light aircraft

0

50

FIGURE 10.33

TABLE 10.9

B707 Helicopter

100 150 time (ms)

200

250

Comparison of impact loads for different aircraft.

Examples of Aircraft and Peak Impact Loads

Aircraft

Mass (kg)

Length (m)

Velocity V0 (m/sec)

Peak Load (MN)

Duration (msec)

Aust. SUPAPUP light aircraft Westland Sea King helicopter Boeing 707-320 Phantom F4 aircraft Boeing 767-300 ER Supersonic Concorde

340 9,500 91,000 22,000 187,000 138,000

5.7 17 40 19.2 54.9 62.2

51.3 63.9 103.6 210 140 344

4.6 19.6 92 145 320 568

111 266 386 91 362 181

Predicted Load-time history 250

233

F (t) in MN

200 Air Frame 150 Engine 100 50

Total impact

49.1

0 0

20

40 60 time in ms

80

100

FIGURE 10.34 Impact loads of Phantom F4 aircraft. (Source: Data from Kinsella, K. and Jowett, J. 1981. The Dynamic Load Arising from a Crashing Military Combat Aircraft, Safety and Reliability Directorate, Wigshaw, U.K. With permission.)

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10.7 10.7.1

10-51

Floor Vibration Introduction

Annoying ﬂoor vibrations may be caused by occupant activities. Walking, dancing, jumping, aerobics, and audience participation at music concerts and sporting events are some prime examples of occupant activities that create ﬂoor vibrations. The operation of mechanical equipment is another cause for concern. Heating, ventilation, and air-conditioning systems, if not properly isolated, can cause serious vibration problems. The current trend towards longer spans and lighter ﬂoor systems has resulted in a signiﬁcant increase in the number of ﬂoor vibration complaints by building owners and occupants. Most of the sources contributing to reported human discomfort rest on the ﬂoor system itself. However, human activities or machinery off a ﬂoor can cause signiﬁcant ﬂoor vibrations. On more than one occasion, aerobics on one ﬂoor of a high-rise building has been reported to cause vibration discomfort on another level in the building. The vibrations caused by automobiles on parking levels below have been reported to disrupt sensitive laboratory work on upper ﬂoors. Other equipment and activities off the ﬂoor that can contribute to a ﬂoor vibration problem are ground or air trafﬁc, drilling, the impact of falling objects, and other construction-related events. When the natural frequency of a ﬂoor system is close to a forcing frequency and the deﬂection of the system is signiﬁcant, motion will be perceptible, and perhaps even annoying. Perception is related to the activity of the occupants: a person at rest or engaged in quiet work will tolerate less vibration than a person performing an active function, such as dancing or aerobics. If a ﬂoor system dissipates the imparted energy in a very short period of time, the motion is likely to be perceived as less annoying. Thus, the damping characteristics of the system affect acceptability. In design guidelines for ﬂoor vibration analysis, limits are stated as a minimum natural frequency of a structural system. These limits depend on the permissible peak accelerations (as a fraction of gravitational acceleration) on the mass affected by an activity, the environment in which the vibration occurs, the effectiveness of interaction between connected structural components, and the degree of damping, among other factors. Recently, excessive ﬂoor vibrations have become a common problem due to a decrease in the natural frequency at which buildings vibrate due, in turn, to increased ﬂoor spans and a decrease in the amount of damping and mass used in standard construction practice, because of the availability of stronger and lighter materials. Some methods have been developed in the recent past to check the ﬂoor vibrations of structures. These methods are summarized in this section. More details can be found in the texts given in the list of references.

10.7.2

Types of Vibration

10.7.2.1

Walking

A walking person’s foot touching the ﬂoor causes a vibration of the ﬂoor system. This vibration may be annoying to other persons sitting or lying in the same area, such as an ofﬁce, a church, or a residence. Although more than one person may be walking in the same area at the same time, their footsteps are normally not synchronized. Therefore, the analysis is based on the effect of the impact of the individual walking. 10.7.2.2

Rhythmic Activities

In some cases, more than a few people may engage in a coordinated activity that is at least partially synchronized. Spectators at sporting events, rock concerts, and other entertainment events often move in unison in response to music, a cheer, or other stimuli. In these cases, the vibration is caused by many people moving together, usually at a more or less constant tempo. The people disturbed by the vibration may be those participating in the rhythmic activity, or those in nearby part of the structure engaged in a more quiet activity. The people engaged in the rhythmic activity

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have higher level of tolerance for the induced vibrations, while those nearby will have a lower level of tolerance. 10.7.2.3

Mechanical Equipment

Mechanical equipment may produce a constant impulse at a ﬁxed frequency, causing the structure to vibrate. 10.7.2.4

Analysis Methods

Because the nature of the input varies for these three types of loads, each of the three requires a somewhat different solution. However, all cases require knowledge of an important response parameter of the ﬂoor system, its natural frequency of vibration, and all three analysis methods are based on ﬁnding a required minimum frequency.

10.7.3

Natural Frequency of Vibration

The natural frequency of a ﬂoor system is important for two reasons. It determines how the ﬂoor system will respond to forces causing vibrations. It is also important in determining how human occupants will perceive the vibrations. It has been found that certain frequencies set up resonance with internal organs of the human body, making these frequencies more annoying to people. Figure 10.35 shows the human sensitivity over a range of frequencies during various activities. The human body is most sensitive to frequencies in the range of 4 to 8 Hz. This range of natural frequencies is commonly found in typical ﬂoor systems. Recommended acceleration limits for vibrations due to rhythmic activities are given in Table 10.10. Rhythmic activities, outdoor footbridges

10.00 5.00

Indoor footbridges, shopping malls, dining and dancing

Peak Acceleration (% gravity)

3.00 1.50

Offices, residences, churches

1.00

Operating rooms

0.50 0.25

ISO Baseline curve of RMS acceleration for human reaction

0.10 0.05 1

2

4 8 Frequency (Hz)

12

20

FIGURE 10.35 Recommended permissible peak vibration acceleration levels acceptable for human comfort while in different environments. (Source: Data from Mast, R.F., Vibration of precast prestressed concrete ﬂoors, PCI J., Nov– Dec, 2001. With permission.)

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TABLE 10.10 Recommended Acceleration Limits for Vibrations Due to Rhythmic Activities Occupancies Affected by the Vibration

Acceleration Limit (%g)

Ofﬁce and residential Dining and weightlifting Rhythmic activity only

0.4–0.7 1.5–2.5 4–7

Source: Data from Alen, D.E., Building vibration from human activities, Concr. Int., 66–73, 1990.

10.7.3.1

Computing the Natural Frequency

The natural frequency of a vibrating beam is determined by the ratio of its stiffness to its mass (or weight). The deﬂection of simple-span beam is also dependent on its weight and stiffness. A simple relationship exists between the self-weight deﬂection and the natural frequency of a uniformly loaded simple-span beam on rigid supports: qﬃﬃﬃﬃﬃ fn ¼ 0:18 g=Dj ð10:47Þ where fn ¼ natural frequency in the fundamental mode of vibration g ¼ acceleration due to gravity Dj ¼ instantaneous simple-span deﬂection of ﬂoor panel due to dead load plus actual live load 10.7.3.2

Computing Deﬂection

The equation for the deﬂection Dj for a uniformly loaded simple-span beam is Dj ¼

5wl 4 384EI

ð10:48Þ

where l ¼ span length of member I ¼ gross moment of inertia, for prestressed concrete members Many vibration problems are more critical when the mass (or weight) is low. For continuous spans of equal length, the natural frequency is the same as for simple spans. This may be understood by examining Figure 10.36. For static loads, all spans deﬂect downward simultaneously, and continuity signiﬁcantly reduces the deﬂection. But for vibration, one span deﬂects downward while the adjacent spans deﬂect upward. An inﬂection point exists at the supports, and the deﬂection and natural frequency are the same as for a simple span. For unequal continuous spans, and for partially continuous spans with supports, the natural frequency may be increased by a small amount.

Simple Span

Inflection point at support

Continuous Spans FIGURE 10.36

Natural frequency of simple and continuous spans.

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Vibration Monitoring, Testing, and Instrumentation

Damping

Damping determines how quickly a vibration will decay and die out. This is important because human perception and tolerance of vibration or motion is dependent on how long it lasts. Damping of a ﬂoor system is highly dependent on the nonstructural items (partitions, ceilings, furniture, and other items) present. The modal damping ratio of a bare structure undergoing low-amplitude vibrations can be very low, on the order of 1%. Nonstructural elements may increase this damping ratio up to 5% (see Table 10.11). It must be appreciated that the results of a vibration analysis are highly inﬂuenced by the choice of the assumed damping, which can vary widely. 10.7.3.4

Resonance

Resonance occurs when the frequency of a forcing input nearly matches the natural frequency of a system. In order to avoid excessive ampliﬁcation of vibration, the natural frequency must be higher than the frequency of the input forces by an amount related to the damping of the ﬂoor system.

10.7.4

Vibration Caused by Walking

Vibrations caused by walking can often be objectionable in lighter constructions of wood or steel. Because of the greater mass and stiffness of concrete ﬂoor systems, vibrations caused by walking are seldom a problem in these systems. However, when designing concrete ﬂoor systems of long span, the serviceability requirement on vibrations may become critical. 10.7.4.1

Minimum Natural Frequency

People are most sensitive to vibrations when engaged in sedentary activities while seated or lying. Much more vibration is tolerated by people who are standing, walking, or active in other ways. Thus, different criteria are given for ofﬁces, residences, and churches than for shopping malls and footbridges. An empirical formula, based on the resonant effects of walking, has been developed to determine the minimum natural frequency of a ﬂoor system needed to prevent disturbing vibrations caused by walking: fn $ 2:861 ln where

K bW

ð10:49Þ

K ¼ a constant, given in Table 10.11 b ¼ modal damping ratio W ¼ weight of area of ﬂoor panel affected by a point load

10.7.5

Design for Rhythmic Excitation

Rhythmic excitation may occur when a group of people exercise or respond to a musical beat. Because a group is acting in unison at a constant frequency, the input forces are much more powerful than those produced by random walking. Resonance can occur when the input frequency is at or near the TABLE 10.11

Values of K and b

Occupancies Affected by the Vibration Ofﬁces, residences, churches Shopping malls Outdoor footbridges

b

K (kN) 58 20 8

a

0.02 ; 0.03b; 0.05c 0.02 0.01

a For ﬂoors with few nonstructural components and furnishings, open work areas, and churches. b For ﬂoors with nonstructural components and furnishings, and cubicles. c For ﬂoors with full-height partitions.

Source: Data from Alen, D.E. and Murray, T.M., Design criterion for vibrations due to walking, Am. Inst. Steel Const. Eng. J., Fourth Quarter, 117–129, 1993

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fundamental frequency of vibration, and so the fundamental frequency of the ﬂoor must be sufﬁciently higher than the input frequency as to prevent resonance. 10.7.5.1

Harmonics

A harmonic of frequency is any higher frequency that is equal to the fundamental frequency multiplied by an integer. For instance, if the frequency of an input excitation is 2.5 Hz, the harmonics are 2.5 £ 2 ¼ 5 Hz, 2.5 £ 3 ¼ 7.5 Hz, and so on. If the fundamental frequency of a ﬂoor system is equal to a harmonic of the exciting frequency, resonance may occur. This process is less efﬁcient than one which is in resonance striking at each cycle of vibration. Nevertheless, the 2.5 Hz forcing frequency can cause resonance in the 5 Hz fundamental frequency due to the input force striking every second cycle in the fundamental frequency. Higher harmonics should not be confused with higher modes of vibration. The second mode of vibration of a simple span has a frequency four times the fundamental frequency. This high a frequency is almost never excited. Harmonics refers to the forcing frequency, compared with the fundamental mode of vibration. 10.7.5.2

Minimum Natural Frequency

The following design criterion for minimum natural frequency for a ﬂoor subjected to rhythmic excitation is based on the dynamic response of the ﬂoor system to dynamic loading. The objective is to avoid the possibility of being close to a resonant condition: sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ k ai wp fn $ f 1 þ ð10:50Þ a0 =g wt where f ¼ forcing frequency ¼ (i) ( fstep) fstep ¼ step frequency i ¼ number of harmonic ¼ 1, 2, 3 k ¼ a dimensionless constant (1.3 for dancing, 1.7 for a lively concert or sport events, 2.0 for aerobics) ai ¼ dynamic coefﬁcient a0 =g ¼ ratio of peak acceleration limited to the acceleration due to gravity Wp ¼ effective distributed weight per unit area of participants Wt ¼ effective total distributed weight per unit area of participants (weight of participants 1 3 5 2 4 plus weight of ﬂoor system). The natural frequency of the ﬂoor system, fn, can be found as discussed previously.

10.7.6 Example — Vibration Analysis of a Reinforced Concrete Floor A concrete ﬂoor of a tall building is analyzed in this example. The plan view and structural conﬁguration of the building are shown in Figure 10.37. Perimeter columns are spaced at 12 m centers and are connected by spandrel beams to support the facade. The example ﬂoor will be used for aerobic exercises and needs to be checked for vibration. Aerobic exercises are usually undertaken in the range of 2 to 2.75 Hz, with a maximum value in the order of 3.0 Hz. Ideally aerobic

12 m

FLOOR PLAN

FIGURE 10.37

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Structural conﬁguration.

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exercise ﬂoors should be designed so that the ﬂoor’s natural frequency exceeds the third harmonic by a factor of 1.2, resulting in fn . 1:2 £ 3 £ 2:75 ¼ 9:9 Hz. This is not always achievable in practice, especially for long span ﬂoors that have the natural frequency in the range from 4 to 8 Hz. Hence, a ﬂoor with natural frequency greater than 7.5 Hz is considered a minimum standard, although in some cases ﬂoor vibrations may be quite noticeable. The modal analysis of the ﬂoor system was carried out with the assumed damping factor b of 2% (see Table 10.11). It was found that the ﬂoor natural frequency is 6.75 Hz, which may result in some problems in ﬂoor vibration. To reduce the vibration problem the following approaches can be used: *

*

*

Reduce mass (normally not very effective) Increase damping (e.g., using dampers) Reduce vibration transmission (stiffening joists at columns may reduce transmission)

Bibliography AASHTO Guide Speciﬁcations for Seismic Isolation Design. American Association of State Highway and Transportation Ofﬁcials (AASHTO), Washington, 1991. Alen, D.E., Building vibration from human activities, Concr. Int., 66 –73, 1990. Alen, D.E. and Murray, T.M., Design criterion for vibrations due to walking, Am. Inst. Steel Const. Eng. J., Fourth Quarter, 117 –129, 1993. Asian Model Concrete Code (ACMC), International Committee on Concrete Model Code for Asia (ICCMC), 2001. AS/NZS 1170.2, Structural Design Actions — Wind, Standards Australia/New Zealand, 2002. Baker, W.E., Cox, P.A., Westine, P.S., Kulesz, J.J., and Strehlow, R.A. 1983. Explosion Hazards and Evaluation, Elsevier, New York. Biggs, J.M. 1964. Introduction to Structural Dynamics, McGraw-Hill, New York. Booth, E., Ed. 1994. Concrete Structures in Earthquake Regions, Longmans, Harlow, Essex. Brode, H.L., Numerical solution of spherical blast waves, J. Appl. Phys., 26, 766–775, 1955. Burdekin, M., Ed. 1996. Seismic Design of Steel Structures after Northridge and Kobe, SECED/Institution of Structural Engineers, Rotterdam. Casciati, F. 1996. Active control of structures in European seismic areas. In Proceedings of the Eleventh World Conference on Earthquake Engineering, Pergamon Press, Oxford. Chen, P.W. and Robertson, L.E., Human perception thresholds of horizontal motion, J. Struct. Div., 92, 8, 1681 –1695, 1972. Chen, S.S. 1987. Flow-Induced Vibration of Circular Cylindrical Structures, Hemisphere, Washington. Chopra, A.K. 1981. Dynamics of Structures, A Primer, Earthquake Engineering Research Institute, Oakland, CA. Chopra, A.K. 2001. Dynamics of Structures: Theory and Applications to Earthquake Engineering, Prentice Hall, Upper Saddle River, NJ. Chopra, A.K. and Goel, R.K., Capacity –demand-diagram methods based on inelastic design spectrum, Earthquake Spectra, 15, 4, 637 –656, 1999. Clough, R.W. and Penzien, J. 1993. Dynamics of Structures, McGraw-Hill, New York. Collins, J.A. 1993. Failure of Materials in Mechanical Design, 2nd ed., Wiley Interscience, New York. Comite´ Euro-International du Be´ton 1990. CEB-FIP Model Code 1990, Redwood Books, Trowbridge, Wiltshire. Conrath, E.J., Krauthammer, T., Marchand, K.A. and Mlakar, P.F. 1995. Structural Design for Physical Security — State of the Practice Report, American Society of Civil Engineers, ASCE, New York. DAHS CWE 1998. Technical Manual — Design and Analysis of Hardened Structures to Conventional Weapons Effects, U.S. Army Corps of Engineers (CEMP-ET), Washington. Davenport, A.G., Gust loading factors, J. Struct. Div., 93, 11 –34, 1967. Deaves, D.M. and Harris, R.I. 1978. A Mathematical Model of the Structure of Strong Winds, Report 76, Construction Industry Research and Information Association, London, U.K.

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DOE/TIC-11268. 1992. A Manual for the Prediction of Blast and Fragment Loadings on Structures, U.S. Department of Energy, Washington. Dowling, A.R. and Harding, J. 1967. Tensile properties of mild steel under high strain rates. In Proceedings of the First HERF Conference, University of Denver, Denver. Ellis, B.R. and Crowhurst, D. 1991. The response of several LPS maisonettes to small gas explosions. IStructE/BRE Seminar: Structural Design for Hazardous Loads: The Role of Physical Tests. ENV. 1998. Design Provisions for Earthquake Resistance of Structures, CEN (European Committee for Standardisation), Brussels (various dates), ENV 1994-8. ESL-TR-87-57. 1989. Protective Construction Design Manual, U.S. Air Force Engineering and Services Center. Fajfar, P., A nonlinear analysis method for performance-based seismic design, Earthquake Spectra, 16, 3, 573–592, 2000. FEMA 306/307/308 - NEHRP. 1998. Evaluation and Repair of Earthquake Damaged Concrete and Masonry Buildings, USA. FEMA 273/274 - NEHRP. 2000. Guidelines for Seismic Rehabilitation of Buildings, 1997 Edition, FEMA 356/357, USA. FEMA 350-354. 2000. FEMA 368/369 - NEHRP. 2000 Recommended Provisions for Seismic Regulations for New Buildings and Other Structures, USA. Grote, D., Park, S., and Zhou, M., Dynamic behaviour of concrete at high strain rates and pressures, J. Impact Eng., 25, 869 –886, 2001. Gulkan, P. and Sozen, M.A., Inelastic responses of reinforced concrete structures to earthquake motions, J. Am. Concrete Inst., 71, 12, 604–610, 1974. Gupta, A.K. 1990. Response Spectrum Method in Seismic Analysis and Design of Structures, Blackwell, Oxford. Holmes, J. 2001. Wind Loading of Structures, Spon Press, London. Irwin, A.W., Human response to dynamic motion of structures, Struct. Engineer, 56A, 1978. Izzudin, B.A. and Smith, D.L. 1996. Response of offshore structures to explosion loading. In Proceedings of the Sixth Offshore and Polar Engineering Conference, pp. 323– 330, Los Angeles, CA. Kabori, T. 1996. Active and hybrid structural control research in Japan. In Proceedings of the Eleventh World Conference on Earthquake Engineering, Pergamon Press, Oxford. Kar, K., Impactive effects of tornado missiles and aircraft, J. Struct. Div., 105 (11), 2243 –2260, 1979. Kinsella, K. and Jowett, J. 1981. The Dynamic Load Arising from a Crashing Military Combat Aircraft, Safety and Reliability Directorate, Wigshaw, U.K. Krauthammer, T. 1994. Buildings subjected to short duration dynamic effects. In Proceedings of First Cradington Conference, November, Building Research Establishment, Bedford, England. Longinow, A. and Mniszewski, K.R. 1996. Protecting buildings against vehicle bomb attacks. In Practice Periodical on Structural Design and Construction, pp. 51 –54, ASCE, New York. Maguire, J.R. and Wyatt, T.A. 1999. Dynamics — An Introduction for Civil and Structural Engineers, Thomas Telford, London. Malvar, L.J., Review of static and dynamic properties of steel reinforcing bars, ACI Mater. J., 95(5), 609–616, 1998. Mast, R.F., Vibration of precast prestressed concrete ﬂoors, PCI J., Nov–Dec, 2001. Mays, C.G. and Smith, P.D. 1995. Blast Effects on Buildings, TTL, London. Melbourne, W. 1988, Deﬁnition of wind pressure on tall buildings, In Proceedings of the Third International Conference on Tall Buildings, Chicago, USA. Melbourne, W. 1992. Towards an engineering wind model. In Course Notes on Wind Engineering, Monash University, Melbourne. Mendis, P.A. and Ngo, T. 2002. Assessment of tall buildings under blast loading and aircraft impact. In Toward a Better Built Environment, Innovation, Sustainability and Information Technology, International Association of Bridge and Structural Engineering, Australia.

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Mills, C. A. 1987. The design of concrete structure to resist explosions and weapon effects. In Proceedings of the First International Conference on Concrete for Hazard Protections, pp. 61 –73. Milne-Thompson, L.M. 1968. Theoretical Hydrodynamics, McMillan, New York. Muga, B. and Wilson, J. 1970. Dynamic Analysis of Ocean Structures, Plenum Press, New York. Newmark, N.M. and Hansen, R.J. 1961. Design of blast resistant structures. In Shock and Vibration Handbook, Vol. 3, Harris, C.M. and Crede, C.E., Eds., McGraw-Hill, New York. Ngo, T., Mendis, P., Hongwei, M., and Mak, S. 2004a. High strain rate behaviour of concrete cylinders subjected to uniaxial compressive impact loading. In Proceedings of 18th Australasian Conference on the Mechanics of Structures and Materials, Perth, Australia. Ngo, T., Mendis, P., and Itoh, A. 2004b. Modelling reinforced concrete structures subjected to impulsive loading using the Modiﬁed Continuum Lattice Model. In Proceedings of 18th Australasian Conference on the Mechanics of Structures and Materials, Perth, Australia. Norris, G.H., Hansen, R.J., Holly, M.J., Biggs, J.M., Namyet, S., and Minami, J.K. 1959. Structural Design for Dynamic Loads, McGraw-Hill, New York. Pappin, J.W. 1991. Design of foundation and soil structures for seismic loading. In Cyclic Loading of Soils, M.P. O’Reilly and S.F. Brown, eds., pp. 306– 366, Blackie, London. Park, R. and Paulay, T. 1975. Reinforced Concrete Structures, Wiley, New York. Paulay, T. and Priestley, M.J.N. 1992. Seismic Design of Reinforced Concrete and Masonry Buildings, Wiley, New York. Paulay, T. 1993. Simplicity and conﬁdence in seismic design. In Fourth Mallet-Milne Lecture, Wiley, Chichester. Penelis, G. and Kappos, A. 1996. Earthquake Resistant Concrete Structures, E. & F.N. Spon, London. Priestley, M.J.N. and Kowalsky, M.J., Direct displacement-based seismic design of concrete buildings, Bull. N.Z. Soc. Earthquake Eng., 33 (4), 421 –444, 2000. Sachs, P. 1978. Wind Forces in Engineering, Pergamon Press, Oxford. Soong, T.T. 1996. Active research control in the US. In Proceedings of the 11th World Conference on Earthquake Engineering, Pergamon Press, Oxford. Soong, T. and Costantinou, M., Eds. 1994. International Centre for Mechanical Sciences, Springer, Wien. Tamura, T., Reliability on CFD estimation for wind– structure interaction problems, J. Wind Eng. Ind. Aerodyn., 81, May– Jul, 117 –143, 1999. Tiejens, O.G. and Prandtl, L. 1957. Applied Hydro & Aeromechanics, Dover Publications, New York. TM 5-1300. 1990. The Design of Structures to Resist the Effects of Accidental Explosions, Technical Manual. U.S. Department of the Army, Navy, and Air Force, Washington DC. TM 5-855-1. 1986. Fundamentals of Protective Design for Conventional Weapons, U.S. Department of the Army, Washington. Uniform Building Code, Vol. 2. Structural Engineering Design Provisions, International Conference Building Ofﬁcials, Whittier, 2000. Vickery, B.J., On the reliability of gust loading factors, Civil Eng. Trans., 13, 1 –9, 1971. Wolf, J.P. 1985. Dynamic Soil–Structure Interaction, Prentice Hall, Englewood Cliffs, NJ. Wolf, J.P. 1994. Foundation Vibration Analysis Using Simple Physical Models, Prentice Hall, Englewood Cliffs, NJ. Wood, H.O. and Neumann, Fr., Modiﬁed Mercalli intensity scale of 1931, Bull. Seis. Soc. Am., 21, 277 –283, 1931. Woodson, S.C. 1993. Response of slabs: in plane forces and shear effects. In Structural Concrete Slabs under Impulsive Loads, T. Krauthammer, Ed., pp. 51 –68. Research Library, U.S. Army Engineer Waterways Experiment Station, Vicksburg, MS. Yamada, M., Goto, T. 1975. The criteria to motions in tall buildings. In Proceedings of Pan Paciﬁc Tall Buildings Conference, pp. 233–244, Hawaii. Zukas, J.A. 1990. High Velocity Impact Dynamics, Wiley, New York. Zukas, J.A., Nicholas, S.T. and Swift, H.F. 1982. Impact Dynamics. Wiley, New York.

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11

Seismic Base Isolation and Vibration Control 11.1 Introduction ........................................................................ 11-1 From Ductility Design to Base Isolation and Control Design The Importance of Reducing Seismic Input and Response

†

11.2 Seismic Base Isolation ......................................................... 11-4

Hirokazu Iemura Kyoto University

Sarvesh Kumar Jain

Madhav Institute of Technology and Science

Mulyo Harris Pradono Kyoto University

Historical Development of Base Isolation † Basic Principle † Issues in Seismic Base Isolation † Seismic Isolation Devices † Design of Isolation Devices † Veriﬁcation of Properties of Isolation Systems † Analysis of Base-Isolated Structures † Experimental Methods for Isolated Structures † Implementation of Seismic Isolation † Performance during Past Earthquakes

11.3 Seismic Vibration Control ................................................. 11-33 Historical Development of Seismic Vibration Control † Basic Principles † Important Issues in Vibration Control † Vibration-Control Devices † Control Algorithm † Experimental Performance Veriﬁcation † Implementations

Summary This chapter presents seismic vibration control of civil engineering structures. It is divided into two main sections. The ﬁrst part of the chapter deals with vibration control by seismic base isolation, whereas the second part covers methods of response control that use passive energy dissipation, active control, semiactive control, and hybrid control, respectively. Each part starts with a brief description of the historical development of these methods, followed by basic principles and important issues in their implementation. Thereafter, devices used for structural response control, their design methods, and recommended experimental procedures for veriﬁcation of their properties and analytical modeling are discussed. Various methods generally used for analysis of such structures are then discussed in detail followed by a brief description of the implementation of these methods for various types of structures. In addition, performance of existing structures during past earthquakes is also included, to highlight the effectiveness of these methods during real earthquakes. Further information on the general topic of this chapter is found in Chapter 12 and Chapter 13.

11.1

Introduction

Seismic isolation and vibration-control systems are relatively new and sophisticated concepts that require more extensive design and detailed analysis than do most conventional seismic designs of structures. In general, these systems will be most applicable to structures whose designers seek superior earthquake performance. Seismic base isolation and passive energy-dissipation systems are viable design strategies that have already been used for seismic protection of a number of structures. Other special seismic protective system techniques such as active control, semiactive control, hybrid combinations of active and passive devices, and tuned mass and liquid dampers may also provide practical solutions in the 11-1 © 2007 by Taylor & Francis Group, LLC

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near future. An innovative challenge is highly expected in this ﬁeld for the seismic safety enhancement of civil structures.

11.1.1

From Ductility Design to Base Isolation and Control Design

The conventional method of seismic design mainly deals with increasing capacity. The approach is based on designing a strong and ductile structure Plastic hinges in a ductile structure, (see Figure 11.1), which can take care of the inertial enabling the whole forces generated by the earthquake shaking. The structural system to approach results in increasing the size of structural absorb seismic energy members and connections, and providing additional bracing members and shear walls, or other stiffening members. The increase in stiffness Ground excitation then attracts more seismic forces and in turn requires further strengthening, which becomes FIGURE 11.1 Schematic of a structure with ductile uneconomical. Therefore, the conventional pracmembers. tice permits safe design of a structure on the premise that inelastic action in a ductility-based designed structure will dissipate signiﬁcant energy and enable it to survive a severe earthquake without collapse. The conventional designs may permit some structural damage because of inelastic deformation in the members and also in nonstructural elements. Contents of structure can get damaged due to large interstory drift and high-ﬂoor accelerations. It is difﬁcult to control structural damage and it may be dangerous in unexpected strong seismic events. It has been observed that, in the event of major seismic events, structures based on the conventional design methods suffered damage, experienced high-ﬂoor accelerations, and resulted in disruption of essential services such as transportation, communication, and so on. Thus, for the class of structures like nuclear power plants, museums, hospital buildings, buildings with artifacts, important bridges, and such structures located in high-seismicity regions, this ductility-based design is not suitable. The need to minimize earthquake damage in critical and important structures prompted civil engineers to search for other methods of earthquake-resistant designs, which can not only protect structures from earthquake motions but also keep them functional during and after strong earthquakes. To this end, base isolation and structural control methods are found to be a solution. Base isolation has the capability to reduce the seismic response of a structure by isolating it from the ground shaking (Figure 11.2a). An isolation system reduces the transmission of ground vibration, thus enabling the structure to experience less shaking from the ground. Therefore, structural damage and occupants’ inconvenience can be minimized using this technique. However, at the expense of safety and the convenience of structure, the bearings undergo signiﬁcant drift during large earthquakes that may disrupt the function of the bearings themselves and supply lines of services such as water and gas. Another way of reducing seismic response is by using the structural control method. It has the capability of modifying the structural properties, such as stiffness, mass, and damping, and providing passive or active counterforces. Figure 11.2b shows the schematic diagram of the structural control method in a civil structure. It shows some examples of devices generally used for applying control forces. The seismic safety enhancement of structures using the structural control method can be categorized as active and passive systems. There are also hybrid systems that represent combinations of active and passive, and semiactive systems to represent active controller that employs controllable passive devices.

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11-3

Moving Mass Control

Joint Damper

Variable Damping Hysteretic Type Damping Active Varying Stiffness

Structure response

Base Isolation Bearings Ground shaking

(a) FIGURE 11.2

(b)

Schematic of a civil structure with (a) isolation bearings and (b) the structural control method.

Owing to changes in code provisions or upgradation of seismic zones, many structures come into the category of “unsafe” and require retroﬁtting. Response control strategies are found to be easier than other options, economical, and are often the only alternative for such cases.

11.1.2

The Importance of Reducing Seismic Input and Response

As mentioned above, by using the conventional method of seismic design, the design may permit some structural damage because of inelastic deformation in the members, and also in nonstructural elements, during large earthquakes. The ductility enables the structure as a whole to absorb the seismic energy. Once the structural response goes deeply into the plastic range during a large earthquake, structures may not be operational or repairable. If the seismic input to the structure and structural response can be reduced, then the structural damage can be minimized. For higher reliability of structures even under very severe earthquake motion, structural control techniques that effectively reduce seismic force to structures are developed. The fast development of technology, particularly in the ﬁelds of electronics and computer science, has provoked the researchers in some centers worldwide to intensify development of a new concept with the new philosophy of seismic design. Generally, this concept is known as a design of intelligent structures or smart structures. Owing to the experience of severe damage due to the Kobe earthquake, public demand for seismic performances of civil infrastructures became relatively clear in Japan. Civil infrastructures are constructed with the tax paid by the public, so a collapse or near collapse with unrepairable damage cannot be accepted, even under a very rare earthquake loading. Infrastructures are also expected to serve as public tools to help rehabilitate the affected society. For this purpose, infrastructures have to be repaired in a relatively short time, even though their functions are temporarily terminated due to severe earthquake loading.

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Vibration Monitoring, Testing, and Instrumentation

Earthquake Performance Level Fully Operational Near collapse Operational Life safe Frequent (T= 43 years) Earthquake Occasional Design (T= 72 years) Level Rare (T= 475 years)

Unacceptable Performance (for New Construction)

Very rare (T= 970 years)

FIGURE 11.3

Public demand for seismic performance of infrastructure.

The public demand for seismic performance objectives of infrastructures shows that structural damage has to be limited even against very rare earthquake loading (Figure 11.3). The ﬁgure shows that civil infrastructures must be fully operational during and after frequent, weak earthquakes. They also expected to be operational even after very rare, strong earthquakes. To achieve the objectives, new technologies are to be developed that can result in the desired performance.

11.2 11.2.1

Seismic Base Isolation Historical Development of Base Isolation

Seismic base isolation is not a very new idea. More than a century ago, John Milne, a professor of engineering in Japan, built a small house of wood and placed it on ball bearings to demonstrate that a structure could be isolated from earthquake shaking (Housner et al., 1997). In 1891, after the Narobi earthquake ðM ¼ 8:0Þ; Kawai, a Japanese person, proposed a base-isolated structure with timber logs placed in several layers in the longitudinal and transverse direction (Izumi, 1988; see Figure 11.4). In 1906, Jacob Bechtold of Germany applied for a U.S. patent in which he proposed to place building on rigid plate, supported on spherical bodies of hard material (Buckle and Mayes, 1990). In 1909, a medical doctor from England, Calentarients, applied for patents for his invention comprising FIGURE 11.4 Base isolation by timber logs. (Source: isolation system for earthquake-proof building JSSI, Introduction of Base Isolated Structures, Japan (see Figure 11.5). He proposed separating a Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With building from its foundation with a layer of sand permission.) or talc (Kelly, 1986). The Imperial Hotel in Tokyo, constructed in 1921, was intended to ﬂoat on an underlying layer of mud. The building was founded on an 8-ft thick layer of ﬁrm soil under which exists a 60- to 70-ft thick layer of soft mud. The building was

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11-5

FIGURE 11.5 Calentarients’ idea of seismic base isolation. (Source: JSSI, Introduction of Base Isolated Structures, Japan Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With permission.)

highly decorative with many appendages. The soft mud acted as isolation system and the building survived the devastating 1923 Tokyo earthquake (Kelly, 1986; Buckle and Mayes, 1990). Attempts were made in the 1930s to protect the upper ﬂoors of multistory buildings by designing very ﬂexible ﬁrst-story columns. In a later modiﬁcation of the ﬂexible ﬁrst-story columns approach, it was proposed that the ﬁrst-story columns should be designed to yield during an earthquake to produce an energy-absorbing action. However, to produce enough damping, several inches of displacement is required, and a yielded column has greatly reduced buckling load, proving the concept to be impractical. It was then proposed that, if the soft ﬁrst story is constructed underground, then energy dissipaters can be installed at the top of this story (which approximates ground level) that prevent the superstructure from moving too far, and dissipate the energy of ground motion before it enters the building. The superstructure, from the ﬁrst ﬂoor up, can be an economically braced, nonductile concrete frame requiring no internal shear walls (Arnold, 2001). To overcome the inherent dangers of soft supports at the base, many types of roller-bearing systems have been proposed. The rollers and the spherical bearings are very low in damping and have no inherent resistance to lateral load, and therefore some other mechanism that provides wind restraint and energy-absorbing capacity is needed. A long duration between two successive earthquakes can result in the cold welding of bearings and plates, thus causing the system to become rigid after a time. Therefore, the application of rolling supports was restricted to the isolation of special components of low or moderate weight (Caspe, 1984). Parallel to the development of the soft ﬁrst-story approach, the ﬂexibility of natural rubber was also seen as another solution for increasing the ﬂexibility of the system. In 1968, large blocks of hard rubber, 54 in number, were used to isolate the three-story Heinrich Pestalozzi School in Skopje, Republic of Macedonia. The building is constructed of reinforced concrete shear walls. This is the ﬁrst building for which rubber bearings were used as base isolation against strong earthquakes. These rubber blocks are unreinforced and bulge sideways under the weight of this concrete structure (see Figure 11.6). Owing to having the same stiffness of the isolation system in all the directions, the building bounces and rocks FIGURE 11.6 Unreinforced rubber blocks. (Source: backwards and forwards (Jurukovski and Rakice- Ohashi, U.G. Earthquakes and Base Isolation, Pub. Asakura, Tokyo, 1995. With permission.) vic, 1995). These types of bearings are unsuitable for the earthquake protection of structures. The subsequent development of laminated rubber bearings has made base isolation a practical reality (Figure 11.7). Later, a large number of isolation devices were developed, and now base isolation has reached the stage of gaining acceptance and replacing the conventional construction, at least for important structures. FIGURE 11.7 Rubber bearings with steel shims.

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11-6

11.2.2

Vibration Monitoring, Testing, and Instrumentation

Basic Principle

Seismic base isolation is basically a lengthening of the fundamental time period of the structure with the help of a specially designed system that is placed between its superstructure and substructure (see Figure 11.8). Besides other advantages, the concept gained widespread acceptance due to the fact that most of the earthquake motions around the world have dominating frequencies in the range of 1.0 to 10 Hz, and the majority of conventionally designed structures also has their fundamental frequency of vibration lying in this range. Owing to this unwanted matching of the frequencies, these structures are subjected to high forces during earthquakes. The application of seismic base isolation shifts the fundamental time period away from the dominating frequencies of earthquake motions and thus detunes the frequencies. In other words, base isolation consists in ﬁltering out high-frequency waves from the ground motion, thereby preventing the transmission of high energy in the structure. The effect of base isolation on reduction in force is shown schematically in Figure 11.9. Under favorable conditions, seismic base isolation can reduce drift to 0.2 to 0.5 of that which would occur if the building were ﬁxed base (Figure 11.10). Reduction in acceleration has more inﬂuence on the force –deﬂection characteristics of the isolation system and may not be as signiﬁcant as the reduction of drift (FEMA 356, 2000). However, the additional ﬂexibility required for this period shift give rise to excessive relative displacement at the isolation level. Additional damping is introduced in the isolation system to limit this displacement response to within feasible limits. Still, it is necessary to provide an adequate seismic gap that can accommodate displacements at the isolation level. Most of the isolators have inherent damping, although sometimes supplemental energy dissipation devices may also be required at the isolation level. Various types of energy dissipation devices like metallic dampers and hydraulic dampers have been developed and can be used for this purpose (Skinner et al., 1993). The isolation damping also suppresses the resonance

FIGURE 11.8

Application of seismic isolation for different structures.

Normalized acceleration response

4.0 Conventionally designed structure

3.0 2.0

V = 2% Seismically isolated structure

1.0 0.0

V =15% 0.0

1.0

2.0

3.0

4.0

Natural period (seconds) FIGURE 11.9

Conceptual diagram for seismic isolation.

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5.0

Seismic Base Isolation and Vibration Control

11-7

resulting due to higher period contents of the earthquake motion. Although damping is useful in reducing the required seismic gap, excess damping may result in an increase in acceleration that may affect the performance of nonstructural elements and contents. Thus, an isolation system should essentially be able to (1) support a structure, (2) provide horizontal ﬂexibility, and (3) dissipate energy. These three functions can be incorporated in a single device or can be provided by means of different components. In addition, it may be necessary to provide buffers, which can limit the isolator displacements during extreme earthquakes.

FIGURE 11.10 Behavior of (a) ﬁxed-base and (b) baseisolated building.

11.2.3

Issues in Seismic Base Isolation

A number of issues for seismic isolation design have been identiﬁed based on experiences of their behavior. Some of the issues that should be considered before choosing the base-isolation approach for a project are touched in the following sections. 11.2.3.1

Performance Criteria

The performance criteria for the structure needs to be established in order to evaluate alternative seismic resisting systems; for example, it must be established whether the structure is required to be functional during and after major earthquakes, or if it is to be preserved for its historical importance. Whether seismic base isolation is a suitable design strategy for a particular project will depend primarily on the performance required. To achieve the fully operational or operational performance level, one can consider seismic base isolation as a possible design strategy, but if life safe is the required structural performance level, it may not be practical to choose seismic base isolation. 11.2.3.2

Type of Structure

Signiﬁcant beneﬁts obtained from isolation exist in structures for which the fundamental period of vibration without base isolation is short, that is, less than 1 sec. Certain structures may not be suitable for base isolation because of their shape; for example, this is true for slender high-rise buildings that have a natural period long enough to attract low earthquake forces without isolation. Therefore, seismic isolation is mostly used for low-rise buildings. Historical buildings, which generally are stiff masonry structures, can be appropriate structures for seismic base isolation. Bridges are the structures for which application of seismic isolation is very convenient. The provision of bearings at the tops of piers adds ﬂexibility to stiffer piers and in turn avoids yielding of piers. It is easy to examine these bearings after a seismic event and replace them if needed. 11.2.3.3

Site Characteristics

In base-isolation design, the basic objective is to ﬁlter out the high frequencies of the ground motion by lengthening the time period of vibration to approximately 2 sec. Thus, conventional base isolation is not suitable for structures on soft soils where the ground motions are dominated by low frequencies. Therefore, a detailed investigation of the site must be carried out before possible isolation can be considered. Another important aspect is near-fault ground motions. Waves from such motions usually have long-period velocity pulses, which impart lot of momentum to the structure. This is particularly damaging to base-isolated structures because it may cause large horizontal base displacements. The displacement can result in instability of the structure, or it

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11-8

Vibration Monitoring, Testing, and Instrumentation

can result in impact with moat walls, which can affect the sensitive equipment housed in the building. An isolator with bilinear force– displacement behavior and a large ratio of yield-force to supported weight can substantially reduce the displacement. The provision of high damping in an isolation system can also work. However, the degree of isolation during relatively frequent earthquakes without near-fault pulses is much reduced due to these provisions (Skinner and McVerry, 1996). 11.2.3.4

Retroﬁt Issues

In selecting a suitable retroﬁt system and properties of seismic isolation system, consideration should be given to the characteristics of the existing building, such as foundation capacity and strength of the superstructure. For retroﬁt of buildings, successful implementation of a seismic isolation system requires that the sequences of temporary bracing, shoring, cutting of existing columns and walls, and installation of isolators be well planned. Base isolation for the retroﬁt of bridges is simpler as they usually have thermal bearings, which can easily be replaced by seismic isolation bearings. The retroﬁtting of monuments or buildings of historical importance requires special efforts to cope with the need of minimum alterations. Provisions must be made to protect them from any seismic event during the retroﬁtting. Also it must be identiﬁed whether workable spaces and access to the work area is available. 11.2.3.5

Design of Building Services

Depending on the base-isolation system, base-isolated structures under earthquake motions can exhibit signiﬁcant base slab displacements due to the low horizontal stiffness of the isolation elements. This may create problems on the supply lines transitioning between the parts of structure below and above isolation level. Therefore, special attention is to be given to installations of building services such as water supply, sewerage, gas, air-conditioning, and so on in order to prevent any damage to these supply lines, which might cause secondary effects. In the case of isolation of two or more structural units founded on a common foundation and connected by expansion joints, special care is needed regarding the proportions of the expansion joint in order to prevent the pounding of buildings during earthquakes. 11.2.3.6

Expected Life of Isolator

The isolation system should remain operational for the expected lifetime of the isolated structure. It should not require frequent maintenance during this period. Although the functioning of an isolator may be required few times during the lifetime of structure, it must perform well at such times. If life of the isolator is less than the life of structure, then it may be necessary to provide a mechanism for the inspection and replacement of the isolation system. Another related aspect is the protection of isolation elements against ﬁre, and measures should to be taken for this. Furthermore, as an isolation system is provided mostly at the base, its resistance to chemical and biological reactions is also important (Jurukovski and Rakicevic, 1995).

11.2.4

Seismic Isolation Devices

The successful seismic isolation of a particular structure is strongly dependent on the appropriate choice of the isolation system. In addition to providing adequate horizontal ﬂexibility and appropriate damping, the isolation system should essentially have the capability of self-centering after deformation, high vertical stiffness to avoid rocking, and enough initial stiffness to avoid frequent vibration from wind and minor seismic events. Different types of isolators have been developed and proposed to achieve these properties, and some of them are discussed below.

