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Mechanical Shock
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Mechanical Vibration and Shock Analysis second edition – volume 2
Mechanical Shock
Christian Lalanne
First published in France in 1999 by Hermes Science Publications © Hermes Science Publications, 1999 First published in English in 2002 by Hermes Penton Ltd © English language edition Hermes Penton Ltd, 2002 Second edition published in Great Britain and the United States in 2009 by ISTE Ltd and John Wiley & Sons, Inc. Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK
John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA
www.iste.co.uk
www.wiley.com
© ISTE Ltd, 2009 The rights of Christian Lalanne to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Lalanne, Christian. [Vibrations et chocs mécaniques. English] Mechanical vibration and shock analysis / Christian Lalanne. -- 2nd ed. v. cm. Includes bibliographical references and index. Contents: v. 1. Sinusoidal vibration -- v. 2. Mechanical shock -- v. 3. Random vibration -- v. 4. Fatigue damage -- v. 5. Specification development. ISBN 978-1-84821-122-3 (v. 1) -- ISBN 978-1-84821-123-0 (v. 2) 1. Vibration. 2. Shock (Mechanics). I. Title. TA355.L2313 2002 624.1'76--dc22 2009013736 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-121-6 (Set of 5 Volumes) ISBN: 978-1-84821-123-0 (Volume 2) Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne.
Table of Contents
Foreword to Series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
List of Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvii
Chapter 1. Shock Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1. Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2. Transient signal . . . . . . . . . . . . . . . . . . . . . 1.1.3. Jerk . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4. Simple (or perfect) shock . . . . . . . . . . . . . . . 1.1.5. Half-sine shock . . . . . . . . . . . . . . . . . . . . . 1.1.6. Versed sine (or haversine) shock. . . . . . . . . . . 1.1.7. Terminal peak sawtooth (TPS) shock (or final peak sawtooth (FPS)) . . . . . . . . . . . . . . . . 1.1.8. Initial peak sawtooth (IPS) shock . . . . . . . . . . 1.1.9. Square shock. . . . . . . . . . . . . . . . . . . . . . . 1.1.10. Trapezoidal shock . . . . . . . . . . . . . . . . . . . 1.1.11. Decaying sinusoidal pulse . . . . . . . . . . . . . . 1.1.12. Bump test . . . . . . . . . . . . . . . . . . . . . . . . 1.1.13. Pyroshock . . . . . . . . . . . . . . . . . . . . . . . 1.2. Analysis in the time domain . . . . . . . . . . . . . . . . 1.3. Fourier transform . . . . . . . . . . . . . . . . . . . . . . 1.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2. Reduced Fourier transform . . . . . . . . . . . . . . 1.3.3. Fourier transforms of simple shocks. . . . . . . . . 1.3.4. What represents the Fourier transform of a shock? 1.3.5. Importance of the Fourier transform. . . . . . . . .
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5 6 7 8 8 9 9 12 12 12 14 14 25 27
vi
Mechanical Shock
1.4. Energy spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1. Energy according to frequency . . . . . . . . . . . . . . . . . . . 1.4.2. Average energy spectrum . . . . . . . . . . . . . . . . . . . . . . 1.5. Practical calculations of the Fourier transform . . . . . . . . . . . . 1.5.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.2. Case: signal not yet digitized . . . . . . . . . . . . . . . . . . . . 1.5.3. Case: signal already digitized . . . . . . . . . . . . . . . . . . . . 1.5.4. Adding zeros to the shock signal before the calculation of its Fourier transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6. The interest of time-frequency analysis . . . . . . . . . . . . . . . . 1.6.1. Limit of the Fourier transform . . . . . . . . . . . . . . . . . . . 1.6.2. Short term Fourier transform (STFT) . . . . . . . . . . . . . . . 1.6.3. Wavelet transform . . . . . . . . . . . . . . . . . . . . . . . . . .
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28 28 29 29 29 29 32
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32 36 36 39 44
Chapter 2. Shock Response Spectrum . . . . . . . . . . . . . . . . . . . . . . .
51
2.1. Main principles . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Response of a linear one-degree-of-freedom system. . . 2.2.1. Shock defined by a force . . . . . . . . . . . . . . . . 2.2.2. Shock defined by an acceleration . . . . . . . . . . . 2.2.3. Generalization . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Response of a one-degree-of-freedom system to simple shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Response spectrum . . . . . . . . . . . . . . . . . . . . 2.3.2. Absolute acceleration SRS . . . . . . . . . . . . . . . 2.3.3. Relative displacement shock spectrum . . . . . . . . 2.3.4. Primary (or initial) positive SRS . . . . . . . . . . . . 2.3.5. Primary (or initial) negative SRS . . . . . . . . . . . 2.3.6. Secondary (or residual) SRS . . . . . . . . . . . . . . 2.3.7. Positive (or maximum positive) SRS . . . . . . . . . 2.3.8. Negative (or maximum negative) SRS . . . . . . . . 2.3.9. Maximax SRS . . . . . . . . . . . . . . . . . . . . . . . 2.4. Standardized response spectra . . . . . . . . . . . . . . . . 2.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Half-sine pulse . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Versed sine pulse . . . . . . . . . . . . . . . . . . . . . 2.4.4. Terminal peak sawtooth pulse . . . . . . . . . . . . . 2.4.5. Initial peak sawtooth pulse . . . . . . . . . . . . . . . 2.4.6. Square pulse . . . . . . . . . . . . . . . . . . . . . . . . 2.4.7. Trapezoidal pulse . . . . . . . . . . . . . . . . . . . . . 2.5. Choice of the type of SRS . . . . . . . . . . . . . . . . . . 2.6. Comparison of the SRS of the usual simple shapes . . . 2.7. SRS of a shock defined by an absolute displacement of the support. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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61 65 65 65 65 66 66 66 67 67 68 69 69 71 72 74 75 77 77 78 79
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80
Table of Contents
2.8. Influence of the amplitude and the duration of the shock on its SRS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9. Difference between SRS and extreme response spectrum (ERS) 2.10. Algorithms for calculation of the SRS . . . . . . . . . . . . . . . 2.11. Subroutine for the calculation of the SRS . . . . . . . . . . . . . 2.12. Choice of the sampling frequency of the signal . . . . . . . . . . 2.13. Example of use of the SRS . . . . . . . . . . . . . . . . . . . . . . 2.14. Use of SRS for the study of systems with several degrees of freedom . . . . . . . . . . . . . . . . . . . . . . . . .
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81 82 82 83 86 90
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92
Chapter 3. Properties of Shock Response Spectra . . . . . . . . . . . . . . . .
95
3.1. Shock response spectra domains . . . . . . . . . . . . . . . . . . . . 3.2. Properties of SRS at low frequencies. . . . . . . . . . . . . . . . . . 3.2.1. General properties . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2. Shocks with zero velocity change . . . . . . . . . . . . . . . . . 3.2.3. Shocks with 'V 0 and ' D z 0 at the end of a pulse . . . . 3.2.4. Shocks with 'V 0 and ' D 0 at the end of a pulse . . . 3.2.5. Notes on residual spectrum . . . . . . . . . . . . . . . . . . . . . 3.3. Properties of SRS at high frequencies . . . . . . . . . . . . . . . . . 3.4. Damping influence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Choice of damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Choice of frequency range . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Choice of the number of points and their distribution . . . . . . . . 3.8. Charts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9. Relation of SRS with Fourier spectrum . . . . . . . . . . . . . . . . 3.9.1. Primary SRS and Fourier transform . . . . . . . . . . . . . . . . 3.9.2. Residual SRS and Fourier transform . . . . . . . . . . . . . . . 3.9.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra. . . . . . . . . 3.10. Care to be taken in the calculation of the spectra . . . . . . . . . . 3.10.1. Main sources of errors . . . . . . . . . . . . . . . . . . . . . . . 3.10.2. Influence of background noise of the measuring equipment . 3.10.3. Influence of zero shift . . . . . . . . . . . . . . . . . . . . . . . 3.11. Use of the SRS for pyroshocks . . . . . . . . . . . . . . . . . . . .
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95 96 96 96 105 108 110 111 114 114 118 118 118 120 120 122
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125 129 129 130 132 135
Chapter 4. Development of Shock Test Specifications . . . . . . . . . . . . .
139
4.1. Introduction. . . . . . . . . . . . . . . . 4.2. Simplification of the measured signal 4.3. Use of shock response spectra . . . . . 4.3.1. Synthesis of spectra . . . . . . . . 4.3.2. Nature of the specification . . . . 4.3.3. Choice of shape . . . . . . . . . . . 4.3.4. Amplitude . . . . . . . . . . . . . .
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139 140 142 142 144 144 146
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Mechanical Shock
4.3.5. Duration . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.6. Difficulties . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Other methods . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.1. Use of a swept sine . . . . . . . . . . . . . . . . . . . . 4.4.2. Simulation of SRS using a fast swept sine . . . . . . 4.4.3. Simulation by modulated random noise. . . . . . . . 4.4.4. Simulation of a shock using random vibration . . . . 4.4.5. Least favorable response technique . . . . . . . . . . 4.4.6. Restitution of an SRS by a series of modulated sine pulses . . . . . . . . . . . . . . . . . . . . . . 4.5. Interest behind simulation of shocks on shaker using a shock spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . .
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146 150 151 152 153 157 158 159
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Chapter 5. Kinematics of Simple Shocks . . . . . . . . . . . . . . . . . . . . . .
167
5.1. Introduction. . . . . . . . . . . . . . . . . . . . . 5.2. Half-sine pulse . . . . . . . . . . . . . . . . . . . 5.2.1. General expressions of the shock motion . 5.2.2. Impulse mode . . . . . . . . . . . . . . . . . 5.2.3. Impact mode . . . . . . . . . . . . . . . . . . 5.3. Versed sine pulse . . . . . . . . . . . . . . . . . 5.4. Square pulse . . . . . . . . . . . . . . . . . . . . 5.5. Terminal peak sawtooth pulse . . . . . . . . . . 5.6. Initial peak sawtooth pulse . . . . . . . . . . . .
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Chapter 6. Standard Shock Machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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167 167 167 170 171 181 183 186 188
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6.1. Main types . . . . . . . . . . . . . . . . . . . . . . . 6.2. Impact shock machines . . . . . . . . . . . . . . . . 6.3. High impact shock machines . . . . . . . . . . . . 6.3.1. Lightweight high impact shock machine . . . 6.3.2. Medium weight high impact shock machine . 6.4. Pneumatic machines. . . . . . . . . . . . . . . . . . 6.5. Specific testing facilities . . . . . . . . . . . . . . . 6.6. Programmers . . . . . . . . . . . . . . . . . . . . . . 6.6.1. Half-sine pulse . . . . . . . . . . . . . . . . . . 6.6.2. TPS shock pulse. . . . . . . . . . . . . . . . . . 6.6.3. Square pulse trapezoidal pulse . . . . . . . . 6.6.4. Universal shock programmer . . . . . . . . . .
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191 193 203 203 204 205 207 208 208 216 223 224
Chapter 7. Generation of Shocks Using Shakers . . . . . . . . . . . . . . . . .
233
7.1. Principle behind the generation of a signal with a simple shape versus time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Main advantages of the generation of shock using shakers . . . . . . . . 7.3. Limitations of electrodynamic shakers . . . . . . . . . . . . . . . . . . . .
233 234 235
Table of Contents
7.3.1. Mechanical limitations. . . . . . . . . . . . . . . . . . . . . . 7.3.2. Electronic limitations . . . . . . . . . . . . . . . . . . . . . . 7.4. Remarks on the use of electrohydraulic shakers . . . . . . . . . 7.5. Pre- and post-shocks . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Requirements . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2. Pre-shock or post-shock . . . . . . . . . . . . . . . . . . . . . 7.5.3. Kinematics of the movement for symmetric pre- and post-shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.4. Kinematics of the movement for a pre-shock or post-shock alone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.5. Abacuses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.6. Influence of the shape of pre- and post-pulses . . . . . . . . 7.5.7. Optimized pre- and post-shocks . . . . . . . . . . . . . . . . 7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2. Influence of the pre- and post-shocks on the time history response of a one- degree-of-freedom system . . . . . . . . . . . 7.6.3. Incidence on the shock response spectrum . . . . . . . . . .
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Chapter 8. Control of a Shaker Using a Shock Response Spectrum . . . . .
271
8.1. Principle of control using a shock response spectrum . 8.1.1. Problems . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2. Parallel filter method . . . . . . . . . . . . . . . . . . 8.1.3. Current numerical methods . . . . . . . . . . . . . . 8.2. Decaying sinusoid . . . . . . . . . . . . . . . . . . . . . . 8.2.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2. Response spectrum . . . . . . . . . . . . . . . . . . . 8.2.3. Velocity and displacement . . . . . . . . . . . . . . 8.2.4. Constitution of the total signal . . . . . . . . . . . . 8.2.5. Methods of signal compensation . . . . . . . . . . . 8.2.6. Iterations . . . . . . . . . . . . . . . . . . . . . . . . . 8.3. D.L. Kern and C.D. Hayes’ function . . . . . . . . . . . 8.3.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2. Velocity and displacement . . . . . . . . . . . . . . 8.4. ZERD function . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2. Velocity and displacement . . . . . . . . . . . . . . 8.4.3. Comparison of ZERD waveform with standard decaying sinusoid. . . . . . . . . . . . . . . . . . . . . . . . 8.4.4. Reduced response spectra . . . . . . . . . . . . . . . 8.5. WAVSIN waveform . . . . . . . . . . . . . . . . . . . . 8.5.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . 8.5.2. Velocity and displacement . . . . . . . . . . . . . . 8.5.3. Response of a one-degree-of-freedom system . . .
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271 271 272 273 278 278 279 282 283 284 290 292 292 292 294 294 295
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x
Mechanical Shock
8.5.4. Response spectrum . . . . . . . . . . . . . . . . . . . . . . . . 8.5.5. Time history synthesis from shock spectrum. . . . . . . . . 8.6. SHOC waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.1. Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2. Velocity and displacement . . . . . . . . . . . . . . . . . . . 8.6.3. Response spectrum . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4. Time history synthesis from shock spectrum. . . . . . . . . 8.7. Comparison of WAVSIN, SHOC waveforms and decaying sinusoid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.8. Use of a fast swept sine. . . . . . . . . . . . . . . . . . . . . . . . 8.9. Problems encountered during the synthesis of the waveforms . 8.10. Criticism of control by SRS . . . . . . . . . . . . . . . . . . . . 8.11. Possible improvements . . . . . . . . . . . . . . . . . . . . . . . 8.11.1. IES proposal . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.11.2. Specification of a complementary parameter . . . . . . . . 8.11.3. Remarks on the properties of the response spectrum . . . 8.12. Estimate of the feasibility of a shock specified by its SRS . . 8.12.1. C.D. Robbins and E.P. Vaughan’s method . . . . . . . . . 8.12.2. Evaluation of the necessary force, power and stroke . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
. . . . . . .