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Seismic Base Isolation and Vibration Control

11.2.4.1

11-9

Laminated Rubber Bearings

Load (ton)

Rubber bearings offer the simplest method of seismic base isolation and are relatively easy to manufacture. The bearings are made by vulcanization bonding of sheets of rubber to thin steel reinforcing plates. Initially, the main function of the laminated rubber bearings was to provide ﬂexibility for thermal displacements in bridges. Later, similar bearings found application in the isolation of buildings from vibration due to underground railways, and these bearings have FIGURE 11.11 Laminated rubber bearing. performed well over a substantial period of time (Kelly, 1990). The bearings are very stiff in the 20 vertical direction and ﬂexible in the horizontal direction. High vertical stiffness of these bearings is achieved through the laminated construction of the bearing using steel plates. The cross section of a typical rubber bearing is shown in Figure 11.11. The most common elastomers used in elastomeric bearings are natural rubber, neoprene rubber, 0 butyl rubber, and nitrile rubber. The mechanical (tear strength, high strain fatigue resistance, creep resistance) and low-temperature properties of natural rubber are superior to those of most synthetic elastomers used for seismic isolation bearings. Therefore, natural rubber is the most −20 frequently recommended material for use in elastomeric bearings, followed by neoprene. Butyl 0 100 −100 rubbers are suitable for low-temperature appliDisplacement (mm) cations and nitrile rubber has limited application in offshore oil structures (Taylor et al., 1992). FIGURE 11.12 Load – displacement loops for highThe damping ratio (i.e., the fraction of critical damping rubber bearing. (Source: Tanzo, W. et al. Res. damping) achieved from natural rubber is low, in Rpt. 92-ST-01, Kyoto Univ., 1992. With permission.) the order of 0.02 to 0.04, and therefore it is unusual to use it without some other element that is able to provide increased damping. In order to achieve better performance in a single unit, rubber used in the bearing is a compound formed with some ﬁller agents. This compounding results in desired properties, such as (1) high damping and (2) high horizontal stiffness at low values of shear strain. The damping ratio (i.e., the fraction of critical damping) achieved is in the order of 0.10 to 0.20. These high-damping rubber bearings, originally developed in England, found several applications in Japan and United States. A number of ﬁllers are employed, such as metal oxides, clay, and cellulose, but the ﬁller that is most commonly used in seismic isolation bearings is carbon black (Taylor et al., 1992). Force–displacement behavior of these bearings depends upon the type of compounding. Figure 11.12 shows the results of cyclic loading test conducted on a four-layer highdamping rubber bearing specimen (Tanzo et al., 1992). In the experiment, the tests were carried out up to 200% (96 mm) shear strain. Vertical load for the tests was kept as 40 tonf (64 kgf/cm2). The application of steel shims in laminated rubber bearings provides necessary vertical stiffness, but at the same time makes these isolators heavy and expensive. Recently, Kelly (2001) proposed a seismic isolation system for developing countries, in which steel plates are replaced by ﬁber mesh. The ﬁber-reinforced isolator is expected to be signiﬁcantly lighter and could lead to a much less laborintensive manufacturing process.

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11-10

Lead Rubber Bearing

This isolation system consists of a cylinder of lead enclosed in a rubber bearing. The system is also known as the NZ bearing system, and its components are shown in Figure 11.13. The lead plug produces a substantial increase in damping, from about 3% of critical damping in the natural rubber to 10 to 15%, and also increases the resistance to frequently occurring loads such as minor earthquakes or wind. The reason for choosing lead is that it is a crystalline material, and under normal conditions it changes its crystal structure under deformation but almost instantly regains its original crystal structure when the deformation ceases. Thus, repeated yielding due to cyclic loading does not cause fatigue. Lead yields in shear at the relatively low stress of about 8 to 10 N/ mm2, and therefore produces stable hysteretic behavior and dissipates signiﬁcant energy in strong ground motion. Load –displacement behavior of a lead rubber bearing is shown in Figure 11.14 (JSSI, 1995). The hysteretic behavior of lead rubber bearings can be treated as bilinear, with initial stiffness in the range of 9 to 16 times the postyield stiffness. These bearings provide an economical and effective solution, incorporating period shifting, increased damping, and high stiffness at low strains, and providing vertical support in a single device (Skinner et al., 1993). It has found several applications in new constructions as well as for retroﬁtting of buildings and bridges in different parts of the world.

11.2.4.3

Friction-Based Systems

FIGURE 11.13 Lead rubber bearing. (Source: Skinner, R.I. et al. An Introduction to Seismic Isolation, Wiley, Chichester, UK, 1993. With permission.)

15

Stress (kg/cm2)

11.2.4.2

Vibration Monitoring, Testing, and Instrumentation

0

−15 −300

0 Shear strain (%)

300

FIGURE 11.14 Behavior of lead rubber bearing under cyclic loading. (Source: JSSI, Introduction of Base Isolated Structures, Japan Society of Seismic Isolation, Ohmsa, Tokyo, 1995. With permission.)

In this class of isolators, the superstructure is allowed to slide during major seismic events. The structure slides whenever the lateral force exceeds the friction force at the sliding interface. The horizontal friction force at the sliding surface offers resistance to motion and dissipates energy. Pure sliding systems have no inherent natural period and therefore are insensitive to variations in the frequency content of ground excitation. The acceleration at the base of the structure is limited to the coefﬁcient of friction at the sliding interface. Thus, by keeping this coefﬁcient of friction low, the acceleration felt by the structure can be reduced. Systems based on a Teﬂon (PTFE) and stainless steel interface have the potential to provide high levels of protection against ﬂoor acceleration because of the low friction of the materials. However, the friction coefﬁcient cannot be reduced arbitrarily, as the sliding displacement may exceed the acceptable value. Another feature of this isolation system is that the frictional force is proportional to the vertical load coming on the bearing, and therefore the center of mass and the center of resistance of the sliding support coincide. As a result, the torsional effects produced by the asymmetry of building are diminished.

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11-11

The main drawback of the system is the absence of restoring force due to which a large residual displacement from the original position of the structure may be left after a major earthquake event. There are other practical problems that can affect the efﬁciency of sliding isolation, such as cold welding, freezing, and deterioration of the sliding surfaces. Changes may occur in the friction coefﬁcient due to ageing, environmental attack, temperature vari- FIGURE 11.15 Schematic diagram of friction penduation, or wear during use (Skinner et al., 1993). lum system. As the sliding depends upon the coefﬁcient of friction, the system may require regular inspection to maintain its coefﬁcient of friction. The unsatisfactory predictability of friction coefﬁcient and the absence of any centering force suggest that Teﬂon bearings should be used as seismic isolators only in combination with some other centering devices like steel dampers or rubber bearings (Priestley et al., 1996). Based on this concept another system called the resilient friction bearing (R-FBI) system was developed (Mostaghel and Khodaverdian, 1987). The system consists of several concentric layers of Teﬂon-coated friction plates that are in friction contact with each other and a central core of rubber. The rubber provides the resilient force while the friction forces dissipate the energy. An effective mechanism to provide recentering force by gravity has been utilized in a friction pendulum system (FPS). In this system, the sliding surface takes a concave spherical shape so that the sliding and recentering mechanisms are integrated in one unit. As the name indicates, there are two mechanisms that are employed to achieve isolation, namely, sliding friction and pendulum motion. The internal components consist of a stainless steel concave surface upon which slides a stainless steel articulated slider surfaced with a high-load capacity and a low friction bearing material composite (see Figure 11.15). The radius of curvature determines the sliding or isolation period of the system. The period of the structure supported on the FPS is independent of the structure mass and therefore the period does not change if the structure weight changes or is different than assumed. This results in better control over the response of the systems, like liquid storage tanks in which weight varies in time because of variable liquid storage level. However, since the restoring force is linearly proportional to the sliding displacement, in case of high-intensity earthquakes or a low coefﬁcient of friction, the additional sliding introduces additional energy in the structure. To overcome this problem, Pranesh and Sinha (2002) proposed the variable frequency pendulum isolator, in which the geometry of the concave surface is designed such that its frequency decreases with the increase in sliding displacement and asymptotically approaches zero at very large displacement. 11.2.4.4

Other Systems

A base-isolation system, which has found many applications in Japan, comprises of low-damping rubber bearings with yielding metal devices. The yielding of metals provides the necessary dissipation of energy. The commonly used dampers are shown in Figure 11.16 (Aoyagi et al., 1988; Takayama et al., 1988; Yasaka et al., 1988). To standardize the designs of the sensitive structures of nuclear power plants for regions of different seismicity, an isolation system called the EDF (Electricite de France) system was developed. This is a laminated rubber bearing with friction plate at the top. An attractive feature of the EDF isolator is that, for the lower amplitude ground excitations, the lateral ﬂexibility of the rubber provides the base isolation. At a high level of excitation, however, sliding will occur assuring maximum acceleration transmissibility of mg, where m is the coefﬁcient of friction between materials of sliding surface (Constantinou, 1994).

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11-12

Vibration Monitoring, Testing, and Instrumentation

FIGURE 11.16

(a) Elasto-plastic steel damper; (b) lead damper; (c) steel rod damper.

FIGURE 11.17 Elements of a TASS system. (Source: Kawamura, S. et al. 1998. Proc. IX World Conf. on Earthquake Engineering, pp. 735 –740. Tokyo, Japan. With permission.)

Taisei Corporation, Japan, developed the TASS (TAISEI Shake Suppression) system (Figure 11.17), which is composed of PTFE-elastomeric bearings and neoprene springs (Kawamura et al., 1988). PTFE-elastomeric bearings support the vertical load of the superstructure and reduce the horizontal seismic forces by sliding against severe earthquake motion. Horizontal springs provide weak lateral stiffness and restrain displacement. The TASS system never resonates to any type of excitation, stably supports superstructure, and limits the horizontal force transmitted to the superstructure to that equal to friction force. In situations where circumstances may result in substantial tension forces on the bearings, it has generally been accepted that elastomeric, sliding, or roller isolators alone are not suitable. For this, a system that controls uplift was developed. The system consists of rubber bearings with a central hole to accommodate a tension device (Kelly and Chalhoub, 1990). Logiadis et al. (1996) proposed a prestressed bearing isolation system in which vertical prestressed tendons are located in pairs, symmetrical on both sides of each bearing to avoid tension in the rubber bearing.

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11-13

Tarics (1996) developed a composite isolator to provide equal protection in minor, moderate, or major earthquakes (see Figure 11.18). The composite seismic isolator has two distinctly different stiffnesses, which are activated by the displacement demand. During minor or moderate earthquakes, the upper isolator provides ﬂexibility. In the event of major earthquake, the higher stiffness of the lower isolator prevents excessive displacement. This isolator seems to be a promising alternative for near-fault ground FIGURE 11.18 Composite isolator and its force – motions. With similar objectives, Shimoda et al. displacement diagram. (Source: Tarics, A.G. Proc. XI (1992) proposed lead rubber bearings with a World Conf. on Earthquake Engineering, 1996. With stepped lead plug. Several other systems, for permission.) example, the sleeved pile system, GERB, alexisismon, and so on, were developed by different researchers (Kelly, 1986; Buckle and Mayes, 1990) and the search for new isolation systems, which can give better performance under different conditions, is still continuing.

11.2.5

Design of Isolation Devices

As discussed above, several types of isolators have been proposed so far. Only a few bearings, namely, laminated rubber bearings, lead rubber bearing, FPS bearings, and Teﬂon –steel sliding bearings with a restoring device, have been developed to the stage of practical application. These isolators are in general observed to be effective in reducing seismic response; however, their effectiveness depends on the situation and design constraints. For example, rubber bearings are most suitable for high-frequency motions, while friction-type isolators may also be applied at the site where ground motion can have lowfrequency content. Buildings with sophisticated electronic equipment and loose contents should preferably be isolated with rubber bearings to avoid transmission of high frequencies to the superstructure. In the case where, for some reason, it is not possible to provide large seismic gaps or for the conditions where light weight structure is to be isolated, the FPS-type isolators are more suitable. The main requirements for the design of a base-isolation system are (1) the ability to sustain gravity loads, (2) low horizontal stiffness that can lengthen the fundamental time period to the desired value, (3) large vertical stiffness to minimize ampliﬁcation in vertical direction and complications due to rocking, (4) energy dissipation capacity to keep displacements at the isolation level within acceptable limits, and (5) sufﬁcient initial stiffness to avoid unwanted vibrations due to wind loads and frequent minor seismic events. The design procedure of most widely used isolation systems, namely, high-damping rubber bearings, lead rubber bearings, and the FPS, is discussed in the following sections.

11.2.5.1

High-Damping Rubber Bearing

The design of high-damping laminated rubber bearings involves determining the plan size, the number of rubber layers, the thickness of each rubber layer, and steel shim. The steps involved in the design of high-damping rubber bearings are as follows: 1. Specify the design vertical load on the bearing ðWÞ; the design period for the isolated structure ðTI Þ; and site conditions. The load, W; is computed for the dead load and live load combination. The design period, TI ; depends on site conditions. Its minimum value is usually taken as three times the fundamental time period of the structure when it is ﬁxed base.

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Vibration Monitoring, Testing, and Instrumentation

2. Find corresponding horizontal stiffness of the bearing, Kh ; using Kh ¼

W ð2p=TI Þ2 g

ð11:1Þ

3. Assign the values to quantities such as the maximum permissible shear strain ðgÞ; Young’s modulus ðEÞ; and the shear modulus ðGÞ for the rubber compound. Generally, g is considered to be in the range of 100 to 150%. G can be taken from design speciﬁcations provided by the manufacturer. The value of G for this range of shear strains varies from 0.69 to 0.86 MPa, depending on the rubber composition (Kelly, 1997). 4. Specify the effective equivalent damping in the isolation system and allowable pressure on the bearing ðpÞ for vertical loads. The allowable pressure ðpÞ for bearings with metal reinforcement can be taken as about 6.9 N/mm2 (Kelly, 1997). Damping can be taken as 15 to 20%, subject to veriﬁcation. 5. Using a response spectrum or code formulae, ﬁnd the maximum horizontal displacement, that is, the design displacement ðDÞ: This depends on the spectral coefﬁcient (i.e., the site condition), design period, and damping. 6. The plan area of the rubber ðAÞ in bearing is primarily controlled by the design vertical load ðWÞ and allowable pressure ðpÞ; and is determined by the expression A ¼ W=p

ð11:2Þ

7. The total thickness of rubber, tr ; in the bearing is determined from the following equation: tr ¼ GA=Kh

ð11:3Þ

Total thickness, tr ; should not be less than D=g: 8. Evaluate the shape factor ðSÞ for the desired value of ratio of vertical stiffness ðKv Þ and horizontal stiffness ðKh Þ using following expressions: 6S2 ¼ Kv =Kh

for circular bearings

6:73S2 ¼ Kv =Kh

for square bearings

ð11:4aÞ ð11:4bÞ

The minimum recommended value for the stiffness ratio is 400. These expressions are fairly accurate for shape factors of ten or less. However, for higher values of S; the effect of compressibility should also be taken into account (Kelly, 1997). 9. Using a basic deﬁnition of shape factor as the ratio of the cross-sectional area of the bearing to the force-free area of a single layer of rubber, the thickness of each rubber layer can now be determined by t ¼ f=4S for circular bearings ð11:5aÞ t ¼ b=4S

for square bearings

ð11:5bÞ

where f is the diameter for circular bearing and b is the side for square bearing. Knowing the values of tr and t; the number of rubber layers ðnÞ can be computed. 10. The thickness of steel shims provided for laminated construction should be in the range of one tenth to one eighth of an inch (Kelly, 1997). The material properties for rubber compounds used for isolation devices can be related to the rubber hardness. Properties for the range of rubber hardness normally used in bearing are listed in Table 11.1 (Bridgestone, 1990). The designed bearing should be checked for the following criteria also. 11.2.5.1.1 Stability against Buckling Buckling load may become critical in situations where bearings with a high rubber thickness relative to the plan dimension are to be provided. For example, in the case of base isolation for low vertical loads,

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Seismic Base Isolation and Vibration Control TABLE 11.1

11-15

Relationship between Rubber Hardness and Other Properties

Rubber Hardness (IRHD ^2)

Young’s Modulus, E (MPa)

Shear Modulus, G (MPa)

Material Constant, k

Minimum Elongation at Break (%)

35 40 45 50 55 60

1.18 1.50 1.80 2.20 3.25 4.45

0.37 0.45 0.54 0.64 0.81 1.06

0.89 0.85 0.80 0.73 0.64 0.57

650 600 600 500 500 400

rubber bearings are generally slender. Average compressive stress in the bearing ðW=AÞ should not exceed the critical stress ðpcr Þ given by Kelly (1997): pﬃﬃ 2pGS pcr ¼ r ð11:6Þ tr pﬃﬃ where r is radius of gyration (b=2 3 for a square bearing and f/4 for a circular bearing). Maximum shear strain is described as follows. Under vertical loading only, as per AASHTO recommendations for service loads (i.e., for the dead and live load), a factor of safety of three should be adopted for maximum shear strain values. Thus, maximum shear strain ðgs Þ should satisfy the following condition: W 1 gs i:e: 6S # b Ec A 3 where 1b is the minimum tensile strain at which rubber breaks and Ec is compression modulus for bearing and is given by Ec ¼ Eð1 þ kS2 Þ: E and k, for a given value of rubber hardness, can be taken from Table 11.1. Under earthquake loading, the following applies. In the case of ultimate load that includes effect of earthquake loads, AASHTO recommends a factor of safety of 1.33. Thus, maximum shear strain (gu) due to combined effect of compression (gc), torsion (gt), and lateral load (geq) should not exceed 1b =1:33; that is, gc þ geq þ gt # 1b =1:33: geq is given by D=tr ; and gt can be evaluated by

gt ¼

b2 12De 2ttr ðB2 þ L2 Þ

ð11:7Þ

Here, B and L are dimensions of a structure with rectangular plan and e is the eccentricity. gc ð¼ 6SW 0 =Ec Ar Þ is computed for vertical load (W 0 ), which also includes the effect of earthquake load. In this case, the effective area (Ar) is reduced due to lateral displacement (D) and it is the overlapping area of top and bottom in displaced condition. Ar can be determined using the following expressions: Ar ¼ A 1 2

D b

for square bearings

( pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ!) 1 f2 2 D2 2 21 for circular bearings f sin 2 D f2 2 D2 Ar ¼ 2 f

ð11:8aÞ ð11:8bÞ

11.2.5.1.2 Stability against Roll Out Maximum horizontal displacement should not exceed the roll-out displacement (d) given by

d¼

WE b W E þ Kh h

ð11:9Þ

where vertical load (WE) also includes the effect of earthquake load, b is the side for square and the diameter for the circular bearing, and h is the overall height of the bearing (Figure 11.19).

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The check should be made for bearings with bolted connections also, so as to avoid the development of tensile stresses. 11.2.5.2

Lead Rubber Bearings

The design of lead rubber bearings can be split into two parts, namely, (a) the design of the lead plug and (b) the design of the laminated rubber bearing. 11.2.5.2.1 Design of Lead Plug In fact, the lead rubber bearing can be designed effectively by assuming its force – displacement behavior comprises two distinct stiffnesses, namely, the preyielding and postyielding stiffnesses, as shown in Figure 11.20. The characteristic strength, Qd ; for the desired energy dissipation (WD) or the known effective damping (jeff ) can be computed using

FIGURE 11.19 Stability against roll out. (Source: Kelly, J.M. Earthquake Resistant Design with Rubber, Springer, London, 1997. With permission.)

F

Fy

Qd Ke

Qd ¼ WD =4ðD 2 dy Þ ¼ 2pKeff D2 jeff =4ðD 2 dy Þ

dy

ð11:10Þ

Qd and yield displacement (dy) are the unknowns. So for a ﬁrst trial, considering dy is very small as compared with design displacement (D), we have, as a ﬁrst approximation Qd ¼ WD =4D

Kp

ð11:11Þ

FIGURE 11.20 bearing.

Keff D Displacement

Design parameters for lead rubber

Using this approximate value for Qd ; the postyield stiffness (Kp) can be evaluated by Keff D ¼ Qd þ Kp D

ð11:12Þ

This may lead to modiﬁed value of yield displacement, dy ; given by dy ¼ Qd =ðKe 2 Kp Þ

ð11:13Þ

where, depending upon the physical parameters of the laminated rubber bearing, Ke can be assumed to be about ten times the value of Kp : This value of dy can be used in the next trial to compute revised values of Qd ; Kp ; and dy : Trials are to be repeated until dy converges. Now, the yield force can be evaluated by Fy ¼ Ke dy

ð11:14Þ

Thus, the area of lead plug (Ap) can be determined using Ap ¼ Fy =f y ; in which f y is the yield strength of the lead in shear and has a value of 10.5 MPa (Skinner et al., 1993). In design, owing to considerations for the conﬁnement of the plug, a reduced value of 7 to 8 MPa is generally considered.

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11.2.5.2.2 Design of Laminated Rubber Bearing Postyield stiffness of the bearing (Kp) is typically higher than the corresponding stiffness of the bearing without the lead plug (Kh): Kp ¼ f L Kh where f L is a factor that takes into account the effect of lead plug and is always larger than unity. Typically, f L is 1.15 (FEMA 356, 2000). Another expression for computing Kp is given by (Yang et al., 2003) Kp ¼ Kh 1 þ 12

Ap A0

ð11:15Þ

where A0 is the area of the bearing based on allowable normal stress under the vertical load. The ﬁnal value of Kp obtained during design of the lead plug is used in these expressions to ﬁnd Kh : This contribution (Kh) towards combined stiffness (Kp) controls the design of the laminated rubber bearing. The steps from Step 3 onwards, described earlier in the design of high-damping rubber bearings, can be followed to design this laminated rubber bearing. One additional check for the lead plug should be applied: 1:25 #

hp # 5:0 fp

where hp is effective height of the lead plug and fp is its diameter (Yang et al., 2003). 11.2.5.3

Design of Friction Pendulum System

The basic design variables in FPS are (1) the radius of curvature (R), (2) the material friction coefﬁcient ðmÞ; and (3) the plan dimension. The desired isolation period governs the radius of curvature, whereas the plan dimension is controlled by the design displacement. R for a known value of the design period of the isolated structure (TI) can be computed using R ¼ gðTI =2pÞ2

ð11:16Þ

where g is acceleration due to gravity. The coefﬁcient of friction ðmÞ; along with the displacement, controls the energy dissipation. The effective damping (b) for this system is given by expression

b¼

2 m p m þ D=R

ð11:17Þ

The desired damping, depending upon the properties of the system, may vary from 10% to 20% of critical damping. As the design displacement (D) depends upon damping (b), it becomes a trial-anderror process and requires few iterations to get the values of b, m, and D for desired response. The effective stiffness of the isolation system at design displacement can be evaluated by Keff ¼

11.2.6

mW W þ D R

ð11:18Þ

Veriﬁcation of Properties of Isolation Systems

As they are the most critical part of the base-isolated structure, the quality of isolation bearings is of paramount importance. Looking into uncertainties in the manufacture, the deformation characteristics and damping values of the isolation system must be veriﬁed by tests. Therefore, the selected units of the isolation system have to undergo testing for examining quality control and determining their actual load –deformation behavior. Force–deformation characteristics of the isolation system obtained from tests are used for analyzing the base-isolated structures. These tests also serve as a means to verify the

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ability of the isolation bearings to withstand long-term vertical loading and numerous cycles of shear displacement during design basis and maximum capable earthquake. The primary function of an isolation system is called on during major earthquakes only and, for most of its design life, it is subjected to nonseismic loads. Therefore its behavior for such loads should also be veriﬁed by tests. 11.2.6.1

Tests for Isolation Systems

System characterization tests, prototype tests, and quality control tests are the three levels of tests that are recommended for an isolation system. The ﬁrst tests provide the fundamental characteristics of the isolation system; the second allow one to know the design parameters of the isolation system fabricated for the project; and the third verify the quality control and consistency achieved in manufacturing. These tests are brieﬂy discussed in the following sections (Taylor et al., 1995). 11.2.6.1.1 System Characterization Tests These tests are performed to evaluate the basic characteristics of the isolation systems, especially its dependence on temperature, frequency, bilateral loading, wear and fatigue, and so on. The tests are also intended to provide the ultimate and reserve capacity of a device for various loading conditions. These tests are conducted only once for a system of a given design, material, and construction. However, they should be repeated if there are major changes. These tests are formally not required by codes; however, in practice, they are conducted for a new isolation system and the test results provide some idea about the suitability of the isolation system for a particular project. 11.2.6.1.2 Prototype Tests These tests are conducted to verify the design properties of an isolation system prior to its construction. Effective stiffness, energy dissipation capacity, stability of hysteretic behavior (in a check for system degradation), and stability at maximum seismic displacement are the properties generally veriﬁed through these tests. The stability of an isolation system is veriﬁed for maximum and minimum vertical load conditions. Tests are also carried out for nonseismic loads. In the case of isolation systems for buildings, the major nonseismic force is wind load, and for bridges it may be thermal displacements and braking or centrifugal forces. If the isolation system is frequency-dependent, then prototype tests should be performed dynamically for a range of frequency that represents full-scale prototype loading rates. Similarly, if the behavior of the isolation system is signiﬁcantly different for unilateral and bilateral loadings, then the isolation system should be tested for different combinations of displacements in two orthogonal directions. Specimens in these tests are subjected to extreme conditions and therefore are not to be used for construction. 11.2.6.1.3 Quality Control Tests These tests are carried out to verify the quality and consistency of the manufacturing process, and to measure the as-built properties of the isolation system prior to installation. The quality control tests are divided into production tests and completed unit tests. Production tests are carried out on materials or components that are used in the making of the isolation unit. In the case of elastomeric bearings, the elastomer is tested for hardness, tensile strength and elongation at break, bond strength, compression set, low-temperature properties, high-temperature aging, and ozone resistance. Elements used for a sliding system are supposed to be tested for surface roughness, trueness of surface, interface material properties, bearing pad attachment, and sliding interface attachment. All completed units are tested for (1) sustained compression, and (2) combined compression and shear. The purpose of the sustained compression test is to verify (1) the quality of the bond between the elastomer and the steel for laminated elastomeric bearings and (2) the bond between the bearing liner and the metal backing plate for sliding bearings. A combined compression and shear test is conducted to verify that actual values of effective stiffness and damping are in close agreement with design values. After quality control testing, laminated elastomeric bearings are visually inspected for rubber-to-steel bond, surface cracks, laminate placement, and permanent deformation, while sliding bearings should be inspected for bearing liner-to-metal bond,

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scoring of the stainless steel plate, leakage, and permanent deformation. In the case that the bearings do not meet the requirements, they should be rejected. 11.2.6.2

Uniform Building Code Recommendations

Uniform Building Code (UBC, 1997) recommends that the deformation characteristics and damping values of the isolation system used in the design and analysis of seismic-isolated structures is based on the sequence of cyclic loading tests of a selected sample of the components (Figure 11.21). First, the specimen is tested under a cyclic load corresponding to the design wind load. Sub- FIGURE 11.21 Schematic diagram for cyclic loading sequently, it is tested, for a prescribed number of test of bearing. cycles, with cyclic displacements varying from one ﬁfth of the design displacement to the total maximum design displacement. Vertical load (dead load plus 50% of the live load) for all the above tests is prescribed as the average load on all isolators of common type and size. If an isolator unit is also a vertical load-carrying element, the cyclic load tests should be performed for maximum and minimum vertical loads, which includes the effects of an earthquake overturning evaluated corresponding to the test displacement. 11.2.6.2.1 Units Dependent on Loading Rates If there is greater than 10% difference in effective stiffness values at the design displacement (i) obtained by performing test on an isolator unit at a frequency equal to the inverse of the effective period TI of the isolated structure, and (ii) obtained by performing test on the same unit at any frequency in the range of 0.1 to 2.0 times the inverse of TI, then the force– deﬂection properties of the isolator unit are considered to be dependent on rate of loading. In such cases, the tests are prescribed to be performed at a frequency equal to the inverse of TI. 11.2.6.2.2 Units Dependent on Bilateral Load If the bilateral and unilateral force–deﬂection properties have greater than a ^10% difference in effective stiffness at the design displacement, the force–deﬂection properties of an isolator unit are be considered to be dependent on bilateral load. In such cases, the tests speciﬁed above shall be augmented to include bilateral load increments of the total design displacement 0.25 and 1.0, 0.50 and 1.0, 0.75 and 1.0, and 1.0 and 1.0. 11.2.6.2.3 System Adequacy The performance of the test specimens is assessed as adequate if the following conditions are satisﬁed: 1. The load –displacement curves for all tests have a positive incremental load-carrying capacity. 2. The difference between the average values of effective stiffness of the two test specimens of a common type and size does not exceed the prescribed value of 10%. 3. The effective stiffness of each test specimen for each cycle of test is within ^10% of the average value of effective stiffness for the specimen. 4. There is no greater than a 20% change in the initial effective stiffness of each test specimen over the prescribed number of cycles. 5. There is no greater than a 20% decrease in the initial effective damping of each test specimen over the prescribed number of cycles. 6. All specimens of vertical load-carrying elements remain stable at the total maximum displacement for the prescribed vertical load.

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Vibration Monitoring, Testing, and Instrumentation

Analysis of Base-Isolated Structures

Depending upon the code requirements, one can choose the static or dynamic analysis procedure for a particular project. Regarding the static analysis method, codes generally have some bindings, whereas time-history analysis can be used for all types of situations. In UBC (1997), minimum values of some design parameters to be used in dynamic analysis are deﬁned in terms of percentage of values obtained by static analysis and thus most of the steps of static analysis are necessary for a designer, even if the dynamic analysis procedure is followed. These analysis methods require the modeling of the base-isolation system and the superstructure. In most of the cases, the linear model for the superstructure of a base-isolated building is enough. However, the isolation systems generally require nonlinear model. 11.2.7.1

Modeling of a Superstructure

The superstructure of a base-isolated building can be modeled as rigid blocks, a two-dimensional and a three-dimensional linear model, or nonlinear models. The rigid block assumption is the simplest one, which is used in the design of isolation system, for preliminary analysis and feasibility studies. Moreover, it can also be used as a tool to crosscheck the results obtained from other rigorous analysis methods. Assuming the superstructure as a single-degree-of-freedom (single-DoF) model and the isolation system as linear spring plus linear viscous damper, Kelly (1990) developed FIGURE 11.22 A two-DoF model of base-isolated linear theory for a base-isolated structure (Figure building. (Source: Kelly, J.M. Earthquake Spectra, 6(2), 11.22). In general, an isolation system aims at 223 –244, 1990. With permission.) providing signiﬁcant protection not only to the building but also to nonstructural components and contents. Therefore, a three-dimensional linear model for the superstructure is generally sufﬁcient for analyzing a base-isolated structure. However, for the situations when the response to extreme earthquakes must be investigated, nonlinear models for superstructure should be used. In case of the analysis of seismically isolated bridges, it may be important to take into account the pier masses and their own modes of vibration. The pier can be modeled using a linear beam element with mass, and the deck can be modeled with linear beams with mass. However, for very squat piers, the shear ﬂexibility should also be considered. In the case of a regular structure, the coupling effect of the deck can be neglected, and a bend can be considered independent of the others. However, if a stiff deck is supported on piers of different ﬂexibility, the structure should be modeled as a linear multi-DoF system. To ascertain the response for extreme loading conditions, the piers should be modeled with nonlinear models; however, the deck can still be simulated with linear elements. In such cases, proper account should be made for change in the effective stiffness of piers due to cracks (Priestley et al., 1996). In the case of equipment, the modeling depends upon the level of protection; however, the rigid block assumption for equipment generally sufﬁces. Liquid storage containers should be modeled with proper consideration to the sloshing affect of liquid during base motion. 11.2.7.2

Modeling of Isolation System

The isolation system can be modeled as a linear or nonlinear element. The linear model of an isolation system consists of effective stiffness and effective damping at the design displacement. These values are derived from force– displacement test data for the isolation system. However, except in preliminary designs, the isolation system is generally modeled as nonlinear. Commonly used isolation systems, namely, high-damping rubber bearings, lead rubber bearings, friction bearings, and FPSs, can be modeled by bilinear model (Figure 11.23). However, for more precise results, the isolation elements should be modeled by their actual force–displacement behavior, for example, in the modeling of

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high-damping rubber bearings, which in extreme conditions follow a trilinear model with stiffening at larger strains. The model used for analysis should be veriﬁed with force – displacement characteristics obtained by prescribed tests, discussed in Section 11.2.6. Moreover, the isolation system should be modeled with considerations for the following requirements (UBC, 1997):

F

F

msW

msW d

d

(a)

(b)

1. Account for the spatial distribution of FIGURE 11.23 Analytical models for isolators: (a) pure isolator units. friction system; (b) friction pendulum system. 2. Calculate translation, in both horizontal directions, and the torsion of the structure above the isolation interface, considering the most disadvantageous location of mass eccentricity. 3. Assess overturning or uplift forces on individual isolator units. 4. Account for the effects of vertical load, bilateral load, and/or the rate of loading if the force – deﬂection properties of the isolation system are dependent on one or more of these attributes. 11.2.7.3

Static Analysis Procedure

This procedure may be sufﬁcient for analyzing a base-isolated structure that satisﬁes certain code requirements related to the height of a building, soil conditions, geometry of a structure, effective period of base-isolated building, force– deﬂection, and restoring-force characteristics of the isolation system, and so on. The procedure involves the use of simple expressions for computing design displacements (D) and lateral forces. Using the response spectrum or code formula, the design displacement is computed for a target period of the isolation system, its effective damping, and site conditions. Since time period and effective damping itself depends on design displacement, it becomes an iterative procedure. Appropriate values of damping and the target time period can be assumed for the ﬁrst iteration. Force–displacement data for the isolation system, obtained from tests, is used to determine effective stiffness ðkeff Þ and damping at design displacement. The design lateral seismic force for the isolation system and structural elements at or below the isolation system is simply evaluated as Vb ¼ keff D

ð11:19Þ

Total lateral force for the structure above the isolation system is given by Vs ¼

keff D RI

ð11:20Þ

The factor RI is based on the type of lateral force resisting system used for the structure above the isolation level. The value of Vs is not to be taken as less than the following (UBC, 1997): 1. The lateral seismic force required for a ﬁxed-base structure of the same weight, W, and a period equal to the isolated period 2. The base shear corresponding to the design wind load 3. One and a half times the lateral seismic force required to fully activate the isolation system The total lateral force shall be distributed over the height of the structure in accordance with the formula W x hx Fx ¼ X Vs ð11:21Þ n WI hI i¼1

where Wx is weight at level x and hx is the distance of level x from the base. Fx ; the lateral force at level x, is applied over the area of the building at that level in accordance with the mass distribution. The maximum

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displacement of the isolation system is calculated in the most critical direction and for maximum capable earthquake. The total design and maximum displacement of the elements of the isolation system should include the additional displacement due to actual and accidental torsion, calculated by taking into account the spatial distribution of the lateral stiffness of the isolation system and the most disadvantageous location of mass eccentricity. 11.2.7.4

Dynamic Analysis Procedure

The method involves response spectrum or time-history analysis, and is essential when, owing to code requirements, it is not possible to use the static analysis procedure alone. However, for the situations when (1) the structure above isolation system may reach the nonlinear range for the design earthquake motion, (2) the soil is soft or site conditions require site speciﬁc evaluation, or (3) the isolation system does not meet certain prescribed code criteria, even the use of the response spectrum method is not sufﬁcient and the time-history analysis procedure becomes essential. 11.2.7.4.1 Response Spectrum Method This method of analysis is very convenient for the laminated rubber bearing-type isolation system. A design spectrum is constructed for the design basis and the maximum capable earthquake. Total design displacement of the isolation system and the lateral forces and displacements of the isolated structure are computed for the design basis earthquake. Maximum displacement of the isolation system should be obtained for the maximum capable earthquake. In this method of analysis, damping for fundamental mode in the direction of interest is generally restricted to the effective damping of the isolation system or 30% of the critical damping, whichever is less. Damping for higher modes is consistent with those appropriate for the response spectrum analysis of similar ﬁxed-base structure. To take into account the effect of bidirectional excitation, the total maximum displacement includes simultaneous excitation of the model by 100% of the most critical direction of the ground, and not less than 30% of the ground in the orthogonal direction (UBC, 1997). The maximum displacement of the isolation system is calculated as the vector sum of the two orthogonal displacements. The total lateral force shall be distributed over the height of the structure in accordance with Equation 11.21. 11.2.7.4.2 Time-History Analysis This method of analysis can be used for base-isolated structures under all types of situation. The method is necessary for conditions where (1) isolation systems have a sliding system, (2) the structure is founded on very soft soil, (3) the isolation system allows low displacement, or (4) the force–displacement behavior of the isolation system depends on (a) the rate of loading, (b) the vertical load, or (c) the bilateral load. Appropriate horizontal ground motion time histories are selected, and their magnitudes, fault distances, and source mechanism should be consistent with those that control the design basis (or the maximum capable) earthquake. Where appropriate recorded ground motion time-history pairs are not available, appropriate simulated ground motion time-history pairs may be used. Each pair of time histories is applied simultaneously to the model, considering the most disadvantageous location of mass eccentricity. The maximum displacement of the isolation system is calculated from the vectorial sum of the two orthogonal displacements at each time step. 11.2.7.5

Software for Analysis

The need for an isolation system that is stiff under low levels of frequently occurring loads but ﬂexible under higher levels necessarily leads to a nonlinear system. The force–displacement behavior of most of the isolators can be modeled as bilinear. However, in the case of high-damping rubber bearings, in extreme conditions, a trilinear model with stiffening at large shear strains better represents the behavior. These nonlinear force–displacement relationships of the bearings need the use of specialized computer programs to analyze base-isolated structures. The program should be capable of analyzing base-isolated

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buildings with isolator models made of combination of discrete nonlinear elements and superstructure models that are fully elastic or that permit some localized nonlinear behavior. Several programs are available for this purpose. The computer code NPAD has plasticity-based nonlinear elements that can be used to model certain types of elastomeric bearings. The program simulates the superstructure using a three-DoF per ﬂoor model and can analyze base-isolated structures with linear superstructure (Tospelas et al., 1994). The justiﬁcation behind the approach of considering superstructure as linear is that seismic base isolation is generally provided to reduce the force transmitted to the superstructure to the point where it can be assumed to remain elastic. The computer program SADSAP has the capability to analyze three-dimensional structures with localized nonlinearities. The nonlinear elements available in the software can be combined to simulate the behavior of various types of isolation systems. The program may also be useful in analyzing base-isolated buildings with nonlinear bracing elements provided for supplemental damping (Kelly, 1997). Some general-purpose programs such as the DRAIN series and ANSR have the capability of nonlinear analysis of two- and three-dimensional structures and have provisions that can be used to simulate seismic isolators, which exhibit bilinear behavior (Tospelas et al., 1994). Some commonly used ﬁnite element analysis software such as ABAQUS, ANSYS, and COSMOS also have the capability for analyzing complex superstructures on nonlinear isolators (Kelly, 1997). However, sliding bearings cannot be modeled precisely with these programs. SAP2000 (2002) can be used to analyze these structures and also has the facility to model friction bearings. The 3D-BASIS group of programs is a group of special purpose programs developed for nonlinear dynamic analysis of three-dimensional base-isolated structures, and has been used for the analysis of many base-isolated structures (Constantinou, 1994). The programs assume that the superstructure is elastic and the isolation system is nonlinear. Models used for modeling the isolation elements can also capture precisely frictional behavior. It is not possible to account for stiffening of laminated rubber bearings at higher strain levels using the programs discussed above. The program LPM (lumped parameter model), originally developed for nonlinear three-dimensional analysis of masonry structures, has the capability to model this stiffening and at the same time can also incorporate a nonlinear element in the superstructure (Kelly, 1997).

11.2.8

Experimental Methods for Isolated Structures

Although analytical simulation has progressed tremendously through the use of powerful computers utilizing accumulated theoretical and experimental data on structures, it is not yet possible to analyze and predict the inelastic response of base-isolated structures with total conﬁdence. The responses of base-isolated structures are very much dependent on the characteristics of the input earthquake motion, isolation system, and the supported structures. Experimental tests have been used to verify and calibrate mathematical models, as well as to provide data for the design of structures. However, extensive tests are needed to study the complex behavior of base-isolated structures under realistic conditions that are likely to occur during earthquakes. Different test methods such as quasistatic testing, shaking table testing, online hybrid testing, and substructured online testing have been proposed. A typical problem that appears when performing quasistatic tests on an isolated structure is the alteration of the restoring forces due to strain-rate effect existing for this material. This problem does not exist for shaking table tests performed at the real-size and real-time scale. Shaking table tests of base-isolated specimens have been used successfully for investigating the behavior. However, it requires a large-capacity table for testing a full-scale specimen. The involvement of high cost for large-size shaking tables and the limited capacity of available shaking tables necessitate the structure to be drastically scaled down. This scaling of a structure presents some limitations for the extrapolation of the results to the real scale. As a promising option, the online hybrid test method seems to be well suited for obtaining the seismic response of isolated structures thanks to available quasistatic loading equipment and highspeed computers. However, this technique involves the fabrication of a complete structural prototype

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FIGURE 11.24 Substructured hybrid testing for seismically isolated structures. (Source: Tanzo, W. et al. Res. Rpt. 92-ST-01, Kyoto Univ., 1992. With permission.)

for proper modeling. Thus, test models must be simpliﬁed, for example, into a single-DoF model, and/or specimens have to be reduced in scale. By incorporating substructure concepts into the online hybrid test technique, a substructured online hybrid test method can be used, in which a complete system is considered to be divided into analytical substructures and experimental substructures (Tanzo et al., 1992). In the case of seismically isolated structures, inelastic deformations are designed to occur only in isolation system. The isolation system, which is generally difﬁcult to model mathematically, is taken as experimental substructures (Figure 11.24). The remaining part of the structure is taken as analytical substructures in which presently available analytical models are used to describe their restoring-force characteristics. For the experimental substructure, restoring-force information is directly measured from a specimen loaded according to its current deformation state. The substructured online hybrid test procedure for the earthquake response of base-isolated structure is explained in Figure 11.25.

11.2.9

Implementation of Seismic Isolation

11.2.9.1

New Construction

11.2.9.1.1 Buildings In the early stages of the development of seismic isolation, the main target of the earthquake-resistant design of the structure was the prevention of collapse. However, later, other additional considerations, such as the value of structure or its content, have exerted their importance on the seismic design of structures. For example, facilities of postearthquake importance such as ﬁre stations, police stations, hospitals, communication centers, and so on, have a critical role and are required to remain operational immediately after a major seismic event. Similar reasoning applies to museums housing unique artifacts or buildings that are architecturally important. The required low level of structural and nonstructural damage may be achieved by using an isolation system, which limits structural deformations and ductility demands to low values. The isolation system can be provided at different levels. However, while deciding the isolation level, considerations should be given for (1) the seismic gap, (2) the continuity of supply lines, stairways, and elevators, and (3) details of cladding below the isolation level. In addition, provisions for (1) access to bearings for inspection and replacement, (2) backup system for vertical loads, and (3) full diaphragm for the uniform distribution of lateral load to individual bearings also need to be considered. Figure 11.26 shows typical positions for bearings in buildings (Mayes and Naeim, 2001). The ﬁrst building constructed using modern isolation bearings was a government facility, the William Clayton Building in Wellington, New Zealand, which was completed in 1981. After that, a number of seismically isolated buildings have been constructed there, and in most of them the isolation system used is lead rubber bearings. The most widespread use of seismic isolation systems is in Japan. The ﬁrst seismically isolated structure to be completed in Japan was the Yachiyodai Residential Dwelling, a two-story building completed in 1982. Better performance of seismically isolated buildings during the

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FIGURE 11.25 Procedure for substructured online test method for a base-isolated structure. (Source: Tanzo, W. et al. Res. Rpt. 92-ST-01, Kyoto Univ., 1992. With permission.) © 2007 by Taylor & Francis Group, LLC

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FIGURE 11.26 Typical positions of isolation level for buildings. (Source: Mayes, R.L. and Naeim, F. 2001. The Seismic Design Handbook, 2nd ed., Kluwer, Boston. With permission.)

Kobe earthquake of 1995 triggered the rapid increase in number of seismic isolation projects. Before the Kobe earthquake, the number of such buildings in Japan was 82; this suddenly increased to about 650 within 4 years of the earthquake. Besides rubber bearings, there have been some applications of frictional sliding systems also for seismic isolation of buildings (Nishitani, 2000). The ﬁrst newly constructed baseisolated building in the United States was the Foothill Communities Law and Justice Center in California, which was completed in 1986. Until 1990 there were four isolated buildings. After the good performance of base-isolated buildings during the Northridge earthquake, the number increased and, at the end of 1998, there were approximately 40 isolated buildings completed or in construction in the United States (Clark et al., 2000). Some buildings in other countries, including Italy, France, Canada, Mexico, China, England, Russia, Armenia, Indonesia, Iran, Chile, and India, now use these systems. Until now, the emphasis on base-isolation applications has mostly been for structures with sensitive and expensive contents or high-risk structures like nuclear power plants and computer centers. However, there is an increasing interest in applying this technology to public housing and schools in developing countries (Fuller and Muhr, 1995). Most of the base-isolation projects have made use of laminated rubber bearings and lead rubber bearings. In Japan, laminated rubber bearings are often used with energy-dissipating devices, both with and without provision for recentering. 11.2.9.1.2 Bridges In most cases, bridges are strategic structures and require a higher degree of protection to ensure their functionality after a seismic event. For many simple bridges, it has been found that seismic isolation of the superstructure gives improved seismic resistance, often at a reduced cost, while it is also effective for thermal expansion of the superstructure (Skinner et al., 1993). The aim of seismically isolating bridge superstructures is usually to protect the piers and their foundations, and sometimes to protect the abutments also. With this approach, bearings are placed between the superstructure and the top of the substructure (the piers and abutments). Under normal conditions, they behave like regular thermal bearings. However, in the event of a strong earthquake, they add ﬂexibility to the structure by elongating its period and dissipating energy. In addition, most of the mass of a bridge is concentrated at the deck level, which is inherently strong and can be assumed to be rigid. This permits the superstructure to oscillate at a lower frequency than its piers, which results in the reduction or elimination of deformation of the substructure components beyond their elastic range, particularly at locations that are difﬁcult to inspect or repair (e.g., the piles). Superstructure isolation systems are designed, as far as is practical, to provide moderate ﬂexibility, high damping, torsional balance, and an appropriate distribution of seismic loads between the superstructure supports. Isolated bridges in the United States are generally designed for the effects of a design-basis earthquake; however, the components of the isolation system are provided with increased displacement capacity and are tested for the effects of maximum capable earthquake. This design philosophy is based on observations of bridge failures in past earthquakes, which were primarily due to excessive bearing displacement and the loss of bearing support rather than the collapse of substructure. Providing for strong restoring force in this design philosophy prevents the accumulation of signiﬁcant permanent displacements and allows for a relatively reliable estimation of peak bearing displacement and substructure force.

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Isolation systems adopted by Italian engineers restrict the transmission of force to elements of the substructure to a predetermined level, which is independent of seismic action. Thus, the Italian engineers chose to limit the force transmitted to the substructure to a desired level, at the expense, however, of large variation in peak bearing displacements and the development of permanent displacements. In Japan, menshin (reduction of response) design is followed for the design of bridges. Although menshin design is closely related to the seismic isolation design, the natural period of a bridge is not forcibly elongated in this method, because there are various restrictions in increasing the natural period. Instead of elongating the period, the emphasis in the menshin design is on increasing the energy-dissipating capability and distribution of lateral forces to as many substructures as possible in order to decrease the lateral forces for the design of substructures (Iemura, 1994a, 1994b). It is common to restrain the transverse displacement of bridge isolators, and therefore many bridges in Japan are isolated in longitudinal direction only. The application of seismic isolation for bridges started in the early 1970s. Bridges isolated in New Zealand before 1978 used metallic dampers for seismic isolation. Later, lead rubber bearings replaced the metallic dampers for their seismic isolation. Italian engineers also started the application of modern technology of seismic isolation for bridges in the mid-1970s and now have a long list of isolated bridges. The ﬁrst seismically isolated bridge in Japan was the Hokuso line viaduct in Chiba Prefecture. The viaduct was completed in 1990 and used lead rubber bearings for its isolation system. The United States also obtained its ﬁrst seismically isolated new bridge in 1990, when two bridges, namely, Sexton Creek Bridge in Illinois and Toll Plaza Road Bridge in Pennsylvania, were completed. Lead rubber bearings were used as the isolation system for both the bridges (Skinner et al., 1993). 11.2.9.2

Seismic Retroﬁt

Postearthquake analysis of recent earthquakes around the world shows that the damage was especially concentrated in the buildings and bridge structures that were not designed for earthquake loads or were designed by following the old seismic design guidelines. Moreover, a signiﬁcant proportion of the damage was due to nonstructural elements and the contents housed in the buildings (Kelly, 1997). In order to prevent the recurrence of damage in such seismically deﬁcient structures, it is vital to understand their vulnerability and retroﬁt them for expected future seismic events. In the case of important structures, retroﬁt techniques are required to provide a solution that can also lead to continued functioning during and after major earthquakes, protection of nonstructural components and contents, and, sometimes, to preserve their historic character. 11.2.9.2.1 Buildings Conventional retroﬁtting has been the most common way of enhancing the performance of existing, seismically deﬁcient structures. However, to maintain the usability of important structures after a big earthquake and to protect the valuable contents housed in the buildings, conventional methods can help a little. Simple strengthening or stiffening of a structure through reducing inelastic response displacement, however, may increase the response acceleration and the forces. Some of the other demerits of this approach are (i) the large amount of labor required to increase the strength and ductility of critical sections distributed throughout the complex structural system and (ii) damage may occur due to inelastic behavior of structural elements during a major seismic event (Iemura and Adachi, 2003). Besides this, conventional methods of retroﬁtting often require restricting the use of the building during the retroﬁt. Some buildings, housing important activities, cannot afford to have their routine usage stopped and therefore require retroﬁtting through an earthquake-resistance system that enables continued building use even during the retroﬁt operation. Seismic isolation has proved to be very effective not only in enhancing the safety of existing seismically deﬁcient buildings, but also to preserve the function and protect the contents. Unlike the conventional approach, this retroﬁt method aims to reduce the earthquake force to a level lower than the strength of the existing structure (Figure 11.27). Seismic isolation is the method that can mitigate both interstory drift and high-ﬂoor accelerations simultaneously, and it has been used in retroﬁtting of important buildings, buildings housing valuable contents, and structures of postearthquake importance.

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Safety margin

Required strength

Required strength

Seismic isolation Safety margin Lack of strength

Acceleration (Strength) Lack of strength

Fundamental Period

Fundamental Period Before retrofit Retrofit by seismic isolation

Before retrofit After retrofit (b)

(a)

Note: Region above the required strength curve indicates safe zone.

FIGURE 11.27 Concept of retroﬁtting methods: (a) conventional; (b) seismic isolation. (Source: Sugano, S., Proc. XII World Conf. on Earthquake Engineering, 2000. With permission.)

Moreover, as the modiﬁcation or demolition of building features is minimized, the method has found application in many monuments of historical importance, protecting them from future earthquakes (Buckle, 1995). Recently, the old masonry Chapel of Rikkyo (St Paul’s) University, Tokyo, constructed in 1920, was retroﬁtted with seismic isolation to preserve this historical asset to the university (Seki et al., 2000). Retroﬁtting using seismic isolation of the 72-year-old Clock Tower, a symbol of Kyoto University, Japan, is presently underway (Ogura et al., 2003). The seismic isolation method is found to be more appropriate for the retroﬁtting of seismically vulnerable buildings that are required to function even during retroﬁt operation. To meet such a demand, the retroﬁtting method should affect minimum alterations to the existing structures and should produce a low level of noise and vibration. The technique has successfully been applied to a number of such buildings. In the retroﬁtting of Rankine Brown Building of Victoria University, Wellington, New Zealand, which houses the main library and administration of the university, one of the main requirements was the building’s uninterrupted usage during the retroﬁt operation (Robinson, 2003). Meeting a similar challenge, the government ofﬁce of Toshima Ward, in Tokyo, has been retroﬁtted with rubber and sliding isolators below the existing foundation and was used as usual by the ofﬁcials and citizens even during the retroﬁt works. Similarly, a building of Nihon University in Tokyo, Japan, in which seismic isolation was applied at the top of the columns of basement ﬂoor, was available for research and education during the retroﬁt operation (Kawamura et al., 2000). Although seismic isolation reduces earthquake forces, it does not eliminate them. Consequently, the strength and ductility of an existing structure must at least be sufﬁcient to resist the reduced earthquake forces that remain even after isolation. Further, the requirement of space for the seismic gap and provisions for services, elevators, and so on to follow large displacements at isolation level are some issues that need prior attention. Although not applicable to all structures and all site conditions, seismic isolation still has great potential to reduce the risk in existing buildings and bridges that are not capable of resisting expected future earthquakes. This method, in the beginning, was used generally for preservation of cultural assets, but now has many applications in the retroﬁtting of other buildings also. 11.2.9.2.2 Bridges A large number of old bridges were constructed with little or no consideration for seismic requirements. Some of them have been retroﬁtted, mostly by conventional methods. However, the remaining bridges are waiting for the enhancement of their strength to meet the latest code requirements. In existing bridges, bearings have generally been provided to accommodate longitudinal thermal movements between the superstructure and the supports. Therefore, little structural modiﬁcation is required to use seismic isolation as a retroﬁt method. A common structural conﬁguration, a continuous deck supported

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on bearings at the tops of piers, attracts the option of seismic upgrading of existing bridge structures by seismic isolation and is often practical and relatively inexpensive. Seismic isolation has been used for the retroﬁtting of number of bridges in New Zealand. During the early period of its application in the United States, seismic isolation was mainly used for the retroﬁtting of bridges and the isolation system used was mainly lead rubber bearings (Skinner et al., 1993). However, to replace an existing bearing with an isolation bearing, enough space is required between the deck and pier top. All the jacks used for replacing the bearings should be driven by the same hydraulic pump and raised at the same time to prevent stressing any cross members between girders. The dampers can be connected to the substructure using heavy-duty mechanical or chemical anchorages, with the other end bolted to the girders. Since an earthquake can shift a bridge in any direction, universal joints are recommended on each end of the damping device to prevent the shaft from being bent during movements other than axial ones. The damper should be set to the bridge in its midstroke position or a position that takes full advantage of the free movements (Hipley, 1997). In situations where rigid connection of footing and piers has made the bearing capacity of existing piles inadequate for the new seismic requirements, seismic isolation bearings can be used at the base of columns. Sometimes, it may be costly to retroﬁt a seismically deﬁcient pier and foundation by conventional methods due to submerged conditions that may require construction of cofferdams. In such cases, seismic isolation can reduce the demand on the seismically deﬁcient piers by redistributing the force and transferring it to the abutments, which are signiﬁcantly less expensive to retroﬁt. Seismic retroﬁts are intended to bring bridges up to reasonable earthquake standards; however, in some old bridges, the retroﬁt method is also required to have the least effect possible on the historic appearance and seismic isolation is the appropriate alternative for this requirement. 11.2.9.3

Other Implementations

Apart from in buildings and bridges, seismic isolation has found application in other structures. It has been used for a number of reactor units in France for which the site safe shutdown earthquake acceleration was 0.2g. An isolation system of neoprene pads with topping of lead –bronze alloy limits the acceleration to a value of 0.2g. A similar system was used for reactor units in Koeberg, South Africa, where the site safe shutdown earthquake acceleration was 0.3g (Tajiran, 1998). Seismic isolation has a lot of potential for the protection of sensitive equipment that can malfunction when the acceleration reaches beyond its allowable limit. Communication centers, hospitals, computer centers, and other facilities housing such sensitive equipment can use seismic isolation to remain functional even after a major seismic event. Proposed systems for the seismic isolation of sensitive equipment are to (1) base isolate the entire building or a portion of the building, (2) isolate essential ﬂooring systems only (raised ﬂoors), and (3) isolate individual pieces of equipment by placing them on separate isolation bearing systems (see Figure 11.28). In Japan, there are many buildings that are using the raised ﬂoor system to protect computers and other sensitive equipment. Seismic isolation is found to be very useful for semiconductor facilities, in which cost of the delicate items is as high as 75% of the total investment (Amick et al., 1998). Liquid storage tanks are important components of chemical factories and are also used in increasing

FIGURE 11.28 Seismic isolation for protection of equipment. Unreinforced rubber blocks. (Source: Ohashi, U.G. Earthquakes and Base Isolation, Pub. Asakura, Tokyo, 1995. With permission.)

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FIGURE 11.29

Three-dimensional base-isolation system for Moroku Bosatsu statue.

TABLE 11.2 Details of Isolation System and Dynamic Characteristics of the Structure

Elastic support Damping mechanism Natural frequency (Hz)

Horizontal Direction

Vertical Direction

Coil spring Friction damper 0.25–0.35

Coil spring Air damper 2.0–2.5

number to store liqueﬁed natural gas. The FPS was applied for seismic isolation of a large-size tank in Greece, and steel –rubber bearings were used for another tank in Korea, storing liqueﬁed natural gas (Tajiran, 1998). Seismic base isolation can also be used for the seismic protection of antique pieces. Iemura and coworkers designed a special seismic isolation system to protect a wooden carved statue of Moroku Bosatsu (Maitreya). It is an antique cultural artifact, considered to be the ﬁrst wooden carved statue in Japan. The isolation system provides protection in all three directions FIGURE 11.30 Overall size of the three-dimensional (Figure 11.29). The details of the isolation system base-isolation system. are provided in Table 11.2. The overall size of the isolation system is shown in Figure 11.30. The shaking table test was also carried out to verify the effectiveness of the isolation system. Results of the experiment are presented in Table 11.3 and clearly show that the isolation system is very effective in reducing the accelerations.

11.2.10

Performance during Past Earthquakes

Analytical models and experimental tools have been developed by researchers to simulate the behavior of a structure under earthquake motion. However, none of these methods can validate the behavior of a structure as does a real earthquake. In fact, many of the concepts of earthquake-resistant design are

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Results of Shaking Table Tests of a Three-Dimensional Base Isolation

Direction

Level Acceleration (gal) Shaking Table

NS EW Vertical

795 505 314

Above Base-Isolation System 166 136 250

Relative Displacement of Base Isolation (cm) 15.3 10.0 1.1

the outcome of the analysis of observations made or recorded during past earthquakes. In the case of base isolation, as it is a relatively newer approach, it can help to assess the modeling techniques, existing analysis and design procedures, and methods followed in construction. Such analyses may help in evaluating the effectiveness of base isolation and also exposing weak aspects of designs. Strong motionrecording instrument networks have been installed on some of the base-isolated structures in different parts of the world in order to record the response of those structures during earthquakes. During the Northridge, Kobe, and Turkey earthquakes, a few base-isolated structures had their isolation system reaching nonlinear range, and the recorded performance of these structures can give a fair idea of the behavior of base-isolated structures during strong earthquake motions.

11.2.10.1

Performance of Buildings

The Northridge earthquake of January 1994 saw thousands of conventionally designed buildings and their contents suffer extensive damage and be closed for several days. However, two base-isolated buildings, namely, University of Southern California Hospital building and Fire Command and Control Facility, performed well during the earthquake. During the event, the seven-story baseisolated hospital building experienced a peak ground acceleration of 0.37g. The recorded peak acceleration at the roof was 0.21g. Analytical study showed that presence of the isolation system reduced the peak story shear at the base to about one third of that if the building had its base ﬁxed (Nagarajaiah et al., 1996). “Workers in a large supply room within the hospital reported no disruption of supplies, or material falling off shelves, as a result of the earthquake. In contrast a pharmacy at the ground level in an adjacent medical building with ﬁxed-base reported substantial disruption of the supplies” (EERI, 1996). The two-story base-isolated Fire Command and Control Facility could not perform as per expectation due to minor negligence in detailing the seismic gap. The building is base-isolated by high-damping rubber bearings and experienced peak ground acceleration of 0.19g. The recorded response shows the presence of sharp acceleration spikes that were due to pounding. However, the analytical study by Nagarajaiah et al. (2001) shows that, even with pounding, maximum base story shear of the base-isolated building was half that of the ﬁxed-base case and without pounding it would have been one fourth. In January 1995, the unexpected earthquake of magnitude 7.2 in Kobe shocked the engineering community. The 7 sec of strong shaking caused a heavy loss of life (over 5000 people died) and a catastrophic loss to infrastructure. The strong earthquake tested two existing base-isolated buildings in the region, namely, the six-story building of West Japan Postal Savings Computer Center (WJPSCC) and the three-story building of Technical Research Institute of Matsumura Gumi Corporation (TRIMGS). The six-story WJPSCC building that is base-isolated with elastomeric isolators, lead rubber bearings, and steel coil dampers experienced peak ground acceleration of 0.306g. A nearby six-story ﬁxed-base building experienced peak ground acceleration of 0.27g. However, during the event, the maximum acceleration recorded at the roof of the ﬁxed-base building was about nine times of that recorded at the roof of the WJPSCC building. Similarly, a

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comparison of the roof acceleration of the three-story base-isolated TRIMGS building with that of a nearby three-story ﬁxed-base building reveals that, due to base isolation, the peak accelerations at the roof were reduced by a factor of approximately 4.9 in the north–south direction and 2.5 in the east –west direction. It was later realized that the performance of this base-isolated building might be adversely affected due to freezing cold, when the rubber has a tendency to become stiffer (Izawa et al., 1996). 11.2.10.2

Performance of Bridges

In the year 1999, Turkey faced two major earthquakes, the ﬁrst one on 17th August and the second on 12th November. At the time of the events, there were two base-isolated structures in the region, namely, Bolu Viaduct and Bolu Bridge. The structures were designed for a peak ground acceleration of 0.4g with isolation system comprising of a friction system and crescent moon-type hysteretic elements. During the earthquake of August 17, 1999, when the estimated ground acceleration at the Bolu viaduct site was 0.39g in the longitudinal direction and 0.31g in the transverse direction, the viaduct survived without any damage. The residual displacement was of the order of 1 mm only. However, the second event produced near-fault, or rather on-fault, pulse-type motion with a peak ground acceleration at the viaduct site in the order of 1.0g. The unexpected level of ground acceleration caused heavy damage to the viaduct. It was reported that the sliding bearings fell down from most of the piers before any cyclic movement. This performance again raised the issue of feasibility of base isolation for such near-fault pulse-type motions. During the same event, Bolu Bridge experienced the estimated peak ground motion of 0.65g, much higher than the design value. However, the only signiﬁcant damage suffered by the bridge was the failure of some elements of the expansion joints because they were designed for a movement of 100 mm only (corresponding to the earthquake with a return period of 50 years). After the earthquake, the bridge showed a residual displacement in longitudinal direction of 45 mm and in transversal direction varying from 60 to 100 mm (Marioni). The Eel River Bridge (U.S.), after being retroﬁtted with lead laminated rubber bearings, was hit by the magnitude 7.0 Cape Mendocino (Petrolia) earthquake in the year 1992. The peak ground acceleration recorded at a nearby recording station was 0.55g. The bridge suffered structural damage limited to the spalling of concrete at joints. Another base-isolated bridge, Sierra Point overpass (U.S.), was subjected to the 1989 Loma Prieta earthquake when peak ground acceleration at the base of bridge columns was 0.090g. The bridge, ﬁrst constructed in 1956, was later retroﬁtted with lead rubber bearings in 1985. However, the abutments were not modiﬁed for the larger clearance required in isolation. This design weakness was exposed during the Loma Prieta earthquake and caused signiﬁcant ampliﬁcation in the steel superstructure of the bridge. However, no damage was sustained (Lee et al., 2001). Rangitaiki River Bridge at Te Teko, New Zealand, with lead rubber bearings on each pier and plain rubber bearings on abutments, suffered minor damage and a small permanent displacement during the 1987 Bay of Plenty earthquake. The peak acceleration, recorded at about 11 km from the bridge site was 0.33g. It was observed that owing to the inadequate fastening details for the isolators, one of the two bearings on abutment was dislocated (McKay et al., 1990). Bai-Ho bridge (Taiwan), a three-span, continuous nonprismatic prestressed concrete girder bridge, was subjected to the magnitude 6.0 Gia-I earthquake in year 1999. The bridge is seismically isolated in its longitudinal direction with lead rubber bearings; however, shear keys and specially designed steel rods are installed on both abutments to restrict the transverse movement of the superstructure. The peak acceleration recorded on the deck in longitudinal direction was slightly higher than that in the foundation while, owing to restraints, the peak acceleration in the transverse direction was 2.5 times than that of foundation (Lee et al., 2001). During the Kobe earthquake, the base-isolated Matsunohama Bridge had good recorded performance; however, the ground acceleration was not profound enough to test the effectiveness of the isolation system (Izawa et al., 1996).

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In summary, analysis of the records obtained from a variety of base-isolated structures during real earthquakes presents some very interesting facts. In general, it was observed that, during strong earthquakes, numerous conventionally designed buildings suffered damage and remained closed for several days; however, the base-isolated buildings in the same region not only survived but continued to operate. On the other hand, in some cases, strong earthquakes exposed the lack of attention to minor construction details and the effectiveness of base isolation for near-fault earthquakes. Better performance of base-isolated structures during the Northridge and Kobe earthquakes have now persuaded the structural engineers and owners and triggered a rapid increase in the number of baseisolated buildings. In Japan alone, the number of buildings of this kind constructed during 1995 was almost the same as that of those constructed during the period from 1985 to 1994.

11.3 11.3.1

Seismic Vibration Control Historical Development of Seismic Vibration Control

It is worth noting that a structural vibration control system for reducing seismic responses was developed several centuries ago in Japan, by an unknown engineer. This was applied to the construction of a Gojunoto (pagoda). The Gojunoto was ﬁve stories tall and was constructed of closely ﬁtting mortised wooden beams and Wooden pagoda columns (Figure 11.31). During an earthquake, the vibrations of such a vertical cantilever structure would produce bending moments that could not be resisted by tension at the mortised joints. To Suspended overcome this weakness, a long wooden pole was pole suspended freely from the upper part of the pagoda so that it could undergo pendular vibrations if the pagoda was excited into motion by an earthquake. The weight of the pole exerted a compressive prestress on the pagoda, thus increasing the bending resistance. The bottom of the pole extended into a cylindrical hole in the ground that was of a larger diameter than the pole. Thus, when the pagoda was excited into vibrations by an earthquake, some of the vibrational energy would be transferred into oscillations of the pole and the impact of the pole on the sides of the hole would dissipate energy. In 1969, Gupta and Chandrasekaran (1969) FIGURE 11.31 Sketch of a wooden pagoda with studied the effectiveness of a number of tuned suspended pole that is free to undergo pendular vibration. (Source: Tanabashi, R., Proc. II World Conf. mass dampers (TMDs) put in a single-DoF on Earthquake Engineering, 1960. With permission.) system to reduce seismic responses. The TMD system has both linear and elasto-plastic restoring-force characteristics and viscous-type damping. The study was shown in the Fourth World Conference on Earthquake Engineering (WCEE) in Santiago, Chile. In the Fifth WCEE in 1973, Skinner et al. (1973) presented their studies on energy absorption devices for earthquake-resistant structures. The devices were based on the plastic deformation of mild steel. In 1977, Ohno et al. (1977) introduced optimum tuning of the dynamic damper for civil structures under earthquake

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excitations. Skinner et al. (1977) showed their Floor Force works on studying hysteretic dampers for Fy increasing structural earthquake resistance, and Friction Tyler (1978) demonstrated the usefulness of damper Deflection ﬂexural yielding of steel plate energy-dissipation Column devices in New Zealand. In the seventh WCEE in 1980, Sladek and Klingner (1980) reported their studies on tuned(b) mass dampers for seismic response reduction in (a) civil structures, whereas Keightley (1980) presented his work on a dry friction damper for multistory FIGURE 11.32 (a) Friction dampers in a framed structures. Figure 11.32 shows the schematic of a structure; (b) hysteretic loop of the damper. (Source: framed structure with additional friction dampers Keightley, W.O. Proc. VII World Conf. on Earthquake Engineering, 1980. With permission.) and the hysteretic loop of the damper. In 1984, Pall (1984) numerically studied the performance of a framed building equipped with friction devices. The performance was found to be superior to that of a conventional building. Scholl (1984) introduced brace dampers. The stiffness and damping of the damper can be set to alter the stiffness and damping in a structure. It was typically installed at the cross points in braced-frame structure systems. Liu et al. (1984) investigated the effectiveness of an active mass-damper control system for controlling coupled lateral –torsional motions of tall buildings subjected to strong earthquakes. A special themed session named “Seismic response control of structural systems” was held at the Ninth WCEE in 1988. Some studies on passive systems were reported such as: a hysteretic damper that is put between two adjacent structures (Kobori et al., 1988); a multiple passive TMD (Clark, 1988); viscoelastic materials (Bergman and Hanson, 1988); viscous damping walls (Arima et al., 1988); and magnetic damping (Shimosaka et al., 1988). Also, some studies on active control were reported, such as an experimental study on active mass dampers (Aizawa et al., 1988) and air injection for suppressing sloshing in a liquid tank (Hara and Saito, 1988; Sogabe et al., 1988). Since then, the active vibration control of buildings and other civil structures has attracted a worldwide growing interest as an innovative technology in earthquake engineering. During the Tenth WCEE in 1992, more thorough studies on structural control were reported such as: predictive control of structures with reduced number of sensors and actuators (Lopez-Almansa et al., 1992); applicability of vibration control to nonlinear structure (Shimada et al., 1992); full-scale implementation of active control (Soong et al., 1992); effects of soil–structure interaction (SSI) on actively controlled structure (Wong and Luco, 1992); the time-delay problem (Hou and Iwan, 1992); and fuzzy logic control (Tani et al., 1992). In this conference, the decision to form an international association and to hold a world conference dedicated to structural control was made. The International Association for Structural Control (IASC) was formed in 1993. The efforts led to the successful First World Conference on Structural Control in 1994 in Pasadena, California (IASC, 1994). It was followed by the Second World Conference on Structural Control in 1998, which was held in Kyoto, Japan (IASC, 1999). The Third World Conference on Structural Control was held in April 2002, in Como, Italy (IASC, 2003). The conferences showed increasing interest among researchers and an increase of innovations and applications of structural control technologies in civil structures.

11.3.2

Basic Principles

Structural vibration control basically involves the regulation of structural properties in order to achieve structural a desirable response to a given external load and modiﬁcation of the excitation. The regulation of structural properties includes the modiﬁcation of mass, damping, and stiffness of the structure so that

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it can respond more favorably to the external excitation. Moreover, it is also possible to reduce the level of excitation transmitted to the structure. Structural control can be deﬁned as a mechanical system that is installed in a structure to reduce structural vibrations during loadings such as strong earthquakes. 11.3.2.1

Passive Control System

The basic function of a passive control system is to absorb or consume a portion of the input energy, thereby reducing energy-dissipation demand on primary structural members and minimizing the possible structural damage. In the passive system, controlling forces develop at the locations of installation of the mechanism itself. The energy necessary for generation of these forces is provided through the motion of the mechanism during the dynamic excitation. The relative motion of the mechanism deﬁnes the amplitude and the direction of the controlling force. Passive control has four main advantages: (1) it is usually relatively inexpensive; (2) it consumes no external energy (energy may not be available during a major earthquake); (3) it is inherently stable; and (4) it works even during a major earthquake. In the passive system area, structural response control can be divided into two groups. The ﬁrst group uses supplemental damping to reduce structural responses by conversion of kinetic energy to heat. The second group uses supplemental oscillators to reduce the structural response by transferring energy among vibration modes. The ﬁrst group includes devices that operate on principles such as frictional sliding, the yielding of metals, phase transformation in metals, deformation of viscoelastic solids or ﬂuids, and ﬂuid oriﬁcing. The second group includes dynamic vibration absorbers such as TMDs and tuned liquid dampers. The major difference between viscous/viscoelastic devices and friction/yielding devices is the maximum force that each device will develop during an earthquake. The maximum earthquake forces developed in viscous/viscoelastic devices are determined by the maximum displacements and velocities across these devices. The maximum earthquake forces in a friction/yielding device equals the design friction force/design yield force plus strain hardening. Thus, the maximum earthquake forces are more easily controlled in the friction/yielding devices. The effect of increasing damping in a structure can be appreciated by studying the earthquake response spectrum. The earthquake response spectrum of a quantity is a plot of the peak value of the response quantity as a function of the natural vibration period of the system. Each such plot is for single-DoF system having a ﬁxed damping ratio, and several such plots for different values of damping ratio are included to cover the range of damping values. The most commonly used spectra in earthquake engineering are the absolute acceleration response spectrum, the velocity response spectrum, and the displacement response spectrum. For more information on how to develop earthquake response spectrum, readers are referred to Chopra (1995). Figure 11.33 shows the response spectra of measured ground acceleration during the Kobe, Japan, earthquake on January 17, 1995. The ground acceleration north–south (NS) component was recorded at the Kobe Marine Meteorological Observatory in Chuo-ku. As is shown in the ﬁgure, increasing the damping ratio from 5 to 30% decreases the response spectra. Therefore, on a level of ordinary structure that has 5% of damping, the response can be reduced further by increasing the structural damping. In the area of passive systems, a variety of mechanical energy dissipaters to increase structural damping has been developed and tested in the laboratory and, in some cases, in actual structural applications, such as bracing systems, friction dampers, viscoelastic dampers, and other mechanical dampers. The principle of the TMD dates back to the 1940s when Den Hartog (1947) reintroduced the dynamic absorber invented by Frahm in 1909. Figure 11.34 shows the principle of the TMD. A large system under alternating force P0 sin vt is represented by a K–M system where K and M are the stiffness and the mass of the system, respectively. The vibration absorberpconsists of a comparatively small vibratory system ﬃﬃﬃﬃﬃ with stiffness k and mass m. The natural frequency, k=m; of the attached absorber is chosen to be equal to the frequency, v, of the disturbing force. It has been shown (Den Hartog, 1947) that the main mass, M, does not vibrate at all and that the small system, k–m, vibrates in such a way that its spring force is at all instants equal and opposite to P0 sin vt: Thus, there is no net force acting on M and therefore that mass

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11-36

Vibration Monitoring, Testing, and Instrumentation

FIGURE 11.33

Effect of damping on earthquake response spectra.

does not vibrate sﬃﬃﬃﬃ rﬃﬃﬃﬃﬃ K k m ; m¼ ð11:22Þ vM ¼ ; vm ¼ M m M

x1 K

M

P0 sinwt k

x2 m

It is useful to introduce the new notations, above. Assume that m ¼ 0:2; and vM ¼ vm : For this FIGURE 11.34 Additional vibratory k –m system to a condition, the resonance curve of the system is larger K –M system to reduce vibration of the larger illustrated in Figure 11.35. At v ¼ vm or v=vm ¼ system. 1; the motion of the main mass ceases altogether (Chopra, 1995). Since the system has two DoF, two resonant frequencies exist and the response is unbounded at those frequencies. If the exciting frequency, v, is close to the natural frequency, vm, of the attached absorber, and the operating restrictions make it possible to vary either one, the vibration absorber can be used to reduce the response amplitude of the main system to near zero. It is worth noting that the larger the attached absorber mass, m, the smaller the absorber displacement, ðx2 2 x1 Þ; relative to the displacement of main mass, M. However, a large absorber mass presents a practical problem. At the same time, the smaller is the mass, m, the narrower is the operating frequency range of the absorber (Chopra, 1995). 11.3.2.2

Active Control System

Active control systems are used to control the response of structures to internal or external excitation, such as machinery or trafﬁc noise, wind, or earthquakes, where the safety or comfort level of the

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occupants is of concern. Active control makes use 8 of a wide variety of actuators, which may employ x1 hydraulic, pneumatic, electromagnetic, or motor- P /K o driven ball-screw actuation. An essential feature of active control is that external power is used to effect the control action. This makes the system vulnerable to power failure, which is always a 1 0 possibility during a strong earthquake. 01 0.8 1 1.25 ω/ωm For the actuators properly to apply the desired forces, sensors are placed in the structure FIGURE 11.35 Resonance curve for the system in in order to measure structural response. The Figure 11.34 that shows response amplitude vs. exciting sensors convey response information to a central frequency (the dashed curve indicates negative x or 2 computer that then uses this information to phase opposite to excitation) for m ¼ 0:2 and vM ¼ vm . calculate the desired actuator forces. The advan- (Source: Chopra, A.K. 1995. Dynamics of Structures, tage of an active control system is that it attains Theory and Applications to Earthquake Engineering, excellent control results. However, there are many Prentice Hall, New York. With permission.) drawbacks to using the active control system. It is relatively expensive to design and to operate due to Excitation Structure Response the large amount of power needed. It has the potential to destabilize the structural system. Control Furthermore, it tends to take up more space than passive control devices. Sensors Controller Sensors A fully active structural control system has the basic conﬁguration as shown schematically in Figure 11.36. The structural control system in the FIGURE 11.36 Schematic diagram of active control ﬁgure basically consists of sensors, controllers, system. and actuators. Sensors are used to measure either external excitations or structural responses, Desired Control or both. Controllers process the measured inforOutput output input Controller Plant mation and compute necessary control forces needed based on a given control algorithm. Actuators are used to produce the required forces and are FIGURE 11.37 An open-loop control system; the usually powered by external energy sources. controller applies control input without knowing the According to the characteristics of the controlplant output. ling effects, active control can be classiﬁed into two categories: open-loop control and closed-loop control. A control system in which the control input is applied without the knowledge of the plant output is called an open-loop control system. Figure 11.37 shows a block diagram of an open-loop control system, where the subsystems (controller and plant) are shown as rectangular blocks. Open-loop control will be successful only if the controller has a reasonably good prior knowledge of the behavior of the plant, which can be deﬁned as the relationship between the control input and the plant output. Mathematically, the relationship between the output of a linear plant and the control input can be described by a transfer function. However, in actuality, the presence of plant behavior uncertainty is unavoidable. Therefore, it is clear that an open-loop control system is unlikely to be successful. In the case that the controller adjusts the control input according to the actual observed output, the system is called a closed-loop system. In this system, the control input is a function of the plant’s output. Since in a closed-loop system the controller is constantly in touch with the actual output, it is likely to succeed in achieving the desired output even in the presence of uncertainty in the linear plant’s behavior (the transfer function). The mechanism by which the information about the actual output is conveyed to the controller is called feedback. On a block diagram, the path from the plant output to the controller is called a feedback loop. A block diagram of a possible closed-loop system is given in Figure 11.38.

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Vibration Monitoring, Testing, and Instrumentation

Desired output +−

Error

Controller

Control input

Plant

Output

Feedback loop FIGURE 11.38 Example of a closed-loop control system with feedback; the controller applies a control input based on the plant output.

When the desired output is a constant, the resulting controller is called a regulator. If the desired output changes with time, the corresponding control system is called a tracking system. In any case, the principal task of a closed-loop controller is to minimize the “error” as quickly as possible. 11.3.2.3

Semiactive Control System

Semiactive control falls between passive and active on the control spectrum. A semiactive control system is similar to an active system in that the system uses sensors and controllers, and operates on external power. However, the source of external energy is used only for adjustment of the mechanical characteristics of the system (Rakicevic and Jurukovski, 2001). The inherent beneﬁt of a semiactive control device is that the mechanism used does not require large amounts of external power. Many semiactive devices can be powered by batteries protecting them from sudden power loss during earthquakes. Semiactive control systems are basically derived from passive systems; they are modiﬁed in such a way that they enable adjustment or correction of their mechanical characteristics. A typical strategy for a semiactive control system is that an “ideal” actively controlled device is ﬁrst assumed and appropriate primary controller designs for this device are designed. Then, a secondary controller is used that clips the optimal control force so it is dissipative in a manner consistent with the physical nature of the device. This strategy has been widely used (e.g., Dyke et al., 1996a, 1996b, Jung et al., 2001). Spencer and Sain (1997) found that many active control systems for civil engineering applications operate primarily to modify structural damping. They claimed that preliminary studies indicate that appropriately implemented semiactive systems perform signiﬁcantly better than passive devices and have the potential to achieve the majority of the performance of fully active systems, thus allowing for the possibility of effective response reduction during a wide array of dynamic loading conditions. In other words, semiactive control devices offer the adaptability of active control devices without requiring the associated large power sources (Spencer and Sain, 1997). Moreover, according to presently accepted deﬁnitions (Housner et al., 1997), a semiactive control device is one that cannot inject mechanical energy into the controlled structural system (including the structure and the control device), but has properties that can be controlled optimally to reduce the responses of the system. Therefore, in contrast to active control devices, semiactive control devices do not have the potential to destabilize (in the bounded input/bounded output sense) the structural system. Semiactive control devices are often viewed as controllable passive devices. Semiactive control technologies have recently been widely investigated in terms of the reduction of the dynamic response of structures subjected to earthquake and wind excitations (Housner et al., 1997; Spencer et al., 1997; Patten, 1998; Kurata et al., 1999; He et al., 2001; Iemura et al., 2001; Jung et al., 2001). Various semiactive devices have been proposed that utilize forces generated by surface friction or viscous ﬂuids to dissipate vibratory energy in a structural system. 11.3.2.4

Hybrid Control System

The common usage of the term “hybrid control” implies the combined use of active and passive control systems. For example, this may be a structure equipped with distributed viscoelastic damping

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supplemented with an active mass damper on the top of the structure, or a base-isolated structure with actuators actively controlled to enhance performance. The method, which consists of using both passive and active devices, should utilize the merits of both passive and active methods and avoid the demerits of these methods. Thus, higher levels of performance should be achievable. Additionally, the resulting hybrid control system can be more reliable than a fully active system because, should the active control malfunction, the minimum seismic protection of the structure can be done by the passive control. 11.3.2.5

Categorization of Basic Principles

From the above group of structural control systems, there are basically ﬁve principles concerning earthquake-induced structural response control that are important to consider. They are listed below: 1. Cutting off the input energy from the earthquake ground motion; examples: (a) Floating structures (b) Frictional structures 2. Isolating the natural frequencies of the structures from the predominant seismic power components; examples: (a) Base-isolated structures (b) Long period structures 3. Providing nonlinear structural characteristics and establishing a nonstationary state nonresonant system; examples: (a) Inelastic structures (b) Varying stiffness and damping structures 4. Utilizing energy absorption mechanism; examples: (a) Viscous damper (b) Viscoelastic damper (c) Inelastic behavior 5. Supplying control force to suppress the structural response; examples: (a) Active mass damper (b) Active tendon (c) Joint damper

11.3.3

Important Issues in Vibration Control

11.3.3.1

Soil–Structure Interaction

Seismic vibration control of a civil structure deals with methods to suppress the response in a structure subjected to earthquake excitation. To control the structural dynamic response, the structural system model must be known. Modeling of a structure is relatively straightforward, using a ﬁnite number of DoF, because the dimensions of the structure are ﬁnite. However, in general, civil structures are supported on surrounding soil from which the tremor excites the structure. This makes the structure interact with the surrounding soil. Therefore, it is very important to include SSI for controlling the seismic response. SSI will result in a structural response that may be quite different from the structural response computed for a ﬁxed-base building. The frequency of vibration of the structure may be lower because of the interaction. The change in frequency may also affect the response of the overall structure or its substructures or components. Moreover, soils are notoriously nonlinear when subjected to strong ground motions at the level of engineering interest (Marshall, 2001). Damping of the ﬁnal system increases because of the radiation of energy of the propagating waves away from the structure (Wolf, 1985).

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11.3.3.2

Vibration Monitoring, Testing, and Instrumentation

Device –Structure Interaction and Time Delay

Other important effects come from the device–structure dynamic interaction. Dyke et al. (1995) studied the role of control–structure interaction in mitigating dynamic response. Their ﬁndings show that accounting for control– structure interaction and actuator dynamics in the design process can improve the performance and robustness of a control system. Time delay between control command and actual control force also causes problems. A study by Hou and Iwan (1992) showed the problems of time delay in the vibration control of structures. Also, Agrawal and Yang (2000) claimed that applications of unsynchronized control forces due to time delay may result in a degradation of the control performance, and it may even render the controlled structure unstable. Therefore, they provide a state-of-the-art review for available methods of time-delay compensation. 11.3.3.3

Design Guidelines

Since structural control is a new concept in civil structures, design guidelines are now being developed to provide the design engineers with tools for the safe design or seismic rehabilitation of structures. For a passive control system, the American Society of Civil Engineers has prepared for the Federal Emergency Management Agency a “Prestandard and Commentary for the Seismic Rehabilitation of Buildings” (ASCE, 2000), in which requirements for the systematic rehabilitation of buildings using energydissipation systems is set forth. In the standard, analysis and design criteria for passive energy dissipation systems are provided. In this prestandard, two sections deal with structural control technology. First, is “Passive Energy Dissipation Systems,” which contains (1) General Requirements, (2) Implementation of Energy Dissipation Devices, (2) Modeling of Energy Dissipation Devices, (3) Linear Procedures, (4) Nonlinear Procedures, (5) Detailed Systems Requirements, (G) Design Review, and (H) Required Test of Energy Dissipation Devices. Each subsection is explained in detail in the prestandard for the users. However, in the second section, “Other Response Control Systems,” it is mentioned that analysis and design of response control other than in the passive systems above shall be reviewed by an independent engineering review panel. This is because the technology of active, semiactive, and hybrid control is not sufﬁciently mature and the necessary hardware is not sufﬁciently robust to warrant the preparation of general guidelines for the implementation of the technology.

11.3.4

Vibration-Control Devices

11.3.4.1

Passive Control Systems

11.3.4.1.1 Metallic Yielding Dampers One of the effective mechanisms useful for the dissipation of energy input to a structure from an earthquake is through the inelastic deformation of metals. The idea of utilizing added metallic energy dissipaters within a structure to absorb a large portion of the seismic energy began with the conceptual and experimental work of Kelly et al. (1972) and Skinner et al. (1975). Devices considered by them included torsional beams, ﬂexural beams, and U-strip energy dissipaters as shown schematically in Figure 11.39. During the following years, considerable progress has been made in the development of metallic dampers. For example, many new designs have been proposed, including the hourglass or X-shaped and triangular plate dampers as shown in Figure 11.40. The X-shaped steel plates form devices called ADAS (Added Damping and Stiffness; see Perry et al., 1993), because the devices essentially add stiffness as well as damping to the element where they are installed. For example, if they are installed between two adjacent ﬂoors, then it will increase the stiffness and damping between those two ﬂoors. Increasing the stiffness will generally attract more seismic forces. However, since the devices have much lower yielding forces than the elements where they are installed, then the postyield stiffness of the devices is dominant during a strong earthquake.

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Force

Force Force Force

Force (a)

(b)

Force

FIGURE 11.39 Schematic of metallic dampers: (a) torsional beam and (b) U-strip, and the directions of alternating forces. (Source: Skinner, R.I. et al. Earthquake Eng. Struct. Dyn., 3, 287 –296, 1975. With permission.)

Force Beam of Upper Floor

Force

Braced to Beam of Lower Floor

Force (a)

(b)

Force

FIGURE 11.40 Schematic of (a) X-shaped (Source: Tsai, K.-C. et al. Earthquake Spectra, 9, 505 –528, 1993. With permission.) and (b) triangular plate dampers, and the directions of alternating forces. (Source: Perry, C.L. et al. Earthquake Spectra, 9, 559– 579, 1993. With permission.)

The devices dissipate energy through the ﬂexural yielding deformation of mild steel plates. Each device consists of a series of steel plates arranged in parallel, with boundary elements at the top and bottom to establish end ﬁxity, which bend in double-curvature ﬂexure when subjected to lateral loading. The direction of the lateral loadings is shown in Figure 11.41. The plates are cut in an hour-glass shape to match the moment diagram and thus maximize the uniformity of plastiﬁcation over the height of the steel plates. Cyclic loading tests to determine the reliability of the ADAS were performed by Bergman and Goel (1987). The tests demonstrated that the devices maintained stable hysteretic properties, dissipated signiﬁcant energy with no pinching or slip zones, and continued to have these properties until plate fracture at high strain or high numbers of cycles. The schematic hysteretic loop of ADAS

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11-42

Vibration Monitoring, Testing, and Instrumentation

Force Moment diagram

Force FIGURE 11.41 ADAS element: (a) lateral view; (b) longitudinal view with lateral forces; and (c) moment diagram over the height of the steel plates.

devices tested by Bergman and Goel (1987) is shown in Figure 11.42. It is worth noting that the apparent stiffening of the device in the ﬁgure is due to ﬁnite deformation, not strain hardening of the material; therefore, the effects of ﬁnite deformation should be included in the assessment of a design. Tsai et al. (1993) have performed cyclic loading tests on TADAS (Triangular-plate ADAS; see Figure 11.40b). The hysteretic loops of the tested TADAS are similar to those of ADAS, such as the apparent stiffening at large displacement, since the mechanism is the same. Although the slotted pin connection at the apex reduces the axial tension force, some friction at the pin location contributes to the stiffening of hysteretic loops (Soong and Dargush, 1997). 11.3.4.1.2 Friction Dampers Another excellent mechanism for energy dissipation is friction. The mechanism has been used for many years in automotive brakes to dissipate kinetic energy of motion. In friction dampers, stick-slip phenomena must be minimized to avoid introducing high-frequency excitation. 10

8 Force (kips)

(a) −0.5

Displacement (inch)

−8 0.5

(b)

−10 −1.1

Displacement (inch)

1.1

12 Force (kips) (c)

−12 −1.6

Displacement (inch)

1.6

FIGURE 11.42 Schematic hysteresis loop of the ADAS device with maximum displacements of (a) 0.42 in. (10.7 mm); (b) 1.04 in. (26.4 mm); and (c) 1.56 in. (39.6 mm). (Source: Bergman, D. and Goel, S., Rpt. UMCE 87-10, Univ. Michigan, 1987. With permission.)

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Seismic Base Isolation and Vibration Control

brace

11-43

A

C

column

beam

links

slip lap joint with friction pad damper

(a)

B (b)

FIGURE 11.43

D brace

Pall friction damper: (a) set up in a structure; (b) schematic of the damper.