305 306 307 307 310 311 312
. . . . . . . . . . .
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. . . . . . . . . . .
. . . . . . . . . . .
. . . . . . . . . . .
313 314 317 319 323 323 324 329 329 329 331
Chapter 9. Simulation of Pyroshocks . . . . . . . . . . . . . . . . . . . . . . . .
337
9.1. Simulations using pyrotechnic facilities . . . . . 9.2. Simulation using metal to metal impact . . . . . 9.3. Simulation using electrodynamic shakers . . . . 9.4. Simulation using conventional shock machines
. . . .
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. . . .
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. . . .
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. . . .
337 341 342 342
Appendix: Similitude in Mechanics . . . . . . . . . . . . . . . . . . . . . . . . .
345
Mechanical Shock Tests: A Brief Historical Background . . . . . . . . . . .
349
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
351
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
365
Summary of other Volumes in the Series. . . . . . . . . . . . . . . . . . . . . .
369
Foreword to Series
In the course of their lifetime, simple items in everyday use such as mobile telephones, wristwatches, electronic components in cars or more specific items such as satellite equipment or flight systems in aircraft, can be subjected to various conditions of temperature and humidity, and more particularly to mechanical shock and vibrations, which form the subject of this work. They must therefore be designed in such a way that they can withstand the effects of the environmental conditions they are exposed to without being damaged. Their design must be verified using a prototype or by calculations and/or significant laboratory testing. Sizing and testing are performed on the basis of specifications taken from national or international standards. The initial standards, drawn up in the 1940s, were often extremely stringent, blanket specifications, consisting of a sinusoidal vibration, the frequency of which was set to the resonance of the equipment. They were essentially designed to demonstrate a certain standard resistance of the equipment, with the implicit hypothesis that if the equipment survived the particular environment, it would withstand, undamaged, the vibrations to which it would be subjected in service. Sometimes with a delay due to a certain conservatism, the evolution of these standards followed that of the testing facilities: the possibility of producing swept sine tests, the production of narrow-band random vibrations swept over a wide range and finally the generation of wide-band random vibrations. At the end of the 1970s, it was felt that there was a basic need to reduce the weight and cost of on-board equipment and to produce specifications closer to the real conditions of use. This evolution was taken into account between 1980 and 1985 concerning American standards (MIL-STD 810), French standards (GAM EG 13) and international standards (NATO), which all recommended the tailoring of tests. Current preference is to talk of the tailoring of the product to its environment in order to assert more clearly that the environment must be taken into account from the very start of the project, rather than to check the behavior of the material a
xii
Mechanical Shock
posteriori. These concepts, originating with the military, are currently being increasingly echoed in the civil field. Tailoring is based on an analysis of the life profile of the equipment, on the measurement of the environmental conditions associated with each condition of use and on the synthesis of all the data into a simple specification, which should be of the same severity as the actual environment. This approach presupposes a correct understanding of the mechanical systems subjected to dynamic loads and knowledge of the most frequent failure modes. Generally speaking, a good assessment of the stresses in a system subjected to vibration is possible only on the basis of a finite element model and relatively complex calculations. Such calculations can only be undertaken at a relatively advanced stage of the project once the structure has been sufficiently defined for such a model to be established. Considerable work on the environment must be performed independently of the equipment concerned either at the very beginning of the project, at a time where there are no drawings available, or at the qualification stage, in order to define the test conditions. In the absence of a precise and validated model of the structure, the simplest possible mechanical system is frequently used consisting of mass, stiffness and damping (a linear system with one degree of freedom), especially for: – the comparison of the severity of several shocks (shock response spectrum) or of several vibrations (extreme response and fatigue damage spectra); – the drafting of specifications: determining a vibration which produces the same effects on the model as the real environment, with the underlying hypothesis that the equivalent value will remain valid on the real, more complex structure; – the calculations for pre-sizing at the start of the project; – the establishment of rules for analysis of the vibrations (choice of the number of calculation points of a power spectral density) or for the definition of the tests (choice of the sweep rate of a swept sine test). This explains the importance given to this simple model in this work of five volumes on Vibration and Mechanical Shock: Volume 1 of this series is devoted to sinusoidal vibration. After several reminders about the main vibratory environments which can affect materials during their working life and also about the methods used to take them into account,
Foreword to Series
xiii
following several fundamental mechanical concepts, the responses (relative and absolute) of a mechanical one-degree-of-freedom system to an arbitrary excitation are considered, and its transfer function in various forms are defined. By placing the properties of sinusoidal vibrations in the contexts of the real environment and laboratory tests, the transitory and steady state response of a single-degree-offreedom system with viscous and then with non-linear damping is evolved. The various sinusoidal modes of sweeping with their properties are described, and then, starting from the response of a one-degree-of-freedom system, the consequences of an unsuitable choice of the sweep rate are shown and a rule for the choice of this rate deduced from it. Volume 2 deals with mechanical shock. This volume presents the shock response spectrum (SRS) with its different definitions, its properties and the precautions to be taken in calculating it. The shock shapes most widely used with the usual test facilities are presented with their characteristics, with indications how to establish test specifications of the same severity as the real, measured environment. A demonstration is then given on how these specifications can be produced with classic laboratory equipment: shock machines, electrodynamic exciters driven by a time signal or by a response spectrum, indicating the limits, advantages and disadvantages of each solution. Volume 3 examines the analysis of random vibration which encompasses the vast majority of the vibrations encountered in the real environment. This volume describes the properties of the process, enabling simplification of the analysis, before presenting the analysis of the signal in the frequency domain. The definition of the power spectral density is reviewed, as well as the precautions to be taken in calculating it, together with the processes used to improve results (windowing, overlapping). A complementary third approach consists of analyzing the statistical properties of the time signal. In particular, this study makes it possible to determine the distribution law of the maxima of a random Gaussian signal and to simplify the calculations of fatigue damage by avoiding direct counting of the peaks (Volumes 4 and 5). The relationships that provide the response of a degree of freedom linear system to a random vibration are established. Volume 4 is devoted to the calculation of damage fatigue. It presents the hypotheses adopted to describe the behavior of a material subjected to fatigue, the laws of damage accumulation and the methods for counting the peaks of the response (used to establish a histogram when it is impossible to use the probability density of the peaks obtained with a Gaussian signal). The expressions of mean damage and of its standard deviation are established. A few cases are then examined using other hypotheses (mean not equal to zero, taking account of the fatigue limit, non-linear accumulation law, etc.). The main laws governing low cycle fatigue and fracture mechanics are also presented.
xiv
Mechanical Shock
Volume 5 is dedicated to presenting the method of specification development according to the principle of tailoring. The extreme response and fatigue damage spectra are defined for each type of stress (sinusoidal vibrations, swept sine, shocks, random vibrations, etc.). The process for establishing a specification from the lifecycle profile of the equipment is then detailed taking into account the uncertainty factor (uncertainties related to the dispersion of the real environment and of the mechanical strength) and the test factor (function of the number of tests performed to demonstrate the resistance of the equipment). First and foremost, this work is intended for engineers and technicians working in design teams responsible for sizing equipment, for project teams given the task of writing the various sizing and testing specifications (validation, qualification, certification, etc.) and for laboratories in charge of defining the tests and their performance following the choice of the most suitable simulation means.
Introduction
Transported or on-board equipment is very frequently subjected to mechanical shocks in the course of its useful lifetime (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 ignored. The initial work on shocks was carried out in the 1930s on earthquakes and their effect on buildings. This resulted in the notion of the shock response spectrum. Testing on equipment started during World War II. Methods continued to evolve with the increase in power of exciters, making it possible to create synthetic shocks, and again in the 1970s, with the development of computerization, enabling tests to be directly conducted on the exciter employing a shock response spectrum. After a brief recapitulation of the shock shapes most often used in tests and of the possibilities of Fourier analysis for studies taking account of the environment (Chapter 1), Chapter 2 presents the shock response spectrum with its numerous definitions and calculation methods. Chapter 3 describes all the properties of the spectrum 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 the shock. The shock response spectrum is the ideal tool for drafting specifications. Chapter 4 details the process which makes it possible to transform a set of shocks recorded in the real environment into a specification of the same severity, and presents a few other methods proposed in the literature. Knowledge of the kinematics of movement during a shock is essential to the understanding of the mechanism of shock machines and programmers. Chapter 5
xvi
Mechanical Shock
gives the expressions for velocity and displacement, according to time, for classic shocks, depending on whether they occur in impact or impulse mode. Chapter 6 describes the principle of the shock machines currently most widely used in laboratories and their associated programmers. To reduce costs by restricting the number of changes in test facilities, specifications expressed in the form of a simple shock (half-sine, rectangle, sawtooth with a final peak) can occasionally be tested using an electrodynamic exciter. Chapter 7 sets out the problems encountered, stressing the limitations of such means, together with the consequences of modification, that have to be made to the shock profile, on the quality of the simulation. Determining a simple-shaped 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 occasionally 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. Chapter 8 describes the principles of the composition of the equivalent shock, gives the shapes of the basic signals most often used, with their properties, and emphasizes the problems that can be encountered, both in the constitution of the signal and with respect to the quality of the simulation obtained. Pyrotechnic devices or equipment (cords, valves, etc.) are very 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. Their simulation in the laboratory requires specific means, as described in Chapter 9. Containers must protect the equipment carried in them from various forms of disturbance related to handling and possible accidents. Tests designed to qualify or certify containers include shocks that are sometimes difficult or even impossible to produce given the combined weight of the container and its content. One relatively widely used possibility consists of performing shocks on scale models, with scale factors of the order of 4 or 5, for example. This same technique can be applied, although less frequently, to certain vibration tests. At the end of this volume, the Appendix summarizes the laws of similarity adopted to define the models and to interpret the test results.
List of Symbols
The list below gives the most frequent definition of the main symbols used in this book. Some of the symbols can have another meaning locally which will be defined in the text to avoid any confusion.
A T b
Maximum value of a t Component of shock x t Amplitude of compensation signal Indicial admittance Parameter b of Basquin’s
c C
relation N V C Viscous damping constant Basquin’s law constant
a max a t Ac
F t Frms Fm g h
b
b
d t
D D f0 e E ERS E t f f0
C) (N V Displacement associated with a t Diameter of programmer Fatigue damage Neper’s number Young’s modulus or energy of a shock Extreme response spectrum Function characteristic of swept sine Frequency of excitation Natural frequency
h t H HR
H
i IPS : k K
A rms Am A t A t
External force applied to system Rms value of force Maximum value of F t Acceleration due to gravity Interval ( f f0 ) or thickness of the target Impulse response Drop height Height of rebound Transfer function 1 Initial peak sawtooth : Imaginary part of X Stiffness or coefficient of uncertainty Constant of proportionality of stress and deformation Rms value of A t Maximum of A t Generalized excitation (displacement) First derivative of A t
xviii
A t L L : m n
nFT N p q0 q 0 q T q T q T Q Q p r(t) Re Rm R :
: s S SRS STFT S t td ti tr tR T T0 TPS u t
Mechanical shock
Second derivative of A t Length Fourier transform of A t Mass Number of cycles undergone by test-bar or material Number of points of the Fourier transform Number of cycles to failure Laplace variable or percentage of amplitude of shock Value of q T for T 0 Value of q T for T 0 Reduced response First derivative of q T Second derivative of q T Q factor (quality factor) Laplace transform of q T Time window Yield stress Ultimate tensile strength Fourier transform of the system response : Real part of X Standard deviation Area Shock response spectrum Short term Fourier transform Power spectral density Time Decay time to zero of shock Fall duration Rise time of shock Duration of rebound Vibration duration Natural period Terminal peak sawtooth Generalized response
u t u t vf vi vR v t
V xm x t x t
x t x rms x m X m
: X y t
y t y t
zm zs zsup
First derivative of u t Second derivative of u t Velocity at end of shock Impact velocity Velocity of rebound Velocity x t or velocity associated with a t Fourier transform of v t Maximum value of x t Absolute displacement of the base of a one-degree-offreedom system Absolute velocity of the base of a one-degree-offreedom system Absolute acceleration of the base of a one-degree-offreedom system Rms value of x t Maximum value of x t Amplitude of Fourier : transform X Fourier transform of x t Absolute response of displacement of mass of a one-degree-of-freedom system Absolute response velocity of the mass of a one-degreeof-freedom system Absolute response acceleration of mass of a one-degree-of-freedom system Maximum value of z t Maximum static relative displacement Largest value of z t
List of Symbols
z t
z t z t
D Gt G t 'V I I : K Kc
O
/ p S
Relative response displacement of mass of a one-degree-of-freedom system with respect to its base Relative response velocity Relative response acceleration Coefficient of restitution Temporal step Dirac delta function Velocity change Dimensionless product f0 W Phase Damping factor of damped sinusoid Relative damping of compensation signal Reduced excitation Laplace transform of O 3.14159265...
T Td Tm T0 U V Vcr Vm W W1 W2 W rms Zc Z0 : [
xix
Reduced time ( Z0 t ) Reduced decay time Reduced rise time Value of T for t W Density Stress Crushing stress Maximum stress Shock duration Pre-shock duration Post-shock duration Rms duration of a shock Pulsation of compensation signal Natural pulsation (2 S f0 ) Pulsation of excitation (2 S f ) Damping factor
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Mechanical shock
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Chapter 1
Shock Analysis
1.1. Definitions 1.1.1. Shock Shock is defined as a vibratory excitation with a duration between once and twice the natural period of the excited mechanical system. Example 1.1. Figures 1.1 and 1.2 show accelerometric signals recorded during an earthquake and during the functioning of a pyrotechnic device.
Figure 1.1. Example of shock
2
Mechanical Shock
Figure 1.2. Acceleration recorded during an earthquake
Shock occurs when a force, position, velocity or acceleration is abruptly modified and creates a transient state in the system being considered. The modification is usually regarded as abrupt if it occurs in a time period which is short compared to the natural period concerned (AFNOR definition) [NOR 93]. 1.1.2. Transient signal This concerns a vibratory signal of short duration (a fraction of a second up to several dozen seconds) – mechanical shock – but also a phase between two different states or a state of short duration, as with the functioning of airbrakes on an aircraft.