Furthermore, friction materials should have a Force (kN) (lbs) consistent coefﬁcient of friction over the intended 2000 8.9 life of the device (Housner et al., 1997). 0.7 (in) −0.7 The Pall device (Pall and Marsh, 1982) is one of Displacement the friction dampers that can be installed in a −17.78 17.78 (mm) structure in an X-braced frame as illustrated in −2000 −8.9 Figure 11.43. The mechanism of the damper is that, when there is axial tension force in the bracing system, as shown in the ﬁgure by outward arrows, FIGURE 11.44 Schematic of force – displacement then plates A and B are moving outward from each hysteresis loop of the Pall friction damper. (Source: other. This movement is resisted by some friction Filiatrault, A. and Cherry, S. Earthquake Spectra, 3, force at the slip lap joint between plates A and B. 57– 78, 1987.) This movement also causes plates C and D to move toward each other because of the existence of the links. This movement is also resisted by some frictional force between plates C and D. Without the existence of the links, the bracing may buckle because of axial compression force. The friction level between plates is designed so that the plates will not slip to each other during wind storms or moderate earthquakes. Under severe loading conditions, the devices slip in order to dissipate energy so that structural response can be reduced. The force– displacement relationship of Pall dampers has been studied extensively. A plot of its typical cyclic response is illustrated in Figure 11.44 (Filiatrault and Cherry, 1987). 11.3.4.1.3 Viscoelastic Dampers The application of viscoelastic dampers to civil Viscoelastic Force/2 material engineering structures seems to have begun in 1969, when approximately 10,000 viscoelastic Force dampers were installed in each of the twin towers Force/2 of the late World Trade Center in New York to Center reduce wind-induced vibrations. An example of a plate Steel flange viscoelastic damper is illustrated in Figure 11.45. The damper consists of a viscoelastic material FIGURE 11.45 Typical viscoelastic-damper conbonded to steel plates. The viscoelastic materials ﬁguration. used in structural application are typically copolymers or glassy substances, which dissipate energy when subjected to shear deformation (Soong and Dargush, 1997). With induced structural vibration, the damper will absorb and dissipate the vibrational

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Vibration Monitoring, Testing, and Instrumentation

1.00 Elastic stiffness

Force (kips) 0.00

Energy dissipated in one cycle of oscillation −1.00 −0.05

Displacement (in)

0.05

FIGURE 11.46 Typical schematic hysteresis loop of viscoelastic damper. (Source: Shen, K.L. and Soong, T.T. J. Eng. Mech. ASCE, 121, 694– 701, 1995. With permission.)

energy by shearing deformation within the viscoelastic material. Heat will be generated in the viscoelastic material and released through the steel members of the damper. A typical hysteresis loop produced by viscoelastic dampers is shown in Figure 11.46. By using viscoelastic dampers, although the structural response is elastic, hysteresis loops are formed because of the viscoelastic material. The area enclosed in the hysteresis loops is the energy dissipated by the viscoelastic dampers during one cycle of oscillation. The behavior of viscoelastic materials under dynamic loading depends on vibrational frequency, strain, and ambient temperature. In general, the relationship between shear strain, gðtÞ; and shear stress, t ðtÞ; under harmonic shear strain with frequency, v; can be expressed as (Zhang et al., 1989) G00 ðvÞ g_ðtÞ ð11:23Þ v G0 ðvÞ and G00 ðvÞ are shear storage modulus and shear loss modulus of the viscoelastic material, respectively. The loss factor is deﬁned by h ¼ G00 ðvÞ=G0 ðvÞ: In general, as the vibrational frequency increases, the values of G0 ðvÞ and G00 ðvÞ become larger. However, if the ambient temperature increases, those values become smaller. Test results of a typical viscoelastic damper averaged over the ﬁrst 20 cycles are shown in Table 11.4. For a viscoelastic damper with shear area, A; and thickness, d; the corresponding force–displacement relationship is FðtÞ ¼ kd ðvÞX þ cd ðvÞX_ ð11:24Þ

tðtÞ ¼ G0 ðvÞgðtÞ þ

in which X and X_ are the relative displacement and velocity of the damper, respectively, and kd ðvÞ ¼

TABLE 11.4

AG0 ðvÞ AG00 ðvÞ cd ðvÞ ¼ d vd

ð11:25Þ

Typical Viscoelastic Damper Properties

Temperature (8C)

Frequency (Hz)

Strain (%)

kd a (lb/in.)

G 0 (psi)

G 00 (psi)

h

24 24 24 24 36 36 36 36

1 1 3 3 1 1 3 3

5 20 5 20 5 20 5 20

2124 2082 4084 3840 880 873 1626 1542

142 139 272 256 59 58 108 103

193 192 324 306 67 65 119 112

1.36 1.38 1.19 1.20 1.13 1.12 1.10 1.09

a The deﬁnition of kd is shown in the text. Source: Data from Chang, K.C. et al. Earthquake Spectra, 9, 371–387, 1993.

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From the above equations, it is clear that a linear system with added viscoelastic dampers remains linear with the dampers contributing to increased viscous damping as well as stiffness. However, it needs to be pointed out that, for viscoelastic material at large strains, there is considerable self-heating due to the large amount of energy dissipated. The heat generated changes the mechanical properties of the material, and the overall behavior becomes nonlinear. This means that a linear analysis utilizing the above equations can only be for approximation of the response. 11.3.4.1.4 Viscous Fluid Dampers The most convenient and common functional output equation for a damper comes from classical system theory, and is that of the so-called “linear” or “viscous” damping element _ FðtÞ ¼ cd XðtÞ

ð11:26Þ

_ are the damping force, damping coefﬁcient, and relative velocity across the in which FðtÞ; cd, and XðtÞ damper, respectively. In mechanical engineering, it is difﬁcult to manufacture a useable ﬂuid-ﬁlled component having a purely viscous output, because even moderate pressure hydraulic ﬂows through a simple oriﬁce follow a very different output equation, in which differential pressure varies with the ﬂuid velocity squared. Therefore, the output of the basic hydraulic damping element is _ _ 2 FðtÞ ¼ sgnðXðtÞÞc d lXðtÞl

ð11:27Þ

As a result of long research in this area, which was started in the 1960s, the most useful dampers being used in buildings today are the so-called “low exponent” type, with an output equation of the form: _ _ a FðtÞ ¼ sgnðXðtÞÞc d lXðtÞl

ð11:28Þ

In most cases, a is an exponent having a speciﬁed value in the range of 0.3 to 1.0. Values of a that have proven to be the most popular are in the range of 0.4 to 0.5 for building designs with seismic input (Taylor, 2002). The design elements of a ﬂuid damper are relatively few. However, the detailing of these elements varies greatly. Typical elements for a ﬂuid damper are shown in Figure 11.47. Typical experimentally measured force–displacement loops are shown in Figure 11.48. 11.3.4.1.5 Tuned Mass Dampers Much of the early development of dynamic vibration absorbers, as mentioned in Section 11.3.2.1, has been limited to the use of dynamic absorbers in mechanical engineering systems in which one operating frequency is in resonance with a machine’s fundamental frequency. However, building structures are subjected to earthquakes which posses many frequency components. The performance of a Piston rod

Chamber 1

Cylinder

Seal retainer

Piston head with orifices

Compressible silicon fluid

Chamber 2

Control valve

Accumulator housing

Rod make-up accumulator

High strength acetal resin seal

FIGURE 11.47 Typical schematic of a ﬂuid damper. (Source: Taylor, D.P. Passive Structural Control Symp., 2002. With permission.)

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Vibration Monitoring, Testing, and Instrumentation

dynamic vibration absorber, referred to as the TMD herein, in complex multi-DoF systems is expected to be different. Consider the resonance curve in Figure 11.49 of a system shown in Figure 11.34, but now with different parameters: m ¼ 0:05; but still vM ¼ vm (see Equation 11.22 for the deﬁnition of these parameters). An additional parameter, the damping ratio of TMD, is shown in Equation 11.29. Where c is damping coefﬁcient of a linear viscous damper installed between structural mass, M; and TMD mass, m; c ð11:29Þ 6TMD ¼ 2mvm

1000 Force (lb)

f = 4 Hz f = 1 Hz

0

−1000 −1.5

FIGURE 11.48

0.0

1.5 Displacement (in)

Typical schematic of ﬂuid damper

When the damping ratio in the TMD equals zero, hysteretic loops. (Source: Constantinou, M.C. and the response amplitude is inﬁnite at the two Symans, M.D. Struct. Des. Tall Build., 2, 93 –132, 1993. resonant frequencies. When the damping ratio With permission.) becomes inﬁnite, the two masses are virtually stuck to each other; the result is a single-DoF system with mass, 1.05 ðMÞ with the amplitude becoming inﬁnite again at a resonant frequency (see Figure 11.49). Therefore, somewhere between these extremes, there must be a value of the TMD damping ratio for which the peak becomes a minimum. Therefore, the objective in adding the TMD herein is to bring the resonant peak of the amplitude to its lowest possible value so that smaller ampliﬁcations over a wider frequency bandwidth can be achieved. There are many types of TMD for implementation and the following are some examples (Figure 11.50). An innovative challenge is highly expected in this ﬁeld. There is another type of TMD that uses liquid as the mass; this damper is called the tuned liquid damper (TLD). The TLD has been used in ships for controlling vibrations because of water waves. The TLD uses water or other liquid as the moving mass and the restoring force is generated by gravity. Energy absorption comes from boundaries between liquid and containers and turbulence in the liquid ﬂow. The basic principle of the TLD in absorbing kinetic energy of the main structure is the same as the TMD. Figure 11.51 shows types of TLD. Favorable properties of TLD compared with TMD are as follows:

*

*

Smooth movement in small vibration is possible because of no mechanical friction. It is reasonable in cost and maintenance because of no complex mechanism. It can be applied easily in two horizontal directions with a single TLD. It can be compact and portable if large numbers are used. 16

8

*

*

ζ TMD = 0

x1 Po/K 8

0

0.32 0.1

4 0 0.6

1

ω/ωm

1.3

FIGURE 11.49 Resonance curve for the system in Figure 11.34 that shows response amplitude vs. exciting frequency for m ¼ 0:05 and vM ¼ vm :

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11-47

(b)

(a)

(c)

(e)

(d)

(f)

FIGURE 11.50 Examples of different types of TMD: (a) pendulum with damper; (b) inverted pendulum with spring and damper; (c) pendulum of which hangers are winded to save space; (d) swinging mass on rotational bearings; (e) sliding mass with spring and damper; (f) mass on rubber bearings. (Source: Iemura, H. Passive and Active Vibration Control in Civil Engineering, Springer, New York, 1994a,b. With permission.) p

(a)

Vibration

(b)

Vibration

FIGURE 11.51 Types of TLD: (a) tuned sloshing damper with meshes and rods; (b) tuned liquid column damper with oriﬁce. (Source: Iemura, H. Passive and Active Vibration Control in Civil Engineering, Springer, New York, 1994a,b. With permission.)

The TLD can be divided into two categories. First, is the sloshing damper as shown in Figure 11.51a. The vibration period is adjusted by the size of the container and the depth of the liquid. The damping capacity is increased by placing meshes or rods in the liquid. The second category is the tuned liquid column damper as shown in Figure 11.51b. The vibration period is adjusted by the shape of the column or the air pressure in the column. The damping capacity is increased by adjusting the oriﬁce in the column, which generates a high turbulence. 11.3.4.2

Active Control System

11.3.4.2.1 Electromagnetic Actuator An electromagnetic actuator is frequently employed where it is necessary to provide a mechanical force depending on an electric current. Figure 11.52 shows the elements of such a device. A magnetic structure supports a circuit of magnetic ﬂux driven by the coil of N turns carrying current i, in ampere. Part of the magnetic circuit is a movable armature that slides smoothly on the support member. An air gap (or possibly a vacuum gap) of length d m is also in the magnetic circuit. Assume that d is small enough for the magnetic ﬂux density to be essentially constant at B Wb/m2 across the face of the armature. The crosssectional area of the armature face is A m2. A tractive force, F, in Newtons, developed on the face of the armature, is related to the area and ﬁeld strength as F¼

AB2 2m0

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ð11:30Þ

11-48

Vibration Monitoring, Testing, and Instrumentation

where m0 is the magnetic permeability of the air gap having a value of 4p £ 1027 H/m (Henries/ meter). The magnetic ﬁeld intensity is proportional to the magnetomotive force N £ i ampere turns. For an efﬁcient magnetic structure, the magnetic ﬂux density is

m B¼ 0N £i d

ð11:31Þ

Magnet flux density = B Webers/meter2 z (t) Armature

F¼

Am0 N 2 2

#

i2 d2

! Newton

ð11:32Þ

Note that F is positive when i is either positive or negative. This means that the force, F, is always tractive if the current, I, is either positive or negative. The maximum value for magnetic ﬂux density, B, which can be realized using modern magnetic materials in normal environments, is approximately 1 Wb/m2 (Clark, 1996). In most cases, it is necessary to apply force in both the positive and negative directions. In this case, the device is constructed with two air gaps having the corresponding traction forces opposed to one another, as shown in Figure 11.53. It is interesting to observe how the net force, F, on the armature depends on z and Di (differential current ¼ iL 2 i0 ¼ iR 2 i0 ; see Figure 11.53 for the parameters) when these are constants. The schematic plot of F vs. z, with Di as a parameter, is shown in Figure 11.54. It is worth noting that the plot is nonlinear. It is also noted from the ﬁgure that, if z is positive, F is also positive. It means that the force developed on the armature due to a small displacement from the equilibrium position is in the direction to increase that small displacement. This characterizes a condition of static instability. Feedback control can be used to stabilize this inherently unstable device when such a device is used as an actuator.

d

Sliding armature

N turns

which, combined with Equation 11.30, gives a useful relationship: "

A

F

i (Amperes)

FIGURE 11.52 Elements of an electromagnet. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.) z(t) Armature

N turns

N turns

iL

2i0

iR

FIGURE 11.53 Push – pull electromagnet. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.) 30

Di = 2A

F (Newton)

Di = 1A Di = 0A

0 Di = −1A −30 −3

i0 = 2 A d0 = 5 mm

Di = −2A 0

z (mm)

3

FIGURE 11.54 Force vs. displacement of push– pull electromagnet with differential current as a parameter. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.)

11.3.4.2.2 Hydraulic Actuators Hydraulic actuators can produce large forces even at large displacements, which is useful for seismic response-control applications. The actuator is usually driven using electrical signals. In other words, an electro-hydraulic device is used to convert the low-powered signals into high-powered hydraulic ﬂuid ﬂow. One example of such a device is an electro-hydraulic servo-valve. The schematic of such a device is shown in Figure 11.55.

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Seismic Base Isolation and Vibration Control

11-49 Hydraulic Return

Single-state spool valve

To push-pull current amplifier

To push-pull current amplifier

Hydraulic Pressure

Actuator Force Low hydraulic pressure

High hydraulic pressure

FIGURE 11.55 Schematic of servo-valve actuator. (Source: Clark, R.N. Control System Dynamics, Cambridge Univ. Press, U.K., 1996. With permission.)

The valve shown in the ﬁgure is a single-stage spool valve. The simple design shown in the ﬁgure exhibits the basic principles, inherent in most types of servo-valves, of conversion from electromagnetic to hydraulic energy; in other words, actuating force. By changing the electric signals going to the spool valve, the actuating force can be generated because the hydraulic pressure acting on the piston in one side is greater than the other side (see Figure 11.55). The actuating force can also be altered in real time, enabling real-time control of seismically excited structures. 11.3.4.3

Semiactive Control System

11.3.4.3.1 Variable-Oriﬁce Hydraulic Dampers A variable damping device used for the semiactive control method can be achieved by using a Controllable Valve controllable, electro-mechanical, variable-oriﬁce valve to alter the resistance to ﬂow of a convenLoad tional hydraulic ﬂuid damper. Figure 11.56 shows a schematic of such a device. The effectiveness of variable-oriﬁce dampers in controlling seismically excited buildings has been demonstrated through both simulation and small scale experimental FIGURE 11.56 Schematic diagram of variable-oriﬁce studies (Hrovat et al., 1983; Mizuno et al., 1992; valve oil damper. Kurata et al., 1994; Patten et al., 1994; Sack et al., 1994; Liang et al., 1995; Niwa et al., 2000). 11.3.4.3.2 Controllable Fluid Dampers Another class of semiactive devices use controllable ﬂuids. The advantage of controllable ﬂuid devices over controllable valve devices is that they contain no moving parts other than the piston. The essential characteristic of controllable ﬂuids is their ability to change reversibly from a free-ﬂowing, linear viscous ﬂuid to a semisolid with a controllable yield strength in milliseconds, when exposed to an electric (for electro-rheological ﬂuid) or magnetic (for magneto-rheological [MR] ﬂuid) ﬁeld (Housner et al., 1997).

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Vibration Monitoring, Testing, and Instrumentation

MR ﬂuids typically consist of micron-sized, magnetically polarizable particles dispersed in a carrier medium such as mineral or silicone oil. When a magnetic ﬁeld is applied to the ﬂuid, particle chains form and the ﬂuid becomes a semisolid and exhibits viscoplastic characteristics. Transition to rheological equilibrium can be achieved in a few milliseconds. A schematic diagram of the controllable ﬂuid damper is shown in Figure 11.57.

Wires to MR fluid electromagnet

Magnetic Choke

Accumulator Load

Load Rod

Piston

FIGURE 11.57 Schematic diagram of magnetorheological damper. (Source: Spencer, B.F. et al. J. Eng. Mech., 123, 1997. With permission.)

11.3.4.3.3 Controllable Friction Dampers Various controllable friction devices have been Upper Plate proposed to dissipate vibratory energy in a Slotted Bolts structural system. Akbay and Aktan (1990, 1991) and Kannan et al. (1995) proposed a variableN(t) N(t) friction device in which the force at the frictional + + interface was adjusted by allowing slippage in − − controlled amounts. A similar device was also studied by Cherry (1994) and Dowdell and Cherry Lower Plate Friction Pad (1994a, 1994b). A recent work by He et al. (2003) studied a semiactive electromagnetic friction damper FIGURE 11.58 Schematic drawing of semiactive (SAEMFD) for controlling seismic responses. electro-magnetic friction damper. (Source: He, W.L. et al. J. Struct. Eng., ASCE, 129, 941 – 950, 2003. With Figure 11.58 shows a schematic diagram of the permission.) SAEMFD. The device consists of a friction pad sandwiched between two steel plates. These three layers are slot-bolted together so that sliding takes place between the steel plates and the friction pad. The friction force between steel plates and the friction pad depends on the coefﬁcient of friction ðmÞ and the normal force NðtÞ: Two insulated solenoids are installed on the outer surfaces of the steel plates and the electric current in these solenoids is regulated such that an electromagnetic attractive force exists between the two solenoids. Hence, the normal force NðtÞ between the steel plates is directly proportional to the square of the current in the solenoids. 11.3.4.4

Hybrid Control System

The hybrid control methods which consist of both passive and active devices have been proposed and implemented, utilizing the merits of both passive and active methods and avoiding the demerits of these methods. Thus, higher levels of performance may be achievable. Additionally, the resulting hybrid control system can be more reliable than a fully active system, although it is also often somewhat more complicated. One example of a hybrid system is a TMD with actuators that is put between the TMD mass and its support so that the effectiveness of the TMD is increased by this technique. Figure 11.59 shows a schematic diagram of a passive TMD, an active AMD (active mass damper), and a hybrid ATMD (active TMD). Another example of hybrid control is a combination of base isolation with some form of active control to limit excessive displacement (Fujii et al., 1992; Kageyama, M. and Yasui, 1992; Feng et al., 1993; Reinhorn and Riley, 1994).

11.3.5

Control Algorithm

11.3.5.1

Active Control System

The most important part of active control is the algorithm, because the control forces are based on this. Research efforts in active structural control have been focused on a variety of control algorithms based

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Seismic Base Isolation and Vibration Control Auxiliary Mass

11-51

Auxiliary Mass

m2

Auxiliary Mass

m2

m2 Controller

Spring

Damper

m1

Actuator

Controller

G

m1

G

m1 Sensor

Passive

FIGURE 11.59

Active

Sensor

Hybrid

Schematic diagram of a passive, an active, and a hybrid mass damper system.

on several control design criteria. An example of a famous control algorithm is linear optimal control because all of the control design parameters can be determined for multiinput and multioutput systems. Also, the control allows us to formulate directly the performance objectives of a control system. The adjective optimal above means that a control system can be designed to meet the desired performance objectives with the smallest control energy, which is the energy associated with generating the control inputs. Such a control system that minimizes the cost associated with generating control inputs is called an optimal control system. The optimal control system directly addresses the desired performance objectives, while minimizing the control energy, by formulating an objective function that must be minimized in the design process. If the transient energy of a system is the total energy of the system when it is undergoing the transient response, then the successful control system must have the capability to decay quickly the transient energy to zero. By including the transient energy and the control energy in the objective function, both parameters can be minimized. The objective function for the optimal control problem is a time integral of the sum of transient energy and control energy expressed as a function of time. The general, optimal control formulation for regulators can be explained as follows. Consider a structure (Figure 11.60) under dynamic loading that is represented by Equation 11.33: M€xðtÞ þ C_xðtÞ þ KxðtÞ ¼ DuðtÞ þ EfðtÞ

ð11:33Þ

where M, C, K are, respectively, the n £ n mass, damping, and stiffness matrices, and xðtÞ is the ndimensional displacement vector, fðtÞis an r-vector representing applied load or external excitation, and uðtÞ is the m-dimensional control force vector. The n £ m matrix, D, and n £ r matrix, E, are location matrices that deﬁne locations of the control force and the excitation, respectively. The equation can be rewritten using the state-space representation in the form z_ ðtÞ ¼ AzðtÞ þ BuðtÞ þ HfðtÞ; zð0Þ ¼ z0

ð11:34Þ

where " zðtÞ ¼

xðtÞ x_ ðtÞ

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# ð11:35Þ

11-52

Vibration Monitoring, Testing, and Instrumentation

xn mn

un

kn

cn

x2 m2 x1 m1

k2

c2

k

c1

FIGURE 11.60

f2

u2 u1

fn

f1

m1  m2 M=  ...  mn  c1 + c2  −c 2 C=    

−c2

k1 + k2  −k K= 2   

−k2 ... ...

... ...

         −cn   cn 

... ... −cn

... ... −kn

   −kn   kn 

Typical civil structure with its dynamic properties.

is the 2n-dimensional state vector

" A¼

is the 2n £ 2n system matrix, and

" B¼

0

#

I

ð11:36Þ

2M21 K 2M21 C #

0 M21 D

" and H ¼

I M21 E

# ð11:37Þ

are 2n £ m and 2n £ r location matrices specifying, respectively, the locations of controllers and external excitation in the state space. In Equation 11.36, 0 and I denote the null matrix and the identity matrix of appropriate dimensions, respectively. For simplicity, assume that we have a linear time invarying plant, as shown in Equation 11.34 above, and suppose we would like to design a full-state feedback regulator for the plant such that the control input vector is given by uðtÞ ¼ 2GxðtÞ

ð11:38Þ

where G is a feedback gain matrix. The control law given by the above equation is linear. Since the plant is also linear, the closed-loop control system is linear. The control energy, CE, can be expressed as CE ¼ uT ðtÞRuðtÞ

ð11:39Þ

where R is a square, symmetric matrix called the control cost matrix. Such an expression for control energy is called a quadratic form, because the scalar function in Equation 11.39 contains quadratic functions of the elements of uðtÞ: The transient energy, TE, can also be expressed in a quadratic form as TE ¼ xT ðtÞQxðtÞ

ð11:40Þ

where Q is a square, symmetric matrix called the state weighting matrix. The objective function can then be written as follows: ðtf ð11:41Þ Jðt; tf Þ ¼ ½xT ðtÞQxðtÞ þ uT RuðtÞ dt t

where t and tf are the initial and ﬁnal times, respectively; that is, the control begins at t ¼ t and ends at t ¼ tf ; where t is the variable of integration.

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11-53

The optimal control problem consists of solving for the feedback gain matrix, G, such that the scalar objective function, Jðt; tf Þ; given by Equation 11.41, is minimized. Hence, the optimal control problem solves a regulator gain matrix, G, which minimizes Jðt; tf Þ; subject to the constraint given by Equation 11.34. MATLABw Control System Toolbox (MathWorks, 1998) provides a function “lqr” for the solution of the linear optimal control problem shown above (also see Appendix 32A). By using this command, the gain matrix, G, can easily be obtained. For a numerical simulation, the matrix, G, is used for obtaining optimal forces, uðtÞ; as shown in Equation 11.38. The MATLAB Control System Toolbox user manual is a good reference (MathWorks, 1998) for applying this command. For a real application in a structure, the optimal forces, uðtÞ; must be converted to real forces by actuating devices. Actuating devices are covered in Section 11.3.4. In the above explanation, the linear optimal controller has been derived with full-state feedback that minimizes a quadratic objective function. The controller robustness to process and measurement noise can only be indirectly ensured by iterative techniques. There are more advanced topics in modern control that directly address the problem of robustness by deriving controllers that maintain system response and error signals to within prescribed tolerances. One example is the H1 (pronounced H-inﬁnity) optimal control design technique. The reader may refer to control design textbooks, such as that written by Tewari (2002). 11.3.5.2

Semiactive Control System

Because of the intrinsically nonlinear nature of semiactive control devices, development of control strategies that are practically implementable and can fully utilize the capabilities of these unique devices is an important and challenging task. Various nonlinear control strategies have been developed to take advantage of the particular characteristics of semiactive devices, including bang-bang control (Mukai et al., 1994; McClamroch and Gavin, 1995), clipped optimal control (Dyke et al., 1996a), bistate control (Patten et al., 1994), fuzzy control methods (Sun and Goto, 1994), and adaptive nonlinear control (Kamagata and Kobori, 1994). Caughey (1993) proposed a variable stiffness system that employed a semiactive implementation of the Reid (1956) spring as a structural element. He et al. (2001) proposed a resetting semiactive stiffness damper used for controlling seismically excited cablestayed bridges. Iemura and Pradono (2003) introduced a pseudonegative stiffness control algorithm used for producing artiﬁcially rigid –perfectly plastic force–deformation hysteretic loops by using controllable damper. 11.3.5.2.1 Common Control Schemes for Controllable Dampers An example will be given here for common control schemes for a controllable damper. Examples of a controllable damper are variable-oriﬁce damper and MR ﬂuid dampers. Both types of dampers are covered in Section 11.3.4. The strategy of a clipped-optimal control algorithm (Dyke et al., 1996a, 1996b) for seismic protection using MR ﬂuid dampers is as follows. First, an “ideal” active control device is assumed, and an appropriate primary controller for this active device is designed. Then a secondary bangbang-type controller causes the smart damper to generate the desired active control force, so long as this force is dissipative. The primary controller can be one of active control algorithms shown above. For the general smart damping device, the secondary control strategy is given by ( fsa;i ¼

fa;i ; fa;i £ x_ dev , 0 0; otherwise

ð11:42Þ

where fsa,i is the control force of the ith MR ﬂuid damper, fa,i is the desired control force for the ith device, and x_ dev is the velocity across the ith damper. Since the controllable damper is an energy-dissipative device that cannot add mechanical energy to the structural system, special care must be taken in the design of the primary controller so that the desired control force, fa,i, is dissipative during the majority of the seismic event.

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11.3.5.2.2 Special Control Schemes for Controllable Damper The term “special” here refers to control schemes that have certain objectives for the hysteretic loops of the controllable dampers. The control schemes are not centralized, so that each damper is controlled separately in one structure or, in other words, one controller is for one damper. The advantage is that should one controller malfunction, this controller will not affect the other controller. One example is given by He et al. (2001). A resetting semiactive stiffness damper (RSASD) is used to control the peak seismic response of a structure. The RSASD consists of a cylinder-piston system with an on –off valve in the bypass pipes connecting two sides of the cylinder. Basically, the damper is similar to a variable-oriﬁce oil damper as shown in Figure 11.58; however, the oriﬁce valve is replaced by an on – off valve. This damper serves as a stiffness element in which the stiffness is provided by the bulk modulus of the ﬂuid in the cylinder when the valve is closed. When the valve is open, the piston is free to move, and the hydraulic damper provides only a small damping, without stiffness. Such a stiffness damper can be operated in the resetting mode. During the operation in this mode, the valve is always closed. The energy is then stored in the hydraulic oil of the damper in the form of potential energy. At an appropriate time, the valve is pulsed to open and close quickly. At that moment, the piston is at the resetting position, and the energy stored in the hydraulic damper is released and converted into the head loss across the damper. Hence, by pulsing the valve at appropriate times, structural response can be reduced by drawing energy from the system (He et al., 2001). Another example is that of Kurino et al. (2003). They presented a device developed for an actual application whose system employs a decentralized control algorithm that uses information only from built-in sensors. The pseudonegative stiffness control algorithm (Iemura and Pradono, 2003) is also intended for controlling a variable damper based only on the displacement and velocity sensors located within the damper. The purpose is to control the variable damper’s hysteresis loop. An example of the application of this control algorithm to a bridge model is shown in Section 11.3.7.2.

11.3.6

Experimental Performance Veriﬁcation

11.3.6.1 Shaking Table Tests of a Flexible Structural Model with TMD, AMD, and ATMD For practical implementation of TMD, AMD, and ATMD for structures, it is important to ﬁnd the efﬁciency of each control system for random excitations. The author has made an analytical and numerical studies on the efﬁciency of different control methods. To verify the results, a three-DoF structural frame model (Figure 11.61) with and without control devices is tested a on shaking table at Kyoto University, Japan (Iemura et al., 1992). Natural periods for modes 1, 2, and 3 are 0.6578, 0.2580, and 0.1568 sec, respectively. The relevant participation factors for each mode are 1.2204, 0.3493, and 20.1341, respectively. The moving mass and other mass of the TMD are 3.5 and 5.5 kg, respectively. The spring constant is 0.581 kgf/cm. The damping ratio is 25.06%. These properties are used for the experiment. The masses of the TMD consist of the AC servo-motor, moving mass, driving guides, and velocity meter. At the time of the experiment of the TMD, the moving mass was ﬁxed and the TMD

Velocity Meter Velocity Meter

Motor Moving Mass

Velocity Meter

Velocity Meter

Velocity Meter Shaking Table

FIGURE 11.61 TMD/ATMD.

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Three-DoF experimental model with

Seismic Base Isolation and Vibration Control

11-55

was hung from the third ﬂoor. In order to work as the hybrid type ATMD, the moving mass was driven by the motor. For the pure active control experiment, the motor and the moving mass are set directly on the third ﬂoor. It was clearly found that the second mode response is not reduced by the TMD but is effectively reduced by the AMD and ATMD. The TMD is effective only in the ﬁrst mode frequency range, while active control force can cover a wide frequency range. It was also found that the control force of the ATMD is much lower than that of AMD, especially in the ﬁrst mode frequency range. The ﬁrst mode response is reduced mainly by the TMD and the second mode response is reduced by the AMD. This veriﬁes the energy efﬁciency of ATMD. This is the reason that the ATMD concept is now popularly used for practical application. 11.3.6.2

Substructure Hybrid Test

Before introducing additional dampers to a structure for seismic-response reduction, the precise frequency-dependent properties of the damper should be obtained from the damper. These properties will be used for numerical simulation on the effectiveness of the damper in reducing seismic responses. It is quite difﬁcult to work out the appropriate model for a speciﬁed device because of the existence of strong nonlinearity. For a more reliable experimental test of structures with structural control devices, so-called “substructure hybrid experiment” techniques have been developed. The term “substructure hybrid” implies a technique that combines the device loading experiment and numerical simulation of structural response. Why should it be separated? It is because civil structures are relatively large and expensive to construct in a laboratory. Therefore, only the inelastic part is tested experimentally. The elastic part, which is easier to model, is numerically simulated on a computer. Both results are combined in real time at every time step of the simulation. Up until now, various kinds of test methods have been proposed. Most of them can be classiﬁed into three categories from the viewpoint of the loading equipment. 11.3.6.2.1 Hybrid Tests Using Hydraulic Actuator Hydraulic actuators are commonly used for loading experiments. They are advantageous for testing specimens that needs large excitation force and displacements (Tanzo et al., 1992; Igarashi et al., 1993; Igarashi, 1994; Williams and Blakeborough, 2001). Various algorithms and techniques have been proposed in order to conduct precise, real-time experiments such as the “operator splitting numerical integration scheme,” which is suitable for online controlled experiments (Nakashima, 1993). A compensation method based on extrapolation is proposed by Horiuchi and Konno (2001) for the response delay of the actuator. Similar feedforward-based compensation methods are widely used for the numerical algorithm’s development and real-time testing (Nakashima and Masaoka, 1999; Nakashima et al., 1999; Blakeborough et al., 2001). 11.3.6.2.2 Hybrid Tests Using Shaking Table Substructure hybrid loading test systems have been developed for shaking table equipment. Since most shaking tables are driven by hydraulic actuators, algorithms as well as technologies for the hydraulic actuators system are directly applicable to the shaking table test systems. Iemura et al. (2002) introduced the inverted digital ﬁlter of the shaking table for compensating its dynamics, and conducted a real-time hybrid experiment using the electromagnetic mass damper installed in the nonlinear structure. The shaking table test is applicable to a test specimen such as the TMD. 11.3.6.2.3 Hybrid Tests Using Inertia-Force-Driven Loading System This is a newly developed method for hybrid loading test (Toyooka, 2002; Iemura et al., 2003). The system consists of a large size mass, rubber and roller supports, and active mass driver. The schematic ﬁgure of the system is shown in Figure 11.62 and the property of the system is in Table 11.5. The active mass driver is attached to the mass. The test specimen is attached to the mass and the ground. The mass

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 11.62 TABLE 11.5

Test set up of the IFDL system.

Properties of the IFDL System

Weight of slab mass Total stiffness Total damping Natural frequency Equivalent damping Stroke limit

23.853 tonf 344.43 kN/m 6.32 kN/m/sec 0.55 Hz 3.86% ^10 cm

(i.e., the concrete slab) can be excited with large displacement, velocity, and acceleration by making use of the shaker (Iemura et al., 2003). The Inertia-Force-Driven Loading (IFDL) system was developed to allow an economical and accurate loading environment for energy dissipation devices to characterize the dynamic properties and to comprehend the performances of these devices under the realistic loading conditions.

11.3.7

Implementations

11.3.7.1 Semiactive Control of Full-Scale Structures Using Variable Joint Damper System 11.3.7.1.1 Background of Study In order to verify the effectiveness of the application of the semiactive control technique to the joint damper system (JDS), seismic response control tests using full-scale multistory steel-frame structures, excitation devices, and a variable damper are performed at the Disaster Prevention Research Institute, Kyoto University, Japan. The variable damper allows external control of damping force by the electric servo-valve that regulates the oil ﬂow through the cylinder/piston mechanism. The test results show that the variable damper was successfully controlled with high accuracy, as well as having the advantage of JDS application of the semiactive control in reducing the dynamic response of structures over the

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conventional passive control. A comparison of the semiactive control algorithms to JDSs in terms of the feasibility and the advantage in the engineering application is also based on the test results. Extensive research on the semiactive control approach has been conducted in order to reduce the seismic response of structures, induced especially by strong earthquake ground motions. The JDS, which aims to achieve the dynamic response reduction of adjacent structures through the use of connection devices with energy-absorbing capability, has been considered a promising approach to establish effective semiactive structural systems for earthquakes. The purpose of this study is to verify experimentally the effectiveness of the application of semiactive control to the JDS. Seismic response control tests using full-scale multistory frame structures, excitation devices, and a variable damper are performed at the Disaster Prevention Research Institute, Kyoto University. The variable damper allows for external control of the damping force using the electric servovalve that regulates the oil ﬂow through the cylinder/piston mechanism. Two types of semiactive control algorithms were employed, namely the linear quadratic regulator (LQR) control theory and the newly proposed pseudonegative stiffness control for JDSs. Parametersetting strategies for the algorithms are studied prior to the tests through numerical simulations based on the modeling of the full-scale steel frame structures and the control device used in the tests. 11.3.7.1.2 Test System Set Up As shown in Figure 11.63, the test structure of the JDS consists of two full-scale structural steel frames: a ﬁve-story frame (1 £ 2 span) and a three-story frame (1 £ 1 span). The dimensions of both frames are shown in Table 11.6. Natural frequencies are also shown in the table. In this test system, mass-driver devices are used to reproduce the vibration conditions under both sinusoidal and real earthquake inputs. One-directional horizontal earthquake excitation is applied. Although three mass-driver systems can be seen in the ﬁgure, two of them are used at the same time. The accuracy of the simulated response using the mass driver devices has been veriﬁed by a series of research Variable Damper

Velocity Sensors

Shaker

Data Acq. PC FIGURE 11.63

Shaker Control PC

Damper Control PC

Schematic diagram of joint damper system.

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Vibration Monitoring, Testing, and Instrumentation TABLE 11.6

Test Frames and Mass-Drivers

Height (m) Weight (tonf) First mode natural frequency (Hz) Mass driver

Five-Story

Three-Story

17.22 163.1 1.78 5 ton mass at the fourth ﬂoor

10.65 62.1 2.41 2 ton mass at third ﬂoor

conducted prior to this study. Velocities, relative displacements, and absolute accelerations of all ﬂoors are measured through instrumentation. All measured responses are sent to a digital signal processingbased system for the feedback control. As the control device in the JDS, a variable damping device (variable damper) is used in the test system. The variable damper is installed at the third story of the ﬁve-story frame so as to connect the two frames. The mechanism of the variable damper is similar to that shown in Figure 11.56. It is a semiactive hydraulic damper consisting of a cylinder/piston mechanism ﬁlled with oil, double rods that connect the frames, a by-pass pipe that contains a ﬂow control valve, and an accumulator that keeps the by-pass line pressure constant. The opening ratio of the ﬂow control valve can be changed by a servo-controller using an external signal. The ﬂow volume through the valve can be regulated to control the pressure loss. The delay time for the opening ratio control is sufﬁciently short to allow real-time control. 11.3.7.1.3 Control Algorithm In this study, three types of control algorithms are used: linear viscous damper control, LQR control (discussed in Section 11.3.5), and pseudonegative stiffness (PNS) control. The linear viscous damper control algorithm is intended to be the reference response in the case of passive control, and the effectiveness of the semiactive control is demonstrated by comparing the other two cases with the linear viscous damper control case. In linear viscous control algorithm, the damping force demanded of the variable damper, Fd ðtÞ; is Fd ðtÞ ¼ Cc vr ðtÞ

ð11:43Þ

where Cc is the connecting damping coefﬁcient and vr is the relative velocity of the damper position. Realtime control of the valve opening ratio is required to generate the demanded control force, Fd, with the variable damper, even for this simplest control algorithm. The LQR control theory is used as a semiactive control algorithm in this study, as extensively used in past studies. In the LQR control algorithm, the optimal control force, Fd, is regarded as the demanded force and the variable damper is controlled to track the demanded force as close as possible within the constraint, depending on the piston velocity. The control gain parameters are chosen on the basis of numerical simulation in consideration of the capacity of the variable damper. The control force is calculated in the following manner: Fd ðtÞ ¼ 2GxðtÞ

ð11:44Þ

where G is the optimal gain matrix given by the LQR theory and xðtÞ is the state vector for the structural frames. If the main purpose of a JDS is the response reduction of the upper ﬂoors in the adjacent structures, the most interesting feature of the system is obtained by connecting them at lower stories with a negative stiffness element. Although this characteristic has been reported theoretically and analytically in many studies, there are many problems left in applying the active control device at the present time, mainly owing to the difﬁculty in realizing the negative stiffness with a passive control device. However, semiactive devices such as the variable damper can generate an apparent negative stiffness by controlling the damping. Therefore, taking into account JDS and negative stiffness, a new, simple control algorithm to realize pseudonegative stiffness with a damping element is proposed in this study. To generate the

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negative stiffness, the demanded force of variable damper Fd is deﬁned as follows: Fd ðtÞ ¼ Kc xr ðtÞ þ Cc vr ðtÞ

ð11:45Þ

where Kc and Cc are the connecting stiffness (negative value) and the damping coefﬁcient, respectively. The relative displacement and velocity of the damper piston, respectively, are xr and vr. It follows that this algorithm simulates the state in which the two frames are connected by a negative stiffness and a positive damping elements. The eigenvalue analysis is used to determine the value of Kc and Cc. This pseudonegative stiffness algorithm has a great advantage in practical application. Most of the previously proposed control algorithms require direct measurement of the structural system to produce feedback and to calculate the demanded control force. That means that a considerable number of sensors should be installed in the object structure. Considering practical application, it is difﬁcult to install many sensors because of the economical disadvantage. On the other hand, for the pseudonegative stiffness algorithm, only the relative displacement and velocity are used for feedback and a sensor is required only at the damper. In addition to the simplicity in installation, the required parameters are limited to the connecting stiffness and damping. 11.3.7.1.4 Test Results The response of variable damper to sinusoidal input with 1.8 Hz (max 10 gal), which is approximately the ﬁrst resonance frequency in the connected state, is shown in Figure 11.64. The ﬁve-story top-ﬂoor velocity response in the LQR control theory and pseudonegative stiffness are smaller than that in the viscous damper; especially; the peak value in pseudo negative stiffness is improved at 25% compared with that in the viscous damper, though the velocities at the top story of the three-story frame are almost equal for all control algorithms. On the other hand, the LQR control theory can reduce the peak response of both frames as compared with the viscous damper. When the object is to moderate the response of total system, the LQR theory is the most effective algorithm of the three being compared. Based on the test, semiactive controls based on the LQR control theory and pseudonegative stiffness can reduce the peak responses of the total system more effectively than viscous damper-type passive control. For the earthquake excitation (El-Centro 1940 N–S and Kobe 1995 N–S, scaled to 20 gal max), the inﬂuence of the friction of the variable damper appears in the dynamic characteristics of the variable damper in every control algorithm because of the relatively small responses. The relative velocity and displacement responses of the variable damper in both semiactive controls are larger than that in the viscous damper. Judging from the test result, it is conﬁrmed that the variable damper is controlled effectively in the different control algorithm for real earthquake inputs. With respect to the velocity response of the top story of the three-story frame, the responses are not very different when different control algorithms are used. On the other hand, for the ﬁve-story top-ﬂoor response, both semiactive controls can reduce the response more effectively than the passive control to both real

FIGURE 11.64 Variable damper responses in the sinusoidal excitation test: (a) at the top story of ﬁve-story fame; (b) at the top story of three-story fame.

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Vibration Monitoring, Testing, and Instrumentation

FIGURE 11.65 (a) Viscous-type control, (b) LQR-type control, and (c) pseudonegative stiffness-type control under El-Centro 1940 N– S scaled to 20 gal max.

earthquake excitations. Semiactive controls are more effective than passive control to reduce the response of the top story of the ﬁve-story frame in the earthquake excitation cases. Figure 11.65 shows the hysteretic loops resulting from viscous-type, LQR-type, and pseudonegative stiffness-type controls under El-Centro 1940 N–S, scaled to 20 gal max. It is obvious that the pseudonegative hysteretic loop can be achieved experimentally by using a variable damper. 11.3.7.2 Application of Structural Control Technologies to Seismic Retroﬁt of a Cable-Stayed Bridge 11.3.7.2.1 Background of Study Owing to severe damage to bridges caused by the Hyogo-ken Nanbu earthquake in 1995, very high ground accelerations (level II design) are now required in the new bridge design speciﬁcation set in 1996, in addition to the relatively frequent earthquake motion (level I design) by which old structures were designed and constructed. Hence, the seismic safety of cable-stayed bridges that were built before the present speciﬁcation has to be reviewed and seismic retroﬁt has to be done, if it is found necessary. In order to study the effectiveness of passive and semiactive control on the seismic retroﬁt of a cable-stayed bridge, numerical analyses on a model is carried out. An existing cable-stayed bridge that has ﬁxed-hinge connections between the deck and towers is modeled and its connections are replaced with isolation bearings and dampers. The isolation bearings are assumed to be elastic or hysteretic type. The dampers are linear and variable type. The objective is to increase the damping ratio of the bridge by using passive and semiactive control technologies. The calculation of the structural damping ratio at the main mode is feasible, as the passive or semiactive control method produces certain hysteretic loops under

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FIGURE 11.66

11-61

Side view of the Tempozan Bridge.

harmonic motion, and the main mode has an effective modal mass that is larger than 90% of the total mass. SSI effects on the structural damping ratio are also studied. The Tempozan Bridge (Hanshin Highway Public Corporation, 1992), built in 1988, is a three-span continuous steel cable-stayed bridge that is situated on reclaimed land and crosses the mouth of the Aji River, Osaka, Japan. The total length of the bridge is 640 m with a center span of 350 m, and the lengths of the side spans are 170 and 120 m (Figure 11.66). The main towers are A-shaped to improve the torsional rigidity. The cable in the superstructure is a two-plane fan pattern multicable system with nine stay cables each plane. The bridge is supported on a 35 m thick soft layer and the foundation consists of cast-in-place RC piles of 2 m in diameter. The main deck is ﬁxed at both towers to resist horizontal seismic forces. The bridge is relatively ﬂexible, with a predominant period of 3.7 sec. As to the seismic design in the transverse direction, the main deck is ﬁxed at the towers and the end piers. Figure 11.67 shows the original design spectrum used for designing the bridge and the new design spectrum speciﬁed in the bridge design speciﬁcation set in 1996 for level I and level II earthquakes (Japan Roadway Association, 1996). A level II earthquake has type I (interplate type)

Absolute Acceleration (gal)

10000

1000

100 New Design Spectrum Level II (Type I) New Design Spectrum Level II (Type II) New Design Spectrum Level I Original Design Spectrum

10

0.1

1 Natural Period (sec.) FIGURE 11.67

Design spectra for bridges.

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10

11-62

Vibration Monitoring, Testing, and Instrumentation

and type II (intraplate type). As can be seen in the ﬁgure, the new design spectrum shows higher acceleration response in all period ranges than the original one. 11.3.7.2.2 Basic Concept of Seismic Retroﬁt If the deck is connected with very ﬂexible bearings to the towers, the induced seismic forces will be kept to minimum values, but the deck may have a large displacement response. On the other hand, a very stiff connection between the deck and the towers will result in a lower deck displacement response but will attract much higher seismic forces during an earthquake. This is the case in the original bridge structure, the Tempozan Bridge. Therefore, it is important to replace the existing ﬁxed-hinge bearings with special bearings or devices at the deck-tower connection both to reduce seismic forces and to absorb large seismic energy and reduce the response amplitudes. Additionally, energy-absorbing devices may also be put between the deck-ends and piers; however, this will attract relatively large lateral force of the piers, and therefore this kind of method has been avoided for this bridge at this time. The bridge model that represents the existing Tempozan Bridge is termed the “original bridge model.” The bridge model with the spring and damper (viscous, hysteretic, and semiactive) between the deck and the towers is termed the “retroﬁtted bridge model.” The original and retroﬁtted bridge models are shown in Figure 11.68. The original structure system has ﬁxed-hinge connections between the towers and the deck and rollers connection between the deck-ends and piers, so that the deck longitudinal movement is constrained by the towers (Figure 11.68a). For the retroﬁtted bridge, the isolation bearings and dampers connect the deck to the towers (Figure 11.68b). The cables are modeled by truss elements. The towers and deck are modeled by beam elements, and the isolation bearings are modeled by spring elements. The models were analyzed by a commercial ﬁnite element program (Prakash and Powell, 1993). The moment –curvature relationship of the members is calculated based on the sectional properties of members and material used. 11.3.7.2.3 Modal Shape Analysis The ﬁrst modes of the structures are interesting here because these modes have the largest contribution to the longitudinal movement of the bridge. The mode shapes of the original bridge and the retroﬁtted bridge are shown in Figure 11.69. The ﬁrst mode shape of the original structure is shown in Figure 11.69a. The natural period ðTÞ of this mode is 3.75 sec (frequency ¼ 0.266 Hz), which is close to the design value for the bridge (3.7 sec, frequency ¼ 0.270 Hz; Hanshin Highway Public Corporation, 1992). This ﬁrst mode shape has the effective modal mass as a percentage of the total mass of 84%. For the retroﬁtted structure, the stiffness of the bearings is an important issue, as large stiffness produces a large bearing force. However, very ﬂexible connections produce a large displacement response. Therefore, based on a study on a simpliﬁed model of the bridge under seismic motion, the bearing stiffness

Fixed Hinge

(a)

(b) FIGURE 11.68

Isolation Bearings + Passive or Semi Active Dampers

Cable-stayed bridge models: (a) original structure system; (b) retroﬁtted structure system.