Figure 1.3. Example of transient signal
Shock Analysis
3
1.1.3. Jerk A jerk is defined as the derivative of acceleration with respect to time. This parameter thus characterizes the rate of variation of acceleration with time. 1.1.4. Simple (or perfect) shock This is a shock whose signal can be represented exactly in simple mathematical terms, e.g. half-sine, triangular or rectangular shock. 1.1.5. 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. The excitation, zero for t 0 and t ! W , can be written in the interval (0, W), in the form x t
x m sin : t
[1.1]
where x m is the amplitude of the shock and W its duration. The pulsation is equal to S : . W
Figure 1.4. Half-sine shock
Expression [1.1] becomes, in generalized form, A t
A m sin : t .
4
Mechanical Shock
For an excitation of the type force A(t) A(t)
x t Z 02
F t k
and for an acceleration,
, etc.
In reduced (dimensionless) form, and with the notations used in Volume 1, Chapter 3, the definition of shock can be: O T
Note that h
h
sin h T
[1.2]
:
S T0
Z0
W 2S
.
T0
[1.3]
2W
1.1.6. Versed sine (or haversine) shock
Figure 1.5. Period of a sine curve between two minima
This is a simple shock for which the acceleration curve according to time has the shape of a period of a sine curve between two minima.
Shock Analysis
5
Figure 1.6. Haversine shock pulse
Versed sine1 (or haversine2) shape can be represented by x t x t
xm § 2S ¨ 1 cos 2 © W 0
· t¸ ¹
½ for 0 d t d W ° ¾ elsewhere °¿
Generalized form °A t ® °A t ¯
Am § 2S · ¨1 cos t ¸ 2 © W ¹ 0
for 0 d t d W elsewhere
[1.4]
Reduced form °O T ® ° ¯O T
1 §¨ T · 1 cos 2S ¸ ¨ 2© T0 ¸¹ 0
for 0 d T d T0 elsewhere
1.1.7. Terminal peak sawtooth (TPS) shock (or final peak sawtooth (FPS)) 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.
1 One minus cosine. 2 One half of one minus cosine.
6
Mechanical Shock
Figure 1.7. Terminal peak sawtooth pulse
Terminal peak sawtooth shock pulse can be described by x t
xm
x t
0
t W
½ for 0 d t d W ° ¾ elsewhere °¿
[1.5]
It can be written in a generalized form: A t ® ¯A t
Am
for 0 d t d W
0
elsewhere
and a reduced form: O t 1 ® ¯ A t 0
for 0 d T d T0 elsewhere
1.1.8. Initial peak sawtooth (IPS) shock
This is a simple shock for which the acceleration-time curve has the shape of a triangle, where acceleration instantaneously increases up to a maximum, and then decreases to zero.
Shock Analysis
7
Figure 1.8. IPS shock pulse
Analytical expression of the initial peak sawtooth is of the form: x t x t
½ for 0 d t d W ° ¾ elsewhere °¿
t· § xm¨¨ 1 ¸¸ W¹ © 0
[1.6]
It can be written in a generalized form: ° A t ® ° A t ¯
t· § A m¨¨ 1 ¸¸ W¹ © 0
for 0 d t d W elsewhere
and a reduced form: T °O T 1 T0 ® °O T 0 ¯
for 0 d T d T0 elsewhere
1.1.9. Square 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.
8
Mechanical Shock
Figure 1.9. Square shock pulse
This shock pulse is represented by x t ® ¯ x t
xm
for 0 d t d W
0
elsewhere
[1.7]
It can be written in a generalized form: A t ® ¯A t
Am
for 0 d t d W
0
elsewhere
and a reduced form: O t 1 ® ¯ A t 0
for 0 d T d T0 elsewhere
1.1.10. Trapezoidal shock
This is a simple shock for which the acceleration-time curve grows linearly up to a given value remains constant for a certain time after which it decreases linearly to zero. 1.1.11. Decaying sinusoidal pulse
A pulse comprised of a few periods of a damped sinusoid, characterized by the amplitude of the first peak, the frequency and damping: x( t )
x m exp( 2 S [ f t ) sin( 2 S f t )
[1.8]
Shock Analysis
9
This form is interesting as it represents the impulse response of a one-degree-offreedom system to a shock. It is also used to constitute a signal of a specified shock response spectrum (shaker control from a shock response spectrum). 1.1.12. Bump test
A bump test is a test in which a simple shock is repeated many times (AFNOR) [NOR 93], [DEF 99], [IEC 87]. Standardized severities are proposed. Example 1.2.
The GAM EG 13 (first part – Booklet 43 – Shocks) standard proposes a test characterized by a half-sine: 10 g, 16 ms, 3000 bumps (shocks) per axis, 3 bumps per second [GAM 86]. 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 to those resulting from repetitive shocks likely to be 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. 1.1.13. Pyroshock
The aerospace industry uses many pyrotechnic devices, such as explosive bolts, pyrotechnic shut-off valves, cutting cords, etc. During their functioning, these devices generate mechanical shocks which are characterized by very strong levels of acceleration at very high frequencies. This can be dangerous to structures, but more often to electrical equipment. These shocks were not taken into account up until about the 1960s. It was thought that, despite their large amplitude, their duration was much too short to cause damage to materials. Several incidents occurred on missiles due to this way of thinking. A survey by C. Moening [MOE 86] shows that the failures observed in American launchers between 1960 and 1986 can be divided as follows: due to vibrations: 3; due to pyroshock: 63.
10
Mechanical Shock
We could be tempted to explain this division by the greater severity of shocks. The study by C. Moening shows that this has nothing to do with it. Instead, the causes were: in part, the difficulty of evaluating these shocks a priori; often these stresses were not taken into account during the conception and the absence of rigorous testing specifications.
Figure 1.10. Example of pyroshock
These shocks have the following general characteristics: the acceleration levels are very significant; the amplitude of the shock is not simply linked to the amount of explosive [HUG 83b]; reducing the charge does not consequently reduce the shock; the amount of metal cut by a cord, for example, is a more significant factor; the signals are oscillatory; in the near field, close to the source (material about 15 cm from the detonation point of the device, or about 7 cm for less powerful pyrotechnic devices), the shock effects are essentially linked to the propagation of a stress wave in the material;
Shock Analysis
11
beyond this (far field) the shock propagates by attenuating and the wave combines with the damped oscillatory response of the structure to its resonant frequencies; then only this response remains3 (see Table 1.1); the shocks have very close components along the three axes. Due to their high frequency, these shocks can cause damage to electronic components; a priori estimation of shock levels is neither easy nor precise.
Field
Distance from the source
Devices generating intense shocks
Shock amplitude Frequency
Near field (propagation of a stress wave)
< 7.5 cm
< 15 cm
> 5,000 g up to 300,000 g
Far field (propagation of a stress wave + structural response)
> 7.5 cm
> 100,000 Hz > 15 cm
1,000 g to 5,000 g > 10,000 Hz
Table 1.1. Characteristics of each of the areas of intensity of pyroshocks
These characteristics make them difficult to measure, necessitating sensors capable of accepting amplitudes of 100,000 g, frequencies being able to go over 100 kHz, with significant transversal components. They are also difficult to simulate. The definition of these areas can vary according to the references. For example, IEST proposes a classification according to the method of simulating them in a laboratory [BAT 08]: for the near field: controlling the frequency up to and above 10,000 Hz with amplitudes higher than 10,000 g; intermediate field: frequencies between 2,000 Hz and 10,000 Hz with amplitudes lower than 10,000 g; far field: frequencies below 3,000 Hz, amplitudes lower than 1,000 g.
3 We sometimes define a third field, the intermediate field (material at a distance of between 15 cm and 60 cm for pyrotechnic devices generating intense shocks, between 7 cm and 15 cm for less severe devices), in which the effects of the near field wave are not yet negligible and combine with a damped oscillatory response of the structure to its resonant frequencies, the far field, where only this latter effect persists.
12
Mechanical Shock
1.2. Analysis in the time domain
A shock can be described in the time domain by the following parameters: – the amplitude x t ; – duration W; – the form. The physical parameter expressed in terms of time is, in a general way, an acceleration x t , but can also be a velocity v t , a displacement x t or a force F t . In the first case, which we will particularly consider in this volume, the velocity change corresponding to the shock movement is equal to 'V
³0 x( t ) dt W
[1.9]
1.3. Fourier transform 1.3.1. Definition
The Fourier integral (or Fourier transform) of a function x t of the real absolute integrable variable t is defined by : X
³f x t e i : t dt f
[1.10]
: is generally complex and can be written by separating the The function X real and imaginary parts : and : :
: X
: i :
[1.11]
: X
: e i I : X m
[1.12]
or
with 2 : X m
and
2 2 : :
[1.13]
Shock Analysis
tan I
13
[1.14]
: is the Fourier spectrum of x t , X 2 : the energy spectrum and Thus, X m m I : is the phase.
The calculation of the Fourier transform is a one-to-one operation. By means of the inversion formula or Fourier reciprocity formula, it is shown that it is possible to : using the express in a univocal way x t according to its Fourier transform X relation x t
1
³ 2 S f
f
: ei : t d: X
[1.15]
: is itself an absolute integrable function over all the (if the Fourier transform X domain).
NOTES: 1. For :
X 0
0 f
³f x t dt
The ordinate at f = 0 of the Fourier transform (amplitude) of a shock defined by an acceleration is equal to the velocity change 'V associated with the shock (area under the curve x t ).
2. The following definitions are also sometimes found [LAL 75]: 1 f i: t dt °° X : 2 S f x t e ® f °x t X : e i : t d : °¯ f
³
[1.16]
³
° X : ° ® 1 °x t ° 2S ¯
1 2S f
³f
f
³f x t e
i: t
X : e i : t d :
dt
[1.17]
14
Mechanical Shock f x t e 2 S i : t dt °° X : f ® f °x t X : e 2 S i : t d : °¯ f
³
³
[1.18]
In this last case, the two expressions are formally symmetric. The sign of the exponent of the exponential function is sometimes also selected to be positive in the X : and negative in the expression for x t . expression for 1.3.2. Reduced Fourier transform
The amplitude and phase of the Fourier transform of a shock of a given shape can be plotted on axes where the product f W (W = shock duration) is plotted on the abscissa and on the ordinate, for the amplitude, the quantity A f x m W . In the following section, we draw the Fourier spectrum by considering simple shocks of unit duration (equivalent to the product f W ) and of the amplitude unit. It is easy, with this representation, to recalibrate the scales to determine the Fourier spectrum of a shock of the same form, but of arbitrary duration and amplitude. 1.3.3. Fourier transforms of simple shocks
1.3.3.1. Half-sine pulse The amplitude, phase, and real and imaginary parts of the Fourier transform of the half-sine are given by the following relations.
Figure 1.11. Real and imaginary parts of the Fourier transform of a half-sine pulse
Shock Analysis
Phase:
Amplitude [LAL 75]: X m
2 W x m cos S f W S
2
1 4 f W
2
I
[1.19]
S > f W 1 k @
[1.20]
(k positive integer) Imaginary part:
Real part: X m
15
W x m 1 cos 2 S f W
2
S 1 4 f W
2
[1.21]
X m
W x m sin 2 S f W
2
S 1 4 f W
2
[1.22]
Figure 1.12. Amplitude and phase of the Fourier transform of a half-sine shock pulse
1.3.3.2. Versed sine pulse
Figure 1.13. Real and imaginary parts of the Fourier transform of a versed sine shock pulse
16
Mechanical Shock
Amplitude:
Phase: sin S f W
X m
2
2 S f W 1 f W
2
Real part:
X m
x m sin 2 S f W
2
4 S f 1 f W
2
I
[1.23]
S > f W 1 k @ [1.24]
Imaginary part: x m cos 2 S f W 1 [1.25] X [1.26] m 2 2 4 S f 1 f W
>
@
Figure 1.14. Amplitude and phase of the Fourier transform of a versed sine shock pulse
1.3.3.3. TPS pulse We obtain, from [1.5]: – amplitude: X m
x m 2
4S f W
>
@
4 S f W 4 sin S f W sin S f W 2 S f W cos S f W [1.27] 2
2
2
2
Shock Analysis
17
Figure 1.15. Real and imaginary parts of the Fourier transform of a TPS shock pulse
– phase:
2 S f W cos 2 S f W sin 2 S f W
tan I
[1.28]
2 S f W sin 2 S f W cos 2 S f W 1
– real part:
X m
>
@
x m cos 2 S f W 2 S f W sin 2 S f W 1 2
2
4 S f W
[1.29]
– imaginary part:
X m
>
x m 2 S f W cos 2 S f W sin 2 S f W 2
2
4S f W
@
[1.30]
18
Mechanical Shock
Figure 1.16. Amplitude and phase of the Fourier transform of a TPS shock pulse
1.3.3.4. IPS pulse The Fourier transform calculated using relation [1.6] has the following characteristics: – amplitude: X m
x m 2
4S f W
>
4 S f W 4 sin S f W sin S f W 2 S f W cos S f W 2
2
2
2
@
[1.31]
– phase: tan I
sin 2 S f W 2 S f W 1 cos 2 S f W
[1.32]
Shock Analysis
19
Figure 1.17. Real and imaginary parts of the Fourier transform of an IPS shock pulse
– real part:
X m
>
x m 1 cos 2 S f W 2
@
[1.33]
2
4 S f W
Figure 1.18. Amplitude and phase of the Fourier transform of an IPS shock pulse
– imaginary part:
X m
>
x m sin 2 S f W 2 S f W 2
2
4 S f W
@
[1.34]
20
Mechanical Shock
1.3.3.5. Arbitrary triangular pulse Acceleration increases linearly from zero to a maximum value, then decreases linearly to zero. Let us set tr as the rise time and td as the decay time. Fourier transform
Amplitude: X m
2 x m
4 S t r W t r 2
t 2r sin 2 S f W W 2 sin 2 S f t r
^
`
W t r sin 2 S f t r sin 2 S f W sin 2 >S f W t r @
[1.35]
Phase: tanI
t r sin 2 S f W W sin 2 S f t r
W >cos 2 S f t r 1@ t r >cos 2 S f W 1@
[1.36]
Real part:
X m
x m >W cos 2 S f t r W t r cos 2 S f W t r @ 4 S 2 t r W t r
Figure 1.19. Real and imaginary parts of the Fourier transform of a triangular shock pulse
[1.37]
Figure 1.20. Real and imaginary parts of the Fourier transform of a triangular shock pulse
Shock Analysis
21
Imaginary part:
X m
x m >t r sin 2 S f W W sin 2 S f t r @
4 S 2 t r W t r
[1.38]
Figure 1.21. Amplitude and phase of the Fourier transform of a triangular shock pulse
Figure 1.22. Amplitude and phase of the Fourier transform of a triangular shock pulse
22
Mechanical Shock
1.3.3.6. Square pulse We obtain, from [1.7]:
Figure 1.23. Real and imaginary parts of the Fourier transform of a square shock pulse
– amplitude: x m sin S f W
X m
[1.39]
Sf
– phase: S > f W 1 k @
I
[1.40]
– real part:
X m
x m sin 2S f W
[1.41]
2Sf
– imaginary part:
X m
>
@
x m cos 2S f W 1 2Sf
[1.42]
Shock Analysis
23
Frequency (Hz)
Figure 1.24. Amplitude and phase of the Fourier transform of a square shock pulse
1.3.3.7. Trapezoidal pulse Let us set tr as the rise time of acceleration from zero to the constant value x m and td as the decay time to zero. Fourier transform Amplitude:
X m
: W t d ª 2 : tr sin sin 2 2 x m « 2 2 « :2 « t 2r W t d 2 ¬«
: W td W td tr : tr sin sin cos : 2 2 2 2 tr W td
1
º2 » » » »¼
[1.43]
Phase: sin : W sin : t d tan I
sin : t r
tr W td cos : t r 1 cos : t d cos : W tr W td
[1.44]
24
Mechanical Shock
Real part:
X m
x m 2
ª cos 2 S f t r 1
2«
4 S f ¬«
tr
cos 2 S f t d cos 2 S f W º » W td ¼»
[1.45]
Imaginary part:
X m
ª sin 2 S f t r sin 2 S f W sin 2 S f t d º » « tr W td 4 S 2 f 2 ¬« ¼»
x m
Figure 1.25. Real and imaginary parts of the Fourier transform of a trapezoidal shock pulse
Figure 1.26. Amplitude and phase of the Fourier transform of a trapezoidal shock pulse
[1.46]
Shock Analysis
25
1.3.4. What represents the Fourier transform of a shock? Each point of the amplitude spectrum of the Fourier transform of a shock has an ordinate equal to the largest response of a filter, having as a central frequency the abscissa of the considered point divided by the size of the filter. Example 1.3. Let us take a half-sine shock of 500 m/s2, 10 ms and the amplitude of its Fourier transform (Figure 1.27). Let us choose a point of this spectrum, at the frequency 58 Hz. The amplitude is equal to 2.29 m/s2/Hz at this frequency.
Figure 1.27. Amplitude of the Fourier transform of a half-sine shock 500 m/s2, 10 ms
Let us now consider the response of a rectangular filter with a central frequency of 58 Hz and of size 'f = 2 Hz (Figure 1.28). The maximum response of this filter (which occurs after the end of the shock) is equal to 4.58 m/s2. This value divided by the bandwidth of the filter is equal to the value read on the spectrum of Figure 1.27.
26
Mechanical Shock
Figure 1.28. Response of a filter with a central frequency of 58 Hz, of bandwidth 'f = 2 Hz, to a half-sine shock 500 m/s2, 10 ms
Figure 1.29 shows the response of a filter with a central frequency of 58 Hz and of a bandwidth equal to 1 Hz. We check that the largest peak of this response corresponds directly to the value read on the spectrum of Figure 1.27 at this frequency. The result can become less precise when the bandwidth of the filter increases.
Figure 1.29. Response of a filter with a central frequency of 58 Hz, of bandwidth 'f = 1 Hz, to a half-sine shock 500 m/s2, 10 ms
Shock Analysis
27
1.3.5. Importance of the Fourier transform The Fourier spectrum contains all the information present in the original signal, in contrast, we will see, to the shock response spectrum (SRS). It is shown that the Fourier spectrum R : of the response at a point in a : of the input shock and the structure is the product of the Fourier spectrum X transfer function H : of the structure: R :
: H : X
[1.47]
The Fourier spectrum can thus be used to study the transmission of a shock through a structure, the movement resulting at a certain point being then described by its Fourier spectrum. This relation can also be used to determine the transfer function between two points of a structure using the measures under shock: H (f )
(f ) Y (f ) X
[1.48]
It is desirable that the amplitude of the Fourier transform of the excitation has a value far removed from zero over all considered frequency domains, in order to avoid a division by a very small value which could artificially lead to a peak on the transfer function. The shock shapes such as the half-sine or square are thus prohibited. The response in the time domain can be also expressed from a convolution utilizing the “input” shock signal according to the time and the impulse response of the mechanical system considered. An important property is used here: the Fourier transform of a convolution is equal to the scalar product of the Fourier transforms of the two functions in the frequency domain. It could be thought that this (relative) facility of change in domain (time frequency) and this convenient description of the input or of the response would make the Fourier spectrum method one frequently used in the study of shock, in particular for the writing of test specifications from experimental data. These mathematical advantages, however, are seldom used within this framework because when we want to compare two excitations, we run up against the following problems: – the need to compare two functions. The Fourier variable is a complex quantity which thus requires two parameters for its complete description: the real part and the
28
Mechanical Shock
imaginary part (according to the frequency) or the amplitude and the phase. These curves in general are very little smoothed and, except in obvious cases, it is difficult to decide on the relative severity of two shocks according to frequency when the spectrums overlap. In addition, the phase and the real and imaginary parts can take positive and negative values and are thus not very easy to use to establish a specification; – the signal obtained by inverse transformation generally has, a complex form impossible to reproduce with the usual test facilities, except, with certain limitations, on electrodynamic shakers. The Fourier transform is not used for the development of specifications or the comparison of shocks. The one-to-one relation property and the input-response relation [1.47] make it a very interesting tool to control shaker shock whilst calculating the electric signal by applying these means to produce a given acceleration profile at the input of the specimen, after taking into account the transfer function of the installation. 1.4. Energy spectrum 1.4.1. Energy according to frequency The energy spectrum ES is defined here by:
ES
³ > FT(f )@ f
2
0
df
[1.49]
where FT(f) is the amplitude of the Fourier transform of the signal. This spectrum can be used, for example, to determine the frequency below which the signal includes p percent of the total energy:
p%
³ > FT(f )@ 100 ³ > FT(f )@ fp %
2
0 f max
2
0
df df
[1.50]
Shock Analysis
29
1.4.2. Average energy spectrum The Fourier transform of a shock that has a random character, such as pyroshocks, only gives one image of the frequency content of the shocks of this family amongst many others. It is possible to describe this frequency content statistically using a power spectral density function defined by an average of the squares of transform amplitudes calculated from several measurements. If n is the number of measurements, the average energy is given by: E(f )
1
n
¦ FTi (f ) 2
ni 1
[1.51]
where FTi(f) is the amplitude of the Fourier transform of measurement i calculated for measurement frequencies. This spectrum is called the “energy spectrum” or the “transient autospectrum” by analogy with the autospectrum calculated from several samples of a random signal (Volume 3). 1.5. Practical calculations of the Fourier transform 1.5.1. General Among the various possibilities of calculation of the Fourier transform, the Fast Fourier Transform (FFT) algorithm of Cooley–Tukey [COO 65] is generally used because of its speed (Volume 3). It should be noted that the result issuing from this algorithm must be multiplied by the duration of the analyzed signal to obtain the Fourier transform. 1.5.2. Case: signal not yet digitized Let us consider an acceleration time history x t of duration W which one wishes to calculate the Fourier transform with n FT points (power of 2) until the frequency f max . According to Shannon’s theorem (Volume 1), it is enough that the signal is sampled with a frequency f samp. 2 f max , i.e. that the temporal step is equal to:
30
Mechanical Shock
't
1
1
f samp.
2 f max
[1.52]
The frequency interval is equal to: 'f
f max
[1.53]
n FT
To be able to analyze the signal with a resolution equal to 'f , it is necessary that its duration is equal to: T
1
[1.54]
'f
yielding the temporal step: 't
1
1
2 f max
2 'f n FT
T n FT
[1.55]
If n is the total number of points describing the signal: T
n 't
[1.56]
and we must have: n
[1.57]
2 n FT
yielding: T
n 't
2 n FT
1
n FT
2 f max
f max
[1.58]
The duration T needed to be able to calculate the Fourier transform with the selected conditions can be different from the duration W from the signal to be analyzed (for example, in the case of a shock). It cannot be smaller than W (if not 1 shock shape would be modified). Thus, if we set 'f 0 , the condition T t W leads W 1 1 to t , i.e. to: 'f 'f 0
Shock Analysis
'f
f max n FT
d
1
31
[1.59]
W
If the calculation data ( n FT and f max ) lead to a too large value of 'f , it will be necessary to modify one of these two parameters to satisfy the above condition. If it is necessary that the duration T is larger than W, zeros must be added to the signal to analyze the difference between W and T, with the temporal step 't . The computing process is summarized in Table 1.2. Data: – Characteristics of the signal to be analyzed (shape, amplitude, duration) or one measured signal not yet digitized. – The number of points of the Fourier transform ( n FT ) and its maximum frequency ( f max ). f samp.
't 'f
2 f max
1 2 f max f max n FT
n
2 n FT
T
n 't
Condition to avoid the aliasing phenomenon (Shannon’s theorem) . If the measured signal can contain components at frequencies higher than f max , it must be filtered using a low-pass filter before digitization. To take account of the slope of the filter beyond f max , it is preferable to choose f samp. 2.6 f max (Volume 1). Temporal step of the signal to be digitized (time interval between two points of the signal). Frequency interval between two successive points of the Fourier transform. Number of points of the signal to be digitized. Total duration of the signal to be treated.
If T > W, zeros must be added between W and T. If there are not enough points to correctly represent the signal between 0 and W, f max must be increased. f max 1 The condition 'f d must be satisfied (i.e. T t W ): W n FT – if f max is imposed, take n FT (power of 2) t W f max . – if n FT is imposed, choose f max d
n FT W
.
Table 1.2. Computing process of a Fourier transform starting from a non-digitized signal
32
Mechanical Shock
1.5.3. Case: signal already digitized If the signal of duration W is already digitized with N points and a step 'T, the calculation conditions of the transform are fixed: f max
n FT d
1
[1.60]
2 't
N 2
(nearest power of 2)
and n
2 n FT
This number shall be a power of two. It is thus necessary, in order to respect this condition, to reduce the duration of the shock if possible without degrading the signal, and add several zeros to the end of the signal. If however we want to choose a priori f max and n FT , the signal must be resampled and if required zeros must be added using the principles in Table 1.2. The frequency step of the transform is thus equal to 'f
f max n FT
.
1.5.4. Adding zeros to the shock signal before the calculation of its Fourier transform
The Fourier transform calculated in the conditions of the previous section is often very difficult to read, the useful part of the spectrum being squashed against the amplitude axis and defined by very few points. Adding zeros to the signal to be analyzed before calculating its Fourier transform enables us to obtain a defined curve with many more points and which is much smoother.
Shock Analysis
33
Example 1.4. Let us consider a half-sine shock 500 m/s2, 10 ms. In order to simplify things, let us assume that it has been digitized with 256 points (power of two). Using the previous relations, we determine the following calculation conditions: W 0.01 | 3.91 105 s ; temporal step: Gt n 256 n 128 ; number of points of the transform: n FT 2 1 1 maximum frequency: f max 12,800 Hz ; 2 Gt 2 3.91 105
frequency step: 'f
f max n FT
12,800 128
100 Hz.
The Fourier transform obtained is defined by 128 points spaced every 100 Hz between 0 Hz and 12,800 Hz (Figure 1.30), thus with very few points (5) in the most interesting frequency band (0 Hz to 500 Hz) (Figure 1.31).
Figure 1.30. Amplitude of the Fourier transform (without addition of zeros)
34
Mechanical Shock
Figure 1.31. Amplitude of the Fourier transform between 0 Hz and 500 Hz (without addition of zeros)
Two solutions present themselves to increase the number of points in the useful frequency band. The first consists simply of adding zeros to the end of the shock in order to artificially increase the number of points that define the signal (the total number of points always being equal to a power of two). Since nFT = n/2, we thus increase the number of frequency points by the same ratio. Example 1.5.
Let us look again at the signal defined by 256 points and add 256 zeros 31 times. The signal obtained is thus made up of 5,376 points, the temporal step remaining the same (Gt = 3.91.10–5 s). The total duration is thus equal to 21 u 0.01 s, i.e. 0.21 s. The number of frequency points changes from 128 to (8192 / 2 =) 4096 spread between 0 Hz and 12,800 Hz (this maximum frequency does not change since the frequency step is maintained). The increased amplitude of the Fourier transform between 0 and 500 Hz, now made up of 160 points, is much more smooth (Figure 1.32). The addition of zeros does not bring any additional information. It simply allows us, through the artificial
Shock Analysis
35
increase of the length of the signal (T > W), to reduce the frequency step of the transform ('f = 1/T).
Figure 1.32. Amplitude of the Fourier transform of the signal with the addition of zeros, between 0 Hz and 500 Hz
The disadvantage of this process is however the necessity of calculating and storing a large number of points (4,096 in our example) in one file of which only a small part will be used (here 160). The second solution consists of resampling the signal and adding zeros. Let us assume that we know the frequency range that we are interested in, from 0 Hz to 500 Hz in our example, and let us choose the number of frequency points (or the frequency step, but it must lead to a number of points equal to a power of two): f max
n FT
500 Hz ; 128.
The data of the maximum frequency of the transform fixes the temporal step: Gt
1 2 f max
1 2 x 500
0.001 s
36
Mechanical Shock
The number of points to define the signal (n = 2nFT = 256) leads to the duration of the signal: T = 256 x 0.001 = 0.256 s, a duration that is larger than that of the half-sine shock. It is thus necessary, using these elements: to resample the signal by interpolation with a step Gt = 0.001 s (instead of the initial step 3.91.10–5 s); then to complete the signal with zeros until the total duration is equal to 0.256 s. The calculation leads to a curve that is very close to that in Figure 1.32, but here all the points are useful. If we can imagine the calculation of the Fourier transform before the digitization of the signal, the same procedure can be followed to determine the digitization frequency and the signal length to be used. 1.6. The interest of time-frequency analysis 1.6.1. Limit of the Fourier transform
The spectral components of a shock can be analyzed using its Fourier transform, given by the relation: FT f
f
³ f
x t e 2
Si f t
dt
[1.61]
The Fourier transform: indicates the degree of similarity of the entire signal with a sine curve of frequency f; describes the frequency content of the signal in its totality. A variation of the frequencies during the duration of the shock cannot be seen.
Shock Analysis
37
Example 1.6. Figure 1.33 shows a signal made up of 4 sine curves with the same amplitude in series, with successive frequencies equal to 20 Hz, 50 Hz, 100 Hz and 150 Hz.
Figure 1.33. Signal made up of 4 sine curves in series (20 Hz, 50 Hz, 100 Hz and 150 Hz)
Figure 1.34. Fourier transform of a signal made up of a series of four sine curves
The transform of this signal (Figure 1.34) does indeed show four lines at these frequencies, but we are not given any information on the moment they appeared.
38
Mechanical Shock
It is very close to the Fourier transform (Figure 1.36) of a signal made up of the sum of these four sine curves (Figure 1.35), which are present here throughout the entire duration of the signal.