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11-63

1st mode shape, frequency = 0.266 Hz

Fixed deck-tower connections

(a)

1st mode shape, frequency = 0.158 Hz

Flexible deck-tower connections (b) FIGURE 11.69

First mode shapes of (a) original structure and (b) retroﬁtted structure.

that produces retroﬁtted main period ðT 0 Þ 1.7 times the original main period ðTÞ was chosen. This bearing stiffness makes the energy-absorbing devices work well in reducing seismic-induced force and displacement. The main natural period ðT 0 Þ of the retroﬁtted bridge then becomes 6.31 sec and the effective modal mass as a percentage of total mass is 92%. It is clear from the ﬁgures that smaller curvatures are found at the towers and the decks of the retroﬁtted structure than in the original structure. This shows that the retroﬁtted structure is expected to produce smaller moments at the towers and the decks than the original structure during a seismic attack. 11.3.7.2.4 Time-History Analysis The models were analyzed by a commercial ﬁnite element program (Prakash and Powell, 1993), which produces a piece-wise dynamic time history using Newmark’s constant average acceleration ðb ¼ 1=4Þ integration of the equations of motion, governing the response of a nonlinear structure to a chosen base excitation. The input earthquake motions were type I-III-3, I-III-2, and I-III-1 earthquakes, which are artiﬁcial acceleration data used for design in Japan for soft soil condition. Those data are intended to be type I (interplate type). With numerical comparison (Figure 11.67), type I earthquake motion gives higher effect to the bridge than type II motion, in longer period range. Table 11.7 shows the seismic response effects because of different kinds of bearings and dampers: ﬁxedhinge bearings for the original bridge model; elastic bearings, elastic bearings plus viscous dampers, and hysteretic bearings for the retroﬁtted bridge models. The input earthquake was type I-III-3 earthquake data and was in the longitudinal direction. From the table, it is clearly seen that if only elastic bearings are used for seismic retroﬁt, then the sectional forces are reduced to about 40% of the original ones. However, the displacement response is increased to 176% of the original one. By adding viscous dampers to the elastic bearings, the sectional forces can be reduced to about 25% of the original ones, and the displacement response is reduced to 63% of the original. Thus, the viscous dampers together with bearings work to reduce the seismic response of retroﬁtted bridge. The structural damping ratio is calculated as 35%. If hysteretic bearings are used for seismic retroﬁt, the sectional forces are reduced to about 29% of the original ones and the displacement response is reduced to 67% of the original one. The equivalent structural damping ratio is calculated as 13.1% by using pushover analysis to obtain a hysteretic loop at the main mode. The hysteretic bearings are modeled by a bilinear model, and the second stiffness of the hysteretic bearings is 0.03 times the initial stiffness and produces a ﬁrst mode natural period of 6.31 sec.

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11-64

Vibration Monitoring, Testing, and Instrumentation

TABLE 11.7

Maximum Earthquake Responses and Damping Ratios in Longitudinal Direction

Items

Original Structure

Deck displacement (m) Tower momenta (MN m) Tower axial forcea (kN) Cable force (kN) Bearing forceb (kN) Deck moment (MN m) Deck axial force (kN) Damping ratio (%) Natural period (sec) a b c

Retroﬁtted Structure Elastic Bearings

2.37 3,100 48,000 24,000 94,000 370 56,000 2 3.75

Elastic Bearings þ Viscous Damping

4.17 2,000 15,000 3,440 44,000 95 21,000 2 6.31

1.50 900 15,000 4,000 17,000 75 11,000 35 6.73

Hysteretic Bearings 1.58 900 21,000 5,000 25,000 95 15,000 13.1 3.86 and 6.31c

Base of tower AP3. At connection between deck and tower AP3. Initial and postyield stiffness.

11.3.7.2.5 Soil–Structure Interaction Effect One method to study the SSI effects is to take into Kx account the effects of ﬂexible foundations and the Cr K C r x radiation of energy from foundations. In this method, the cable-stayed bridge is idealized as in Figure 11.70 (Kawashima and Unjoh, 1991). The a a elastic half space subsoil supporting the foundation was assumed to be an elastic half space. The subsoil was assumed to be elastic with no energy dissipation. The foun- FIGURE 11.70 Cable-stayed bridge model with ﬂexdation was idealized as a rigid massless circular ible foundation and energy radiating from foundation. plate. The radius of the rigid circular plate was simply assumed so that it gives the same surface area as the foundation. Dynamic stiffness of the foundation was assumed in a frequency independent form: pGs a2 Vs p ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 8Gs a3 0:25p 2ð1 2 y Þ=ð1 2 2y ÞGs a4 Kr ¼ Cr ¼ 3ð1 2 y Þ Vs Kx ¼

8Gs a 22y

Cx ¼

ð11:46Þ ð11:47Þ

in which Kx and Cx represent the spring and damping coefﬁcient for sway motion, and Kr and Cr represent those for rocking motion. Vs and a represent shear wave velocity of subsoils and the radius of foundation, respectively. The result shows that SSI increases the natural period and the damping ratio of the original structure. However, the damping ratio of the retroﬁtted structure is reduced and the effectiveness of the bearings and dampers in reducing seismic responses is also reduced (Table 11.8). This is mainly because the SSI model introduces ﬂexibility at the base. A ﬂexible base will reduce the frequency of the structure. A smaller frequency will reduce the effectiveness of viscous damping devices in absorbing earthquake energy. Moreover, a ﬂexible base will increase the elastic strain energy of the structure that reduces the damping ratio. If the SSI model possesses an elemental damping ratio, as is a usual case for the soil, the structural damping ratio will also be inﬂuenced by the SSI-model damping characteristics. 11.3.7.2.6 Semiactive Control The semiactive control herein uses the pseudonegative stiffness control algorithm (Iemura and Pradono, 2003) so that the sum of the damper force and bearing force (plus other connecting stiffness forces) are expected to produce a hysteresis loop that is as close as possible to that of rigid –perfectly plastic

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TABLE 11.8

11-65

Maximum Earthquake Responses and Damping Ratios (SSI Included)

Items

Original Structure

Retroﬁtted Structure Elastic Bearings þ Viscous Damping

Deck displacement (m) Tower momenta (MN m) Tower axial forcea (kN) Cable force (kN) Deck moment (MN m) Deck axial force (kN) Foundation displacement (m) Damping ratio (%) Natural period (sec) a b

2.78 1,500 36,200 12,300 228 31,900 0.171 3.1 5.04

Hysteretic Bearings

2.57 800 12,500 3,010 58 9,100 0.079 23 7.66

2.77 882 19,800 4,470 86 12,300 0.093 9.3 5.13 and 7.46b

Base of tower AP3. Initial and postyielding stiffness.

force –deformation characteristics (Figure 11.71a). Moreover, no residual displacement is expected at the bearings after an earthquake attack, because the hysteresis loop is velocity dependent. Figure 11.71 shows ideal and realistic force – deformation characteristics of the variable damper that can produce artiﬁcial rigid –perfectly plastic force–deformation characteristics by using variable damper. One algorithm that can approach the hysteretic loop in Figure 11.71b requires the following variable-damper force (Iemura et al., 2001): Fd;t ¼ Kd ut þ Cd u_ t

F u F F

u u

(a)

F

Connecting Stiffness + Variable Damper = Total

F + F

u u u

(b)

FIGURE 11.71 (a) Ideal and (b) realistic hysteretic loops produced by variable damper.

ð11:48Þ

where Kd is connecting stiffness (negative value) and Cd is damping coefﬁcient (positive value). The algorithm is practical because only displacement and velocity sensors are placed in the dampers. Therefore, each damper can have its own controller. Should a malfunction happen in one damper or controller, it will not affect the other dampers or controllers. This algorithm produces the hysteretic loop shown in Figure 11.72b under harmonic motion. It is clear from the ﬁgure that the variable damper is superior to the linear viscous damper, because the maximum variable damper plus the connecting-stiffness force can be set to be equal to the maximum connecting-stiffness force (Figure 11.72b). One can calculate that the damping ratio of the hysteresis loop in Figure 11.72b is 53.4%. For the same damping ratio, the hysteresis loop in Figure 11.72a produces a total force 1.46 times larger than the connecting-stiffness force (Iemura and Pradono, 2003). The connecting stiffness between the deck and the tower of the retroﬁtted cable-stayed bridge comes from the contribution of cable stiffness, upper tower stiffness, and bearing stiffness. The cable-stayed bridge model with isolation bearings and variable dampers controlled with the pseudonegative stiffness algorithm was analyzed by a program developed by the authors under the MATLAB (MathWorks, 2000) and SIMULINK (MathWorks, 1999) environments. The program produces a piece-wise dynamic time-history, using Newmark’s constant average acceleration ðb ¼ 1=4Þ integration of the equations of motion, governing the response of a nonlinear structure to a chosen base excitation. The input motions were type I-III-1, I-III-2, and I-III-3 earthquakes, which are artiﬁcial acceleration data used for design in Japan (Japan Roadway Association, 1996). The results show that the application of the pseudonegative stiffness control algorithm is effective in reducing seismic response of the bridge model. Figure 11.73 shows the base shear-deck displacement

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Vibration Monitoring, Testing, and Instrumentation

Linear damping + connecting stiffness Linear damping

Variable damping + connecting stiffness

Variable damping

F

F

u

u

Connecting stiffness (a) FIGURE 11.72

(b) Hysteresis loops for (a) linear viscous damping and (b) pseudonegative stiffness damping.

80000

40000 0 −3.0 −2 .0 −1.0

0.0

1.0

2 .0

3.0

Base Shear (kN)

Base Shear (kN)

80000

40000

−3.0 −2.0 −1.0

0.0

1.0

2.0

3.0

−40000

− 40000

−80000

−80000 (a)

0

Deck Displacement (m)

Deck Displacement (m)

(b)

(a)

40000

40000

20000

20000

0 −1.5 −1.0 −0.5

0.0

0.5

1.0

−20000 − 40000 Bearing Displacement (m)

1.5

Damping Force (kN)

Damping Force (kN)

FIGURE 11.73 Base shear vs. deck displacement relationship of a cable-stayed bridge model with (a) linear dampers (b) pseudonegative stiffness dampers (type I-III-1 earthquake).

(b)

−1.5 −1.0

0 −0.5 0.0

0.5

1.0

1.5

−20000 −40000 Bearing Displacement (m)

FIGURE 11.74 Damping force vs. damping displacement relationship of a cable-stayed bridge model with (a) linear dampers, (b) pseudonegative stiffness dampers (type I-III-1 earthquake).

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40000 20000

0 −1.5 −1.0 − 0 .5

0.0

0.5

1.0

− 20000

1.5

11-67

Damping + Bearing Force (kN)

Damping + Bearing Force (kN)

Seismic Base Isolation and Vibration Control

Bearing Displacement (m)

20000 0

−1.5 −1.0 −0.5

− 40000 (a)

40000

0.0

0.5

1.0

1.5

−20000 − 40000

(b)

Bearing Displacement (m)

FIGURE 11.75 Damping plus bearing force vs. damping displacement relationship of a cable-stayed bridge model with (a) linear dampers, (b) pseudonegative stiffness dampers (type I-III-1 earthquake).

relationship for both bridges, with a linear damper and pseudonegative stiffness damper, respectively, under type I-III-1 earthquake input. The bridge model with pseudonegative stiffness dampers shows lower base shear than that of the bridge with linear damper. Figure 11.74 shows the hysteretic loops produced by both linear dampers and pseudonegative stiffness dampers (at tower AP3). The damping force produced by the pseudonegative stiffness damper is larger than that of the linear damper. However, the total force of damping plus the isolation bearing is lower for the pseudonegative stiffness damper (Figure 11.75). Therefore, the base shear of the cable-stayed bridge model is lower for the pseudonegative stiffness dampers.

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Sogabe, K., Goto, Y., Nakamura, T., and Okada, H. 1988. Basic study on active suppression method of sloshing of liquid in tanks during earthquakes by air injection, Vol. V, pp. 877 –882. In Proceedings of Ninth World Conference on Earthquake Engineering, Japan. Soong, T.T. and Dargush, G.F. 1997. Passive Energy Dissipation Systems in Structural Engineering, Wiley, New York. Soong, T.T., Reinhorn, A.M., Wang, Y.P., Lin, R.C., and Riley, M. 1992. Full-scale implementation of active structural control, Vol. 4, pp. 2131 –2136. In Proceedings of Tenth World Conference on Earthquake Engineering, Madrid, Spain. Spencer, B.F., Dyke, S.J., Sain, M.K., and Carlson, J.D., Phenomenological model for magnetorheological dampers, J. Eng. Mech., 123, 1997. Spencer, B.F. and Sain, M.K., Controlling buildings: a new frontier in feedback, Spec. Issue IEEE Control Syst. Mag. Emerging Technol., 17, 19– 35, 1997. Sugano, S. 2000. Seismic rehabilitation of existing concrete buildings in Japan. In Proceedings of XII World Conference on Earthquake Engineering, Auckland, New Zealand, Paper No. 2324. Sun, L. and Goto, Y. 1994. Applications of Fuzzy Theory to variable dampers for bridge vibration control, pp. 31 –40. In Proceedings of First World Conference on Structural Control, Los Angeles, CA, WP1. Tajiran, F.F. 1998. Base isolation design for civil components and civil structures. In Proceedings of Structural Engineers World Congress, San Francisco, CA, July 1998. Takayama, M., Wada, A., Akiyama, H., and Tada, H. 1988. Feasibility study on base isolated building, pp. 669 –674. In Proceedings of IX World Conference on Earthquake Engineering, Tokyo, Japan, Vol. 5. Tanabashi, R. 1960. Earthquake resistance of traditional Japanese wooden buildings, Vol. 1, pp. 151–163. In Proceedings of Second World Conference on Earthquake Engineering, Japan. Tani, A., Kawamura, H., and Watari, Y. 1992. optimal adaptive and predictive control of seismic structures by Fuzzy Logic, Vol. 4, pp. 2155–2160. In Proceedings of Tenth World Conference on Earthquake Engineering, Madrid, Spain. Tanzo, W., Yamada, Y., and Iemura, H. 1992. Substructured Computer-Actuator Hybrid Loading Tests for Inelastic Earthquake Response of Structures, Research Report No. 92-ST-01, Department of Civil Engineering, Kyoto University. Tarics, A.G. 1996. Composite seismic isolator and method. In Proceedings of XI World Conference on Earthquake Engineering, Acapulco, Mexico, Paper No. 1895. Taylor, D.P. 2002. History, design, and applications of ﬂuid dampers in structural engineering, pp. 17–34. Passive Structural Control Symposium, December 13 –14, 2002, Tokyo Institute of Technology, Japan. Taylor, A.W., Lin, M., and Martin, J.W., Performance of elastomers in isolation bearings: a literature review, Earthquake Spectra, 8(2), 279–303, 1992. Taylor, A.W., Shenton, H.W., and Chung, R.M. 1995. Standards for the testing and evaluation of seismic isolation systems, pp. 39 –43. In Proceedings of the ASME/JSME Pressure Vessels and Piping Conference, Honolulu, Hawaii, Vol. 319. Tewari, A. 2002. Modern Control Design with MATLAB and SIMULINK, Wiley, England. Tospelas, P.C., Nagarajaiah, S., Constantinou, M.C., and Reinhorn, A.M., Nonlinear dynamic analysis of multiple building base isolated structures, Comput. Struct., 50(1), 47 –57, 1994. Toyooka, A. 2002. Development of the inertia force driven hybrid loading system and pseudonegative stiffness control method of a MR damper. A dissertation submitted to the Faculty of Engineering of Kyoto University in partial fulﬁllment of the requirements for the degree of Doctor of Engineering, Kyoto, Japan, December 2002. Tsai, K.-C., Chen, H.-W., Hong, C.-P., and Su, Y.-F., Design of steel triangular plate energy absorbers for seismic-resistant construction, Earthquake Spectra, 9, 505 –528, 1993. Tyler, R.G., Tapered steel cantilever energy absorbers, Bull. N.Z. Natl. Soc. Earthquake Eng., 11, 1, 1978. UBC (Uniform Building Code). 1997. International Conference of Building Ofﬁcials, Whittier, CA.

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Williams, M.S. and Blakeborough, A., Laboratory testing of structures under dynamic loads: an introductory review, R. Soc. Lond., 359, 1651– 1669, 2001. Wolf, J.P. 1985. Dynamic Soil –Structure Interaction, Prentice Hall, New York. Wong, H.L. and Luco, J.E. 1992. Effects of soil–structure interaction on the seismic response of structures subjected to active control, Vol. 4, pp. 2137 –2142. In Proceedings Tenth World Conference on Earthquake Engineering, Madrid, Spain. Yang, Y.-B., Chang, K.-C., and Yau, J.-D. 2003. Base isolation. In Earthquake Engineering Handbook, 1st ed., W.F. Chen and C. Scawthorn, eds., pp. 17.1–17.31. CRC Press, Boca Raton, FL. Yasaka, A., Koshida, H., and Iizuka, M. 1988. Base isolation system for earthquake protection and vibration isolation of structures, Vol. 5, pp. 699–704. In Proceedings of IX World Conference on Earthquake Engineering, Tokyo, Japan. Zhang, R.H., Soong, T.T., and Mahmoodi, P., Seismic response of steel frame structures with added viscoelastic dampers, Earthquake Eng. Struct. Dyn., 8, 389 –396, 1989.

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12

Seismic Random Vibration of Long-Span Structures 12.1 Introduction ........................................................................ 12-2 Basic Concepts of Random Vibration Structural Seismic Analysis

†

Three Methods for

12.2 Seismic Random-Excitation Fields .................................... 12-11 Power Spectral Density of Spatially Varying Ground Acceleration † Several Coherence Models † Generation of Ground Acceleration Power Spectral Density Curves from Acceleration Response Spectrum Curves † Seismic Equations of Motion of Long-Span Structures † Seismic Waves and Their Geometrical Expressions

12.3 Pseudoexcitation Method for Structural Random Vibration Analysis ............................................................... 12-16 Structures Subjected to Stationary Random Excitations Structures Subjected to Nonstationary Random Vibration † Precise Integration Method

†

12.4 Long-Span Structures Subjected to Stationary Random Ground Excitations ............................................. 12-27 The Solution of Equations of Motion Using the Pseudoexcitation Method † Numerical Comparisons with Other Methods

12.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations ............................................. 12-34

Jiahao Lin Dalian University of Technology

Yahui Zhang

Dalian University of Technology

Modulation Functions † The Formulas for Nonstationary Multiexcitation Analysis † Expected Extreme Values of Nonstationary Random Processes † Numerical Comparisons with the Corresponding Stationary Analysis

12.6 Conclusions ......................................................................... 12-39

Summary Particular considerations must be made during the design of long-span bridges with regard to safety during earthquakes. These include: the wave-passage effect caused by the different times at which seismic waves arrive at different supports; the incoherence effect due to loss of coherency of the motion caused by either reﬂections and refractions of the waves in the inhomogeneous ground medium or the difference in the manner of superposition of waves from an extended source arriving at various supports; and the local effect because of the differences in soil conditions at different supports and the manner in which these inﬂuence the amplitude and frequency content of the bedrock motion. This chapter deals with the random vibration approach to analyzing these structures, which is based on a statistical characterization of the set of motions at the supports. This approach is particularly suitable for dealing with the above spatially varying input motions. The computational problems may be largely overcome by 12-1 © 2007 by Taylor & Francis Group, LLC

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the pseudo excitation method. This approach is presented here. Numerical comparisons are given to show the accuracy of the method and its capability of dealing with the spatial effects and nonstationary effects.

12.1

Introduction

Seismic computations of long-span structures have long been an issue of great concern. Such computations are usually executed numerically using schemes in the time domain (i.e., time history). For short-span bridges, all supports can be assumed to move uniformly and the response-spectrum method (RSM) is a suitable computation tool. For long-span bridges, however, various spatial effects such as the wave-passage effect, the incoherence effect, the local site effect, and so on, may be important. Such spatial effects cannot be dealt with directly by the conventional RSM. Instead, the time-history method (THM) is the most widely used method for these systems. The time-history scheme requires solving the dynamic equations for a number of seismic acceleration samples. The results are then processed statistically to produce the quantities required by the designs. This process is rather complex and requires a considerable computational effort. As a result, more efﬁcient and effective methods are under investigation. Seismic motions are random (stochastic) in nature (Housner, 1947). Spatial effects of long-span bridges can be analyzed using the random-vibration approach. In the last two decades, many scholars and experts (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al., 1990; Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) have made great progress in promoting the seismic random analysis of long-span structures and its engineering applications. Although available computational methods still need further improvement both in precision and efﬁciency, the random-vibration approach, as a theoretically advanced tool, has been gradually accepted by the earthquake engineering community. For example, it has been adopted by the European Bridge Code (European Committee for Standardization, 1995). Developed recently (Lin, 1992; Lin et al., 1994a, 1995a, 1995b, 1997a, 1997b; Lin and Zhan, 2003), the pseudoexcitation method (PEM) is an accurate and highly efﬁcient approach to the stationary and nonstationary random seismic analysis of long-span structures. For typical three-dimensional ﬁnite element models of long-span bridges with thousands of degrees of freedom (DoF) and dozens of supports, when using 100 to 300 modes for mode-superposition analysis, the seismic responses can be implemented quickly and accurately on a standard personal computer. Numerical results show that the wave-passage effect is of particular importance for the seismic analysis of long-span bridges, and the incoherence effect is of comparatively less importance. The details will be given in this chapter. The PEM has been successfully applied to some practical engineering analyses (Wang et al., 1999; Liu and Liu, 2000; Xue et al., 2000; Fan et al., 2001), and has been proven to be quite effective.

12.1.1

Basic Concepts of Random Vibration

12.1.1.1

Stationary Random Process

The probabilistic properties of stationary random processes are independent of time. A random process is said to be strictly stationary (or strongly stationary) if its probability density function does not change with time. However, such a condition is very difﬁcult to satisfy in practical engineering problems. Therefore, a wide-sense stationary (or weakly stationary) process is deﬁned for which only the mean value and autocorrelation function of the process are not permitted to vary with time. A random variable x is said to be Gaussian-distributed if its probability density can be written in the form ! 1 ðx 2 xÞ2 pðxÞ ¼ pﬃﬃﬃﬃ exp 2 ð12:1Þ 2s 2 s 2p in which s is the standard deviation of x; and the variance is given by ð1 ðx 2 xÞ2 pðxÞdx s2 ¼ 21

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ð12:2Þ

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12-3

p(x)

0.5

s = 0.5

0.4 0.3 0.2

s = 1.0 s = 2.0

0.1 −4

−3

−2

FIGURE 12.1

0.0 −1 0

x 1

2

3

4

5

6

7

8

Probability density functions of Gaussian random variables.

Probability density functions of typical Gaussian random processes are shown in Figure 12.1, in which x is the mean value given by the abscissa (horizontal coordinate) of the peak value. A smaller s corresponds to a narrower and higher peak. For a random process xðtÞ; if the joint probability density function of its values xðt1 Þ; xðt2 Þ; …; xðtn Þ at n arbitrary time instants is Gaussian, then xðtÞ is said to be a Gaussian random process. Since the joint probability density function depends only on the mean values and covariances of the n values, a weakly stationary Gaussian random process is also strongly stationary. If a stationary random process has statistical properties that can be computed by taking the time average of an arbitrary sample over a sufﬁciently long period, the process is said to be ergodic. A typical seismic ground motion record is usually assumed to be a Gaussian and ergodic stationary random process. Its expected value (i.e., mean value) can be computed by ðþ1 E½xðtÞ ¼ xðtÞ ¼ xðtÞpðx; tÞdx ð12:3Þ 21

in which pðx; tÞ is the probability density function of xðtÞ: In order to investigate the relation between the values of a random process xðtÞ at two different times, the autocorrelation function of xðtÞ is deﬁned as Rxx ðtÞ ¼ E½xðtÞxðt þ tÞ ¼ lim

T!1

1 ðT=2 xðtÞxðt þ tÞdt T 2T=2

ð12:4Þ

A stationary random process, denoted as xðtÞ; is not absolutely integrable in a region of t [ ð21; 1Þ: Therefore, a subsidiary function xT ðtÞ is deﬁned: ( xðtÞ when 2 T=2 # t # T=2 xT ðtÞ ¼ ð12:5Þ 0 elsewhere Obviously, xT ðtÞ is absolutely integrable within t [ ð21; 1Þ: Therefore, its Fourier transformation can be computed by 1 ð1 XT ðvÞ ¼ ð12:6Þ x ðtÞexpðjvtÞdt 2p 21 T Let Sxx ðvÞ ¼ lim

T!1

1 lX ðvÞl2 T T

Equation 12.7 is the deﬁnition of the auto-PSD (power spectral density) function of xðtÞ:

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ð12:7Þ

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Vibration Monitoring, Testing, and Instrumentation

Note that the repeated subscripts in Rxx or Sxx can be represented by just one; that is, they can be denoted as Rx or Sx : When xðtÞ is a zero-mean stationary random process, its variance is given by ð1 Sxx ðvÞdv s2x ¼ ð12:8Þ 21

Figure 12.2 gives the auto-PSD curves of four typical stationary random processes, which show the energy distribution with frequency for each kind of random process. The energy of a narrowband random process is concentrated within a narrow frequency band (see Figure 12.2b) whereas the energy of a wideband random process is distributed over a rather wide frequency range, as shown in Figure 12.2c. The energy of a white noise process is distributed uniformly over an inﬁnite region, v [ ð21; 1Þ; as shown in Figure 12.2d. Using such a random process model results in mathematical convenience. However, the white noise process does not physically exist. A single harmonic random wave has nonzero values only at two isolate frequencies ^v0 (see Figure 12.2a), and its initial phase angle w is usually regarded as uniformly distributed over ½0; 2pÞ:

x(t) = x0 sin(w 0t+f)

Sxx(w)

t

−w 0

w0

ω

(a) x(t)

Narrow-banded

Sxx(w)

t

ω

(b) x(t)

Wide-banded

Sxx(w)

ω

t (c) x(t)

White noise Sxx(w) t

(d) FIGURE 12.2

Auto-PSD curves of four typical stationary random processes.

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ω

Seismic Random Vibration of Long-Span Structures

12-5

The auto-PSD Sxx ðvÞ of a stationary random process xðtÞ has the following properties: 1. Sxx ðvÞ is a nonnegative real number; that is Sxx ðvÞ $ 0

ð12:9Þ

This can be judged from the deﬁnition of auto-PSD functions, that is, Equation 12.7. 2. Sxx ðvÞ is an even function; that is Sxx ðvÞ ¼ Sxx ð2vÞ

ð12:10Þ

This can also be deduced from Equation 12.7. 3. The auto-PSDs of the derivatives of xðtÞ can be computed from Sxx ðvÞ directly by Sx_ x_ ðvÞ ¼ v2 Sxx ðvÞ;

Sx€ x€ ðvÞ ¼ v4 Sxx ðvÞ

ð12:11Þ

12.1.1.1.1 Wiener–Khintchine Theorem Wiener and Khintchine proved that for an arbitrary stationary random process xðtÞ; its auto-PSD Sxx ðvÞ and autocorrelation function Rxx ðtÞ are a Fourier transform pair; that is 1 ð1 R ðtÞexpð2jvtÞdt ð12:12Þ Sxx ðvÞ ¼ 2p 21 xx Rxx ðtÞ ¼

ð1 21

Sxx ðvÞexpðjvtÞdv

ð12:13Þ

According to this theorem, if either of Sxx ðvÞ or Rxx ðtÞ has been found, the other can be directly obtained. If stationary random processes xðtÞ and yðtÞ are both ergodic, then their cross-correlation function can be computed in terms of their sample functions x^ ðtÞ and y^ ðtÞ: 1 ðT=2 Rxy ðtÞ ¼ lim x^ ðtÞ^yðt þ tÞdt ð12:14Þ T!1 T 2T=2 Ryx ðtÞ ¼ lim

T!1

1 ðT=2 y^ ðtÞ^xðt þ tÞdt T 2T=2

ð12:15Þ

Also, their cross-PSD functions can be deﬁned by means of the Fourier transforms of the corresponding cross-correlation functions: 1 ð1 R ðtÞexpð2ivtÞdt ð12:16Þ Sxy ðvÞ ¼ 2p 21 xy Syx ðvÞ ¼

1 ð1 R ðtÞexpð2ivtÞdt 2p 21 yx

ð12:17Þ

For more details, see Lin (1967). 12.1.1.2

Nonstationary Random Process

Nonstationary random processes are generally short in duration. Their basic characteristic is that the statistical properties vary signiﬁcantly with time. An example is the process of a typical earthquake record, for which the medium ﬂat segment is often regarded as a stationary random process in order to simplify the structural analysis. However, such simpliﬁcation sometimes causes signiﬁcant errors. For instance, some long-span bridges are very ﬂexible, with fundamental periods of approximately 15 to 20 sec. The period of the strong earthquake portion of a typical earthquake record is only approximately 20 to 30 sec. For such slender long-span bridges, the seismic excitations exhibit clear nonstationary characteristics. In order to avoid computational complexities in the structural analyses, such excitations are usually assumed to be stationary random processes. This chapter shows that the analysis of such nonstationary random responses is made very simple by using PEM.

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Vibration Monitoring, Testing, and Instrumentation

Nonstationary random processes are not ergodic because their statistical properties vary with time. In earthquake engineering, the evolutionary random process deﬁned by Priestly (1967) has been investigated extensively. It is expressed in terms of the Riemann–Stieltjes integration as ð1 Aðv; tÞexpðivtÞdaðvÞ f ðtÞ ¼ ð12:18Þ 21

in which aðvÞ satisﬁes the relations xðtÞ ¼

ð1 21

expðivtÞdaðvÞ

ð12:19Þ

E ½dap ðv1 Þdaðv2 Þ ¼ Sxx ðv1 Þdðv2 2 v1 Þdv1 dv2

ð12:20Þ

Here, xðtÞ is a zero-mean stationary random process, with auto-PSD Sxx ðvÞ; Aðv; tÞ is a deterministic slowly varying nonuniform modulation function, and d is a Dirac delta function. The variance of f ðtÞ is ð1 ð1 s2f ðtÞ ¼ Sff ðvÞdv ¼ lAðv; tÞl2 Sxx ðvÞdv ð12:21Þ 21

21

The PSD of f ðtÞ as given by Sff ðv; tÞ ¼ lAðv; tÞl2 Sxx ðvÞ

ð12:22Þ

is known as an evolutionary power spectral density function. Responses of structures subjected to nonstationary random excitations expressed by Equation 12.18 are not easy to compute. Therefore, the nonuniform modulation assumption is often replaced by a uniform modulation assumption; that is, the nonuniform modulation function Aðv; tÞ is replaced by a uniform modulation function gðtÞ: Thus, Equation 12.18 reduces to ð1 f ðtÞ ¼ gðtÞexpðivtÞdZðvÞ ¼ gðtÞxðtÞ ð12:23Þ 21

x(t)

(a)

g(t) (b)

f(t) = g(t)x(t)

(c) FIGURE 12.3

A uniformly modulated evolutionary random excitation f ðtÞ:

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Seismic Random Vibration of Long-Span Structures

12-7

Equation 12.18 and Equation 12.23 are known as the nonuniformly modulated and uniformly modulated evolutionary random processes, respectively. Figure 12.3 shows a stationary random process xðtÞ and the corresponding uniformly modulated evolutionary random excitation f ðtÞ with a given modulation function gðtÞ:

12.1.2

Three Methods for Structural Seismic Analysis

12.1.2.1

Response Spectrum Method

The equations of motion of a linear multi-DoF structure subjected to a ground acceleration excitation x€ g ðtÞ can be written as M€y þ C_y þ Ky ¼ 2Me€xg ðtÞ

ð12:24Þ

in which M, C, and K are the n £ n mass, damping, and stiffness matrices of the structure, and e is the index vector of inertia forces. For short-span structures, all supports can be assumed to move uniformly with the same ground acceleration x€ g ðtÞ: If the structure under consideration has a very large number of DoF, Equation 12.24 can be solved by using the mode-superposition scheme. First, the lowest q natural angular frequencies vj ð j ¼ 1; 2; …; q; q ,, nÞ and the corresponding n £ q mass normalized mode matrix F should be extracted. Then, yðtÞ can be decomposed in terms of these modes: yðtÞ ¼ FuðtÞ ¼

q X j¼1

uj wj

ð12:25Þ

With proportional damping assumed, Equation 12.24 can be decoupled into q single-DoF equations u€ j þ 26j vj u_ j þ v2j uj ¼ 2gj x€ g ðtÞ

ð12:26Þ

in which 6j is the jth damping ratio and gj is the jth modal participation factor

gj ¼ wTj Me

ð12:27Þ

According to the response spectrum theory, the solution of Equation 12.26 is uj ¼ gj aj g=v2j

ð12:28Þ

in which g is the gravity acceleration and aj is the value of the ground acceleration response spectrum (ARS) at frequency vj : If the kth element of y, denoted as yk ; is required, then the kth elements of all yj ð j ¼ 1; 2; …; qÞ are taken to compose a vector yk ; which is then used in the computation of the response (or demand) yk : qﬃﬃﬃﬃﬃﬃﬃﬃﬃ yk ¼ yTk Rc yk ð12:29Þ Here, Rc is the correlation matrix representing the degree of correlation between all participating modes. Based on random-vibration theory, Wilson and Kiureghian (1981) have derived the expression for its elements as pﬃﬃﬃﬃﬃ 8 zi zj ðzi þ rzj Þr3=2 Rij ¼ ð12:30Þ ð1 2 r2 Þ2 þ 4zi zj rð1 þ r2 Þ þ 4ðz2i þ z2j Þr 2 in which r ¼ vj =vi : This is the widely used Complete Quadratic Combination (CQC) algorithm in the RSM. If the correlation coefﬁcients between all modes are neglected, that is, if Rij ¼ dij (Dirac delta function), then Rc becomes a unity (identity) matrix and Equation 12.29 reduces to the square root of the sum of squares (SRSS) algorithm. The RSM, as outlined above, is very popular in the seismic analysis of short-span structures. Some extensions have been published (Lee and Penzien, 1983; Dumanoglu and Severn, 1990; Lin et al., 1990;

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Berrah and Kausel, 1992; Kiureghian and Neuenhofer, 1992; Ernesto and Vanmarcke, 1994) in order to deal with the seismic analysis of long-span structures. However, the efﬁciency and accuracy still need further improvement before they can be widely accepted in engineering practice. 12.1.2.2

Time-History Method

Assume that all supports move uniformly with the same acceleration x€ g ðtÞ; which is now given in a discrete numerical form. Equation 12.24 can now be solved using the Newmark method, the Wilson-u method (Clough and Penzien, 1993), or the precise integration method (Zhong and Williams, 1995). In these THMs, the structural parameters can be modiﬁed at any time. Therefore, this method is good for nonlinear problems for which structural parameters often vary with time, for example in seismic elastoplastic analysis. A major disadvantage of THMs is that the computational results rely heavily on the selected ground acceleration records. In general, a number of records must be selected for structural analyses, and statistical results are then used in the designs. In order to reduce the computational effort, usually only about three to ten records are used for statistical purposes. When the wave passage effect needs to be taken into account, the same ground acceleration record is applied to different supports with time lags and this generates x€ b on the right-hand side of Equation 12.78. If the incoherence effect between the supports must also be considered, then the process for generating x€ b becomes rather complicated (Deodatis, 1990). In fact, real records of this type are difﬁcult to ﬁnd. 12.1.2.3

Random Vibration Method

The random vibration approach is appealing for seismic random analysis of long-span structures. Previously, because of its high complexity and low efﬁciency, it was not accepted as a method of analysis by practicing engineers. However, this situation has changed considerably in recent years. Let us still begin with Equation 12.24, which we can also apply to structures subjected to uniform stationary random ground excitations. Now x€ g ðtÞ is a zero-mean Gaussian stationary random process with a known auto-PSD Sa ðvÞ representing acceleration excitations uniformly applied to all supports of the structure. By means of the modal superposition scheme, that is, Equation 12.25 to Equation 12.27, the traditional CQC method can be established (Clough and Penzien, 1993): Syy ðvÞ ¼

q X q X j¼1 k¼1

gj gk wj wTk Hjp ðvÞHk ðvÞSa ðvÞ

ð12:31Þ

in which wj and gj are the jth mode and the jth modal participation factor, and Hj ¼ ðv2j 2 v2 þ 2i6j vvj Þ21

ð12:32Þ

is the jth frequency-response function. For a real long-span bridge, the number of structural DoF n usually ranges from 103 to 104, and the numbers of v and q typically range from 102 to 103. Equation 12.31 includes all quadratic terms of the participating modes, and it must be repeatedly computed for dozens or hundreds of frequencies. Although it is a simple form of excitation, the computational effort is still considerable. Therefore, in engineering practice, the following SRSS method obtained by neglecting all j – k terms in Equation 12.31, is generally used in place of the above CQC method: Syy ðvÞ ¼

q X j¼1

g2j wj wTj lHj ðvÞl2 Sa ðvÞ

ð12:33Þ

This is frequently recommended in academic literature. The SRSS formula is an approximation of Equation 12.31 that neglects the cross-correlation terms between participating modes, thereby reducing the computational effort to about 1/q of that required by Equation 12.31. However, this approximation can be used only for lightly damped structures for which the participating frequencies must be sparsely spaced. For most structures (in particular their three-dimensional structural models), some participating frequencies are often closely spaced. Hence, the applicability of the SRSS approximation is

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12-9

somewhat questionable. It will be seen in the next section that PEM will produce results identical to those from Equation 12.31 with much less computational effort. The random-vibration analysis outlined above is executed in terms of power spectral densities in the frequency domain and therefore it is also referred to as the power-spectrum method. A diagonal element Sjj in the PSD matrix represents the auto-PSD of a random response j: Assume that this response is signiﬁcant only within the frequency domain v [ ½vL ; vU : Thus, the ith spectral moment of j can be computed by ð1 ðvU li ¼ 2 vi Sjj ðvÞdv < 2 vi Sjj ðvÞdv ð12:34Þ vL

0

The PSD values at the negative frequencies do not have any intuitive physical signiﬁcance and so the single-sided PSD Gxx ðvÞ is deﬁned for applications in many engineering ﬁelds: ( 2Sxx ðvÞ v $ 0 ð12:35Þ Gxx ðvÞ ¼ v,0 0 Thus, Equation 12.34 becomes

li ¼

ð1 0

vi Gjj ðvÞdv <

ð vU vL

vi Gjj ðvÞdv

ð12:36Þ

For general multiple input (xðtÞ ¼ {x1 ðtÞ; x2 ðtÞ; …; xn ðtÞ}T ) and multiple output (yðtÞ ¼ {y1 ðtÞ; y2 ðtÞ; …; ym ðtÞ}T ) problems (or MIMO problems), the response (i.e., output) PSD matrix SyyðvÞ can be computed using the excitation (i.e., input) PSD matrix Sxx ðvÞ: Syy ðvÞ ¼ Hp Sxx ðvÞHT

ð12:37Þ

in which H is the frequency-response function matrix. Also, the cross-PSD matrices between the excitations and responses can be computed from Sxy ðvÞ ¼ Sxx ðvÞHT

ð12:38Þ

Syx ðvÞ ¼ Hp Sxx ðvÞ

ð12:39Þ

Equation 12.37 to Equation 12.39 have simple forms and are comparatively convenient for engineering applications. However, they must be executed for dozens or hundreds of discrete frequencies. For complex structures, such matrix operations may require extensive effort. PEM, which will be introduced in the next section, is a better alternative than these equations. If the ﬁrst N modes are used in the modal superposition analysis, numerical tests for seven bridges show that taking ½vL ; vU ¼ ½0:7v1 ; 1:2vN seems to be a good choice for the integration interval, where v1 and vN are the ﬁrst and the Nth natural angular frequencies of the structure. It is inconvenient for engineers to take such spectral moments for practical designs. However, some approaches have been suggested to estimate structural responses (or demands) in terms of these spectral moments. Two popular approaches are described next. 12.1.2.3.1 Davenport Approach With the seismic excitations assumed to be zero-mean stationary Gaussian processes, an arbitrary linear response of the structure subjected to such excitations, denoted yðtÞ; will also possess the same probability characteristic. It is also assumed that if a given barrier (threshold) a is sufﬁciently high, the peaks of yðtÞ above this barrier will appear independently. Let NðtÞ be the number of upcrossing of a within the time interval ð0; t ; then NðtÞ will be a Poisson process with a stationary increment (Davenport, 1961). Denote the extreme value of yðtÞ; that is, the maximum value of all peaks by their absolute values, within the earthquake duration ½0; Ts as ye ; and the standard deviation of yðtÞ as sy : Deﬁne h as the dimensionless parameter of ye ; and n as the mean zero-crossing rate, which can be

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Vibration Monitoring, Testing, and Instrumentation

expressed as

h ¼ ye =sy ;

pﬃﬃﬃﬃﬃﬃﬃ n ¼ l2 =l0 p

ð12:40Þ

Based on these assumptions, the probability distribution of h can be derived as PðhÞ ¼ exp ½2nTs expð2h2 =2Þ ;

h.0

ð12:41Þ

The expected value of h; known as the peak factor, is approximately given by EðhÞ < ð2 ln nTs Þ1=2 þ g ð2 ln nTs Þ1=2

ð12:42Þ

and its standard deviation is

sh < p 12 ln nTs

1=2

ð12:43Þ

in which g ¼ 0:5772 is the Euler constant, while the expected value of ye is approximately E½ye ¼ E½h sy

ð12:44Þ

This quantity is the demand usually required by engineers. 12.1.2.3.2 Vanmarcke Approach In the preceding paragraph, the barrier a was assumed to be sufﬁciently high. Therefore, the peaks of yðtÞ above this barrier will appear independently, and NðtÞ can be regarded as a Poisson process. Vanmarcke (1972) considered that the barrier a should not be very high. Therefore, the Poisson process assumption should be replaced by the two-state Markov process assumption and the probability distribution of h becomes # " !# " pﬃﬃﬃﬃﬃ 1 2 exp 2 p=2q1:2 h h2 ð12:45Þ PðhÞ ¼ 1 2 exp 2 exp 2nTs expðh2 =2Þ 2 1 2 in which n and Ts have the same meanings as the above, while the shape factor for the response PSD is

d0 ¼ 1 2 l21 =ðl0 l2 Þ

1=2

ð12:46Þ

Here, d0 is a bandwidth parameter with values ranging from zero to one. For a narrowband process, d0 is close to zero. Based on the probability distribution function shown in Equation 12.45, Kiureghian (1980) proposes the following approximate expressions for the peak factor EðhÞ and standard deviation sh when 10 # nt # 1000 and 0:11 # q # 1; which are of interest in earthquake engineering: g ð12:47Þ EðhÞ ¼ ð2 ln ne Ts Þ1=2 þ ð2 ln ne Ts Þ1=2 8 1:2 5:4 > < 2 ne Ts . 2:1 13 þ ð2 ln ne Ts Þ3:2 ð12:48Þ sh ¼ ð2 ln ne Ts Þ1=2 > : 0:65 ne Ts # 2:1 in which

(

ne ¼

ð1:63q0:45 2 0:38Þn0

d0 , 0:69

n0

d0 $ 0:69

ð12:49Þ

Gupta and Trifunac (1998) made numerical experiments to compare the above two models using 1000 simulated time-history excitations. Their research shows that for most practical purposes in earthquake engineering studies, the effect of the dependence among level crossings is not signiﬁcant. 12.1.2.4

Comparisons of the Three Seismic-Analysis Methods

The RSM is the most popular method for the seismic analysis of short-span structures. Some extensions have been made to allow this method to be used in the seismic analysis of long-span structures.

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However, the accuracy and efﬁciency still need further improvement for practical applications. The THM can be used in the seismic analysis of long-span structures without theoretical difﬁculties. However, its major disadvantages are that it requires a good selection of the ground acceleration record samples and its computational cost is very high. The random-vibration method is appealing because of its statistical nature. In the past, the random vibration computation of complex structures has been very costly. This issue has received much attention in the last two decades. As will be described below, PEM has remarkably improved this situation. Now, long-span structures with thousands of DoF and dozens of supports can be computed far more quickly and accurately on a personal computer.