Figure 1.35. Signal made up of the sum of 4 sine curves from Figure 1.33 over a duration of 1 second
Figure 1.36. Fourier transform of a signal made up of the sum of four sine curves
It thus seems useful to look for a means of analysis enabling us to distinguish between these two situations.
Shock Analysis
39
1.6.2. Short term Fourier transform (STFT)
A first idea can consist of calculating the Fourier transform of signal samples isolated using a “sliding” window (Figure 1.37) and plotting these different transforms according to time. The result can be expressed in the form of a 3D amplitude-time-frequency diagram (STFT) [BOU 96] [GAD 97] [WAN 96].
Figure 1.37. Sliding window
The STFT is given by the relation: STFT f , W
³
f
f
ª¬ x t r t W º¼ e 2 S i f
t
dt
[1.62]
where x( t ) is the signal to be analyzed and r(t) is the rectangular window. Thus, the analysis is with constant bandwidth. Not very sensitive to noise, it is often used to identify harmonics in a signal.
40
Mechanical Shock
Example 1.7.
Linear swept sine between 20 Hz and 200 Hz in 2 seconds.
Figure 1.38. Modulus of the STFT of a swept sine (20 Hz to 200 Hz in 2 s)
Example 1.8.
STFT of a signal made up of 4 successive sine curves from Figure 1.33.
Figure 1.39. STFT of 4 sine curves from Figure 1.33
In this representation, we obtain an image that is symmetric with respect to the frequency axis. Consequently, eight peaks appear in this example.
Shock Analysis
41
Uncertainty principle (Heisenberg)
The shorter the duration of the window, the better the temporal resolution, but the worse the frequency resolution. We cannot know with precision which frequencies exist at a given moment. We only know the content of the signal: in a certain frequency band; in a given time interval. A window of a finite length: only covers part of a signal; leads to a bad frequency resolution. With a window of infinite length, the frequency resolution is perfect, but we do not have any temporal information. The shorter the duration of the window, the better the temporal resolution and the worse the frequency resolution. Disadvantages of a rectangular window: it shows the lobes in the frequency domain; of a finite duration, it leads to a poor frequency resolution (Fourier transform:
e
jZ t
).
The Gabor window (Gaussian function) minimizes this inequality: r t
1
2 S
14
b
2 2 e t 4 b
[1.63]
Other windows can be used for the same purpose (for example, Hamming, Hanning, Nuttall, Papoulis, Harris, triangular, Bartlett, Bart, Blackman, Parzen, Kaiser, Dolph, Hanna, Nutbess, spline windows, etc.).
42
Mechanical Shock
Example 1.9.
In this example we have chosen the Hamming window.
Figure 1.40. Hamming window used for the STFT of Figure 1.39
Figures 1.41 and 1.43 show the STFT obtained with a window 10 times smaller and 10 times larger, respectively. We note that a window of short duration enables us to date the changes in frequency very well, but leaves a certain imprecision on the frequency value (the peaks are large). On the other hand, a large window enables us to read the frequency with precision, but the peaks overlap and we cannot date the changes in frequency precisely.
Figure 1.41. STFT calculated with a window 10 times smaller (Figure 1.42)
Shock Analysis
Figure 1.42. Window 10 times smaller than that of Figure 1.40
Figure 1.43. STFT calculated with a window 10 times larger (Figure 1.44)
43
44
Mechanical Shock
Figure 1.44. Window 10 times larger than that of Figure 1.40
1.6.3. Wavelet transform
The transformation into wavelets enables us to avoid this difficulty. The principle of calculating this transform resembles that of STFT, in that it consists of multiplying the different parts of the signal by a function of a given form, the wavelet [BOU 96] [GAD 97] [WAN 96]. There are however two important differences: here we do not calculate the Fourier transform of the signal obtained after this windowing (there are thus no negative frequencies); the size of the window is changed for the calculation of the transform of each of the spectral components. The signal is broken down on the basis of elementary signals constructed by the expansion or compression of a “mother” function. The calculation can be explained as follows. 1) Choice of the wavelet, which is expressed in its general form
r(t)
§ t W· <¨¨ ¸¸ s © s ¹
1
[1.64]
This function of time t includes two parameters: constant W which characterizes the position of the wavelet with respect to the signal (translation of the window); constant s, scale parameter, which defines the size of the window.
Shock Analysis
45
Example 1.10.
Gauss wavelet (Figure 1.45): r t
e t
2
2 s2
[1.65]
Morlet wavelet (Figure 1.46):
r(t )
t2
ei a t e 2 s
[1.66]
“Mexican hat” (second derivation of a Gauss function) (Figure 1.47):
r(t)
1 s3
ª t2 « 2s 2 «e 2S « ¬
Figure 1.45. Gauss wavelet
º § t2 ·» ¨ 1¸ » ¨ s2 ¸ © ¹» ¼
Figure 1.46. Morlet wavelet
[1.67]
Figure 1.47. “Mexican hat” wavelet
2) Choice of a wavelet of size s for a first analysis
The wavelet is placed at the beginning of the signal to be analyzed and multiplied by the signal values, as for a windowing. The signal obtained is integrated over its entire duration and divided by 1 / s with a view to normalizing this for all scales:
46
Mechanical Shock
TO
1 s
T
§t W· ¸¸ dt s ¹
³0 x(t) r¨¨©
[1.68]
The result of the calculation is thus a single value.
Figure 1.48. Application of a window at the beginning of a signal to be studied
The calculation is performed again with the same wavelet and the same wavelet size s, after having moved it by one step with respect to the origin of the signal to be analyzed.
Figure 1.49. Application of a window in the middle of a signal to be studied
Shock Analysis
47
When the entire length of the signal has thus been swept, we obtain a first curve for a given value of s.
Figure 1.50. Application of a window at the end of a signal to be studied
3) All of these calculations are reproduced for a second value of s, then a third value, etc., until all the chosen values of s have been used.
Figure 1.51. Application of a larger window to the signal to be studied
48
Mechanical Shock
Figure 1.52. Application of an even larger window to the signal to be studied
4) All the plotted curves according to s delimit a surface, the wavelet transform
This surface is plotted according to two parameters, time and scale s: time corresponds to the position of the window with respect to the signal; scale s characterizes the fineness of the analysis: a small scale shows detailed information, while a large scale gives more global information. The scale changes inversely to the frequency. The wavelet transform is continuous, the time and scale varying theoretically in a continuous manner. We have seen that the resolution of the STFT frequencies is constant in time and frequency and that we cannot have good resolution on both parameters at the same time. The resolution of the wavelet transform has the properties given in Table 1.3. Resolution In scale
In frequency
In time
Small scales
Good
Bad
Good
Large scales
Bad
Good
Bad
Table 1.3. Quality of the wavelet transform resolution
Shock Analysis
49
These properties, added to a lack of sensitivity to noise, make up a good tool for the analysis of shocks made up of low frequency responses, which are damped less quickly than high frequency components. Example 1.11.
The wavelet transform of a signal made up of 4 sine curves (Figure 1.33) is plotted in Figure 1.53.
Figure 1.53. Wavelet transform of a signal made up of 4 sine curves (Figure 1.33)
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Chapter 2
Shock Response Spectrum
2.1. Main principles A shock is an excitation of short duration which induces transitory dynamic stress in structures. These stresses are a function of: – the characteristics of the shock (amplitude, duration and form); – the dynamic properties of the structure (resonance frequencies and Q factors). The severity of a shock can thus be estimated only according to the characteristics of the system which undergoes it. In addition, the evaluation of this severity requires the knowledge of the mechanism leading to a degradation of the structure. The two most common mechanisms are: – exceeding a value threshold of the stress in a mechanical part, leading 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. shock recorded on the landing gear of an aircraft, operation of an electromechanical contactor, etc.), the fatigue damage accumulated in the structural elements can lead to fracture in the long term. We will deal with this aspect later on. The severity of a shock can be evaluated by calculating the stresses on a mathematical or finite element model of the structure and, for example, comparison with the ultimate stress of the material. This is the method used to dimension the structure. Generally, however, the problem is instead to evaluate the relative severity
52
Mechanical Shock
of several shocks (shocks measured in the real environment, measured shocks with respect to standards, establishment of a specification etc.). This comparison would be difficult to carry out if one used a fine model of the structure and besides, this is not always available, in particular at the stage of the development of the specification of dimensioning. One searches for a method of general nature, which leads to results which can be extrapolated to any structure. A solution was proposed by M.A. Biot [BIO 32] in 1932 in a thesis on the study of the effects of earthquakes on buildings; this study was then generalized to analysis of all kinds of shocks. The study consists of applying the shock under consideration to a “standard” mechanical system, which thus does not claim to be a model of the real structure, composed of a support and N linear one-degree-of-freedom resonators, each one comprising a mass m i , a spring of stiffness k i and a damping device ci , chosen ci is the same for all N such that the fraction of critical damping [ 2 ki mi resonators (Figure 2.1).
Figure 2.1. Model of the shock response spectrum (SRS)
When the support is subjected to the shock, each mass m i has a specific
movement response according to its natural frequency f0 i
1
ki
2S
mi
and to the
chosen damping [, while a stress Vi is induced in the elastic element. The analysis consists of seeking the largest stress V m i observed at each frequency in each spring. A shock A is regarded as more severe than a shock B if it
Shock Response Spectrum
53
induces a large extreme stress in each resonator. We then carry out an extrapolation, which is certainly open to criticism, by assuming that, if shock A is more severe than shock B when it is applied to all the standard resonators, then it is also more severe with respect to an arbitrary real structure (which can be non-linear or have a single degree of freedom). NOTE: A study was carried out in 1984 on a mechanical assembly composed of a circular plate on which one could place different masses and thus vary the number of degrees of freedom. The stresses generated by several shocks of the same spectra (in the frequency range including the principal resonance frequencies), but of different shapes [DEW 84], were measured and compared. It was noted that for this assembly, whatever the number of degrees of freedom: – two pulses of simple form (with no velocity change) having the same spectrum induce similar stresses, the variation not exceeding 20% approximately. It is the same for two oscillatory shocks; – the relationship between the stresses measured for a simple shock and an oscillatory shock can reach 2. These results were supplemented by numerical simulation intended to evaluate the influence of non-linearity. Even for very strong non-linearity we did not note, for the cases considered, a significant difference between the stresses induced by two shocks of the same spectrum, but of a different form. A complementary study was carried out by B.B. Petersen [PET 81] in order to compare the stresses directly deduced from a SRS with those generated on an electronics component by a half-sine shock envelope of a shock measured in the environment, and by a shock of the same spectrum made up from WAVSIN signals (Chapter 8) added with various delays. The variation between the maximum responses measured at five points in the equipment and the stresses calculated starting from the shock response spectra does not exceed a factor of 3 in spite of the important theoretical differences between the model of the response spectrum and the real structure studied. A more recent study by D.O. Smallwood [SMA 06] shows that we can find two signals with very close SRS which can lead to responses in a ratio of 1.4 on a system with several degrees of freedom.
For applications deviating from the assumptions of definition of the SRS (linearity, only one degree of freedom), it is desirable to observe a certain prudence if we want to quantitatively estimate the response of a system starting from the spectrum [BOR 89]. The response spectra are used more often to compare the severity of several shocks.
54
Mechanical Shock
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 (Figure 2.2). If the mass-spring-damper system is assumed to be linear, it is then appropriate to compare two shocks by the maximum response stress V m they induce or by the maximum relative displacement z m that they generate, since: Vm
K zm
[2.1]
Figure 2.2. Stress–strain curve
z m is only a function of the dynamic properties of the system, whereas V m is also a function, via K, of the properties of the materials which constitute it. 2
The curve giving the largest relative displacement zsup multiplied by Z0 according to the natural frequency f0 , for a given [ damping, is the SRS. The first work defining these spectra was published in 1933 and 1934 [BIO 33] [BIO 34], then in 1941 and 1943 [BIO 41] [BIO 43]. The SRS, then named the shock spectrum, was presented there in the current form. This spectrum was used in the field of environmental tests from 1940 to 1950: J.M. Frankland [FRA 42] in 1942, J.P. Walsh and R.E. Blake in 1948 [WAL 48], R.E. Mindlin [MIN 45]. Since then, there have been many works which used it as a tool for analysis and for the simulation of shocks [HIE 74] [KEL 69] [MAR 87] [MAT 77].
Shock Response Spectrum
55
2.2. Response of a linear one-degree-of-freedom system 2.2.1. Shock defined by a force
Given a mass-spring-damping system subjected to a force F t applied to the mass, the differential equation of the movement is written as: 2
d z
m
dt
2
c
dz
kz
dt
[2.2]
F( t )
Figure 2.3. Linear one-degree-of-freedom system subjected to a force
where z t is the relative displacement of the mass m relative to the support in response to the shock F t . This equation can be put in the form: 2
d z dt
2
2 [ Z0
dz dt
2
Z0 z
F( t ) m
[2.3]
where: [
c 2
km
[2.4]
(damping factor) and: Z0
k m
(natural pulsation).
[2.5]
56
Mechanical Shock
2.2.2. Shock defined by an acceleration
Let us set as x( t ) an acceleration applied to the base of a linear one-degree-offreedom mechanical system, with y( t ) the absolute acceleration response of the mass m and z t the relative displacement of the mass m with respect to the base. The equation of the movement is written as above: 2
d y
m
dt
2
k ( y x ) c (
dy dt
dx
dt
[2.6]
)
Figure 2.4. Linear one-degree-of-freedom system subjected to acceleration
i.e.: 2
d y dt
2
dy
2 [ Z0
dt
or, while setting z t 2
2
Z0 x( t ) 2 [ Z0
dx dt
[2.7]
y t x t :
d z
dz
dt
dt
2 [ Z0 2
2
Z0 y
2
Z0 z
2
d x dt
2
[2.8]
2.2.3. Generalization
Comparison of differential equations [2.3] and [2.8] shows that they are both of the form:
Shock Response Spectrum 2
d u dt
du
2 [ Z0
2
dt
2
2
Z0 u ( t )
Z0 A( t )
57
[2.9]
where A t and u t are generalized functions of the excitation and response. NOTE: Generalized equation [2.9] can be written in the reduced form: d 2q dT
2
2[
dq q( T ) dT
O( T )
[2.10]
where: q( T )
T
u( T )
[2.11]
Am
Z0 t
O( t )
[2.12]
A (t ) Am
[2.13]
A m = maximum of A t . Resolution Differential equation [2.10] can be integrated by parts or by using the Laplace transformation. We obtain, for zero initial conditions, an integral called Duhamel’s integral: q ( T)
1
³ O(G ) e[ ( TG ) sin 2 0 T
1 [
2
1 [ ( T G ) dG
[2.14]
where G = variable of integration. In the generalized form, we deduce that: u( t )
Z0
³0 A(D) e[Z ( t D) sin Z0 t
2
1 [
0
2
1 [ ( t D ) dD
[2.15]
where D is an integration variable homogenous with time. If the excitation is an acceleration of the support, the response relative displacement is given by:
58
Mechanical Shock
1
z( t )
x( D ) e [Z ( t D ) sin Z0 ³ 2 0 t
Z0 1 [
2
1 [ ( t D ) dD
0
[2.16]
and the absolute acceleration of the mass by: Z0
y( t )
1 [
³0 x(D) e[ Z t
2
0
( t D ) ª
2
1 2 [ ¬«
sin Z
0
2
1 [ ( t D)
º 2 2 2 [ 1 [ cos Z0 1 [ ( t D ) » dD ¼
[2.17]
Application
Let us consider a package intended to protect a material from mass m and comprising a suspension made up of two elastic elements of stiffness k and two dampers with damping constant c. k
1.2337 106 N/m
c 1.57 10 3 N.s/m m
100 kg
Figure 2.5. Model of the package
Figure 2.6. Equivalent model
Shock Response Spectrum
59
We want to determine the movement of mass m after free fall from a height of h 5 m, assuming that there is no rebound of the package after the impact on the ground and that the external frame is not deformable (Figure 2.5). This system is equivalent to the model in Figure 2.6. We have (Volume 1, Chapter 4): / p
Q p
2
p 2 [ p 1 mg
A T
q0
and: q 0
q 0
K
mg
, zs
K
p q 0 q 0 2 [ p
[2.18]
2
p 2 [ p 1 mg
, O T
K zs
mg mg K K
1
0
dz T
1
dT zs
2gh
1
2mgh
Z0 zs
zs
K
dz
Z0 zs dt
K
2mgh
2Kh
mg
K
mg
q 0 2 p 2 [ p 1 p p 2 [ p 1
Q p
1
p
1
e
1
2
p2[ 2
p 2 [ p 1 [T
1 [
2
[2.19]
[2.20]
Q p
q T
q 0 2
p 2 [ p 1
ª 2 2 2 º «¬[ sin 1 [ T 1 [ cos 1 [ T»¼ q 0
e
[T
2
2
1 [
sin 1 [ T
[2.21]
60
Mechanical Shock
where: T
K
Z0 t
t
m
and: C
[
2
C
Km
2 m Z0
With the chosen numerical values, it becomes: 2.47 10 6 100
Z0
Z0
f0
2S
| 0.1
2 2 100 157.16
z t
z t
z t
| 25 Hz
2 3.14 103
[
157.16 Hz
mg K
q t
g 2
Z0
q t
[Z t g ° e 0 ®1 2 2 Z0 ¯° 1 [
º ª 2 2 2 «¬ [ sin 1 [ Z0 t 1 [ cos 1 [ Z0 t »¼
½ ° sin 1 [ Z0 t ¾ 2 °¿ 1 [
2gh e zs Z0
[2.22]
[Z0 t
2
[Z t ½ g ° e 0 ª º° 2 2 2 ®1 q [ sin 1 [ Z0 t 1 [ cos 1 [ Z0 t »¾ 2 2 ¬« 0 ¼° Z0 °¯ 1 [ ¿ [2.23]
Shock Response Spectrum
q 0
2Kh
2 2.47 106 5
mg
100 9.81
61
| 158.677
z t | 3.97 10 4 e 15.7 t 0.06327 sin 156.37 t 3.97 10 4 cos 156.37 t
From this it is easy to deduce the velocity z t and the acceleration z t from successive derivations of this expression. The first term corresponds to the static deformation of the suspension under load of 100 kg.
2.2.4. Response of a one-degree-of-freedom system to simple shocks Half-sine pulse Let us set T0
Z0 W (W = shock duration) and h
S T0
.
0 d T d T0 q T
ª h2 1 § [T 2 · « sin h T 2 [ ¨ cos h T e cos 1 [ T¸ 2 2 2 ¹ © h 2 h 4 [ 1 «¬ h h
e
[T
1[
q T
2
1 2 [
2
h
2
A T
sin
º 2 1 [ T» » ¼
[2.24]
[2.25]
T t T0 q T
A T A T T0
Versed sine pulse 0 d T d T0
[2.26]
62
Mechanical Shock
D
q T
2S T0
ª 1 D2 2[ 2 2 2 [ T « cos D T sin D T 4 [ 1 D e cos 1 [ T 2 D 2 2 N «¬ D
1
[
N
D
e
2
[ T
1 [
2
º 2 2 2 12 [ 3 D 5 sin 1 [ T» » ¼
1 D
2 2
2
4[ D
[2.27]
2
T t T0
q T
A T A T T0
[2.28]
Square pulse 0 d T d T0 q T
1
e
[T
1 [
ª 2 2 2 «¬ 1 [ cos 1 [ T [ sin 1 [
2
º T» { 1 A T [2.29] ¼
T t T0 q T
A T A T T0
[2.30]
IPS pulse 0 d T d T0 q T
ª
1 e
[ T«
« ¬
2
cos 1 [ T
º 2 sin 1 [ T» 2 » 1 [ ¼
[
2 ª 1 ° 2 [ 1 2 [ T « 2 ®T 2 [ e 2 [ cos 1 [ T sin 1 [ 2 « T0 °¯ 1 [ ¬
º½ ° T»¾ »° ¼¿
[2.31]
Shock Response Spectrum
q T
A T B T
63
[2.32]
T t T0
q T
A T B T B T T0
[2.33]
TPS pulse 0 d T d T0 q T
ª « T 2 [ 2 [ e [ T cos 1 [2 T 2 [2 1 T0 «¬
1
º 2 sin 1 [ T» { A T 2 » 1[ ¼
e
[ T
[2.34]
T t T0 B T
1 e
[ T T0
q T
cos 1 [
2
T T0 [
e
[ T T0 2
1 [
sin 1 [
2
T T0
A T A T T0 B T
[2.35]
[2.36]
Arbitrary triangular pulse 0 d T d Tr q T
1 ª« e [ T sin 1 [ 2 T 2 [ 2 [ e [ T cos 1 [ 2 T 2 [ 2 1 2 Tr « 1 [ ¬
q T
1 Tr
A T
º T» » ¼
[2.37]
[2.38]
T r d T d T0 q T
T t T0
1
Tr
A T
T0
T r T0 T r
A T T0
[2.39]
64
Mechanical Shock
q T
1
A T
Tr
T0
Tr T0 Tr
A T Tr
1
T0 T r
A T T0
[2.40]
Trapezoidal pulse
Figure 2.7. Trapezoidal shock pulse
0 d T d Tr
q T
1 Tr
A T
[2.41]
where: A T
T 2 [ 2 [ e
[T
2
2
cos 1 [ T 2 [ 1
e
[T 2
1[
2
sin 1 [ T
[2.42]
T r d T d Td q T
1 Tr
>A T A T Tr @
[2.43]
Td d T d T0 q T T t T0
1 Tr
>A T A T Tr @
1 T0 Td
A T Td
[2.44]
Shock Response Spectrum
q T
1 Tr
>A T A T T r @
1 T0 Td
A T T d
1 T0 Td
A T T0
65
[2.45]
For an isosceles trapezoid, we set T r T0 Td . If the rise and decay each have a duration equal to 10% of the total duration of the trapezoid, we have T0 . T r T0 T d 10 2.3. Definitions 2.3.1. Response spectrum A curve representative of the variations of the largest response of a linear onedegree-of-freedom system subjected to a mechanical excitation, plotted against its Z0 , for a given value of its damping ratio. natural frequency f0 2S 2.3.2. Absolute acceleration SRS 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 could be measured using an accelerometer fixed to this mass): the response spectrum is then called the absolute acceleration SRS. This spectrum can be useful when absolute acceleration is the easiest parameter to compare with a characteristic value (study of the effects of shock on a man, comparison with the specification of an electronics component, etc.). 2.3.3. Relative displacement shock spectrum In similar cases, we often calculate the relative displacement of the mass with respect to the base of the system, displacement which is proportional to the stress created in the spring (since the system is regarded as linear). In practice, we 2
generally express, in ordinates, the quantity Z0 zsup called the equivalent static acceleration. This product has the dimension of an acceleration, but does not represent the acceleration of the mass except when damping is zero; this term is then strictly equal to the absolute acceleration of the mass. However, when damping is close to the current values observed in mechanics, and in particular when [ 0.05,
66
Mechanical Shock
2 we can assimilate as a first approximation Z0 zsup to the absolute acceleration ysup
of mass m [LAL 75]. Very often in practice, it is the stress (and thus the relative displacement) which seems the most interesting parameter, the spectrum being 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 (it is also a good comparison between the real environment and the test environment) or to dimension a suspension (relative displacement and stress are then useful). 2
The quantity Z0 zsup is termed pseudo-acceleration. In the same way, we call the product Z0 zsup pseudo-velocity. 2
The spectrum giving Z0 zsup versus 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. 2.3.4. Primary (or initial) positive SRS
This is the highest positive response observed during the shock. 2.3.5. Primary (or initial) negative SRS
This is the highest negative response observed during the shock. 2.3.6. Secondary (or residual) SRS
This is the largest response observed after the end of the shock. The spectrum can also be positive or negative.
Shock Response Spectrum
67
Figure 2.8. Definition of primary and residual SRSs
2.3.7. Positive (or maximum positive) SRS
This is the largest positive response due to the shock, without reference to the duration of the shock. It is thus about the envelope of the positive primary and residual spectra. 2.3.8. Negative (or maximum negative) SRS
This is the largest negative response due to the shock, without reference to the duration of the shock. It is in a similar way the envelope of the negative primary and residual spectra.
68
Mechanical Shock
Example 2.1.
Figure 2.9. Shock response spectra of a square shock pulse
2.3.9. Maximax SRS
This is envelope of the absolute values of the positive and negative spectra. Which spectrum is the best? The damage is assumed 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 z m takes place during or after the shock. The most interesting spectra are thus the positive and negative ones, which 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 dissymmetric, behaves differently, for example under different tension and compression. It is, however, useful to know these various definitions in order to be able to correctly interpret the curves published.
Shock Response Spectrum
69
Figure 2.10. Links between the different definitions of SRS
NOTE: Generally, the shock spectrum is made up of the residual spectrum at low frequency. Then, when the frequency is greater, it is composed of the primary spectrum. The frequency domain in which the residual spectrum dominates depends on the duration of the shock. The shorter the duration of the shock, the larger this domain is, in particular when the shocks are pyrotechnic in origin. An important exception is TPS shock, which is only made up of its residual spectrum, which is always larger than its primary spectrum. 2.4. Standardized response spectra 2.4.1. Definition
For a given shock, the spectra plotted for various values of the duration and amplitude are homothetical. It is thus interesting for simple shocks to have a standardized or reduced spectrum plotted in dimensionless coordinates, while plotting on the abscissa the product f0 W (instead of f0 ) or Z0 W and on the ordinate 2
the spectrum/shock pulse amplitude ratio Z0 z m x m , which, in practice, amounts to tracing the spectrum of a shock of duration equal to 1 s and amplitude 1 m/s2.
70
Mechanical Shock
Figure 2.11. Standardized SRS of a half-sine pulse
These standardized spectra can be used for two purposes: – 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). The following figures give the reduced shock spectra for various pulse forms, 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 W (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, replace each value I (= f0 W ) by f0
I W
.
We will see later on how these spectra can be used for the calculation of test specifications.
Shock Response Spectrum
2.4.2. Half-sine pulse
Figure 2.12. Standardized positive and negative relative displacement SRS of a half-sine pulse
Figure 2.13. Standardized primary and residual relative displacement SRS of a half-sine pulse
71
72
Mechanical Shock
Figure 2.14. Standardized positive and negative absolute acceleration SRS of a half-sine pulse
The positive spectrum shows one peak, then tends quite quickly towards the shock amplitude. The negative spectrum tends towards zero when the natural frequency increases, whatever the damping. 2.4.3. Versed sine pulse
It is difficult to produce perfect half-sine shocks on traditional shock machines. In order to obtain this shape, we use a cylinder made from an elastic material (elastomer). The impact of the surface of the table on the plane face of the cylinder creates a wave which can move back and forth several times in the target, thus generating strong superimposed oscillations at the beginning of the signal. In order to suppress them, manufacturers recommend targets with an impact surface that is slightly conical, making it possible to progressively put the programmer in charge. This geometry is efficient, but it deforms the shock profile by rounding the angles at the beginning and end of the signal. The shock obtained thus resembles a so-called versed sine shape, made up of a sine curve period between its two minima (Figure 1.5). The SRS of the versed sine is very close to that of a half-sine (Figure 2.15).
Shock Response Spectrum
Figure 2.15. Standardized positive and negative relative displacement SRS of a versed sine pulse
Figure 2.16. Standardized primary and residual relative displacement SRS of a versed sine pulse
73
74
Mechanical Shock
2.4.4. Terminal peak sawtooth pulse
Figure 2.17. Standardized positive and negative relative displacement SRS of a TPS pulse
Figure 2.18. Standardized primary and residual relative displacement SRS of a TPS pulse
The positive spectrum presents a first peak that is slightly smaller than that of a half-sine spectrum. The positive and negative spectra are symmetric for zero damping. When the damping increases, the amplitude of the negative spectrum
Shock Response Spectrum
75
decreases, the spectrum conserving a significant value that is quite close to constant (for usual damping values, 0.05 in particular).
Figure 2.19. Standardized positive and negative relative displacement SRS of a TPS pulse with non-zero decay time
2.4.5. Initial peak sawtooth pulse
Figure 2.20. Standardized positive and negative relative displacements SRS of an IPS pulse
76
Mechanical Shock
Figure 2.21. Standardized primary and residual relative displacement SRS of an IPS pulse
Figure 2.22. Standardized positive and negative relative displacement SRS of an IPS with zero rise time
Shock Response Spectrum
77
2.4.6. Square pulse
Figure 2.23. Standardized positive and negative relative displacement SRS of a square pulse
The positive spectrum increases up to a value greater that those observed with the preceding shocks and stays at that value. The negative spectrum shows lobes of constant amplitude. This shock is very theoretical, the shock machines not being able, in practice, to create the climbing fronts and the return to zero with an infinite slope. 2.4.7. Trapezoidal pulse
Figure 2.24. Standardized positive and negative relative displacement SRS of a trapezoidal pulse
78
Mechanical Shock
2.5. Choice of the type of SRS
SRS has been the object of many definitions: SRS of relative displacements, SRS of absolute accelerations, and in any case, primary positive and negative SRS, residual positive and negative SRS, etc. Which spectrum should we choose from all these possibilities? The damage is assumed to be proportional to the largest value of the response, or to the amplitude of the spectrum at the considered frequency, it is of little importance to the system whether the maximum z m took place before or after the end of the shock. The most interesting spectra are thus the positive and negative ones, which do not know the position of the largest peak with respect to the end of the shock. These are the spectra that are used most often with the maximax spectrum. The distinction between positive and negative spectra must be made each time that the system is dissymmetric and has, for example, a different behavior in traction and in compression. When the damping is close to the values currently observed in mechanics, and in particular when [ = 0.05, the relative displacement and absolute acceleration spectra 2
are very close and on a first approximation we can liken Z0 zsup to absolute acceleration ysup of mass m [LAL 75]. This property can be observed for example (Figure 2.25) by superimposing the SRS of the half-sine of Figures 2.12 and 2.14. The distinction between these spectra is thus not essential, even if it is desirable to know the nature of the SRS that we have at our disposal.