12.2 12.2.1

Seismic Random-Excitation Fields Power Spectral Density of Spatially Varying Ground Acceleration

For multiple excitation problems, the PSD matrices of ground acceleration excitations have the form 2 3 SX€ 1 X€ 1 ðivÞ SX€ 1 X€ 2 ðivÞ · · · SX€ 1 X€ N ðivÞ 6 7 6 S € € ðivÞ S € € ðivÞ · · · S € € ðivÞ 7 6 X2 X1 7 X2 X2 X2 XN 7 Sxx ðivÞ ¼ 6 ð12:50Þ 6 7 6 7 ··· ··· ··· ··· 4 5 SX€ N X€ 1 ðivÞ SX€ N X€ 2 ðivÞ · · · SX€ N X€ N ðivÞ in which

qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ SX€ k X€ l ðivÞ ¼ rkl ðivÞ SX€ k ðvÞSX€ l ðvÞ

ð12:51Þ

rkl ðivÞ ¼ lrkl ðivÞlexp½iukl ðvÞ

ð12:52Þ

and rkl ðivÞ is the acceleration coherence function between the kth and lth supports. Its norm must satisfy the relation lrkl ðivÞl # 1: The values of SX€ k ðvÞ and SX€ l ðvÞ in Equation 12.51 can be different due to local effects. However, earthquake records show that the structural responses due to the difference between SX€ k ðvÞ and SX€ l ðvÞ are in general of relatively low signiﬁcance. The factor exp½iukl ðvÞ expresses the wave passage effect, which can be further expressed as exp½iukl ðvÞ ¼ exp½2ivdklL =vapp

ð12:53Þ

in which dkl is the horizontal distance between the two supports; dklL is the projection of dkl along the propagation direction of the seismic waves; and vapp is the apparent velocity of seismic waves along the surface. Assume that the time lags between the supports and the origin of the reference coordinate system are t1 ; t2 ; …; tN ; respectively. Without losing generality, let tl $ tk (when l . k). Then dklL =vapp ¼ tl 2 tk ; and so Equation 12.53 becomes exp½iukl ðvÞ ¼ exp½ivðtk 2 tl Þ

ð12:54Þ

The factor lrkl ðivÞl reﬂects the incoherence effect (Kiureghian and Neuenhofer, 1992). Some mathematical models of rkl ðivÞ have been established based on practical earthquake records, which will be shown in the next section. Using Equation 12.51 to Equation 12.54, Equation 12.50 can be written as SðivÞ ¼ Bp JRJB

ð12:55Þ

B ¼ diag½expð2ivt1 Þ; expð2ivt2 Þ; …; expð2ivtN Þ

ð12:56Þ

hqﬃﬃﬃﬃ pﬃﬃﬃﬃ qﬃﬃﬃﬃﬃ i J ¼ diag SX€ 1 ; SX2 € ; …; SX€ N

ð12:57Þ

in which

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Vibration Monitoring, Testing, and Instrumentation

2

1

lr12 l

···

6 6 lr l 1 ··· 6 21 R¼6 6 6 ··· ··· ··· 4 lrN1 l lrN2 l · · ·

12.2.2

lr1N l

3

7 lr2N l 7 7 7 7 ··· 7 5 1

ð12:58Þ

Several Coherence Models

A number of coherence models have been established based on practical earthquake records. Some of them are outlined below. 12.2.2.1

Feng–Hu Model lrkl ðv; dkl Þl ¼ exp½2ðr1 v þ r2 Þdkl

ð12:59Þ

in which r1 and r2 are the coherence parameters. According to the Hai-Cheng earthquake records (China) and the Niigata earthquake records (Japan), the values of these parameters are (Feng and Hu, 1981): Hai-Cheng: r1 ¼ 2 £ 1025 sec/m, r2 ¼ 88 £ 1024 m21 Niigata: r1 ¼ 4 £ 1024 sec/m, r2 ¼ 19 £ 1024 m21 12.2.2.2

Harichandran –Vanmarcke Model

lrkl ðv; dkl Þl ¼ A exp 2

2d 2d ð1 2 A þ aAÞ þ ð1 2 AÞexp 2 ð1 2 A þ aAÞ auðvÞ auðvÞ

ð12:60Þ

in which

uðvÞ ¼ K 1 þ ðv=v0 Þb

21=2

ð12:61Þ

According to the acceleration records of the SMART-1 array (Harichandran and Vanmarcke, 1986), the parameters in Equation 12.60 and Equation 12.61 are: A ¼ 0:736; a ¼ 0:147; K ¼ 5210; v0 ¼ 6:85 rad/sec, b ¼ 2:78: 12.2.2.3

Loh –Yeh Model

" lrkl ðv; dkl Þl ¼ exp 2a

vdkl 2pvapp

# ð12:62Þ

in which a is the wave-number of the seismic waves. According to the 40 acceleration records of the SMART-1 array (Loh and Yeh, 1988), a ¼ 0:125 is proposed. Parameters vapp and dkl have been explained above. 12.2.2.4

Oliveira –Hao–Penzien Model qﬃﬃﬃﬃ qﬃﬃﬃﬃ lrkl ðv; dkl Þl ¼ expð2b1 dklL 2 b2 dklT Þexpð2 a1 dklL 2 a2 dklT ðv=2pÞ2 Þ

ð12:63Þ

in which ai ¼ ð2pai =vÞ þ ðvbi =2pÞ þ ci : dklL and dklT are the projections of dkl along the propagation direction of the seismic waves and along its normal direction, respectively. Based on 17 acceleration records of the SMART-1 array (Oliveira et al., 1991), parameters b1 ; b2 ; a1 ; a2 ; b1 ; b2 ; c1 , and c2 are given for each of these records.

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12.2.2.5

Luco–Wong Model

"

avdkl lrkl l ¼ exp 2 vs

12-13

2

# ð12:64Þ

in which a is the coherence factor, dkl is the horizontal distance between the kth and lth supports and vs is the shear wave velocity (Luco and Wong, 1986). 12.2.2.6

Qu–Wang–Wang Model

T. Qu, J. Wang, and Q. Wang (QWW) (Qu et al., 1996) proposed the following model h i lrkl l ¼ exp 2aðvÞdklbðvÞ

ð12:65Þ

in which aðvÞ ¼ a1 v2 þ a2 ;

bðvÞ ¼ b1 v þ b2

ð12:66Þ

a1 ¼ 0:00001678; a2 ¼ 0:001219; b1 ¼ 20:0055; b2 ¼ 0:7674: This model was established based on the statistics of dozens of records from four closely located arrays with SMART-1 as the leading one. This model has given reasonable results in some applications in China.

12.2.3 Generation of Ground Acceleration Power Spectral Density Curves from Acceleration Response Spectrum Curves In order to carry out the seismic random vibration analysis of important long-span bridges, the local seismic motion PSD (usually the acceleration PSD) must be established by specialists. For bridges of less importance, however, such seismic acceleration PSD can be derived from the local ARS curve. Two methods to perform the transformation are given below. 12.2.3.1

Kaul Method

Kaul (1978) proposed an approximate transformation method that consists of the two equations Sðv0 Þ ¼ r2 ¼ 2 ln

4jR2a ðj; v0 Þ pv0 r2

2

p ln p v0 Ts

ð12:67Þ 21

ð12:68Þ

in which Ra ðj; v0 Þ is the ARS curve of the seismic absolute acceleration when the damping ratio is j and the natural angular frequency is v0 (usually j takes 0.05); Sðv0 Þ is the equivalent PSD curve corresponding to the given ARS curve; Ts is the seismic duration; and p is the probability of the peaks that do not cross with the given positive or negative barriers — usually p ¼ 0:85 is assumed. 12.2.3.2

Iteration Scheme

An iteration-based scheme for the transformation has been proposed (Sun and Jiang, 1990), which produces more accurate equivalent PSD curve than the Kaul method. For a given ARS curve Ra ðj; v0 Þ; in order to transform it into an equivalent PSD curve SðvÞ; specify its initial values S0 ðvi Þ ¼ Si ; for i ¼ 1; 2; …; N: Then the standard deviation of the acceleration response for a SDOF system can be computed from " #1=2 ð1 1 þ 4j2 ðv=v0 Þ2 dv SðvÞ ð12:69Þ s0 ðj; v0 Þ ¼ ð1 2 ðv=v0 Þ2 Þ2 þ 4j2 ðv=v0 Þ2 0 The peak factor r is

.pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r ¼ 2 lnðvTs Þ þ 0:5772 2 lnðvTs Þ

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ð12:70Þ

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Vibration Monitoring, Testing, and Instrumentation

in which Ts is the period of the strong earthquake portion. The average zero-crossing rate is pﬃﬃﬃﬃﬃﬃﬃ approximately v ¼ l2 =l0 p < v0 =p: The acceleration responses computed by means of Equation 12.69 and Equation 12.70 are Am ðj; v0 Þ ¼ r s0 ðj; v0 Þ

ð12:71Þ

The percentage errors between Ra ðj; v0 Þ and Am ðj; v0 Þ can be computed from Eðv0 Þ ¼

lRa ðj; v0 Þ 2 Am ðj; v0 Þl £ 100% Ra ðj; v0 Þ

ð12:72Þ

Compute Eðv0 Þ for each frequency. If Eðv0 Þ is found to be greater than the given tolerance 1 for at least one frequency, modify all PSD values according to the following equation Skþ1 ðvi Þ ¼ Sk ðvi ÞR2a ðj; vi Þ A2m ðj; vi Þ;

i ¼ 1; 2; …; N

ð12:73Þ

and then repeat the computations of Equation 12.69 to Equation 12.72. The above process is continued until Equation 12.72 is satisﬁed at all frequencies.

12.2.4

Seismic Equations of Motion of Long-Span Structures

For long-span structures subjected to differential ground motion, the equations of motion in the global coordinate system (assumed to be ﬁxed to the center of the Earth) can be written in partitioned form as #( ) ( ) #( ) " " #( ) " xs 0 Ks Ksb x_ s Cs Csb x€ s Ms Msb ð12:74Þ ¼ þ þ xb pb x_ b KTsb Kb CTsb Cb x€ b MTsb Mb in which the subscript m represents the master DoF, that is, the support displacements, while the subscript s represents the slave DoF. The absolute displacement vector xs can be decomposed into the two parts xs ¼ y s þ y r

ð12:75Þ

where ys is the quasi-static displacement vector (Clough and Penzien, 1993), which satisﬁes ys ¼ 2K21 s Ksb xb

ð12:76Þ

Substituting Equation 12.75 and Equation 12.76 into Equation 12.74 gives Ms y€ r þ Cs y_ r þ Ks y r ¼ Ms K21 € b þ ðCs K21 xb s Ksb x s Ksb 2 Csb Þ_

ð12:77Þ

It should be pointed out that Equation 12.77 cannot be reduced to the conventional Equation 12.24 when xb represents uniform ground displacements (Clough and Penzien, 1993). This is because Equation 12.74 assumes the damping forces to be proportional to the absolute velocity vector {_xTs ; x_ Tb }T : In order to avoid this inconsistency, the damping forces should be assumed to be proportional to the relative velocity vector {_yTr ; 0}T in Equation 12.74. This leads to the equations Ms y€ r þ Cs y_ r þ Ks yr ¼ Ms K21 €b s Ksb x

ð12:78Þ

x€ b ¼ eb x€ g

ð12:79Þ

for uniform ground motion

Note that the following rigid displacement condition is satisﬁed: " #( ) ( ) es 0 Ks Ksb ¼ T Ksb Kb eb 0

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ð12:80Þ

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12-15

Its second half gives Ksb eb ¼ 2Ks es

ð12:81Þ

Substituting Equation 12.79 into Equation 12.78 and using Equation 12.81 gives Equation 12.24.

12.2.5

Seismic Waves and Their Geometrical Expressions

Seismic waves can be divided into body waves and surface waves. Body waves include longitudinal waves (or pressure waves, primary waves or P waves) and transverse waves (or shear waves, secondary waves or S waves). Surface waves include Rayleigh waves and Love waves. For P waves, the soil particles move parallel to the traveling direction of waves; for S waves, however, their motion is normal to the wave traveling direction (see Figure 12.4). For horizontal shear waves (SH waves), all particles move horizontally. For vertical shear waves (SV waves), all particles move vertically. Assume that both x and y axes lie in the horizontal plane. The angle between axis x and the horizontal traveling direction of these waves is b; as shown in Figure 12.5. Thus, the acceleration components along the coordinate axes can be expressed by the components parallel, or normal, to the wave traveling direction, that is, for P waves x€ i ¼ u€ i cos b; z€i ¼ 0

y€ i ¼ u€ i sin b;

z

y SH

FIGURE 12.4 waves.

.. yi P

z€i ¼ 0

ð12:83Þ

and for SV waves x€ i ¼ 0;

x Particle motion directions for P and S

üi .. xi

.. yi

üj

ð12:82Þ b

y€ i ¼ u€ j cos b;

SV

y

for SH waves x€ i ¼ 2€uj sin b;

P

β

FIGURE 12.5 components.

y€ i ¼ 0;

z€i ¼ u€ k

.. xi

SH x

Transform of ground acceleration

ð12:84Þ

In the equations of motion of an multi-DoF system under uniform ground excitations, that is, Equation 12.24, e is the index vector of inertia forces. Its mathematical expressions for the different waves are: for P waves e ¼ ex cos b þ ey sin b

ð12:85Þ

e ¼ 2ex sin b þ ey cos b

ð12:86Þ

e ¼ ez

ð12:87Þ

for SH waves for SV waves Clearly, for P waves, e ¼ ex when b ¼ 0 and e ¼ ey when b ¼ 908:

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If a structure has N supports, its ground acceleration excitations along the wave traveling direction can be expressed by the N-dimensional vector u€ b ¼ {€u1 ; u€ 2 ; …; u€ N }T

ð12:88Þ

The m-dimensional ground acceleration vector in Equation 12.78 is x€ b ¼ {€x1 ; x€ 2 ; …; x€ m }T

ð12:89Þ

x€ b ¼ EmN u€ b

ð12:90Þ

The relation between these two vectors is in which EmN is a m £ N block-diagonal matrix EmN ¼ diag½eb ; eb ; …; eb

ð12:91Þ

m£N

If only three translations are considered for each support, then m ¼ 3N and each submatrix eb would be 8 9 8 9 8 9 0> cos b > > > 2sin b > > > > > > < = > < = < > = sin b ; cos b and 0 > > > > > > > > > > : ; > : ; : > ; 0 0 1 for the P, SH and SV waves, respectively. Using Equation 12.90, Equation 12.76 and Equation 12.78 can be rewritten as ys ¼ 2K21 s Ksb EmN ub

ð12:92Þ

€b Ms y€ r þ Cs y_ r þ Ks yr ¼ Ms K21 s Ksb EmN u

ð12:93Þ

12.3 Pseudoexcitation Method for Structural Random Vibration Analysis 12.3.1

Structures Subjected to Stationary Random Excitations

12.3.1.1

Single Stationary Random Excitations

The basic principle of the PEM for structural stationary random vibration analysis can be explained by Figure 12.6. Consider a linear system subjected to a zero-mean stationary random excitation xðtÞ

(a)

(b)

(c)

x(t)

Linear Structure

Sxx

Linear Structure

∼ x(t)= Sxx(w)exp(iwt)

FIGURE 12.6

Linear Structure

y(t) z(t) Syy(w) = Hy*(w)Sxx(w)Hy(w) Syz(w) = Hy*(w)Sxx(w)Hz(w) ∼ y = Sxx(w)Hy(w)exp(iwt) ∼z = Sxx(w)Hz(w)exp(iwt)

Basic principle of pseudoexcitation method (stationary analysis).

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(see Figure 12.6a) and with a given PSD Sxx ðvÞ: Suppose that for two arbitrarily selected responses yðtÞ and zðtÞ; the auto-PSD Syy ðvÞ and cross-PSD Syz ðvÞ are desired. Figure 12.6b gives the conventional formulas for computing these PSD functions. Hy ðvÞ and Hz ðvÞ are the frequency-response functions, that is, if xðtÞ is replaced by a sinusoidal excitation expðivtÞ; the harmonic responses of yðtÞ and zðtÞ would be Hy ðvÞexpðivtÞ and Hz ðvÞexpðivtÞ; respectively. Thus, if xðtÞ is replaced by a sinusoidal excitation (Lin et al., 1994b) pﬃﬃﬃﬃﬃﬃﬃﬃ ð12:94Þ x~ ¼ Sxx ðvÞexpðivtÞ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ the responses of yðtÞ and zðtÞ would be y~ ¼ Sxx ðvÞHy ðvÞexpðivtÞ and z~ ¼ Sxx ðvÞHz ðvÞexpðivtÞ (see Figure 12.6c). It can be readily veriﬁed that pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ ð12:95Þ y~ p y~ ¼ Sxx ðvÞHyp ðvÞexpð2ivtÞ Sxx ðvÞHy ðvÞexpðivtÞ ¼ lHy ðvÞl2 Sxx ðvÞ ¼ Syy ðvÞ pﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃ y~ p z~ ¼ Sxx ðvÞHyp ðvÞexpð2ivtÞ Sxx ðvÞHz ðvÞexpðivtÞ ¼ Hyp ðvÞSxx ðvÞHz ðvÞ ¼ Syz ðvÞ

ð12:96Þ

If yðtÞ and zðtÞ are two arbitrarily selected random response vectors of the structure, and y~ ¼ ay expðivtÞ and z~ ¼ az expðivtÞ are the corresponding harmonic response vectors due to the pseudo-excitation (12.94), it can also be proven that the PSD matrices of yðtÞ and zðtÞ are Syy ðvÞ ¼ y~ p y~ T ¼ apy aTy

ð12:97Þ

Syz ðvÞ ¼ y~ p z~ T ¼ apy aTz

ð12:98Þ

This means that the auto- and cross-PSD functions of two arbitrarily selected random responses can be computed using the corresponding pseudoharmonic responses. Now, consider a structure subjected to a single seismic random excitation. Its equations of motion are M€y þ C_y þ Ky ¼ 2Me€xg ðtÞ

ð12:99Þ

in which the ground acceleration x€ g ðtÞ is a stationary random process. Its PSD Sx€ g ðvÞ is known and e is a given constant vector, indicating the distribution of inertia forces. In order to solve Equation 12.99, let the pseudoground acceleration be qﬃﬃﬃﬃﬃﬃﬃﬃ ð12:100Þ x€~ g ðtÞ ¼ Sx€ g ðvÞexpðivtÞ then Equation 12.99 becomes

qﬃﬃﬃﬃﬃﬃﬃﬃ My€~ þ Cy_~ þ K~y ¼ 2Me Sx€ g ðvÞexpðivtÞ

ð12:101Þ

y~ ðtÞ ¼ ay ðvÞexpðivtÞ

ð12:102Þ

and its stationary solution is Using its ﬁrst q normalized modes for mode-superposition, then (Clough and Penzien, 1993) ay ðvÞ ¼

q X j¼1

pﬃﬃﬃﬃﬃﬃﬃﬃ gj Hj wj Sxx ðvÞ

ð12:103Þ

in which vj ; wj ; 6j ; Hj and gj are the jth natural angular frequency, mass normalized mode, damping ratio, frequency-response function, and mode participation factor, respectively. According to PEM, the PSD matrix of y is Syy ðvÞ ¼ y~ p y~ T ¼ apy ðvÞaTy ðvÞ

ð12:104Þ

Substituting Equation 12.102 into Equation 12.104 and expanding it also gives Equation 12.31. This means these two equations are mathematically identical. However, the computational effort required by Equation 12.104 is approximately only 1=q2 of that required by Equation 12.31. Therefore, Equation 12.104 is also known as the fast CQC algorithm (Lin, 1992).

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Example 12.1 Derivation of the Kanai – Tajimi PSD Formula Consider the system shown in Figure 12.7(a). The single-layered homogeneous soil and the super singleDoF structure can be modeled by the system of Figure 12.7(b). Assume that the horizontal acceleration of the bedrock x€ 0 is a stationary random process with white noise spectrum S0 : The ground displacement relative to the bedrock is y, and the displacement of the superstructure (assumed to be an single-DoF system) relative to the ground is x. The effective horizontal shear-resistant stiffness of the soil layer is kg ; while the corresponding damping coefﬁcient is cg : Assuming that m is very small in comparison with the equivalent ground mass mg ; then the equation of motion of the effective ground mass is mg y€ þ cg y_ þ kg y ¼ 2mg x€ 0

ðiÞ

y€ þ 26g vg y_ þ v2g y ¼ 2€x0

ðiiÞ

or

in which v2g ¼ kg =mg ; 26g vg ¼ cg =mg : Now, form the horizontal pseudoacceleration for the bedrock pﬃﬃﬃ x€~ 0 ¼ S0 expðivtÞ

ðiiiÞ

Substituting it into Equation ii gives

pﬃﬃﬃ y€~ þ 26g vg y_~ þ v2g y~ ¼ 2 S0 expðivtÞ

ðivÞ

Its right-hand side is harmonic. Therefore, the stationary solution of y~ is pﬃﬃﬃ 2 S0 y~ ¼ 2 expðivtÞ vg 2 v2 þ 2i6g vg v

ðvÞ

The pseudoabsolute displacement of the ground is x~ g ¼ x~ 0 þ y~

ðviÞ

The corresponding pseudoabsolute acceleration is " # pﬃﬃﬃ pﬃﬃﬃ v2g þ 2i6g vg v v2 €x~ g ¼ x€~ 0 þ y€~ ¼ S0 expðivtÞ 1 þ ¼ S0 expðivtÞ 2 2 2 vg 2 v þ 2i6g vg v vg 2 v2 þ 2i6g vg v

ðviiÞ

Using PEM, the PSD of x€~ g is Sx€ g ðvÞ ¼ x€~ pg x€~ g ¼ S0

v4g þ 462g v2g v2 ðv2g 2 v2 Þ2 þ 462g v2g v2

ðviiiÞ

This is exactly the Kanai –Tajimi ﬁltered white noise spectrum formula (Clough and Penzien, 1993).

x

m k

c

wg (a) FIGURE 12.7

m k

y

x c

y

mg

Vg

kg cg ..

x0

(b)

..

x0

Structural modeling for Kanai – Tajimi PSD formula.

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12.3.1.2

12-19

Multiple Stationary Random Excitations

Consider a linear structure subjected to a number of stationary random excitations, which are denoted as an m-dimensional stationary random process vector xðtÞ with known PSD matrix Sxx ðvÞ: It is a Hermitian matrix and so it can be decomposed, for example, by using its eigenpairs cj and dj ð j ¼ 1; 2; …; rÞ; into Sxx ðvÞ ¼

r X j¼1

dj cpj cTj ðr # mÞ

ð12:105Þ

in which r is the rank of Sxx ðvÞ: Next, form the r pseudoharmonic excitations qﬃﬃﬃ x~ j ðtÞ ¼ dj cj expðivtÞ ð j ¼ 1; 2; …; rÞ

ð12:106Þ

By applying each of these pseudoharmonic excitations, two arbitrarily selected response vectors yj ðtÞ and zj ðtÞ of the structure, which can be displacements, internal forces, or other linear responses, may be easily obtained and expressed as y~ j ðtÞ ¼ ayj ðvÞexpðivtÞ

ð12:107Þ

z~ j ðtÞ ¼ azj ðvÞexpðivtÞ

ð12:108Þ

The corresponding PSD matrices can be computed by means of the following formulas (Lin et al., 1994a; Zhong, 2004): Syy ðvÞ ¼ Syz ðvÞ ¼

r X j¼1 r X j¼1

y~ pj ðtÞ~yTj ðtÞ ¼ y~ pj ðtÞ~zTj ðtÞ ¼

r X j¼1 r X j¼1

apyj ðvÞaTyj ðvÞ

ð12:109Þ

apyj ðvÞaTzj ðvÞ

ð12:110Þ

The method used to decompose Sxx ðvÞ into the form of Equation 12.105 is not unique. In fact, the Cholesky scheme is perhaps the most efﬁcient and convenient way to do it; that is, Sxx ðvÞ is decomposed into Sxx ðvÞ ¼ Lp DLT ¼

r X j¼1

dj lpj lTj

ðr # mÞ

ð12:111Þ

in which L is a lower triangular matrix with all its diagonal elements equal to unity and D is a real diagonal matrix with r nonzero diagonal elements dj : The implementation of Cholesky decomposition for a Hermitian matrix is very similar to that for a real symmetric matrix (Wilkinson and Reinsch, 1971).

Example 12.2

A Massless Trolley Subjected to Excitations with Phase-Lags

The massless trolley shown in Figure 12.8 is connected to two abutments by springs and viscous dashpots (linear viscous dampers) as shown. The abutments move distances x1 ðtÞ and x2 ðtÞ with spectral densities S0 (constant), but x2 ðtÞ ¼ x1 ðt 2 TÞ; in which T is a ﬁxed time difference. The response spectral density of the trolley displacement Syy ðvÞ is to be determined. The equation of motion of the trolley is (Newland, 1975) ðc1 þ c2 Þ_y þ ðk1 þ k2 Þy ¼ k1 x1 þ c1 x_ 1 þ k2 x2 þ c2 x_ 2

ðiÞ

Since the PSD of x1 ðtÞ is S0 ; the pseudoexcitation corresponding to x1 ðtÞ is pﬃﬃﬃ x~ 1 ðtÞ ¼ S0 expðivtÞ

ðiiÞ

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Vibration Monitoring, Testing, and Instrumentation

Because x2 ðtÞ ¼ x1 ðt 2 TÞ; we have pﬃﬃﬃ x~ 2 ðtÞ ¼ S0 exp½ivðt 2 TÞ Clearly

y(t)

ðiiiÞ

pﬃﬃﬃ x_~ 1 ðtÞ ¼ iv S0 expðivtÞ; pﬃﬃﬃ x_~ 2 ðtÞ ¼ iv S0 exp½ivðt 2 TÞ

x2(t)

x1(t)

ðivÞ

k1

k2

c1

c2

Substituting the above equations into Equation i gives the harmonic equation ðc1 þ c2 Þy_~ þ ðk1 þ k2 Þ~y ¼ ½ðk1 þ ivc1 Þ þ ðk2 þ ivc2 Þ expð2ivTÞ expðivtÞ

FIGURE 12.8

Two-phase input trolley.

ðvÞ

Its solution can be readily obtained as k þ ivc1 þ ðk2 þ ivc2 Þexpð2ivTÞ expðivtÞ y~ ¼ 1 k1 þ k2 þ ivðc1 þ c2 Þ

ðviÞ

Hence, Syy ¼ y~ p y~ ¼

k21 þ k22 þ c12 v2 þ c22 v2 þ 2ðk1 k2 þ c1 c2 v2 Þcos vT þ 2ðk1 c2 v 2 k2 c1 vÞsin vT S0 ðk1 þ k2 Þ2 þ ðc1 þ c2 Þ2 v2

ðviiÞ

This result is identical to that given by Newland (1975). However, the process given here is quite simple.

12.3.2

Structures Subjected to Nonstationary Random Vibration

12.3.2.1

Structures Subjected to Uniformly Modulated Evolutionary Random Excitations

12.3.2.1.1 Single Excitation Problems The basic principle of the PEM for nonstationary random vibration analyses can be described by Figure 12.9. Consider a linear system subjected to an evolutionary random excitation (see Figure 12.9a) f ðtÞ ¼ gðtÞxðtÞ

ð12:112Þ

in which gðtÞ is a slowly varying modulation function, while xðtÞ is a zero-mean stationary random process with auto-PSD Sxx ðvÞ: The deterministic functions gðtÞ and Sxx ðvÞ are both assumed to be given.

(a)

(b)

(c)

f(t) = g(t)x(t)

Linear Structure

∼ f(t) = Sxx(w)g(t)exp(iwt)

Linear Structure

Sff (w,t) = g2(t)Sxx(w)

Linear Structure

FIGURE 12.9

y(t) z(t) ~ y(w, t) ~z(w, t) Syy(w, t) = ~y* ~ yT Syz(w, t) = ~ y*~ zT

Basic principle of pseudoexcitation method (nonstationary analysis).

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The auto-PSD of f ðtÞ (see Figure 12.9b) is Sff ðv; tÞ ¼ g 2 ðtÞSxx ðvÞ

ð12:113Þ

In order to compute the PSD functions of various linear responses due to the action of f ðtÞ; we note that the pseudoexcitation has the form pﬃﬃﬃﬃﬃﬃﬃﬃ ð12:114Þ f~ðv; tÞ ¼ gðtÞ Sxx ðvÞexpðivtÞ Now suppose that yðtÞ and zðtÞ are two arbitrarily selected response vectors (see Figure 12.9a) and y~ ðv; tÞ ~ v; tÞ with the and z~ ðv; tÞ are the corresponding transient responses due to the pseudoexcitation fð structure initially at rest (see Figure 12.9b). It has been proven (Lin et al., 1994a; Lin et al., 2004) that the desired PSD matrices of yðtÞ and zðtÞ are Syy ðv; tÞ ¼ y~ p ðv; tÞ~yT ðv; tÞ

ð12:115Þ

Syz ðv; tÞ ¼ y~ p ðv; tÞ~zT ðv; tÞ

ð12:116Þ

as shown in Figure 12.9c. Now, consider the Equation 12.99 of a linear structure subjected to the evolutionary random excitation pﬃﬃﬃﬃﬃﬃﬃﬃ x€ g ðtÞ ¼ gðtÞxðtÞ: The pseudoground acceleration is now x€~ g ðtÞ ¼ gðtÞ Sxx ðvÞexpðivtÞ: Substituting this into the right-hand side of Equation 12.99 gives the deterministic equations pﬃﬃﬃﬃﬃﬃﬃﬃ My€~ þ Cy_~ þ K~y ¼ 2MegðtÞ Sxx ðvÞexpðivtÞ ð12:117Þ For seismic problems, the structure is initially at rest, that is, y~ ¼ y_~ ¼ 0 when t ¼ 0: Thus, the time history of y~ ðv; tÞ can be computed using the Newmark or Wilson-u schemes. Furthermore, any other linear pseudo response vectors, denoted as uð ~ v; tÞ and v~ ðv; tÞ; can be computed from y~ ðv; tÞ: The response PSD matrices of uðtÞ and vðtÞ can also be accurately computed by using their pseudoresponses uð ~ v; tÞ and v~ ðv; tÞ Suu ðv; tÞ ¼ u~ p ðv; tÞu~ T ðv; tÞ

ð12:118Þ

Suv ðv; tÞ ¼ u~ p ðv; tÞ~vT ðv; tÞ

ð12:119Þ

Example 12.3 Single-Degree of Freedom System Subjected to Suddenly Applied Stationary Random Excitation Consider the following single-DoF example, which was ﬁrst given by Caughey and Stumpf (1961) and has been widely used to compare the efﬁciency and precision of various methods: y€ þ 26v0 y_ þ v20 y ¼ f ðtÞ ¼ gðtÞxðtÞ; in which

( gðtÞ ¼

1:0

t$0

0

t,0

yð0Þ ¼ y_ ð0Þ ¼ 0

ðiÞ

ðiiÞ

and xðtÞ is a zero-mean-valued stationary random process with its PSD Sxx ðvÞ given. Now, constitute a pseudo excitation pﬃﬃﬃﬃﬃﬃﬃﬃ f~ ðtÞ ¼ Sxx ðvÞexpðivtÞ ðt $ 0Þ ðiiiÞ Thus, the motion equation that should be solved is pﬃﬃﬃﬃﬃﬃﬃﬃ y€~ þ 26v0 y_~ þ v20 y~ ¼ Sxx ðvÞexpðivtÞ;

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y~ð0Þ ¼ y_~ ð0Þ ¼ 0

ðivÞ

12-22

Vibration Monitoring, Testing, and Instrumentation

The solution of this simple equation is iv þ 1 sin v1 t v1

y~ ðv; tÞ ¼ H expðivtÞ 2 expð21tÞ cos v1 t þ

pﬃﬃﬃﬃﬃﬃﬃﬃ Sxx ðvÞ

ðvÞ

in which H ¼ ðv20 2 v2 þ 2i6v0 vÞ21 ;

1 ¼ 6v0 ;

pﬃﬃﬃﬃﬃﬃﬃﬃ v1 ¼ v0 1 2 62

ðviÞ

Therefore, the auto-PSD of y is Syy ðv; tÞ ¼ y~ p y~ ¼ l~yl2 ¼ lHl2

expð21tÞ

1 sin v1 t þ cos v1 t 2 cos vt v1

v þ expð21tÞ sin v1 t 2 sin vt v1

2

2

ðviiÞ

Sxx ðvÞ

This result is identical to the exact solution given by Caughey and Stumpf (1961), who used a more complicated process and expressed it less concisely as "

2v 0 2v 6 sin v1 t cos v1 t 2 expð1tÞ 2 cos v1 t þ 0 6 sin v1 t cos vt: v1 v1 # 2v ðv 6Þ2 2 v21 þ v2 2 sin v1 t sin vt þ 0 2expð1tÞ sin v1 t Sxx ðvÞ v1 v21 2

Syy ðv; tÞ ¼ lHl ½1 þ expð221tÞ 1 þ

ðviiiÞ 12.3.2.1.2 Multiple Excitation Problems Fully coherent excitations. In order to include the phase-lags between ground excitations, that is, the wave passage effect, the evolutionary random excitation vector fðtÞ to which the structure is subjected should be 9 9 8 8 a1 gðt 2 t1 ÞFðt 2 t1 Þ > > > F1 ðtÞ > > > > > > > > > > > > > > > > > > = < a2 gðt 2 t2 ÞFðt 2 t2 Þ > = > < F2 ðtÞ > ¼ GðtÞfðtÞ ð12:120Þ ¼ fðtÞ ¼ .. > .. > > > > > > . > > > > > . > > > > > > > > > > > ; : ; > : Fn ðtÞ an gðt 2 tn ÞFðt 2 tn Þ in which 2 6 6 6 6 GðtÞ ¼ 6 6 6 4

3

a1 gðt 2 t1 Þ

7 7 7 7 7; 7 7 5

a2 gðt 2 t2 Þ ..

. an gðt 2 tn Þ

9 8 > Fðt 2 t1 Þ > > > > > > > > > > = < Fðt 2 t2 Þ > fðtÞ ¼ .. > > > > . > > > > > > > > ; : Fðt 2 tn Þ

ð12:121Þ

Here, all components of fðtÞ clearly have the same form, although there are time lags tj ðj ¼ 1; 2; …; nÞ between them; gðtÞ is the modulation function; aj ðj ¼ 1; 2; …; nÞ are given real numbers; and tj are given constants. FðtÞ is a stationary random process, of which the auto-PSD SFF ðvÞ is known. The pseudoexcitations corresponding to {fðtÞ} are (Lin and Zhang, 2004) pﬃﬃﬃﬃﬃﬃﬃﬃ ~ ¼ GðtÞVq0 SFF ðvÞexpðivtÞ fðtÞ ð12:122Þ

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in which 2 R0 ¼

q0 qT0

1

6 61 6 ¼6 6 .. 6. 4 1

1

3

···

1

1 .. .

7 ··· 17 7 7 .. .. 7 . .7 5

1

···

1

ð12:123Þ

V ¼ diag½expð2ivt1 Þ; expð2ivt2 Þ; …; expð2ivtn Þ qT0 ¼ {1; 1; …; 1} The excitation PSD matrix can be written as Sff ðv;tÞ ¼ f~p ðtÞf~T ðtÞ ¼ SFF ðvÞGðtÞVp q0 qT0 VT GT ðtÞ ¼ SFF ðvÞ 2 g 2 ðt 2t1 Þ gðt 2t1 Þgðt 2t2 Þexpðivðt1 2t2 ÞÞ 6 6 6 gðt 2t2 Þgðt 2t1 Þexpðivðt2 2t1 ÞÞ g 2 ðt 2t2 Þ 6 6 6 6 .. .. 6 . . 6 4 gðt 2tn Þgðt 2t1 Þexpðivðtn 2t1 ÞÞ gðt 2tn Þgðt 2t2 Þexpðivðtn 2t2 ÞÞ

3 ·· · gðt 2t1 Þgðt 2tn Þexpðivðt1 2tn ÞÞ 7 7 ·· · gðt 2t2 Þgðt 2tn Þexpðivðt2 2tn ÞÞ7 7 7 7 7 .. .. 7 . . 7 5 ·· · g 2 ðt 2 tn Þ ð12:124Þ

~ can be computed using a numerical integration An arbitrarily chosen response y~ k ðv;tk Þ excited by fðtÞ scheme and the results can be expressed as pﬃﬃﬃﬃﬃﬃﬃﬃ ð12:125Þ y~ k ðv; tk Þ ¼ ayk ðv;tÞ SFF ðvÞ in which ayk ðv;tk Þ ¼

ðtk 0

Hðtk 2 tk ÞGðtk ÞVq0 expðivtk Þdtk

ð12:126Þ

Here, ayk ðv;tÞ is the response excited by GðtÞVq0 expðivtÞ and HðtÞ is the impulse-response function matrix. The cross-PSD matrix between yk ðtÞ and yl ðtÞ is Syk yl ðv;tÞ ¼ y~ pk ðv; tÞ~yTl ðv;tÞ

ð12:127Þ

By letting k ¼ l; the auto-PSD matrix of yk ðtÞ can be directly computed by Equation 12.127. Partially coherent excitations. In addition to the time lags between excitations, if the arbitrary coherence between such nonstationary excitations is also taken into account, then the problem is more difﬁcult. According to PEM, however, it is only required that the matrix R0 in Equation 12.123 is changed into 2

1

6 6r 6 21 6 R¼6 . 6 . 6 . 4 rn1

r12 1 .. .

rn2

···

r1n

3

7 r2n 7 7 7 .. 7 .. 7 . 7 . 5 ··· 1

···

ð12:128Þ

in which rij reﬂects the coherence between the excitations at points i and j, and matrix R is usually symmetric and positive deﬁnite or semipositive deﬁnite. Denoting its rank as rðr $ 1Þ; R can be

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expressed as R¼

r X j¼1

aj cpj cTj

ð12:129Þ

This means that the global excitation PSD matrix is decomposed into r matrices with rank unity where the jth of them corresponds to the pseudoexcitation qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ f~j ðtÞ ¼ GðtÞVcj aj SFF ðvÞexpðivtÞ ð12:130Þ If y~ kj ðtÞ and y~ lj ðtÞ are two arbitrary responses due to f~j ðtÞ; then Syk yl ðv; tÞ ¼

r X j¼1

y~ pkj ðtÞ~yTlj ðtÞ

ð12:131Þ

By letting k ¼ l; the auto-PSD matrix of y k ðtÞ can be directly computed by Equation 12.131. 12.3.2.2 Structures Subjected to Nonuniformly Modulated Evolutionary Random Excitations 12.3.2.2.1 Single Excitation Problems Consider a nonuniformly modulated evolutionary random excitation f ðtÞ (Priestly, 1967) ð1 f ðtÞ ¼ Aðv; tÞexpðivtÞdaðvÞ 21

ð12:132Þ

in which Aðv; tÞ is a given nonuniform modulation function and a satisﬁes the equation E ½dap ðv1 Þdaðv2 Þ ¼ Sxx ðvÞdðv2 2 v1 Þdv1 dv2

ð12:133Þ

Here, Sxx ðv1 Þ is the auto-PSD of the stationary random process xðtÞ: The Riemann – Stieltjes integration in Equation 12.132 causes difﬁculties in conventional computations. However, this problem can be conveniently solved using PEM as follows. First, constitute the following pseudoexcitation (Lin et al., 1997a, 1997b): pﬃﬃﬃﬃﬃﬃﬃﬃ f~ ðv; tÞ ¼ Aðv; tÞ Sxx ðvÞexpðivtÞ ð12:134Þ Second, replace gðtÞ in Equation 12.117 by Aðv; tÞ to yield

pﬃﬃﬃﬃﬃﬃﬃﬃ My€~ þ Cy_~ þ K~y ¼ 2MeAðv; tÞ Sxx ðvÞexpðivtÞ

ð12:135Þ

Third, with the structure initially at rest, compute the time history, y~ ðv; tÞ; for any arbitrary frequency v; and the required time histories uð ~ v; tÞ and v~ ðv; tÞ: Finally, the PSD matrices of uðtÞ and vðtÞ can be computed from Suu ðv; tÞ ¼ u~ p ðv; tÞu~ T ðv; tÞ

ð12:136Þ

Suv ðv; tÞ ¼ u~ p ðv; tÞ~vT ðv; tÞ

ð12:137Þ

Evidently, the PEM is nearly identical for uniformly or nonuniformly modulated evolutionary random excitations. The unique difference is the use of either Aðv; tÞ or gðtÞ in the pseudoexcitation expressions, see Equation 12.114 and Equation 12.134. 12.3.2.2.2 Multiple Excitation Problems For multiple nonstationary random excitation problems, the PEM-based analysis process for uniformly modulated evolutionary random excitations can be immediately extended to that for nonuniformly modulated evolutionary random excitations if the modulation function gðtÞ is replaced by Aðv; tÞ:

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All other formulas remain exactly the same. The difference in computational effort from using different modulation functions is negligible.

12.3.3

Precise Integration Method

12.3.3.1

Precise Integration of Exponential Matrices

Structural equations of motion, for example, Equation 12.99, can be written as M€y þ G_y þ Ky ¼ fðtÞ

ð12:138Þ

in which M, G, and K are given time-invariant n £ n matrices, respectively, and fðtÞ is the given external force vector. The initial displacement yðtÞ and the initial velocity y_ ðtÞ of the system are speciﬁed. The equation of motion, Equation 12.138, combined with the identity y_ ¼ y_ leads to the ﬁrst-order equations of motion in the state space being v_ ¼ Hv þ r in which

"

H¼

0 B

#

( ) ( ) 0 y ; B ¼ 2M21 K; ; r¼ ; v¼ M21 fðtÞ D y_ I

ð12:139Þ

D ¼ 2M21 G

ð12:140Þ

The homogeneous solution of Equation 12.139 is vh ðtÞ ¼ TðtÞc

ð12:141Þ

TðtÞ ¼ exp ðHtÞ

ð12:142Þ

in which Consider the current integration interval t [ ½tk ; tkþ1 ; t ¼ t 2 tk : When t ¼ 0 or t ¼ tk ; TðtÞ ¼ I and, therefore, c is a constant vector. If the particular solution to Equation 12.139, vp ðtÞ; is temporarily assumed to have been found, then the general solution of Equation 12.139 is vðtÞ ¼ TðtÞðvðtk Þ 2 vp ðtk ÞÞ þ vp ðtÞ

ð12:143Þ

In order to compute TðtÞ accurately, it is desirable to subdivide the step t into m ¼ 2N equal intervals, that is, Dt ¼ t=m ¼ 22N t

ð12:144Þ 26

For application purposes, the use of N ¼ 20 is sufﬁcient, because it leads to Dt < 10 t: Such a small Dt is, in general, much smaller than the highest natural period of any practical discretized system. Using the Taylor series expansion, we have expðH £ DtÞ < I þ Ta0

ð12:145Þ

Ta0 ¼ ðH £ DtÞ þ ðH £ DtÞ2 2! þ ðH £ DtÞ3 3! þ ðH £ DtÞ4 4!