Shock Response Spectrum
79
Figure 2.25. For usual damping values (0 d [ d 0.1), the SRS of relative displacements and absolute accelerations are very close
Very often in practice it is in fact the stress (and thus the relative displacement) that seems to be the most interesting parameter, the spectrum essentially being used to study the behavior of a structure, in order to compare the severity of several shocks, to develop test specifications (this is also a comparison of the severity between the real environment and the simulated environment) or to proportion a suspension (relative displacements and stress are thus useful). The absolute acceleration spectrum can have an interest when the absolute acceleration is the simplest parameter to compare with a characteristic value (study of the effect of shocks on a human, comparison of the specification of electronic equipment, etc.) [HIE 74] [HIE 75]. It is useful to know these different definitions to correctly interpret the published curves. 2.6. Comparison of the SRS of the usual simple shapes
The SRS of these three main shapes of simple shocks are plotted in Figure 2.26 (for Q = 10), assuming that they have the same duration and the same amplitude. We check that the positive SRS of the square shock is always larger, followed by the SRS of the half-sine shock around its peak and of the terminal peak sawtooth.
80
Mechanical Shock
Figure 2.26. Comparison of positive and negative SRS of the proposed shocks in the standards: half-sine, TPS and square (Q = 10)
2.7. SRS of a shock defined by an absolute displacement of the support
While the shock is defined by a displacement imposed on the support, the excitation is generally represented by an (absolute) acceleration. The response is thus characterized either by the absolute acceleration of the mass of a system with one degree of freedom, or by the relative displacement of the mass with respect to the support multiplied by Z02 (section 2.3). It can happen that the “input” shock is a velocity or absolute displacement instead. Taking into account the reduced form of the differential equations established in Volume 1 (Chapter 3), the calculation of the SRS of a shock defined by an absolute displacement is exactly the same as that of a shock characterized by an absolute acceleration if the response is represented by the absolute displacement of the mass instead of its absolute acceleration. This means that in practice the same software can be used in both cases. In a similar way, if the input is an absolute velocity according to time, the software provides a SRS giving the absolute velocity of the mass according to the natural frequency.
Shock Response Spectrum
81
2.8. Influence of the amplitude and the duration of the shock on its SRS
The SRS gives the largest response of a linear system with one degree of freedom. Due to this, its amplitude varies as with that of the shock. Multiplying all of the instantaneous values of a signal by 2 leads to a SRS that is twice as big. When the duration of a shock with a simple shape increases, the low resonant frequencies are much more stressed and the SRS move homothetically towards the y-axis. Example 2.2.
Let us take a half-sine shock of amplitude 50 m/s2 and duration 10 ms. Figure 2.27 shows its SRS as well as those of a half-sine shock of amplitude 100 m/s2 and 25 m/s2 (same duration).
Figure 2.27. Positive SRS of a half-sine shock of duration 10 ms and amplitude equal to 25 m/s2, 50 m/s2 and 100 m/s2 respectively (Q = 10)
The SRS of half-sine shocks with an amplitude of 50 m/s2 and durations equal to 5 ms, 10 ms and 20 ms successively are plotted in Figure 2.28.
82
Mechanical Shock
Figure 2.28. Positive SRS of a half-sine shock with an amplitude of 50 m/s2 and a duration equal to 5 ms, 10 ms and 20 ms (Q = 10)
2.9. Difference between SRS and extreme response spectrum (ERS)
A spectrum known as the extreme response spectrum (ERS) or maximum response spectrum (MRS) and comparable with the SRS is often used for the study of vibrations (Volume 5). This spectrum gives the largest response of a linear onedegree-of-freedom system according to its natural frequency, for a given Q factor, when it is subjected to the vibration under investigation. In the case of the vibrations, of long duration, this response takes place during the vibration: the ERS is thus a primary spectrum. In the case of shocks, we generally calculate the highest response, which takes place during or after the shock. 2.10. Algorithms for calculation of the SRS
Various algorithms have been developed to solve the second order differential equation [2.9] ([COL 90] [COX 83] [DOK 89] [GAB 80] [GRI 96] [HAL 91] [HUG 83a] [IRV 86] [MER 91] [MER 93] [OHA 62] [SEI 91] [SMA 81]). Two algorithms that lead to the most reliable results are those of F.W. Cox [COX 83] and D.O. Smallwood [SMA 81] (section 2.11). Although these calculations are a priori relatively simple, the round robins that were carried out ([BOZ 97] [CHA 94]) showed differences in the results, ascribable
Shock Response Spectrum
83
sometimes to the algorithms themselves, but also to the use or programming errors of the software. 2.11. Subroutine for the calculation of the SRS
The following procedure is used to calculate the response of a linear one-degreeof-freedom system as well as the largest and smallest values after the shock (points of the positive and negative and, primary and residual SRS, relative displacements and absolute accelerations). The parameters transmitted to the procedure are the number of points defining the shock, the natural pulsation of the system and its Q factor, the temporal step (presumably constant) of the signal and the array of the amplitudes of the signal. This procedure can be also used to calculate the response of a one-degree-of-freedom system to an arbitrary excitation and in particular to a random vibration (where one is only interested in the primary response). Procedure for the calculation of a point of the SRS at frequency f0 (GFABASIC)
From F.W. Cox [COX 83] PROCEDURE S_R_S(npts_signal%,w0,Q_factor,dt,VAR xpp()) LOCAL i%,a,a1,a2,b,b1,b2,c,c1,c2,d,d2,e,s,u,v,wdt,w02,w02dt LOCAL p1d,p2d,p1a,p2a,pd,pa,wtd,wta,sd,cd,ud,vd,ed,sa,ca,ua,va,ea ' npts_signal% = Number of points of definition of the shock versus time ' xpp(npts_signal%) = Array of the amplitudes of the shock pulse ' dt= Temporal step ' w0= Undamped natural pulsation (2*PI*f0) ' Initialization and preparation of calculations psi=1/2/Q_factor // Damping ratio w=w0*SQR(1-psi^2) // Damped natural pulsation d=2*psi*w0 d2=d/2 wdt=w*dt e=EXP(-d2*dt) s=e*SIN(wdt) c=e*COS(wdt) u=w*c-d2*s v=-w*s-d2*c w02=w0^2 w02dt=w02*dt ' Calculation of the primary SRS ' Initialization of the parameters
84
Mechanical Shock
srca_prim_min=1E100 // Negative primary SRS (absolute acceleration) srca_prim_max=-srca_prim_min // Positive primary SRS (absolute acceleration) srcd_prim_min=srca_prim_min // Negative primary SRS (relative displacement) srcd_prim_max=-srcd_prim_min // Positive primary SRS (relative displacement) displacement_z=0 // Relative displacement of the mass under the shock velocity_zp=0 // Relative velocity of the mass ' Calculation of the sup. and inf. responses during the shock at the frequency f0 FOR i%=2 TO npts_signal% a=(xpp(i%-1)-xpp(i%))/w02dt b=(-xpp(i%-1)-d*a)/w02 c2=displacement_z-b c1=(d2*c2+velocity_zp-a)/w displacement_z=s*c1+c*c2+a*dt+b velocity_zp=u*c1+v*c2+a responsed_prim=-displacement_z*w02 // Relative displac. during shock x square of the pulsation responsea_prim=-d*velocity_zp-displacement_z*w02 // Absolute response accel. during the shock ' Positive primary SRS of absolute accelerations srca_prim_max=ABS(MAX(srca_prim_max,responsea_prim)) ' Negative primary SRS of absolute accelerations srca_prim_min=MIN(srca_prim_min,responsea_prim) ' Positive primary SRS of the relative displacements srcd_prim_max=ABS(MAX(srcd_prim_max,responsed_prim)) ' Negative primary SRS of the relative displacements srcd_prim_min=MIN(srcd_prim_min,responsed_prim) NEXT i% ' Calculation of the residual SRS ' Initial conditions for the residual response = Conditions at the end of the shock srca_res_max=responsea_prim // Positive residual SRS of absolute accelerations srca_res_min=responsea_prim // Negative residual SRS of absolute accelerations srcd_res_max=responsed_prim // Positive residual SRS of the relative displacements srcd_res_min=responsed_prim // Negative residual SRS of the relative displacements ' Calculation of the phase angle of the first peak of the residual relative displacement c1=(d2*displacement_z+velocity_zp)/w c2=displacement_z a1=-w*c2-d2*c1 a2=w*c1-d2*c2 p1d=-a1 p2d=a2
Shock Response Spectrum
IF p1d=0 pd=PI/2*SGN(p2d) ELSE pd=ATN(p2d/p1d) ENDIF IF pd>=0 wtd=pd ELSE wtd=PI+pd ENDIF ' Calculation of the phase angle of the first peak of residual absolute acceleration b1a=-w*a2-d2*a1 b2a=w*a1-d2*a2 p1a=-d*b1a-a1*w02 p2a=d*b2a+a2*w02 IF p1a=0 pa=PI/2*SGN(p2a) ELSE pa=ATN(p2a/p1a) ENDIF IF pa>=0 wta=pa ELSE wta=PI+pa ENDIF FOR i%=1 TO 2 // Calculation of the sup. and inf. values after the shock at the frequency f0 ' Residual relative displacement sd=SIN(wtd) cd=COS(wtd) ud=w*cd-d2*sd vd=-w*sd-d2*cd ed=EXP(-d2*wtd/w) displacementd_z=ed*(sd*c1+cd*c2) velocityd_zp=ed*(ud*c1+vd*c2) ' Residual absolute acceleration sa=SIN(wta) ca=COS(wta) ua=w*ca-d2*sa va=-w*sa-d2*ca ea=EXP(-d2*wta/w) displacementa_z=ea*(sa*c1+ca*c2)
85
86
Mechanical Shock
velocitya_zp=ea*(ua*c1+va*c2) ' Residual SRS srcd_res=-displacementd_z*w02 // SRS of the relative displacements srca_res=-d*velocitya_zp-displacementa_z*w02 // SRS of absolute accelerations srcd_res_max=MAX(srcd_res_max,srcd_res) // Positive residual SRS of the relative displacements srcd_res_min=MIN(srcd_res_min,srcd_res) // Negative residual SRS of the relative displacements srca_res_max=MAX(srca_res_max,srca_res) // Positive residual SRS of the absolute accelerations srca_res_min=MIN(srca_res_min,srca_res) // Negative residual SRS of the absolute accelerations wtd=wtd+PI wta=wta+PI NEXT i% srcd_pos=MAX(srcd_prim_max,srcd_res_max) // Positive SRS of the relative displacements srcd_neg=MIN(srcd_prim_min,srcd_res_min) // Negative SRS of the relative displacements srcd_maximax=MAX(srcd_pos,ABS(srcd_neg)) // Maximax SRS of the relative displacements srca_pos=MAX(srca_prim_max,srca_res_max) // Positive SRS of absolute accelerations srca_neg=MIN(srca_prim_min,srca_res_min) // Negative SRS of absolute accelerations srca_maximax=MAX(srca_pos,ABS(srca_neg)) // Maximax SRS of absolute accelerations RETURN
2.12. Choice of the sampling frequency of the signal
The SRS is obtained by considering the largest peak of the response of a onedegree-of-freedom system. This response is generally calculated by the algorithms with the same temporal step as that of the shock signal. The sampling frequency must first of all be sufficient to correctly represent the signal itself, and in particular not to truncate its peaks (Figure 2.29). The number of points defined by respecting the Shannon’s theorem is sufficient to correctly restore the frequency components of the shock signal and to calculate its Fourier transform, for example. If this signal is used to calculate a SRS, this number can be insufficient.
Shock Response Spectrum
87
Figure 2.29. A sampling frequency that is too weak affects the signal itself more than the response of a one-degree-of-freedom system with a small natural frequency
When the natural frequency of the one-degree-of-freedom system is small, the detection of the response peaks can be carried out with precision even if the signal digitization frequency is not sufficient to correctly describe the shock (Figure 2.30). The error on the SRS is thus solely linked to the bad digitization of the shock and is translated by an imprecision on the velocity change, i.e. on the slope of the low frequency spectrum.
Figure 2.30. The digitization frequency must be sufficient to represent the response of large frequency one-degree-of-freedom systems
On the other hand, the sampling frequency makes it possible to represent the shock well, but it can be insufficient for the response when the natural frequency of the system is greater than the maximum frequency of the signal. Here the error is linked to the detection of the largest peak of the response, which takes place during the shock (primary spectrum). Figure 2.31 shows the error made in the stringent case more when the points surrounding the peak are symmetric with respect to the peak.
88
Mechanical Shock
If we set: SF
Sample frequency SRS maximum frequency
[2.46]
it can be shown that, in this case, the error made according to the sampling factor SF is equal to [SIN 81] [WIS 83]: eS
ª § S 100 «1 cos¨ ¨S «¬ © F
·º ¸» ¸» ¹¼
Figure 2.31. Error made in measuring the amplitude of the peak
[2.47]
Figure 2.32. Error made in measuring the amplitude of the peak plotted against sampling factor
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 10 can lead to an error of about 5%. The method proposing a parabolic interpolation between the points to assess the value of the maximum does not lead to better results. The definition of the sampling frequency from the maximum frequency of the spectrum is penalizing as regards the computing time. To reduce it, it could be
Shock Response Spectrum
89
interesting to choose a variable sampling frequency according to the natural frequency of system [SMA 02]. NOTE: When error eS is small, for example, less than 14%, relation [2.47] can be simplified to make the calculation of SF easier: SF |
S 2 eS
[2.48]
Respecting the Shannon’s theorem only allows us to correctly calculate the SRS up to the frequency equal to around the sampling frequency fsamp / 10. However, the signal can be reconstructed using the method described in Volume 1 in order to enable us to calculate the SRS up to the desired frequency [LAL 04] [SMA 00]. The process is as follows: sampling the vibratory signal with a sampling frequency equal to twice the maximum frequency fmax of the signal (previously filtered by a low-pass filter of cutoff frequency fmax); for shocks, reconstructing the signal to obtain a sampling frequency 10 times the maximum frequency of the SRS to be calculated (to be distinguished from the maximum frequency of the signal). Example 2.3. The shock signal of Figure 2.33 has been filtered analogically at 10 kHz, then sampled according to the Shannon’s theorem at fsamp = 20 kHz. This shock signal was then reconstructed with 5 times as many points (Figure 2.34) in order to be able 5 x 20, 000 to calculate the SRS up to 10,000 Hz ( ). 10
Figure 2.33. Shock sampled according to Shannon
Figure 2.34. Signal sampled according to Shannon and reconstructed (partial) signal
90
Mechanical Shock
We see in Figure 2.35 that the SRS of the reconstructed shock and of the shock sampled according to Shannon begin to diverge at about 2 kHz, thus fsamp / 10.