ð12:146Þ

in which

Substituting Equation 12.145 into Equation 12.142 gives TðtÞ ¼ ðexpðH £ DtÞÞm ¼ ðI þ Ta0 Þm

ð12:147Þ

Note that I þ Tai ¼ ðI þ Ta;i21 Þ2 ¼ ðI þ 2 £ Ta;i21 þ Ta;i21 £ Ta;i21 Þ;

ði ¼ 1; 2; …; NÞ

ð12:148Þ

so that I þ TaN ¼ ðI þ Ta;N21 Þ2 ¼ ðI þ Ta;N22 Þ4 ¼ · · · ¼ ðI þ Ta0 Þm ¼ TðtÞ

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ð12:149Þ

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Vibration Monitoring, Testing, and Instrumentation

Equation 12.148 and Equation 12.149 suggest the following computing strategy. In order to avoid the loss of signiﬁcant digits in the matrix TðtÞ; it is necessary to compute Ta1 directly from Ta0 ; Ta2 directly from Ta1 ; and so on, by using Tai ¼ 2 £ Ta;i21 þ Ta;i21 £ Ta;i21 ; Then TðtÞ should be computed from

ði ¼ 1; 2; …; NÞ

TðtÞ < I þ TaN

ð12:150Þ ð12:151Þ

In Equation 12.151, the approximation is caused by the truncation of the Taylor expansion of Equation 12.146. It is generally negligibly small because when N ¼ 20; the ﬁrst term ignored by the truncation is of the order OðDt 5 Þ ¼ 10230 Oðt5 Þ; which is on the order of the round-off errors of a typical computer. 12.3.3.2 Particular Solutions and Precise Integration Formulas for Various Forms of Loading 12.3.3.2.1 Linear Loading Form (HPD-L Form) Assume that the loading varies linearly within the time step ðtk ; tkþ1 Þ; that is, r ¼ r0 þ r1 £ ðt 2 t0 Þ

ð12:152Þ

in which r0 and r1 are time-invariant vectors. The particular solution of Equation 12.139 is then (Lin et al., 1995a, 1995b; Zhong, 2004) vp ðtÞ ¼ ðH21 þ ItÞð2H21 r1 Þ 2 H21 ðr0 2 r1 tk Þ

ð12:153Þ

Substituting Equation 12.152 into Equation 12.143 gives the HPD-L (High Precision Direct integrationLinear) integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ þ H21 ðr0 þ H21 r1 ÞÞ 2 H21 ðr0 þ H21 r1 þ r1 tÞ

ð12:154Þ

12.3.3.2.2 Sinusoidal Loading Form (HPD-S Form) If the applied loading is sinusoidal within the time region t [ ðtk ; tkþ1 Þ; then rðtÞ ¼ r1 sin vt þ r2 cos vt

ð12:155Þ

in which r1 and r2 are time-invariant vectors. Substituting Equation 12.155 into Equation 12.139 enables the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) v p ðtÞ ¼ v1 sin vt þ v2 cos vt

ð12:156Þ

in which v1 ¼ ðvI þ H2 =vÞ21 ðr2 2 Hr1 =vÞ

v2 ¼ ðvI þ H2 =vÞ21 ð2r1 2 Hr2 =vÞ

ð12:157Þ

Substituting Equation 12.156 into Equation 12.143 gives the general solution of Equation 12.139, that is, the HPD-S direct integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ 2 v1 sin vtk 2 v2 cos vtk Þ þ v1 sin vtkþ1 þ v 2 cos vtkþ1

ð12:158Þ

The time interval t ¼ tkþ1 2 tk can cover an arbitrary segment, or even many periods, of a sinusoidal wave because, no matter how large the step size may be, exact responses will be obtained provided the matrix TðtÞ has been generated accurately, without any instability occurring. 12.3.3.2.3 Exponentially Decaying Sinusoidal Loading Form (HPD-E Form) Suppose that the applied loading varies according to the following exponentially decaying sinusoidal law within the time region t [ ðtk ; tkþ1 Þ: rðtÞ ¼ expðatÞðr1 sin vt þ r2 cos vtÞ

© 2007 by Taylor & Francis Group, LLC

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ð12:159Þ

Seismic Random Vibration of Long-Span Structures

12-27

in which r1 and r2 are time-invariant vectors. Substituting Equation 12.159 into Equation 12.139 enables the particular solution to be obtained (Lin et al., 1995a, 1995b; Zhong, 2004) as vp ðtÞ ¼ expðatÞðv1 sin vt þ v2 cos vtÞ

ð12:160Þ

in which v 1 ¼ ððaI 2 H2 Þ þ v2 IÞ21 ððaI 2 HÞr1 þ vr2 Þ v 2 ¼ ððaI 2 H2 Þ þ v2 IÞ21 ððaI 2 HÞr2 2 vr1 Þ

ð12:161Þ

Thus, substituting Equation 12.161 into Equation 12.143 gives the general solution of Equation 12.139, that is, the HPD-E direct integration formula vðtkþ1 Þ ¼ TðtÞðvðtk Þ 2 expðatk Þðv1 sin vtk þ v 2 cos vtk ÞÞ þ expðatkþ1 Þðv 1 sin vtkþ1 þ v2 cos vtkþ1 Þ ð12:162Þ The time interval is t ¼ tkþ1 2 tk :

12.4 Long-Span Structures Subjected to Stationary Random Ground Excitations 12.4.1 The Solution of Equations of Motion Using the Pseudoexcitation Method In Equation 12.93, the PSD matrix of u€ b ; that is, SðivÞ in Equation 12.55, is an N-dimensional Hermitian matrix, while R is an N-dimensional real symmetric matrix. Both matrices are usually positive deﬁnite or semipositive deﬁnite. If the rank of R is rðr # NÞ; then by using Equation 12.111 it can be readily decomposed into the product of an N £ r matrix Q and its transposition; that is R ¼ QQT

ð12:163Þ

SðivÞ ¼ Bp JQQT JB ¼ Pp PT

ð12:164Þ

P ¼ BJQ

ð12:165Þ

Thus, Equation 12.55 can be written as in which To solve Equation 12.93, the right-hand side u€ b can be replaced by the pseudoground acceleration €~ ¼ P expðivtÞ U b

ð12:166Þ

Thus, Equation 12.93 becomes the following sinusoidal equations of motion: Ms Y€~ r þ Cs Y_~ r þ Ks Y~ r ¼ Ms K21 s Ksb EmN P expðivtÞ

ð12:167Þ

The stable solution of Equation 12.167 is the pseudorelative displacement vector Y~ r ; whilst the pseudostatic displacement vector Y~ s can be computed by 1 K E P expðivtÞ Y~ s ¼ 2 K21 v s sb mN

ð12:168Þ

Thus, the pseudoabsolute displacement vector is ~ s ¼ Y~ r þ Y~ s X

ð12:169Þ

~ s can be computed from X ~ s by means of a quasiIf necessary, any arbitrary pseudointernal force vector N static analysis. Then, the corresponding PSD matrix is ~ ps N ~ Ts SNs Ns ðvÞ ¼ N

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ð12:170Þ

12-28

Vibration Monitoring, Testing, and Instrumentation

If it is assumed that SX€ 1 ¼ SX€ 2 ¼ · · · ¼ SX€ N ; denoted as Sa ; then Equation 12.165 becomes pﬃﬃﬃ P ¼ Sa BQ

ð12:171Þ

If only the wave passage effect is considered, that is, all lrij l ¼ 1 in Equation 12.58, then matrix Q will reduce to a vector q0 with all its elements unity; that is Q ¼ q0 ¼ {1; 1; …; 1}T

ð12:172Þ

pﬃﬃﬃ P ¼ Sa e 0

ð12:173Þ

e0 ¼ {expð2ivt1 Þ; expð2ivt2 Þ; …; expð2ivtN Þ}T

ð12:174Þ

Thus, Equation 12.171 reduces to in which e0 is a complex vector

Therefore, when only the wave passage effect is considered, Equation 12.167 reduces to pﬃﬃﬃ Ms y€~ r þ Cs y_~ r þ Ks y~ r ¼ Ms K21 s Ksb EmN e0 Sa expðivtÞ

ð12:175Þ

Furthermore, if the structure is subjected to a uniform ground motion, then the vector e0 in Equation 12.174 should be replaced by q0 ; and so Equation 12.175 can be further reduced to pﬃﬃﬃ Ms y€~ r þ Cs y_~ r þ Ks y~ r ¼ Ms K21 ð12:176Þ s Ksb EmN q0 Sa expðivtÞ

12.4.2

Numerical Comparisons with Other Methods

12.4.2.1

Song-Hua-Jiang Suspension Bridge

The Song-Hua-Jiang suspension bridge (see Figure 12.10) is located in Jilin Province of China. Its overall length is 450 m, with a main span of 240 m and a width of 28 m. The ﬁnite element model had 2076 DoF, 445 nodes (including 12 supports) and 574 elements. The static equilibrium position of the bridge included the effects of the initial tensions of the cables. The earthquake action was determined based on the Chinese National Standard (Code for Seismic Design of Buildings GB 50011-2001), which FIGURE 12.10 Song-Hua-Jiang suspension bridge. directly gives the ground ARS curve for the bridge. The PSD curve was obtained in terms of the Kaul method from which the samples of the ground acceleration time-history can be produced (Kaul, 1978). For the analyses associated with the SH and SV waves, 100 modes were used for mode-superposition, whereas for the P waves, only 30 modes were used. The apparent wave speeds used were 3 km/s for P waves and 2 km/s for SH or SV waves. 12.4.2.1.1 All Supports Move Uniformly Figure 12.11(a) gives the axial force distribution of this bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move uniformly. The following four computational models were used: 1. 2. 3. 4.

Response spectrum method Pseudoexcitation method Time-history method using three samples Time-history method using ten samples

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Seismic Random Vibration of Long-Span Structures

RSM

kN

PEM

12-29

THM (3 samples)

THM (10 samples)

4.0E+2

3.0E+2

2.0E+2

1.0E+2

0.0E+0

(a)

0

50

100 RSM

kN

150 PEM

200

250

300

THM (3 samples)

350

400

m

THM (10 samples)

4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0

(b)

0

50

100 RSM

kN

150 PEM

200

250

300

THM (3 samples)

350

400

m

THM (10 samples)

5.0E+2

4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

(c)

0

50

100

150

200

250

300

350

400

m

FIGURE 12.11 Deck force distribution of Song-Hua-Jiang bridge due to uniform ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

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Vibration Monitoring, Testing, and Instrumentation

TABLE 12.1

Central Processing Unit (CPU) Times Required by Different Methods for Stationary Analysis Method Used

RSM (sec)

PEM (sec)

THM (for One Sample) (sec)

Uniform ground motion Wave passage effect Wave passage effect and incoherence effect

80.6

15.1 24.2 209.1

29.9 36.7

Note: the CPU time for mode extraction is not included; extracting 100 modes needs 180.9 sec.

Figure 12.11(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck. All supports move uniformly. The above four computational models were used. Figure 12.11(c) gives the vertical shear force distribution along the deck due to the seismic SV waves traveling along the deck. All supports move uniformly. The above four computational models were also used here. All computations were executed on a P3-750 personal computer. The computation times for different methods are listed in Table 12.1. Figure 12.11 and Table 12.1 show that, when ground motion is assumed uniform, that is, the earthquake spatial effects are not taken into account, the RSM, PEM, and THM (using ten samples) give very close results if the excitations are properly produced. The RSM is the most popular method, but the newly developed PEM may be the more efﬁcient one. The THM needs to be executed for a number of ground acceleration samples and so was inefﬁcient.

12.4.2.1.2 Wave Passage Effect Is Taken into Account Figure 12.12(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridges are assumed to move with certain time lags; that is, the wave passage effect is taken into account. The apparent P wave speed is 3 km/sec. The following four computational models were used: 1. 2. 3. 4.

RSM (uniform ground motion is assumed for comparison only) PEM (wave passage effect is considered) THM (wave passage effect is considered using three ground-acceleration samples) THM (wave passage effect is considered using ten ground-acceleration samples)

Figure 12.12(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck and Figure 12.12(c) gives the corresponding vertical shear-force distribution. The above four computational models were used. Figure 12.12 and Table 12.1 show that when the seismic wave-passage effect is taken into account, that is, the earthquake spatial effects are partly taken into account, the PEM and THM (using ten samples) give very close results. The RSM, which does not consider the wave-passage effect, may give quite different results; these may appear larger or smaller than, or very close to, the results by the more reasonable PEM or THM analyses. Therefore, such computations are necessary for evaluating the seismic spatial effects of long-span structures. The PEM gives the most reliable results with the least computational effort and, therefore, this method is strongly recommended.

12.4.2.1.3 Wave-Passage Effect and Incoherence Effect Are Jointly Taken into Account Figure 12.13(a) gives the axial force distribution of the bridge along the deck due to the seismic P waves, which travel along the longitudinal direction of the deck. All supports of the bridge are assumed to move with certain time lags; that is, the wave passage effect is taken into account. In addition, the incoherence effects are also taken into account. Two coherence models, that is, the Loh –Yeh model

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12-31

FIGURE 12.12 Deck force distribution of Song-Hua-Jiang bridge due to wave passage effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

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Vibration Monitoring, Testing, and Instrumentation

RSM (uniform) PEM (Loh model)

kN

PEM (uniform) PEM (QWW model)

PEM (v=3km/s)

5.0E+2 4.0E+2

3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a)

0

50

100

150

RSM (uniform) PEM (Loh model)

kN

200

250

300

PEM (uniform) PEM (QWW model)

350

400

m

PEM (v=2km/s)

4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0 (b)

0

50

100

150

RSM (uniform) PEM (Loh model)

kN

200

250

300

PEM (uniform) PEM (QWW model)

350

400

m

PEM (v=2km/s)

5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

0

50

100

150

200

250

300

350

400

m

(c)

FIGURE 12.13 Deck force distribution of Song-Hua-Jiang bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

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Seismic Random Vibration of Long-Span Structures

12-33

and the QWW model were used for evaluating such effects. The following ﬁve computational models were used: 1. 2. 3. 4. 5.

RSM (uniform ground motion is assumed) PEM (uniform ground motion is assumed) PEM (wave passage effect is considered) PEM (wave passage effect is considered also using the Loh –Yeh coherence model) PEM (wave passage effect is considered also using the QWW coherence model)

Figure 12.13(b) gives the transverse shear force distribution along the deck due to the seismic SH waves traveling along the deck, and Figure 12.13(c) gives the corresponding vertical shear force distribution due to the seismic SV waves. For both analyses the above ﬁve computational models were used. The above computations as well were executed on a P3-750 personal computer, and the computation times required by different methods are listed in Table 12.1. From Figure 12.13 and Table 12.1, we can conclude that: 1. When ground motion is assumed to be uniform, the RSM and PEM give very close internal force responses (i.e., demands), with the PEM being more efﬁcient. 2. The wave passage effect is an important factor that affects the seismic responses of long-span structures. To execute such seismic analyses, the PEM is not only theoretically quite reasonable, but also very efﬁcient. 3. The incoherence effect appears to diverge when using different coherence models. Herein, the inﬂuence caused by the QWW model is more evident than that caused by the Loh –Yeh model. However, compared with the wave passage effect, the inﬂuence of the incoherence effect is of less importance. 12.4.2.2

San Joaquin Concrete Bridge

San Joaquin Bridge, located in California (see Figure 12.14), is a reinforced concrete bridge built in 2001. Its length is 36 þ 50 þ 50 þ 50 þ 36 ¼ 222 m, its width is 12 m, and the height of all piers is 16.76 m. The ﬁnite element model had 367 nodes (including 10 ground z y nodes) and 366 elements. Its basic natural period x is 0.811s. Twenty modes were used in the modesuperposition analysis with all damping ratios being 0.05. The seismic analysis was carried out using the RSM and PEM, respectively. The RSM 36 m 50 m 50 m 50 m 36 m analysis was conducted according to the CALz TRANS Code (1999) with ARS ¼ 0.2 g, Type D x soil proﬁle and magnitude Mw ¼ 7.0. The equivalent ground-acceleration power spectral density curve was produced by means of the FIGURE 12.14 Structural model of San Joaquin Kaul method. All seismic waves were assumed to bridge. travel along the longitudinal direction of the bridge. The apparent P and S wave speeds were 3000 and 2000 m/s, respectively. The internal forces in the deck (i.e., the axial forces due to P waves, the transverse shear forces due to SH waves, and the vertical shear forces due to SV waves) were all computed using the following computational models: 1. RSM (uniform ground motion is assumed) 2. PEM (uniform ground motion is assumed)

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Vibration Monitoring, Testing, and Instrumentation

3. PEM (wave-passage effect is considered) 4. PEM (wave-passage effect is considered also using the Loh –Yeh coherence model) 5. PEM (wave-passage effect is considered also using the QWW coherence model) The computational results are shown in Figure 12.15(a) –(c). This bridge is not very long. However, similar phenomena to those found for the bridge of Example 12.1 are still found. Clearly, when the ground motion is assumed to be uniform, the RSM and PEM still give very close results. If the wavepassage effect is taken into account, then the internal force distribution with the PEM will change considerably, particularly at the midpoint of the deck. It is known that, for symmetric bridges, the antisymmetric modes will not participate in the symmetric motions under the assumption of uniform ground motion. However, when the wave-passage effect is taken into account, this conclusion does not hold. It is obvious that, even for this shorter bridge, the wave-passage effect seems to be quite signiﬁcant. The incoherence effect is comparatively not so important.

12.5 Long-Span Structures Subjected to Nonstationary Random Ground Excitations A typical strong motion earthquake record consists of three stages. In the ﬁrst stage, the intensity of the ground motion increases, which mainly reﬂects the motion of P waves. The intensity of the ground motion remains the strongest in the second stage, which mainly reﬂects the motion of S waves. The ground motion will die down in the last stage. Such a complete seismic motion is usually regarded as a nonstationary random process. If the nonstationary property is assumed to takes place only for the intensity of the motion, then this random process is regarded as a uniformly modulated evolutionary random process. However, if the shape of the ground motion PSD curve also varies with time (in other words, the intensity and the distribution with frequency of the ground motion energy both depend on time), then the ground motion is regarded as a nonuniformly modulated evolutionary random process. It is usually accepted that when the intensity of the seismic motion in the second stage appears quite stationary while the time interval of this stage is much longer (e.g., three times or over) than the basic period of the structure under consideration, a simpliﬁed, stationary-based random analysis may be acceptable as a substitute of the nonstationary analysis. In fact, the basic periods of many long-span bridges range from 10 to 20 sec, and the stationary portion of a typical strong earthquake is usually less than 1 min, being only 20 to 30 sec in most cases. Therefore, nonstationary analyses are appropriate for such problems. Previously, such nonstationary random analyses have been considered very difﬁcult. However, by using the recently developed PEM, combined with the precise integration method, such analyses have become relatively easy.

12.5.1

Modulation Functions

Some popular uniform modulation functions are listed below: 8 I ðt=t Þ2 0 # t # t1 > > < 0 1 gðtÞ ¼ I0 t1 # t # t2 > > : I0 exp{cðt 2 t2 Þ} t $ t2 ( gðtÞ ¼

1

t$0

0

t,0

gðtÞ ¼ a½expð2a1 tÞ 2 expð2a2 tÞ ; gðtÞ ¼ sin bt

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53191—5/2/2007—12:35—VELU—246505— CRC – pp. 1–41

ð12:177Þ

ð12:178Þ 0 # a1 , a2

ð12:179Þ ð12:180Þ

Seismic Random Vibration of Long-Span Structures

12-35

3

2

2

FIGURE 12.15 Deck force distribution of San Joaquin bridge due to incoherence effect: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

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Vibration Monitoring, Testing, and Instrumentation

Nonuniform modulation models have rarely been investigated. Lin et al. (1997a, 1997b) suggested the following nonuniform modulation model: Aðv; tÞ ¼ bðv; tÞgðtÞ ¼ exp 2h0

vt gðtÞ v a ta

ð12:181Þ

in which gðtÞ is an amplitude modulation function; bðv; tÞ is a frequency modulation function; and va and ta are the reference frequency and time, which are introduced to transform v and t into dimensionless parameters. In principle, va and ta can be arbitrarily selected. Once they have been selected, the factor h0 ðh0 . 0Þ can be adjusted accordingly to make the high-frequency components of the nonstationary random process decay more quickly than the low-frequency components and, thus, simulate the seismic motion more accurately. When h0 ¼ 0; that is, bðv; tÞ ¼ 1; Aðv; tÞ reduces to the uniform modulation function gðtÞ:

12.5.2

The Formulas for Nonstationary Multiexcitation Analysis

For nonuniformly modulated multiexcitation problems, the pseudoexcitation for the corresponding stationary problems, that is, Equation 12.166, is extended to €~ ðv; tÞ ¼ Aðv; tÞPexpðivtÞ U b

ð12:182Þ

in which the kth diagonal element of the N £ N diagonal matrix Aðv; tÞ is the modulation function Ak ðv; tÞ of the excitation which is applied to the kth support of the structure. In the case of uniformly modulated excitations, it is only necessary to replace all the nonuniform modulation functions Ak ðv; tÞ by the uniform modulation functions gk ðtÞ: Other formulae remain entirely unchanged. The N £ r matrix P can be generated by means of Equation 12.165 to Equation 12.174. Each column of €~ ðv; tÞ can be regarded as a deterministic acceleration excitation vector. By substituting it into the U b right-hand side of Equation 12.93 and solving the equations of motion, a column of the matrix Y~ r ðv; tÞ can be produced. Because Aj ðv; tÞ is a time-dependent and slowly varying function, the pseudoground displacement matrix can be computed approximately from ~ b ðv; tÞ ¼ 2 1 U ~€ ðv; tÞ U v2 b

ð12:183Þ

The pseudoquasi-static displacement matrix Y~ s ðv; tÞ can then be computed from Equation 12.92. Then, the PSD matrix of the absolute displacement vector Xs ðv; tÞ is SXs Xs ðv; tÞ ¼ ðY~ r ðv; tÞ þ Y~ s ðv; tÞÞp ðY~ r ðv; tÞ þ Y~ s ðv; tÞÞT

ð12:184Þ

~ e ; has been computed, then the PSD matrix of the If a group of pseudointernal forces, denoted as N corresponding internal forces Ne can be computed from ~ pe N ~ Te SNe Ne ðv; tÞ ¼ N

ð12:185Þ

When the ground acceleration PSD matrix is known, the corresponding pseudoacceleration vector u€~ b is easy to generate according to Equation 12.163 to Equation 12.166. If instead, the ground displacement PSD matrix or velocity PSD matrix is known, then the acceleration PSD matrix can be obtained by multiplying the displacement or velocity PSD matrices by v4 or v2 ; respectively.

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Seismic Random Vibration of Long-Span Structures

12.5.3

12-37

Expected Extreme Values of Nonstationary Random Processes

The evaluation of the peak amplitude responses of structures subjected to nonstationary seismic excitations has also received much attention (Shrikhande and Gupta, 1997; Zhao and Liu, 2001). Previously, only very simple structures could be computed. However, by using the PEM, complicated structures can be analyzed, as is brieﬂy described below. To evaluate the expected extreme value responses of a structure subjected to nonstationary Gaussian excitations, the duration of which the intensity of the excitation peaks exceeds 50% of the maximum peak intensity denoted by ½t0 ; t0 þ t is taken as the equivalent stationary duration in order to use Equation 12.40 to Equation 12.49 to evaluate the desired expected extreme values. Provided that the timedependent PSD of any arbitrary response yðtÞ; that is Syy ðv; tÞ, has been computed over that equivalent duration using the PEM, then the equivalent stationary PSD over that duration is

S0yy ðvÞ ¼

1 ðt0 þt Syy ðv; tÞdt t t0

ð12:186Þ

To compute the extreme value responses based on Equation 12.177, the parameters t0 and t are chosen as t0 ¼ t1

pﬃﬃ 2;

pﬃﬃ t ¼ t2 þ ln 2=c 2 t1 = 2

ð12:187Þ

Thus, the equivalent stationary random responses are obtained and the subsequent processing can still use Equation 12.40 to Equation 12.49.

12.5.4

Numerical Comparisons with the Corresponding Stationary Analysis

The example of the Song-Hua-Jiang suspension bridge of the last section is used here for the seismic nonstationary random vibration analysis. The results are compared with those from the corresponding stationary random-vibration analyses with the ground assumed to move uniformly (i.e., at an apparent wave speed vapp ¼ 1), or to move at a limited apparent wave speed vapp (with the wave-passage effect is taken into account), which is 3 km/sec for P waves and 2 km/sec for S waves. The nonstationary random excitation model zðtÞ ¼ gðtÞxðtÞ was used in which the auto-PSD of xðtÞ is assumed to be identical to that used for the stationary excitation in the preceding section. The frequencydomain parameters also remained the same. The modulation function had the form of Equation 12.177 with t1 ¼ 8:0; t2 ¼ 20:0; and c ¼ 0:20: The duration of the earthquake was t [ ½0; 25 ; and the time stepsize was Dt ¼ 0:5: The nonstationary analysis results are shown in Figure 12.16(a) to (c), and are compared with the results of the corresponding stationary random vibration analyses. Clearly, for such a long-span bridge, the wave passage effect is quite signiﬁcant in its seismic analysis, as seen in Figure 12.11 to Figure 12.13. In addition, whether for uniform ground motion or for differential ground motion (i.e., the wave-passage effect is considered), the nonstationary responses are always smaller than the corresponding stationary responses. The maximum difference between their corresponding peak values may reach up to 23.1% for the present problem, as shown in Table 12.2. For very slender bridges, this nonstationary property will be even stronger. By means of the PEM combined with the precise integration method (its HPD-E form for the modulation function used in this example), such modiﬁcation can be fulﬁlled quickly and conveniently. The computational effort required by the nonstationary analysis is only about 25 min (see Table 12.3).

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Vibration Monitoring, Testing, and Instrumentation kN

Uniform-Nonstationary v=3km/s-Nonstatinary

Uniform-Stationary v=3km/s-Stationary

5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0 (a)

0

50

kN

100

150

200

250

300

Uniform-Nonstationary v=2km/s-Nonstatinary

350

m

400

Uniform-Stationary v=2km/s-Stationary

4.0E+3

3.0E+3

2.0E+3

1.0E+3

0.0E+0 (b)

0

50

kN

100

150

200

250

300

Uniform-Nonstationary v=2km/s-Nonstatinary

350

400

m

Uniform-Stationary v=2km/s-Stationary

5.0E+2 4.0E+2 3.0E+2 2.0E+2 1.0E+2 0.0E+0

0

50

100

150

200

250

300

350

400

m

(c)

FIGURE 12.16 Deck-force distribution of Song-Hua-Jiang bridge due to uniform and differential, and stationary and nonstationary random ground motion: (a) axial force distribution along the deck under P waves; (b) transverse shear force distribution along the deck under SH waves; (c) vertical shear force distribution along the deck under SV waves.

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Seismic Random Vibration of Long-Span Structures TABLE 12.2

12-39

Comparisons of the Expected Extreme Values of Deck Internal Forces

Ground Motion

Internal Forces in the Deck (kN)

Uniform ground motion

Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves

Wave-passage effect

Axial force N due to P waves Transverse shear force Fy due to SH waves Vertical shear force Fz due to SV waves

Distance from the Left End (m)

Nonstationary Responses

Stationary Responses

Increases (%)

6 72 258

274.95 712.19 146.00

337.60 832.47 165.81

22.8 16.9 13.6

420 270 246

264.51 1292.17 134.00

325.61 1515.75 152.89

23.1 17.3 14.1

TABLE 12.3 CPU Times Required by PEM for Nonstationary Analyses (Units: Seconds)

12.6

Modes Used

30

100

Uniform ground motion Wave passage effect Extracting modes

450 543 143

1142 1479 181

Conclusions

For short-span structures, the ground spatial effects are negligible, and the seismic analyses using RSM, PEM, or THM (with a sufﬁcient number of samples) are relatively close to one another provided that the ground accelerations have been produced properly, and so are almost equivalent. Although they have almost the same accuracy level, their efﬁciencies are quite different. Of the three methods, if the structural models are rather complex (e.g., the FEM models have thousands or more DoF and need dozens or hundreds of modes for mode superposition), then the PEM will have the highest computational efﬁciency. For long-span structures, the wave passage effect is an important factor for structural seismic responses. The inﬂuence may produce more conservative, or more dangerous, designs, which is difﬁcult to predict by intuitive experience. Thus, computer-based analysis is a preferable choice. The PEM is comparatively efﬁcient and accurate, and is recommended. If the apparent seismic wave speed is not available, then a few possible speeds can be taken for computation, with the most unfavorable results being used in the practical design. The reasonable selection of an incoherence model for a special region needs further study, but its inﬂuence seems to be much less than that caused by the wave passage effect. Therefore, for less important long-span structures, the incoherence effect can be neglected. The inﬂuence of nonstationarity is also signiﬁcant for very ﬂexible long-span structures, and should be taken into account. The PEM, combined with the precise integration method, provides a powerful tool for such nonstationary analysis.

References Berrah, M. and Kausel, E., Response spectrum analysis of structures subjected to spatially varying motions, Earthquake Eng. Struct. Dyn., 21, 461–470, 1992. Caughey, T.K. and Stumpf, H.J., Transient response of a dynamic system under random excitation, Trans. ASME, JAM, 28, 563–566, 1961. Clough, R.W. and Penzien, J. 1993. Dynamics of Structures, McGraw-Hill, New York. Davenport, A.G., Note on the distribution of the largest value of a random function with application to gust loading, Proc. Inst. Civil Eng., 28, 187–196, 1961.

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12-40

Vibration Monitoring, Testing, and Instrumentation

Deodatis, G., Simulation of ergotic multivariate stochastic processes, J. Eng. Mech., 122, 778 –787, 1996. Dumanoglu, A. and Severn, R.T., Stochastic response of suspension bridges to earthquake forces, Earthquake Eng. Struct. Dyn., 19, 133–152, 1990. Ernesto, H.Z. and Vanmarcke, E.H., Seismic random vibration analysis of multi-support structural systems, J. Eng. Mech., ASCE, 120, 1107 –1128, 1994. European Committee for Standardization, 1995. Eurocode 8, Structures in Seismic Regions — Design Part 2: Bridge, Brussels, Belgium. Fan, L.C., Wang, J.J., and Chen, W., Response characteristics of long-span cable-stayed bridges under non-uniform seismic action, Chin. J. Comput. Mech., 18, 358 –363, 2001 (in Chinese). Feng, Q.M. and Hu, Y.X., Mathematical model of spatial correlative ground motion, Earthquake Eng. Eng. Vib., 1, 1 –8, 1981 (in Chinese). Gupta, I.D. and Trifunac, M.D., A note on statistic of level crossing and peak amplitude in stationary stochastic process, Eur. Earthquake Eng., 12, 52– 58, 1998. Harichandran, R.S. and Vanmarcke, E.H., Stochastic variation of earthquake ground motion in space and time, J. Eng. Mech., ASCE, 112, 154 –175, 1986. Housner, G.W., Characteristic of strong motion earthquakes, Bull. Seism. Soc. Am., 37, 17 –31, 1947. Kaul, M.K., Stochastic characterization of earthquake through their response spectrum, Earthquake Eng. Struct. Dyn., 6, 497 –510, 1978. Kiureghian, A.D., Structural response to stationary random excitation, J. Eng. Mech., Div. ASCE, 106, 1195 –1213, 1980. Kiureghian, A.D. and Neuenhofer, A., Response spectrum method for multi-support seismic excitations, Earthquake Eng. Struct. Dyn., 21, 713–740, 1992. Lee, M. and Penzien, J., Stochastic analysis of structures and piping systems subjected to stationary multiple support excitations, Earthquake Eng. Struct. Dyn., 11, 91– 110, 1983. Lin, Y.K. 1967. Probabilistic Theory of Structural Dynamics, McGraw-Hill, New York. Lin, J.H., A fast CQC algorithm of PSD matrices for random seismic responses, Comput. Struct., 44, 683 –687, 1992. Lin, J.H., Li, J.J., Zhang, W.S., and Williams, F.W., Non-stationary random seismic responses of multisupport structures in evolutionary inhomogeneous random ﬁelds, Earthquake Eng. Struct. Dyn., 26, 135 –145, 1997a. Lin, J.H., Shen, W.P., and Williams, F.W., A high precision direct integration scheme for non-stationary random seismic responses of non-classically damped structures, Struct. Eng. Mech., 3, 215–228, 1995a. Lin, J.H., Shen, W.P., and Williams, F.W., A high precision direct integration scheme for structures subjected to transient dynamic loading, Comput. Struct., 56, 113 –120, 1995b. Lin, J.H., Sun, D.K., Sun, Y., and Williams, F.W., Structural responses to non-uniformly modulated evolutionary random seismic excitations, Commun. Numer. Meth. Eng., 13, 605– 616, 1997b. Lin, J.H., Zhang, Y.H. 2004. Pseudo Excitation Method of Random Vibration, Science Press, Beijing (in Chinese). Lin, J.H., Zhang, W.S., and Li, J.J., Structural responses to arbitrarily coherent stationary random excitations, Comput. Struct., 50, 629–633, 1994a. Lin, J.H., Zhang, W.S., and Williams, F.W., Pseudo-excitation algorithm for non-stationary random seismic responses, Eng. Struct., 16, 270– 276, 1994b. Lin, Y.K., Zhang, R., and Yong, Y., Multiply supported pipeline under seismic wave excitations, J. Eng. Mech., 116, 1094–1108, 1990. Liu, T.Y. and Liu, G.T., Seismic random response analysis of Arch Dams, Eng. Mech., 17, 20–25, 2000, in Chinese. Loh, C.H. and Yeh, Y.T., Spatial variation and stochastic modeling of seismic differential ground movement, Earthquake Eng. Struct. Dyn., 16, 583 –596, 1988. Luco, J.E. and Wong, H.L., Response of a rigid foundation to a spatially random ground motion, Earthquake Eng. Struct. Dyn., 14, 891–908, 1986.

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Seismic Random Vibration of Long-Span Structures

12-41

National Standard of The People’s Republic of China, 2001. Code for Seismic Design of Buildings GB50011-2001, Chinese Architectural Industry Press, Beijing. Newland, D.E. 1975. An Introduction to Random Vibration and Spectral Analysis, Longmans, London. Oliveira, C.S., Hao, H., and Penjien, J., Ground motion modeling for multiple-input structural analysis, Struct. Saf., 10, 79 –93, 1991. Priestly, M.B., Power spectral of non-stationary random responses, J. Sound Vib., 6, 86 –97, 1967. Qu, T.J., Wang, J.J., and Wang, Q.X., A practical power spectrum model for spatially varying seismic motion, J. Seismol., 18, 55–62, 1996. Shrikhande, M. and Gupta, V.K., A generalized approach for the seismic response of structural systems, Eur. Earthquake Eng., 11, 3 –12, 1997. Sun, J.J. and Jiang, J.R., Parameters of Kanai-Tajimi power spectrum according to the response spectrum of Chinese code, Chin. J. World Earthquake Eng., 8, 42 –48, 1990 (in Chinese). Vanmarcke, E.H., Properties of spectral moments with applications to random vibration, J. Eng. Mech., Div. Proc. ASCE, 98, 425 –446, 1972. Wang, G.W., Cheng, G.D., Shao, Z.M., Chen, H.Q. 1999. Optimal Fortiﬁcation Intensity and Reliability of Aseismic Structures, Part 4, Science Press, Beijing, in Chinese. Wilkinson, J.H., Reinsch, G. 1971. Linear Algebra, Oxford University Press, London. Wilson, E.L. and Kiureghian, A.D., A replacement for the SRSS method in seismic analysis, Earthquake Eng. Struct. Dyn., 9, 187 –192, 1981. Xue, S.D., Li, M.H., Cao, Z., and Zhang, Y.G. 2000. Random vibration analysis of lattice shells subjected to multi-dimensional earthquake inputs. In Proceedings of the International Conference on Advances in Structural Dynamics, December 2000, pp. 777–784. Elsevier, Hong Kong. Zhao, F.X. and Liu, A.W., Relationship between power-spectral-density functions and response spectra of earthquake ground motions, Earthquake Eng. Eng. Vib., 21, 30 –35, 2001 (in Chinese). Zhong, W.X. 2004. The Dual System in Applied Mechanics and Optimal Control, Kluwer Academic Publishers, Boston. Zhong, W.X. and Williams, F.W., A precise time step integration method, Proc. Inst. Mech. Engrs, 208C, 6, 427–430, 1995.

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13

Seismic Qualiﬁcation of Equipment 13.1 13.2 13.3

Clarence W. de Silva The University of British Columbia

Introduction ......................................................................... 13-1 Distribution Qualiﬁcation ................................................... 13-1 Drive-Signal Generation Procedures

†

Distribution Spectra

†

Test

Seismic Qualiﬁcation ........................................................... 13-6 Stages of Seismic Qualiﬁcation † Test Preliminaries Single-Frequency Testing † Multifrequency Testing Generation of RRS Speciﬁcations

† †

Summary Product qualiﬁcation is intended for determining the adequacy of a product of good quality for a speciﬁc application or a range of applications. A good example is the seismic qualiﬁcation of a nuclear power plant. This chapter presents test procedures used in product qualiﬁcation, with emphasis on seismic qualiﬁcation. Both single-frequency testing and multifrequency testing are described. Generation of test signal speciﬁcations is outlined.

13.1

Introduction

Vibration testing is used in product qualiﬁcation. Here, the objective is to test the adequacy of a product of good quality for a speciﬁc use in a typical operating environment. Clearly, the nature of testing and the test requirements, including test speciﬁcations, will depend on the type of application and the class of product. In this chapter, we will consider just two types of product qualiﬁcation: distribution qualiﬁcation and seismic qualiﬁcation. Procedures of vibration testing for other types of qualiﬁcation will be similar.

13.2

Distribution Qualiﬁcation

The term “distribution qualiﬁcation” denotes the process by which the ability of a product to withstand a clearly deﬁned distribution environment is established. Dynamic effects on the product due to handling loads, characteristics of packaging, and excitations under various modes of transportation (truck, rail, air, and ocean) must be properly represented in the test speciﬁcations used for distribution qualiﬁcation. If a product fails a qualiﬁcation test, then corrective measures and subsequent requaliﬁcation are necessary prior to commercial distribution. Product redesign, packaging redesign, and modiﬁcation of existing shipping procedures might be required to meet qualiﬁcation requirements. Often, the necessary improvements can be determined by analyzing data from prior tests. Proper distribution qualiﬁcation will result in improved product quality (and associated reliability and performance), reduced wastage and inventory problems, cost-effective packaging, reduced shipping and handling costs, and reduced warranty and service costs. 13-1 © 2007 by Taylor & Francis Group, LLC

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13-2

Vibration Monitoring, Testing, and Instrumentation

TABLE 13.1

Comparison of Test Types

Simultaneous multimodal (multiresonant) excitation possible? Test duration Power requirements Represents a random environment? Test system cost Overtesting possibility

Sine Testing

Random Testing

Narrowband Random Sweep

Broadband Random Sweep

No

Yes

No

Yes

Long Low No Low High

Short High Yes High Low

Long Low Yes Moderate to high High

Moderate High Yes High Low

Random testing can more accurately represent vibrations in distribution environments. Several characteristics make it superior to sine testing. A sine test is a single-frequency test; thus, only one frequency is applied to a test object at a given instant. As a result, failure modes caused by the simultaneous excitation of two or more modes of vibration cannot be realized by sine testing, at least under steady excitations. On the other hand, in random testing, many frequencies are simultaneously applied to the test object. Therefore, conditions are more conducive to multiple-mode excitations and associated complex failures. A comparison of testing with four types of excitation signals is given in Table 13.1.

13.2.1

Drive-Signal Generation

The ﬁrst step in signal synthesis for driving the exciter is to assign independent and identically distributed, random phase angles to the digitized spectral magnitude (spectral lines) of the drive spectrum. The number of lines chosen is consistent with the fast Fourier transform (FFT) algorithm that is employed and the desired numerical accuracy. The inverse Fourier transform is obtained from the resulting discrete, complex Fourier spectrum. In general, the signal thus obtained would not possess ergodicity and Gaussianity. Stationarity can be attained by randomly shifting the signal with respect to time and summing the results. The resulting signal would also be weakly ergodic. Ergodicity is improved by increasing the duration of the signal. To obtain Gaussianity, a sufﬁciently large number of time-shifted signals must be summed as dictated by the central limit theorem. Furthermore, because the magnitude of a Gaussian signal almost always remains within three times its standard deviation (99.7% of the time), Gaussianity can be imposed simply by windowing the time-shifted signal. The amplitude of the window function is governed by the required standard deviation of the drive signal. Unwanted frequency components introduced as a result of sharp end transitions in each time-shifted signal component can be suppressed by properly shaping the window. This process introduces a certain degree of nonstationarity into the synthesized signal, particularly if the windowed signal segments are joined end-to-end to generate the drive signal. A satisfactory way to overcome this problem is to introduce a high overlap from one segment to the next. Because the processing time increases in proportion to the degree of overlap, however, a compromise must be reached. In summary, for a given drive spectral magnitude the drive signal can be synthesized as follows: 1. 2. 3. 4. 5. 6.

Assign independent, identically distributed random phase values to the drive-spectral lines. Perform an inverse Fourier transform of the resulting spectrum using FFT. Generate a set of independent and identically distributed time-shift values. Perform a time-shift of the signal obtained in Step 2 using the values from Step 3. Window the time-shifted signals. Join the windowed signals with a ﬁxed overlap.

The resulting digital drive signal is converted into an analog signal using a digital-to-analog converter (DAC) and passed through a low-pass ﬁlter to remove any unwanted frequency components before it is used to drive the shaker. This procedure is illustrated in Figure 13.1.

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Seismic Qualiﬁcation of Equipment

FIGURE 13.1

13.2.2

13-3

The synthesis of a random drive signal.

Distribution Spectra

The distribution environment to which a product is subjected depends on several factors. In particular, one must consider (1) the nature and severity of handling prior to and during shipment, (2) the mode of transportation (truck, rail, air cargo, ship), (3) the geographic factors, (4) the environmental conditions, (5) the characteristics of the protective packaging used, and (6) the dynamic characteristics of the product itself. These factors are complex and essentially random in nature. Laboratory simulation of such an environment is difﬁcult even if a combination of several types of tests — e.g., vibration, shock, drop, and thermal cycling — is employed. A primary difﬁculty arises from the requirement that test speciﬁcations should be simp1e, yet accurately represent the true environment. The test must also be repeatable to allow standardization of the test procedure and to facilitate evaluation and comparison of test data. Finally, testing must be cost effective. During transportation, a package is subjected to multi-degree-of-freedom (multi-DoF) excitations that can include rectilinear and rotational excitations at more than one location simultaneously. However, test machines are predominantly single-axis devices that generate excitations along a single direction. Thus, any attempt to duplicate a realistic distribution environment in a laboratory setting can prove futile. An alternative might be to use trial shipments. However, because of the random nature of the distribution environment, many such trials would be necessary before the data would be meaningful. Therefore, trial shipments are not appealing from a cost–beneﬁt point of view and also because test control and data acquisition would be difﬁcult. Data from trial shipments are extremely useful, however, in developing qualiﬁcation-test speciﬁcations and in improving existing laboratory test procedures. A more realistic goal of testing would be to duplicate possible failure and malfunction modes without actually reproducing the distribution environment. This, in fact, is the underlying principle of testing for distribution qualiﬁcation. For instance, sine tests can reproduce some types of failure caused during shipment even though the test signal does not resemble the actual dynamic environment; however, random testing is generally superior. Test speciﬁcations are expressed in terms of distribution spectra in distribution qualiﬁcation where random testing is used. Speciﬁcation development begins with a sufﬁcient collection of realistic data.

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13-4

Vibration Monitoring, Testing, and Instrumentation

Sources of data include ﬁeld measurements during trial shipments, computer simulations (e.g., Monte Carlo simulations), and previous speciﬁcations for similar products and environments. For best results, all possible modes of transportation, excitation levels, and handling severities should be included. The data, expressed as power spectral density (PSD), must be reduced to a common scale — particularly with respect to the duration of excitation — for comparison purposes. Scaling can be accomplished by applying a similarity law based on a realistic damage criterion. For example, a similarity law might relate the excitation duration and the PSD level such that the value of a suitable damage function would remain constant. Time-dependent damage criteria are developed primarily on the basis of fatigue– strength characteristics of a test product. Owing to nonlinearities of the environment, spectral characteristics (frequency content) will change with the excitation level. If such changes are signiﬁcant, then they should be properly accounted for. The inﬂuence of environmental conditions, such as temperature and humidity, must be considered as well. The PSD curves conditioned in this manner are plotted on a log –log plane to establish an envelope curve. This curve represents the worst composite environment that is typically expected. The envelope is then ﬁtted with a small number of straight-line segments. At this point, the PSD curve should be scaled so that the root-mean-square (RMS) value is equal to that before the straight-line segments were ﬁtted. The resulting PSD curve can be used as the test speciﬁcation. Test duration can be established from the time-scaling criterion. If the corresponding test duration is excessively long, thereby making the test impractical, then the test duration should be shortened by increasing the test level according to a realistic similarity criterion. Product overtesting can be signiﬁcant only if one reference spectrum is used to represent all possible distribution environments. Shipping procedures should thus be classiﬁed into several groups depending on the dynamic characteristics of the shipping environment and a representative reference spectrum should be determined for each group. If a range of products with signiﬁcantly diverse dynamic characteristics is being qualiﬁed, then reference spectra should be modiﬁed and classiﬁed according to product type. At the testing stage, a reference spectrum must be chosen from a spectral database depending on the product type and applicable shipping procedures. Alternatively, a general composite spectrum can be developed by assigning weights to a chosen set of reference spectra and computing the weighted sum. Vibration levels in land vehicles and aircraft can range up to several kilohertz (kHz). Ships are known to have lower levels of excitation. In general, the energy content in vibrations experienced during the distribution of computer products is known to remain within 20 Hz. Consequently, the test speciﬁcation spectra (reference spectra) used in distribution qualiﬁcation are usually limited to this bandwidth. The typical speciﬁcation curve shown in Figure 13.2 can be speciﬁed simply from the co-ordinates of the break points of the PSD curve. Intermediate values can be determined easily because the break points are joined by straight-line segments on a log –log plane. The area beneath the PSD curve gives the required mean-square value of the test excitation. The square root of this value is the RMS value. It is speciﬁed along with the PSD curve, even though it can be determined directly from the PSD curve. An acceptable tolerance band for the control spectrum — usually ^ 3 db — is also speciﬁed. Test duration should be supplied with the test speciﬁcation.

13.2.3

Test Procedures

Dynamic-test systems with digital control are easy to operate. In menu-driven systems, a routine or mode is activated by picking the appropriate item from a menu that is displayed on the cathode-ray tube screen. The system asks for necessary data, and then necessary parameter values are entered into the system. Typically, the user supplies lower and upper RMS limits for test abort levels, break point co-ordinates of the reference spectrum, and test duration. The tolerance bands for test spectrum equalization and the accelerometer sensitivities are also entered. More than one test setup can be stored, a number being assigned to denote each test. Preprogrammed tests can be modiﬁed using a similar procedure in the edit mode. Any preprogrammed test can be carried out simply by entering the corresponding test number. Computed

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Seismic Qualiﬁcation of Equipment

FIGURE 13.2

13-5

A reference spectrum for the distribution qualiﬁcation of personal computers.

results, such as PSD curves and transmissibility functions, are stored for future evaluation. If desired, these results can be displayed, printed, or plotted with proper annotations and scales while the test is in progress. Main steps of a typical test procedure are as follows: 1. Carefully examine the test object and record obvious structural defects, abnormalities, and hazardous or unsafe conditions. 2. Perform a functional test (i.e., operate the product) according to speciﬁcations and record any malfunctions and safety hazards. Note: The test may be abandoned at this stage if the test object is defective. 3. Mount the test object rigidly on the shaker table so that the loading points and the excitation axis are consistent with standard shipping conditions and the speciﬁed test sequence. 4. Perform an exploratory test at half the speciﬁed RMS level (one fourth the speciﬁed PSD level). Monitor the response of the test package at critical locations including the control sensor location. 5. Perform the full-level test for the speciﬁed duration. Record the response data. 6. Change the orientation in accordance with the speciﬁed test sequence and repeat the test. 7. After the test sequence is completed, carefully inspect the test object and record any structural defects, abnormalities, and safety hazards. 8. Conduct a functional test and record any malfunctions, failures, and safety problems. An exploratory test at a fraction of the speciﬁed test level is required for new product models that are being tested for the ﬁrst time, or for older models that have been subjected to major design modiﬁcations. Three mutually perpendicular axes are usually tested, including the primary orientation (vertical axis) that is used for shipping. If product handling during distribution is automated, then testing only the primary axis is adequate. When multiple tests are required, the test sequence is normally stipulated. If the test sequence is not speciﬁed, then it can be chosen such that the least-severe orientation (orientation least likely to fail) is tested ﬁrst. The test is repeated for the remaining orientations, ending with the most severe one. The rationale is that with this choice of test sequence, the aging of the most severe direction would be maximized, thereby making the test more reliable.

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13-6

Vibration Monitoring, Testing, and Instrumentation

The test report should contain the following: 1. Description of the test object: Serial number, size (dimensions and weight), product function (e.g., system unit, hard drive, power supply, printer, keyboard, mouse, monitor, and ﬂoppy disk drive of a personal computer), and packaging particulars. Descriptive photos are useful. 2. Test plan: Usually standard and attached to the report as an appendix. 3. Test setup: Test orientations, sensor (accelerometer) locations, details of mounting ﬁxtures, and a brief description of the test apparatus. Photos may be included. 4. Test procedure: A standard attachment that is usually given according to corporate speciﬁcations. 5. Test results: Ambient conditions in the laboratory (e.g., temperature, humidity), pretest observations (e.g., defects, abnormalities, malfunctions), test data (e.g., reference spectrum, equalized control spectrum, drive spectrum, response time histories and corresponding spectra, transmissibility plots, coherence plots) and posttest observations. 6. Comments and recommendations: General comments regarding the test procedure and test item and recommendations for improving the test, product or packaging. Names and titles of the personnel who conducted the test should be given in the test report, with appropriate signatures, dates, and location of the test facility. Tests for distribution qualiﬁcation can be conducted on both packaged products and those without any protective packaging, even though it is the packaged product that is shipped. However, the reference spectra used in the two cases are usually not the same. The spectrum used for testing a product without protective packaging is generally less severe. Response spectra that are used for testing an unpackaged product should reﬂect the excitations experienced by the product during packaging.

13.3

Seismic Qualiﬁcation

Frequently, it is necessary to determine whether a given piece of equipment is capable of withstanding a pre-established seismic environment in a speciﬁc application. This process is known as seismic qualiﬁcation. For example, electric utility companies should qualify their equipment for seismic capability before installing it in earthquake-prone geographic localities. Safety-related equipment in nuclear power plants also requires seismic qualiﬁcation. Regulatory agencies usually specify the general procedures to follow in seismic qualiﬁcation. Seismic qualiﬁcation by testing is appropriate for complex equipment, but in such cases, the equipment size is a limiting factor. For large systems that are relatively simple to model, qualiﬁcation by analysis is suitable. Often, however, both testing and analysis are needed in the qualiﬁcation of a given piece of equipment. Seismic qualiﬁcation of equipment by testing is accomplished by applying a dynamic excitation by means of a shaker to the equipment, which is suitably mounted on a test table, and then monitoring structural integrity and functional operability of the equipment. Special attention should be given to the development of the dynamic-test environment, mounting features, the operability variables that should be monitored, the method of monitoring functional operability and structural integrity, and the acceptance criteria used to decide qualiﬁcation. In monitoring functional operability, the test facility would normally require auxiliary systems to load the test object or to simulate in-service operating conditions. Such systems include actuators, dynamometers, electrical-load and control-signal circuitry, ﬂuid-ﬂow and pressure loads, and thermal loads. In seismic qualiﬁcation by analysis, a suitable model is ﬁrst developed for the equipment, and then static or dynamic analysis (or computer analysis) is performed under an analytically deﬁned dynamic environment. The analytical dynamic environment is developed on the basis of the speciﬁed dynamic environment for seismic qualiﬁcation. By analytically or computationally determining system response at various locations, and by checking for such crucial parameters as relative deﬂections, stresses, and strains, qualiﬁcation can be established.

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Seismic Qualiﬁcation of Equipment

13.3.1

13-7

Stages of Seismic Qualiﬁcation

Consider the construction of a nuclear power plant. In this context, the plant owner is the customer. Actual construction of the plant is done by the plant builder, who is directly responsible to the customer concerning all equipment purchased from the equipment supplier or vendor. Often, the vendor is also the equipment manufacturer. The equipment may be purchased by the customer and handed over to the plant builder or directly purchased by the plant builder. Accordingly, the purchaser could be the plant builder or the plant owner. A regulatory agency might stipulate seismic-excitation capability requirements for the equipment used in the plant, or might specify the qualiﬁcation requirements for various categories of equipment. The customer is directly responsible to the regulatory agency for adherence to these stipulations. The vendor, however, is responsible to the plant builder and the customer for the seismic capability of the equipment. The vendor may perform seismic qualiﬁcation on the equipment according to required speciﬁcations. More often, however, the vendor hires the services of a test laboratory, which is the contractor, for seismic qualiﬁcation of equipment in the plant. A reviewer, who is hired by the plant builder or the customer, may review the qualiﬁcation procedure and the report, which are usually developed by the test laboratory by adhering to the qualiﬁcation requirements. Figure 13.3 gives a ﬂowchart for test-object movement and for associated information interactions between various groups in the qualiﬁcation program. A basic step in any qualiﬁcation program is the preparation of a qualiﬁcation procedure. This is a document that describes in sufﬁcient detail such particulars as the tests that will be conducted on the test object, pretest procedures, the nature of test-input excitations and the method of generating these signals, inspection and response-monitoring procedures during testing, deﬁnitions of equipment malfunction, and qualiﬁcation criteria. Where analysis is also used in the qualiﬁcation, the analytical methods and computer programs that will be used should be described adequately in the qualiﬁcation procedure. The qualiﬁcation procedure is prepared by the test laboratory (contractor), equipment particulars are obtained from the vendor or the purchaser, and the purchaser usually supplies the test environment for which the equipment will be qualiﬁed. Before the qualiﬁcation tests are conducted, the test procedure is submitted to the purchaser for approval. The purchaser normally hires a reviewer to determine whether the qualiﬁcation procedure satisﬁes the requirements of both the regulatory agency and the purchaser. There could be several stages of revision of the test procedure until it is accepted by the purchaser upon the recommendation of the reviewer. The approved qualiﬁcation procedure is sent to the test laboratory and qualiﬁcation is performed accordingly. The test laboratory prepares a qualiﬁcation report that also includes the details of static or dynamic analysis when incorporated. The qualiﬁcation report is sent to the purchaser for evaluation. The purchaser might obtain the services of an authority to review the qualiﬁcation report. The report might have to be revised and analysis and tests might have to be repeated before a ﬁnal decision is made on the qualiﬁcation of the equipment. Information ﬂow in a typical qualiﬁcation program is shown in Figure 13.4. Regulatory Agency Equipment To Be Equipment Qualified Test Supplier Laboratory (vendor) (Contractor) (Manufacturer)

Customer (Plant Owner)

Plant Builder Qualified Equipment

FIGURE 13.3

Test-object movement and information interactions in seismic qualiﬁcation.

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13-8

Vibration Monitoring, Testing, and Instrumentation

Equipment Data Equipment Supplier Purchaser

Test Environment

Qualification Procedure Test Laboratory (Contractor) Standards Guides Specification

Qualification Procedure Reviewer Test Report

Approved Test Report

13.3.2

Test Laboratory

Purchaser

Regulatory Agency

FIGURE 13.4

Approved Procedure

Purchaser

Test Report

Information ﬂow in a seismic qualiﬁcation program.

Test Preliminaries Acceleration Amplitude

Seismic qualiﬁcation tests are usually conducted by one of two methods depending on whether singlefrequency or multifrequency excitation inputs are employed in the main tests. The two test categories are (1) single-frequency tests and (2) multifrequency tests. Although the second test method is more common in seismic qualiﬁcation by testing, the ﬁrst method is used under some conditions depending on the nature of the test object and its Frequency mounting features (for example, line-mounted vs. ﬂoor-mounted equipment). Typically, multifre- FIGURE 13.5 A typical required input motion curve. quency excitations are preferred in qualiﬁcation tests, and single-frequency excitations are favored in design-development and quality-assurance tests. In single-frequency testing, amplitude of the excitation input is speciﬁed by a required input motion (RIM) curve, similar to that shown in Figure 13.5. If single-frequency dwells (e.g., sine dwell, sine beat) are employed, then the excitation input is applied to the test object at a series of selected frequency values in the frequency range of interest for that particular test environment. In such cases, dwell times (and number of beats per cycle where sine beats are employed) at each frequency point should be speciﬁed. If a single-frequency sweep (such as a sine sweep) is employed as the excitation signal, then the sweep rate should be speciﬁed. When the single-frequency test-excitation is speciﬁed in this manner, the tests are conducted very much like multifrequency tests. Multifrequency test are normally conducted by employing the response spectra method to represent the test-input environment. Basically, the test object is excited using a signal whose response spectrum, known as the test response spectrum (TRS), envelops a speciﬁed response spectrum, known as the required response spectrum (RRS). Ideally, the TRS should equal the RRS but it is practically impossible to achieve this condition. Hence, multifrequency tests are conducted using a TRS that envelops the RRS so that in signiﬁcant frequency ranges, the two response spectra are nearly equal (see Figure 13.6). However, excessive conservatism, which would result in overtesting, should be avoided. It is usually acceptable to have TRS values below the RRS at a few frequency points. The RRS is part of the data supplied to the test laboratory prior to the qualiﬁcation tests being conducted. Two types of RRS are provided, representing (1) the operating-basis earthquake (OBE), and (2) the safe-shutdown earthquake (SSE). The response spectrum of the OBE represents the most severe motions produced by an earthquake under which the equipment being tested would remain functional without undue risk of malfunction or safety hazard. However, if the equipment is allowed to operate at a

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disturbance level higher than the OBE level for a prolonged period then there would be a signiﬁcant risk of malfunction. The response spectrum of the SSE represents the most severe motions produced by an earthquake that the equipment being tested could safely withstand while the entire nuclear power plant is being shutdown. However, prolonged operation (i.e., more than the duration of one earthquake) could result in equipment malfunction. In other words, equipment is designed to withstand only FIGURE 13.6 The TRS enveloping the RRS in a multifrequency test. one SSE in addition to several OBEs. A typical seismic qualiﬁcation test would ﬁrst subject the equipment to several OBE-level excitations, primarily for aging the equipment mechanically to its end-of-design-life condition, and then would subject it to one SSE-level excitation. When providing RRS test speciﬁcations, it is customary to supply only the SSE requirement. The OBE requirement is then taken as a fraction (typically, 0.5 or 0.7) of the SSE requirement. Test response spectra corresponding to the excitation signals are generated by the test laboratory during testing. The purchaser usually supplies the test laboratory with an FM tape containing frequency components that should be combined in some ratio to generate the test-input signal. Qualiﬁcation tests are conducted according to the test procedure approved and accepted by the purchaser. The main steps of seismic qualiﬁcation testing are outlined in the following subsections.

13.3.3

Single-Frequency Testing

Seismic ground motions usually pass through various support structures before they eventually are transmitted to equipment. For seismic qualiﬁcation of that equipment by testing, in theory we should apply the actual excitations felt by it, and not the seismic ground motions. In an ideal case, the shaker-table motion should be equivalent to the seismic response of the supporting structure at the point of attachment of the equipment. The supporting structure would have a particular frequency-response function between the ground location and the equipment-support location (see Figure 13.7). Consequently, it could FIGURE 13.7 Schematic representation of the ﬁltering be considered a ﬁlter that modiﬁes seismic ground of seismic ground motions by a supporting structure. motions before they reach the equipment mounts. In particular, the components of the ground motion that have frequencies close to a resonant frequency of the supporting structure will be felt by the equipment at a relatively higher intensity. Furthermore, the ground motion components at very high frequencies will be almost entirely ﬁltered out by the structure. If the frequency response of the supporting structure is approximated by a lightly damped simple oscillator, then the response felt by the equipment will be almost sinusoidal, with a frequency equal to the resonant frequency of the structure. When the equipment supporting structure has a very sharp resonance in the signiﬁcant frequency range of the dynamic environment (for example, 1 to 35 Hz for seismic ground motions), it follows from the previous discussion that it is desirable to use a short-duration single-frequency test in seismic qualiﬁcation of the equipment. Equipment that is supported on pipelines (valves, valve actuators, gauges, and so forth) falls into this category and is termed line-mounted equipment.

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Resonant frequency of the supporting structure is usually not known at the time of the seismic qualiﬁcation test. Consequently, single-frequency testing must be performed over the entire frequency range of interest for that particular dynamic environment. Another situation in which single-frequency testing is appropriate is when the test object (equipment) itself does not have more than one sharp resonance in the frequency range of interest. In this case, the most prominent response of the test object occurs at its resonant frequency, even when the dynamic environment is an arbitrary excitation. Consequently, a single-frequency excitation would yield conservative test results. Equipment that has more than one predominant resonance may employ singlefrequency testing provided that each resonance corresponds to a dynamic DoF (e.g., one resonance along each dynamic principal axis), and that cross coupling between these DoF is negligible. In summary, single-frequency testing may be used where one or more of the following conditions are satisﬁed: 1. The supporting structure has one sharp resonance in the frequency range of interest (linemounted equipment is included). 2. The test object does not have more than one sharp resonance in the frequency range of interest. 3. The test object has a resonance in each DoF, but the DoF are uncoupled (for which adequate veriﬁcation should be provided in the test procedure). 4. The test object can be modeled as a simple dynamic system (such as a simple oscillator), for which adequate justiﬁcation or veriﬁcation should be provided. Usually, the required SSE excitation level for a single-frequency test over a frequency range is speciﬁed by a curve such as the one shown in Figure 13.5. This curve is known as the RIM magnitude curve. The OBE excitation level is usually taken as a fraction (typically, 0.5 or 0.7) of the RIM values given for the SSE. For a sine-sweep test, the sweep rate and the number of sweeps in the test should also be speciﬁed. Typically, the sweep rate for seismic qualiﬁcation tests is less than 1 octave/min. One sweep, from the state of rest to the maximum frequency in the range and back to the state of rest, is normally carried out in an SSE test (for example, 1–35 to 1 Hz). Several sweeps (typically, ﬁve) are performed in an OBE test. In an SSE sine-dwell test, the dwell time for each dwell frequency should be speciﬁed. The dwellfrequency intervals should not be high (typically, a half-octave or less). For an OBE test, the dwell times are longer (typically, ﬁve times longer) than those speciﬁed for an SSE test. For an SSE test using sine beats, the minimum number of beats and the minimum duration of excitation (with or without pauses) at each test frequency should be speciﬁed. In addition, the pause time for each test frequency should be speciﬁed when sine beats with pauses are employed. For an OBE test, the duration of excitation should be increased (as in a sine-dwell test). The dwell time at each test frequency should be adequate to perform at least one functional-operability test. Furthermore, a dwell should be carried out at each resonant frequency of the test object as well as at those frequencies that are speciﬁed. Total duration of an SSE test should be representative of the duration of the strong-motion part of a standard safeshutdown earthquake. Sometimes, narrow-band random excitations may be used in situations where single-frequency testing is recommended. Narrowband random signals are those that have their power concentrated over a narrow frequency band. Such a signal can be generated for test-excitation purposes by passing a random signal through a narrow-bandpass ﬁlter. By tuning the ﬁlter to different center frequencies in narrowbands, the test-excitation frequency can be varied during testing. This center frequency of the ﬁlter should be swept up FIGURE 13.8 A typical RRS for a narrow-band and down over the desired frequency range at a excitation test.

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reasonably slow rate (e.g., 1.0 octave/min) during the test. Thus, a multifrequency test with a sharp frequency-response spectrum (the RRS), as illustrated in Figure 13.8, is adequate in cases where singlefrequency testing is recommended. In this case, a requirement that has to be satisﬁed by the testexcitation signal is that its amplitude should be equal to or greater than the zero-period acceleration of the RRS for the test.

13.3.4

Multifrequency Testing

When equipment is mounted very close to the ground under its normal operating conditions, or if its supporting structure and mounting can be considered rigid, then seismic ground motions will not be signiﬁcantly ﬁltered before they reach the equipment mounts. In this case, the seismic excitations that are felt by the equipment will retain broadband characteristics. Multifrequency testing is recommended for seismic qualiﬁcation of such equipment. Whereas single-frequency tests are speciﬁed by means of an RIM curve along with the test duration at each frequency (or sweep rates), multifrequency tests are speciﬁed by means of an RRS curve. The test requirement in multifrequency testing is that the response spectrum of the test excitation (the TRS) felt by the equipment mounts should envelop the RRS. Note that all frequency components of the test excitation are applied simultaneously to the test object. This is in contrast to single-frequency testing, in which only one signiﬁcant frequency component is applied at a given instant. When random excitations are employed in multifrequency testing, enveloping of the RRS by the TRS may be achieved by passing the random signal produced by a signal generator through a spectrum shaper. As the analyzing frequency bandwidth (e.g., one-third octave bands, one-sixth octave bands) decreases, the ﬂexibility of shaping the TRS improves. A real-time spectrum analyzer (or a personal computer) may be used to compute and display the TRS curve corresponding to the control accelerometer signal (see Figure 13.9). By monitoring the displayed TRS, it is possible to adjust the gains of the spectrum-shaper ﬁlter to obtain the desired TRS that would envelop the RRS. Most test laboratories generate their multifrequency excitation signals by combining a series of sine beats that have different peak amplitudes and frequencies. Using the same method, many other signal types (such as decaying sinusoids) may be superimposed to generate a required multifrequency excitation signal. A combination of signals of different types could also be employed to produce a desired test input. A commonly used combination is a broadband random signal and a series of sine beats. In this combination, the random signal is adjusted to have a response spectrum that will envelop the broadband portion of the RRS without much conservatism. The narrow-band peaks of the RRS, which generally will not be enveloped by such a broadband response spectrum, will be covered by a suitable combination of sine beats.

Random Signal Generator

Random Signal

Shaped Signal

Spectrum Shaper

Power Amplifier

Spectrum Control TRS

Spectrum Analyzer

Control Accelerometer Output

FIGURE 13.9

Test Table

Test Input Excitation

Matching of the TRS with the RRS in multifrequency testing.

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By employing such mixed composite signals, it is possible to envelop the entire RRS without having to increase the amplitude of the test excitation to a value that is substantially higher than the ZPA of the RRS. One important requirement in multifrequency testing is that the amplitudes of the test excitation be equal to or greater than the ZPA of the RRS.

13.3.5

Generation of RRS Speciﬁcations

Seismic qualiﬁcation of an object is usually speciﬁed in terms of an RRS. The excitation input that is used in seismic qualiﬁcation analysis and testing should conservatively satisfy the RRS; that is, the response spectrum of the actual excitation input should envelop the RRS (without excessive conservatism, of course). For equipment to be installed in a building or on some other supporting structure, the RRS generally cannot be obtained as the response spectrum of a modiﬁed seismic ground-motion time-history. The supporting structure usually introduces an ampliﬁcation effect and a ﬁltering effect on seismic ground motions. This ampliﬁcation factor alone could be as high as three. Some of the major factors that determine the RRS for a particular seismic qualiﬁcation test are as follows: 1. Nature of the building that will be qualiﬁed 2. Dynamic characteristics of the building or structure and the location (elevation and the like) where the object is expected to be installed 3. In-service mounting orientation and support characteristics of the object 4. Nature of the seismic ground motions in the geographic region where the object would be installed 5. Test severity and conservatism that is required by the purchaser or the regulator agency The basic steps in developing the RRS for a speciﬁc seismic qualiﬁcation application include the following: 1. Development of representative safe-shutdown earthquake (SSE) ground-motion time histories for the building (or support structure) location 2. Development of a suitable building (or support structure) model 3. Response analysis of the building model, using the time histories that are obtained in Step 1 4. Development of response spectra for various critical locations in the building (or support structure), using the response time histories obtained in Step 3 5. Normalization of the response spectra obtained in Step 4 to unity ZPA (that is, dividing by their individual ZPA values) 6. Identiﬁcation of the similarities in the set of normalized response spectra that are obtained in Step 5 and grouping them into a small number of groups 7. Representation of each similar group by a response spectrum consisting of straight-line segments that envelop all members in the group, giving a normalized RRS for each group 8. Determination of scale factors for various locations in the building for use in conjunction with the corresponding normalized RRS curves Representative strong-motion earthquake time histories (SSEs) are developed by suitably modifying actual seismic ground-motion time histories that have been observed in that geographic location (or a similar one), or by using a random-signal-generation (simulation) technique or any other appropriate method. These time histories may be available as either digital or analog records, depending on the way in which they are generated. If computer simulation is used in their development, then a statistical representation of the expected seismic disturbances in the particular geographic region (using geological features in the region, seismic activity data, and the like) should be incorporated in the algorithm. The intensity of the time histories can be adjusted, depending on the required test severity and conservatism. The normalized response spectra are grouped so that the spectra that have roughly the same shape are put in the same group. In this manner, relatively few groups of normal response spectra (normalized) are

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obtained. Then, the response spectra that belong to each group are plotted on the same graph paper. Next, straight-line segments are drawn to envelop each group of response spectra. This procedure results in a normalized RRS for each group of analytical response spectra. The RRS that is used for a particular seismic qualiﬁcation scheme is obtained as follows. First, the normalized RRS corresponding to the location in the building where the object would be installed is selected. The normalized RRS curve is then multiplied by the appropriate scaling factor. The scaling factor normally consists of the product of the actual ZPA value under SSE conditions at that location (as obtained from the analytical response spectrum at that location, for example) and a factor of safety that depends on the required test severity and conservatism. In fact, three RRS curves corresponding to the vertical, east– west and north –south directions might be needed, even for single-degree-of-freedom (single-DoF) seismic qualiﬁcation tests because, by mounting three control accelerometers in these three directions, triaxial monitoring could be accomplished. If only one control accelerometer is used in the test, then only one RRS curve is used. In this case, the resultant of the three orthogonal RRS curves should be used. One way to obtain the resultant RRS curve is to apply the square root of the sum of squares (SRSS) method to the three orthogonal components. Alternatively, the envelope of the three orthogonal RRS curves is obtained and multiplied by a safety factor (greater than unity). Note that more than one building (or even many different geographic locations) could be included in the described procedure for developing RRS curves. The resulting RRS curves are then valid for the collection of buildings or geographic locations considered. When the generality of an RRS curve is extended in this manner, the test conservatism increases. This will also result in an RRS curve with a much broader band. In practice, in a particular seismic qualiﬁcation project, only a few normalized RRS curves are employed. In conjunction with these RRS curves, a table of data is provided that identiﬁes the proper RRS curves and the scaling factors that should be used for different physical locations (for example, elevations) in various buildings that are situated at several geographic locations.

Bibliography Bendat, J.S. and Piersol, A.G. 1971. Random Data: Analysis and Measurement Procedures, WileyInterscience, New York. Brigham, E.O. 1974. The Fast Fourier Transform, Prentice Hall, Englewood Cliffs, NJ. de Silva, C.W. 1983. Dynamic Testing and Seismic Qualiﬁcation Practice, D.C. Heath and Co., Lexington, MA. de Silva, C.W., A dynamic test procedure for improving seismic qualiﬁcation guidelines, J. Dyn. Syst. Meas. Control, Trans. ASME, 106, 143 –148, 1984. de Silva, C.W., Hardware and software selection for experimental modal analysis, Shock Vibration Digest, 16, 3– 10, 1984. de Silva, C.W., Matrix eigenvalue problem of multiple-shaker testing, J. Eng. Mech. Div., Trans. ASCE, 108, 457–461, 1982. de Silva, C.W., Optimal input design for the dynamic testing of mechanical systems, J. Dyn. Syst. Meas. Control, Trans. ASME, 109, 111–119, 1987. de Silva, C.W., Seismic qualiﬁcation of electrical equipment using a uniaxial test, Earthquake Eng. Struct. Dyn., 8, 337–348, 1980. de Silva, C.W., Selection of shaker speciﬁcations in seismic qualiﬁcation tests, J. Sound Vibration, 91, 21– 26, 1983. de Silva, C.W., Shaker test-ﬁxture design, Meas. Control, 17, 152–155, 1983. de Silva, C.W., Henning, S.J., and Brown, J.D., Random testing with digital control-application in the distribution qualiﬁcation of microcomputers, Shock Vibration Digest, 18, 3 –13, 1986. de Silva, C.W., Loceff, F., and Vashi, K.M., Consideration of an optimal procedure for testing the operability of equipment under seismic disturbances, Shock Vibration Bull., 50, 149– 158, 1980.

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de Silva, C.W. 2006. Vibration — Fundamentals and Practice, 2nd Edition, Taylor & Francis, CRC Press, Boca Raton, FL. Ewins, D.J. 1984. Modal Testing: Theory and Practice, Research Studies Press Ltd., Letchworth, England. McConnell, K.G. 1995. Vibration Testing, Wiley, New York. Randall, R.B. 1977. Application of B & K Equipment to Frequency Analysis, Bruel and Kjaer, Naerum, Denmark.

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14

Human Response to Vibration 14.1 14.2 14.3

Clarence W. de Silva The University of British Columbia

14.4

Introduction ...................................................................... 14-1 Vibration Excitations on Humans .................................. 14-2

Excitation Characteristics

Human Response to Vibration ........................................ 14-3

Vibration Perception † Frequency Weighting Curve † Iso-Perception Curves † Detection Threshold Curve

Regulation of Human Vibration ..................................... 14-6

Acceptable Vibration Levels in Buildings Body Vibration † Hand–Arm Vibration

†

Whole-

Summary Human response to vibration is an important practical subject. Vibration may be imparted on a human body from various sources. The response has to be determined based on the vibrations transmitted and received at various parts of the body and how these vibrations are felt and perceived both physiologically and psychologically. From the physiological viewpoint alone, the associated issues are closely related to vibration testing and monitoring. Acceptable vibration levels, frequency ranges, and durations of vibration in different situations are governed by various regulations and standards. The subject of human response to vibration is treated in the present chapter.

14.1

Introduction

Human response to vibration can be both psychological and physiological in nature. Its study is multidisciplinary in nature, involving engineering, biomedical, scientiﬁc, and psychological issues. The subject is complicated by the fact that normally other factors, such as posture, noise, and forces simultaneously affect the human response, and it is virtually impossible to separately identify (resolve) the effect of the vibration alone. Furthermore, vibration has both beneﬁcial as well as harmful effects of vibration. Vibration may be imparted on a human body from sources, such as transit vehicles, hand-held power tools, industrial machinery, and civil engineering structures. Such vibrations may be applied at one or more locations of the body and in one or more directions. The imparted vibration is transmitted to various parts of the body. The vibration excitation may be treated primarily as a force input or a motion input. The vibration felt at a speciﬁc location of the human body is governed by not only the nature of the vibration input, but also the dynamics or transfer functions between the location of interest (output) and the point of application (input). The issues of force transmissibility, motion transmissibility, mobility, and receptance are relevant here. The human response to vibration has to be determined based on the 14-1 © 2007 by Taylor & Francis Group, LLC

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Vibration Monitoring, Testing, and Instrumentation

vibrations received at various parts of the body and how these vibrations are felt and perceived both physiologically and psychologically. Acceptable levels, frequency ranges, and durations of vibration in different situations are governed by various regulations and standards.

14.2

Vibration Excitations on Humans

The human response (or effect) to vibration excitation (cause) depends directly on the characteristics of the vibration excitation itself. Vibration is commonly applied to a human body as: 1. Hand –arm (or hand-transmitted) vibration 2. Whole-body vibration. The ﬁrst category is typical in the operation of hand-held power tools (e.g., power drills and saws) and hand-control devices (e.g., steering wheel and bicycle handlebar). The effective frequencies can be as low as 8 Hz to as high as 1 kHz. The second category is common in transit vehicles (e.g., cars and trains) and the vibration excitation is typically transmitted through a seat. The effective frequency range is typically 1 –100 Hz. It is useful to deﬁne a coordinate system for the purpose of indicating the directions of excitation and body response. For vibration excitations received by human hand, a three-axis translatory Cartesian coordinate system may be deﬁned (see Figure 14.1). Here, the coordinate origin is located at the third knuckle of the hand. The three orthogonal coordinate axes of the Cartesian frame are deﬁned as follows: The x-axis is along the outward normal of the palm. The y-axis extends from the third knuckle towards the thumb. The z-axis is along the middle ﬁnger. Note that rotational vibrations are not considered in this case. For whole-body vibrations, a six-degree-of-freedom coordinate system consisting of three orthogonal (Cartesian) translatory axes (x, y, and z) and three rotational motions (roll, pitch, and yaw) about these Cartesian axes may be deﬁned. It is convenient to deﬁne this six-axis coordinate system with reference to a person seated in a vehicle (see Figure 14.2). In this coordinate system, the x-axis represents the fore-aft direction, the y-axis represents the lateral direction, and the z-axis represents the vertical direction. Pitch, roll, and yaw take the usual meanings as for the motion of a ship, an aircraft, or a ground-transit vehicle. Also, with respect to human head motion, the following interpretations can be given for the rotational motions: Roll: a neutral side-to-side headshake (or Asian/East Indian nod) Pitch: conventional agreeing nod Yaw: conventional disagreeing headshake.

14.2.1 Excitation Characteristics z

x Third Knuckle

y

FIGURE 14.1 vibrations.

A coordinate system for representing hand

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When determining the human response to vibration, it is of fundamental importance to consider the nature of the vibration excitation. The characteristics of the excitation that are particularly important include: 1. Speciﬁc frequency value that is considered 2. Frequency content of the vibration signal (e.g., narrowband, broadband, shock, and sine) 3. Magnitude (level or amplitude) of vibration 4. Direction of application 5. Duration of exposure

Human Response to Vibration

14-3

6. Whether continuous or applied intermittently over short periods.

z Yaw y Pitch x Roll

FIGURE 14.2 A coordinate system for representing whole-body vibrations.

14.3

The ﬁrst three factors of vibration excitation may be represented by a curve of root-mean-square (rms) acceleration of vibration versus frequency. The direction may mean whether the vibration is applied vertically or laterally (horizontally) or rotational vibrations, as deﬁned by a suitable coordinate system. One curve may be used for a given direction and for a speciﬁed duration of excitation.

Human Response to Vibration

When a vibration excitation is applied to a human body, it is transmitted to various parts of the body through the dynamic characteristics or transfer functions in the linear case) of the body. These vibrations are felt through various sensory organs and perceived or assimilated by the brain. The human response to vibration is the “effect,” which may mean a vibration sensation, perception, annoyance, discomfort, various biomechanical effects, and health consequences. In a regulatory sense, the vibration response may be represented either as a threshold curve (as a function of frequency), which should not be exceeded in a given application (e.g., ride quality), or as an excitation curve (as a function of frequency) for equalintensity of vibration perception (which is the inverse of a response curve). The response may be classiﬁed according to its severity. For example: 1. 2. 3. 4.

Sense and feel Distraction and annoyance Discomfort (e.g., poor ride quality) Minor, moderate, or major health consequences.

In a work environment, the categories 2, 3, and 4 will affect the human performance. In particular, category 3 concerns occupational health, and may cause disorders of muscular, nervous, and cardiovascular systems. This category may cause as well damage to bones, joints, and the like in the body. Feeling or sensation of vibration may not necessarily cause discomfort or health consequences at low levels and durations. But low levels of vibration, particularly at low frequencies (say, below 1 Hz) when applied over extended periods may result in discomforts, such as the motion sickness and even health consequences associated with them. Note that motion sickness can be caused by “apparent” or simulated virtual vibrations as well. An average person is known to be most sensitive to horizontal vibrations in the frequencies below 2 Hz and vertical vibrations near 5 Hz. Whole-body response depends on position and pose (e.g., sitting, standing, and sleeping) as well. For example, the response considerations for a seated person are different from those of a standing person. Other factors, such as size, age, alertness, body construction, and ﬁtness of the person need to be considered as well. A human is typically more sensitive (e.g., may exhibit more discomfort) to shock-like excitations than smooth and continuous excitations of the same energy level. Furthermore, where the vibration excitation signal is a single-frequency (i.e., sinusoidal or narrow-band) or a multi-frequency (broadband) can inﬂuence the response.

14.3.1 Vibration Perception When a vibration excitation is applied to a human body (whole-body, hand–arm, etc.), it senses the vibration through several sensory means, such as vision, hearing, sensation of the orientation, and feeling through various parts of the body (e.g., joints, muscles, tendons, abdomen, and nerve endings in the skin). Different parts of the body respond or react to vibration in different ways (e.g., generation of forces,

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Vibration Monitoring, Testing, and Instrumentation Human Body Vibration Excitation

Brain Response Physical + Psychological

FIGURE 14.3

Vibration Perception

Assimilation

Perception of human response to vibration.

motions, stresses, and strains in various body parts and associated sensations; irritation and annoyance due to noise; and discomfort and nausea due to accelerations and decelerations, and oscillations). These responses are both physiological (including biomechanical) and psychological in nature. The perception of vibration will involve assimilation (or intelligent combination or fusion) of these sensory components by the brain. The overall process is schematically shown in Figure 14.3. The assimilation of sensory components in the perception of vibration is schematically shown in Figure 14.4. The excitation-response processes involved in Figure 14.3 may be represented by transfer functions (or frequency transfer functions or frequency response functions, in the frequency domain). The overall excitation-perception process forms the combine transfer function. It should be cautioned here that the processes involved are hardly linear. Consequently, the transfer function representation is not strictly applicable unless one allows for variations in the individual transfer function components and parameters.

14.3.2 Frequency Weighting Curve In the practice of human response to vibration, this term is used to denote the excitation-response transfer function as a function of frequency (i.e., in the frequency domain), properly normalized with respect to a speciﬁed level of perception. It is not a frequency transfer function in a classic, linear sense, however. In fact, it is a representative excitation-perception transfer function for an entire population or group of people. In this sense, it may be termed a “Normalized Standard Perception Function” for vibration excitations on human body. Furthermore, it provides the magnitude information only and no phase information is included. The frequency weighting curve depends on the direction of excitation as well. Standard curves have been generated by various regulatory agencies such as the International Organization for Standardization (ISO). Approximate weighting curves for whole-body vibration in vertical and horizontal directions are shown in Figure 14.5, which approximate the ISO 2631-1 (1997) standards. (Visual)

Seeing

(Auditory)

Hearing

(Vestibular)

Inner Ear Orientation Joints, Muscles Tendons

(Somatic)

Abdomen

Brain

Vibration Perception

(Sensory Fusion, Assimilation)

Nerve Endings in Skin

FIGURE 14.4

Vibration perception through the assimilation of various sensory components.

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Human Response to Vibration

14-5

TF Magnitude (Weighting Factor)

Vertical Horizontal

1.0 0.1 0.01

FIGURE 14.5

1

10

100

Frequency (Hz)

Frequency weighting curves (normalized perception functions) for whole-body vibration.

Even though the concept of frequency transfer function or transmissibility function is used, a frequency weighting curve is not strictly a transfer function in view of the following characteristics: 1. The underline principles are not linear 2. It represents an entire group of people rather than a speciﬁc individual 3. Vibration perception is not a response of a speciﬁc component of the human body to a vibration excitation but rather an assimilation of various sensory effects.

14.3.3 Iso-Perception Curves A curve of equal vibration perception, as the name implies, gives the vibration excitation levels for various frequencies for which the brain perceives the same severity level. In concept, then, this is the inverse of a frequency weighting curve (a transfer function is in fact an iso-excitation curve). A set of isoperception curves may be needed, one curve representing a speciﬁc value of vibration perception, as a function of frequency. As for a frequency weighting curve, an iso-perception curve does not contain any phase information.

14.3.4 Detection Threshold Curve A detection threshold curve is indeed an iso-perception curve. Rather than representing a speciﬁc level of perception, however, it represents the boundary between detection and not detection of vibration. So, it is in fact an iso-detection curve. It is expressed as a function of frequency (as for an iso-perception curve). If the vibration level at a particular frequency is above the threshold curve that vibration level is detected (or perceived), and if it is below the threshold curve, it is not detected. Detection threshold curves for sinusoidal whole-body vibration in vertical and horizontal directions are shown in Figure 14.6. RMS Vibration Acceleration (m/s2)

Vertical Horizontal

0.1 0.01 0.001 0.1

FIGURE 14.6

1.0

10.0

100.0 Frequency (Hz)

Vertical and horizontal detection threshold curves for sinusoidal whole-body vibration.

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14-6

Vibration Monitoring, Testing, and Instrumentation

RMS Acceleration (g)

100

Vertical

10−1

Lateral

25 Min ISO Boundaries

10−2

10−3

10−1

100

101

102

Frequency (Hz)

FIGURE 14.7

Typical threshold curves for ride comfort.

The concept of “detection” may be modiﬁed to “comfort” (as in ride comfort) where the vibration levels below the curve are comfortable and those above the curve are uncomfortable. Similarly, threshold curves may be speciﬁed for conditions, such as “quality” (say, ride quality or environmental quality) or “damage” (i.e., vibrations below the threshold curve will not cause damage, or health consequences, to body and those above the threshold will cause damage). Of course, for a damage curve, the duration of vibration application should be speciﬁed as well. A representative ride quality threshold speciﬁcation for the vertical and horizontal directions of response, for a speciﬁc ride (trip) duration (25 min) is shown in Figure 14.7. Actual responses (in units of acceleration due to gravity, g) inside the vehicle compartment in the two directions (vertical and horizontal) may be measured (or computer simulated) and marked on this plane for different frequencies of excitation (Hz). If the marked points fall below the threshold curve, the regulatory requirements are satisﬁed for the particular trip duration. For computer simulation, the responses have to be determined using a suitable dynamic model of the vehicle. The road excitations may be derived from the road irregularity characteristics and the considered vehicle speed, either in the time domain or the frequency domain. If the excitations (or the responses) are available in the time domain, then they have to be converted into the frequency domain (e.g., Fourier spectrum and power spectral density) before using the speciﬁcations (Figure 14.7), which are given in the frequency domain.

14.4

Regulation of Human Vibration

Acceptable levels, frequency ranges, and durations of vibration in different situations are governed by various regulations and standards. Such regulations are largely application oriented. This topic is addressed now.

14.4.1 Acceptable Vibration Levels in Buildings Buildings will undergo vibrations due to activities, such as seismic excitations, winds, heavy construction, and vehicles. The occupants of the buildings will feel these vibrations and will be affected in various ways. If the normal functions and activities are not affected by these vibrations, they may be considered acceptable. On the other hand, even when the resulting vibrations are not physically damaging to the humans, associated psychological effects and distractions can affect their performance and the peace of mind.

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Human Response to Vibration TABLE 14.1

14-7

Multiplication Factors for Acceptable Continuous Vibration Levels in Buildings

Building Activity Type

Multiplication Factor

High-precision/sensitive operations Residences Ofﬁce buildings Factories

1.0 3.0 4.0 8.0

Vibration levels acceptable for humans inside buildings will depend on several important factors including: 1. 2. 3. 4.

Performed activity Frequency value of vibration Whether continuous or intermittent vibrations Time of the day (e.g., day or night).

Standards are available from various regulator agencies to specify the allowable vibration levels. For example, an acceptable vibration spectrum may be speciﬁed for high-precision and sensitive activities (e.g., surgery and reading) and using this as the reference, multiplication factors may be given for other environments and operations in buildings. A rough indication of this concept is given in Table 14.1, for the case of continuous vibrations. For intermittent vibrations, it may be acceptable to increase the factors for continuous vibrations by approximately 30, except where the factor is 1.0, which should remain unchanged. Furthermore, for residences, the multiplication factor for the night time may be set at half that for the day time.

14.4.2 Whole-Body Vibration The human body is a dynamic system, with inputs or excitations and outputs or responses. However, it is a complex and nonlinear system. When considering the effects of a human to whole-body vibrations, both physiological and psychological effects have to be considered, which will compound the difﬁculty. Various levels of effects are possible; for example, detection, discomfort, pain, and damage. Stress, reduced performance in work, motion sickness, internal injury such as back problems, and other biological effects, such as digestive and reproductive problems can be caused by whole-body vibrations. Effects of whole-body vibration depend on many factors. They include: 1. 2. 3. 4.

Spectral characteristics of the excitation (magnitude, frequency, and phase) Duration of exposure Directions (both translatory and rotatory) and location of application Human posture (e.g., standing or seated) during exposure.

The effects are not the same for every person, and can depend on factors such as the age, sex, physical condition, vision, background, and environment (including temperature, humidity, smell, and noise). A realistic model to represent the human response to vibration can be quite complex, for the reasons mentioned above. Even for representing the mechanical response alone, a distributed-parameter (i.e., continuous) and nonlinear model that includes nonlinear damping, nonlinear ﬂexibility, and nonhomogeneous and anisotropic material properties may be needed. For relatively low-level excitations and frequencies, however, a linear model may be adequate. A lumped-parameter model of this type for determining the single-direction (vertical or lateral) response to a vibration excitation applied at the foot of a subject, in a single direction (vertical or lateral) is shown in Figure 14.8. This model assumes body symmetry. Furthermore, the possible dynamic coupling between two orthogonal directions of motion cannot be represented by this model. Using a model of the type shown in Figure 14.8, various transfer functions (transmissibility functions) from the application point (feet) to a response point (brain, skull, eyeball, neck, abdomen, hips, knees, etc.) can be determined in the usual manner. The resonant frequencies and the corresponding peak

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14-8

Vibration Monitoring, Testing, and Instrumentation yH

magnitudes of these transfer functions will have a direct effect on the severity (both physiological and psychological) of the vibrations as felt and perceived by the human. The resonant frequencies and corresponding transmissibility magnitudes for a “typical” human for several key response locations in the body are given in Table 14.2. In the standing position, the vibration is applied at the feet in the vertical direction. In the seated position, the vibrations are applied on the seat in the vertical position. The given values are very approximate. Apart from the variability of the subjects, the values will depend on factors, such as the clothing, footwear, and the seat dynamics, which will change the damping and ﬂexibility characteristics and will affect the transmissibility functions. Standards such as exposure limits for wholebody vibrations as a function of the excitation frequency and exposure duration for humans (standing or seated, vertical, or lateral) are available from bodies such as the ISO. Particularly useful for whole-body vibrations is the standard ISO 2631.

Head

yr

Upper Torso

ys Shoulder ya

ye Arm

Abdomen

yp

yh Hip

yk

Leg

Foot

14.4.3 Hand– Arm Vibration

Vibration Excitation

Hand – arm vibrations are hand-transmitted vibrations. They are typically caused by holding a vibrating object (or tool) by one hand or both FIGURE 14.8 A linear model for representing the (which is a “tactile” interface between the hand mechanical response of a “symmetric” human to and the tool). Since the human hand is a smaller vibration. and more delicate organ than the entire body, hand –arm vibrations can cause more serious discomfort and injury than what is common with whole-body vibrations. Common injuries due to handtransmitted vibrations by and large remain within the hand itself. Operation of power drills, chain saws, and other hand-held power tools for extended periods of time can cause severe discomfort and injury to ﬁngers, wrist, and shoulders. A repetitive application of even small doses of vibration at the hand or ﬁngers can result in serious harm such as the repetitive strain injury (RSI). Permanent injury, particularly to the joints, muscles, bone, and blood vessels, may be possible when subjected to high levels of vibration for prolonged periods. As much as clothing can damp out the effects of whole-body vibrations, antivibration gloves can reduce the effect of hand-transmitted vibrations. uf

TABLE 14.2

Peak Response of a Human to Vibration Approximate Transmissibility Peak (Resonance) Standing Subject

Seated Subject

Response Location

Frequency (Hz)

Magnitude

Frequency (Hz)

Magnitude

Head Shoulder Hip

4.0 and 20.0 4.0 4.0

1.2 and 0.8 1.5 1.8

6.0 and 15.0 4.0 4.0

1.2 and 0.8 1.5 1.1

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Human Response to Vibration

14-9

The effects of hand–arm vibrations are most serious at frequencies up to approximately 16 Hz, while the tactile perception of vibration through the palms and ﬁngers is possible up to much greater levels of frequency (say, more than 1000 Hz). Furthermore, exposure over a short period (say, less than 30 min) is far less serious than continuous exposure for a long period (say, 8 h, where the same effect as a 30-min exposure may result with just 1/5th of the vibration level). Various standards and regulations are available related to hand-transmitted vibrations. Commonly used is the ISO 5349 from the ISO. In particular, the exposure limit curves (rms acceleration) provided by them are most severe and constant (ﬂat) in the frequency range 8– 16 Hz. From 16 to 1000 Hz, the curve rolls up at the rate of 6 dB/octave. The curve is not deﬁned for frequencies below 8 Hz and above 1 kHz. The same curve is used for hand-transmitted vibrations in any direction, unlike in the case of whole-body vibrations.

Bibliography 1. 2. 3.

Broch, J. T., Mechanical Vibration and Shock Measurements, Bruel and Kjaer, Naerum, Denmark, 1980. De Silva, C. W., Vibration—Fundamentals and Practice, 2nd ed., Taylor & Francis/CRC Press, Boca Raton, FL, 2006. Mansﬁeld, N. J., Human Response to Vibration, CRC Press, Boca Raton, FL, 2005.

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