Figure 2.35. SRS of the shock sampled according to Shannon and SRS of the reconstructed shock
2.13. Example of use of the SRS
Let us consider as an example the case of a package intended to limit acceleration on the transported equipment of mass m to 100 m/s2 when the package itself is subjected to a half-sine shock of amplitude 300 m/s2 and of duration 6 ms (Figure 2.36). We also impose a maximum displacement of the equipment in the package (under the effect of the shock) equal to e = 4 cm (to prevent the equipment coming into contact with the wall of the package). It is assumed that the system made up of mass m of the equipment and the suspension is comparable to a one-degree-of-freedom system with a Q factor equal to Q 5 . We want to determine stiffness k of the suspension to satisfy these requirements when mass m is equal to 50 kg.
Figure 2.36. Model of the package
Shock Response Spectrum
91
Figures 2.37 and 2.38 show the response spectrum of the half-sine shock pulse being considered, plotted between 1 and 50 Hz for a damping of [ 0.10 ( 1 2 Q ). The curve of Figure 2.37 gives zsup on the ordinate (maximum relative displacement of the mass, calculated by dividing the ordinate of the spectrum 2
2
Z0 zsup by Z0 ). The spectrum of Figure 2.38 represents the usual curve
2
Z0 zsup f0 . We could also have used a logarithmic four coordinate spectrum to handle just one curve.
Figure 2.37. Limitation in displacement
Figure 2.38. Limitation in acceleration
Figure 2.37 shows that to limit the displacement of the equipment to 4 cm, the natural frequency of the system must be higher than or equal to 4 Hz. The limitation of acceleration on the equipment with 100 m/s2 also imposes f0 d 16 Hz (Figure 2.38). The range acceptable for the natural frequency is thus 4 Hz d f 0 d 16 Hz. Knowing that: f0
1
k
2S
m
,
we deduce that:
8 S 2 m d k d 32 S 2 m i.e.: 3.16 10 4 N/m d k d 5.05 105 N/m .
92
Mechanical Shock
2.14. Use of SRS 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 one-degree-of-freedom 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. We can read on the SRS the maximum response of each one of these modes, but we do 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 simply known. In addition, the SRS is plotted for a given constant damping over the whole frequency range, whereas this damping varies from one mode to another within the structure. With rigor, it thus appears difficult to use a SRS to evaluate the response of a system presenting more than one mode, but it happens that this is the only possible means. The problem is then 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. Let us consider a non-linear system with n degrees of freedom; its response to a shock can be written as: z t
n
¦ a n I i ³0 h i t D x D dD
i
t
1
where: n
total number of modes; modal participation factor for the mode n;
an h n t x t I
n
D
impulse response of mode n; excitation (shock); modal vector of the system;
variable of integration.
[2.49]
Shock Response Spectrum
93
If one mode (m) is dominant, this relation is simplified according to: z t
I
m
³0 h m t D x D dD t
[2.50]
The value of the SRS to the mode m is equal to:
zm m
ª m max « I tt0 ¬
³0 h m t D x D dDº»¼ t
[2.51]
The maximum of the response z t in this particular case is thus: max> z t @ | I
m
t t0
zm m
[2.52]
When there are several modes, several proposals have been made to limit the value of the total response of the mass j of one of the degrees of freedom starting from the values read on the SRS as follows. A first method was proposed in 1934 by H. Benioff [BEN 34], consisting simply of adding the values to the maxima of the responses of each mode, without regard to the phase. A very conservative value was suggested by M.A. Biot [BIO 41] in 1941 for the prediction of the responses of buildings to earthquakes, equal to the sum of the absolute values of the maximum modal responses: max z j t d t t0
n
¦ ai i
1
i
Ij
zm i
[2.53]
The result was considered precise enough for this application [RID 69]. As it is not very 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 [SHE 66]. In 1958, S. Rubin [RUB 58] carried out a study of undamped two-degrees-offreedom systems in order to compare the maximum responses to a half-sine shock calculated by the modal superposition method and the real maximum responses. This study showed that we could obtain an upper limit of the maximum response of the structure by a summation of the maximum responses of each mode and that, in
94
Mechanical Shock
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%. The errors are largest when the modal frequencies are in different areas of the SRS, for example, if a mode is in the impulse domain and the other in the static domain. If the fundamental frequency of the structure is sufficiently high, Y.C. Fung and M.V. Barton [FUN 58] considered that a better approximation of the response is obtained by making the algebraic sum of the maximum responses of the individual modes: max z j t d t t0
n
¦ ai 1
i
i
Ij
zm i
[2.54]
Clough proposed in 1955, in the study of earthquakes, either to add a fixed percentage of the responses of the other modes to the response of the first mode, or to increase the response of the first mode by a constant percentage [CLO 55]. The problem can be approached differently starting from an idea drawn from probability theory. Although the values of the response peaks of each individual mode taking place at different instants of time cannot, in a strict sense, be treated in purely statistical terms, Rosenblueth suggested combining 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 [MER 62]. This criterion, used again in 1965 by F.E. Ostrem and M.L. Rumerman [OST 65] in 1955 [RID 69], gives values of the total response lower than the sum of the absolute values and provides a more realistic evaluation of the average conditions. This idea can be improved by considering the average of the sum of the absolute values and the square root of the sum of the squares [JEN 58]. We can also choose to define positive and negative limiting values starting from a system of weighted averages. For example, the relative displacement response of mass j is estimated by: n
max z j t
i
1
i
zm
1
p 1
t t0
where the terms z m
n
¦ z2m i p ¦
i
i
[2.55]
are the absolute values of the maximum responses of each
mode and p is a weighting factor [MER 62].
Chapter 3
Properties of Shock Response Spectra
3.1. Shock response spectra domains Three domains can be schematically distinguished in shock spectra: – 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 shock here is of very short duration with respect to the natural period of the system. The system reduces the effects of the shock. The properties of the spectra in this domain will be detailed in section 3.2. – A static domain in the range of the high frequencies, where the positive spectrum tends towards the amplitude of the shock whatever the damping. Everything occurs here as if the excitation were a static acceleration (or a very slowly varying acceleration), the natural period of the system being small compared to the duration of the shock. This does not apply to rectangular shocks or to the shocks with zero rise time. The real shocks necessarily have a rise time different from zero, this restriction remains theoretical. – 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, 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 and terminal peak sawtooth (TPS)). Much larger values are reached in the case of oscillatory shocks, made up, for example, by a few periods of a sinusoid.
96
Mechanical Shock
3.2. Properties of SRS at low frequencies 3.2.1. General properties In this impulse region ( 0 d f 0 W d 0, 2 ): – the form of the shock has little influence on the amplitude of the spectrum. We will see below that only (for a given damping) the velocity change 'V associated with the shock, equal to the algebraic surface under the curve x( t ) , is important; – the positive and negative spectra are, in general, the residual spectra (it is necessary sometimes that the frequency of spectrum is very small and there can be exceptions for certain long shocks in particular). They are nearly symmetric so long as damping is small; 2 – the response (pseudo-acceleration Z0 zsup or absolute acceleration ysup ) is
lower than the amplitude of the excitation; there is an “attenuation”. It is consequently 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: 2
m Z0
k
2
2
4 S f0 m
(with m being the mass of the material to be protected); – the curvature of the spectrum always cancels at the origin ( f 0 [FUN 57].
0 Hz)
The properties of the SRS are often better demonstrated by a logarithmic chart or a four coordinate representation. 3.2.2. Shocks with zero velocity change For the shocks that are simple in shape ( 'V z 0 ), the residual spectrum is larger than the primary spectrum at low frequencies. For an arbitrary damping [, it can be shown that the impulse response is given by: 2
Z0 z( t )
Z0 1[
2
'V e
[Z0 t
sin Z0
2
1 [ t
[3.1]
Properties of Shock Response Spectra
where z t is maximum for t such that
Z0 1 [ 2 t
arc tan
dz( t ) dt
97
0 , i.e. for t such that:
1 [2 [
yielding: § [ 1 [2 ¨ exp¨ arc tan ¨ [ 1 [2 1 [2 ©
'V
z sup
Z0
· § 1 [2 ¸ ¨ sin arc tg ¸¸ ¨¨ [ ¹ ©
· ¸ ¸¸ [3.2] ¹
The SRS is thus equal at low frequencies to: § [ 1 [2 ¨ exp¨ arc tan ¨ [ 1 [2 1 [2 ©
Z0 'V
Z02 z sup
· § 1 [2 ¸ ¨ ¸¸ sin ¨¨ arc tan [ ¹ ©
· ¸ ¸¸ [3.3] ¹
i.e.: 2
Z0 zsup
'V Z0 M( [)
[3.4]
2
d ( Z0 zsup ) dZ0
'V M( [)
[3.5]
If [ o 0 , M( [) o 1 and the slope tends towards 'V . The slope p of the spectrum at the origin is then equal to: 2
p
d ( Z0 zsup ) df0
2S 'V
[3.6]
The tangent at the origin of the spectrum plotted for zero damping in linear scales has a slope proportional to the velocity change 'V corresponding to the shock pulse. If damping is small, this relation is approximate.
98
Mechanical Shock
Example 3.1. Half-sine shock pulse 100 m/s2, 10 ms, positive SRS (relative displacements). The slope of the spectrum at the origin is equal to (Figure 3.1): 120
p
30
4 m/s
yielding: 'V
p
4
2
2S
2S
S
m/s,
a value to be compared with the surface under the half-sine shock pulse:
2 S
100 0.01
2 S
m/s
Figure 3.1. Slope of the SRS at the origin
With the pseudovelocity plotted against Z0 , the spectrum is defined by
Z0 zsup
'V M [
[3.7]
Properties of Shock Response Spectra
99
When Z0 tends towards zero, Z0 zsup tends towards the constant value
'V M [ . Figure 3.2 shows the variations of M [ versus [.
Figure 3.2. Variations of the function M([)
Example 3.2.
TPS shock pulse 100 m/s2, 10 ms. Pseudovelocity calculated starting from the positive SRS (Figure 3.3).
Figure 3.3. Pseudovelocity SRS of a TPS shock pulse
It can be seen that the pseudovelocity spectrum plotted for [ 0.5 at low frequencies (area under TPS shock pulse).
0 tends towards
100
Mechanical Shock
The pseudovelocity Z0 zsup tends towards 'V when the damping tends towards zero. If damping is different from zero, the pseudovelocity tends towards a constant value lower than 'V . 2
The residual positive SRS of the relative displacements (Z0 zsup ) decreases at low frequencies with a slope equal to 1, i.e., on a logarithmic scale, with a slope of 6 dB/octave ( [ 0 ). The impulse absolute response of a linear one-degree-of-freedom system is given by relation [4.74] (Volume 1). It can also be written: Z0
h( t )
1 [
e
2
[Z0 t
sin Za t M
where: Za M
Z0 1 [ arc tan
2
2 [ 1 [2 1 2 [2
[3.8]
If damping is zero: h( t )
Z0 sin Z0 t
'V
³ f x(t)dt ³0 x(t )dt
f
[3.9] W
[3.10]
The “input” impulse can be represented in the form: x( t )
'V G( t )
[3.11]
as long as Z0 W 1 . The response which results is: 2
Z0 z( t )
'V h ( t )
[3.12]
The maximum of the displacement takes place during the residual response, for: h ( t max ) | Z0
[3.13]
Properties of Shock Response Spectra
101
yielding the SRS: S
'V Z0
[3.14]
and: log( Z0 ) log( 'V )
log(S )
[3.15]
A curve defined by a relation of the form y slope n on a logarithmic grid:
af
n
is represented by a line of
n log f log a
log y
[3.16]
The slope can be expressed by a number N of dB/octave according to: N dB/octave
20 log10 2
n
20 n (log10 2 )
N dB/octave | 6 n
[3.17] [3.18]
The undamped SRS plotted on a log-log grid thus has a slope at the origin equal to 1, i.e. 6 dB/octave.
Example 3.3.
Terminal peak sawtooth pulse 10 ms, 100 m/s2 'V
0.5 m/s
Figure 3.4. TPS shock pulse
102
Mechanical Shock
Figure 3.5. Residual positive SRS (relative displacements) of a TPS shock pulse 2
The primary positive SRS Z0 zsup always has a slope equal to 2 (12 dB/octave) [SMA 85]. Example 3.4.
Figure 3.6. Primary positive SRS of a half-sine shock pulse
The relative displacement zsup tends towards a constant value z 0
x m equal to
the absolute displacement of the support during the application of the shock pulse (Figure 3.7). At low resonance frequencies, the equipment is not directly sensitive to accelerations, but to displacement: zsup xm
o1
Properties of Shock Response Spectra
103
Figure 3.7. Behavior of a resonator at very low resonance frequency
The system works as soft suspension which attenuates accelerations with large displacements [SNO 68]. This property can be demonstrated by considering the relative displacement response of a linear one-degree-of-freedom system given by Duhamel’s equation (Volume 1, Chapter 3): z( t )
1
t
Z0 1 [
If Z0 o 0 , e
³ x(D) e[Z ( t D) sinª«¬Z0 2 0
[ Z0 ( t D )
0
º 2 1 [ ( t D ) » dD ¼
o 1 and
ª º 2 2 sin« Z0 1 [ ( t D) » | Z0 1 [ ( t D) ¬ ¼ z( t ) |
1
t
Z0 1 [
z( t ) |
³0 x(D)
z( t ) |
³0 x(D)
t
t
³ x(D) e[Z ( t D) Z0 2 0 0
2
1 [ ( t D ) dD
( t D ) dD t dD
³0 x(D) t
D dD
After integration by parts we obtain: z( t ) | t v ( 0 ) x( t ) x 0
[3.19]
104
Mechanical Shock
If x 0
0 and v 0
0,
z( t ) o x ( t )
[3.20]
The mass m of an infinitely flexible oscillator, and therefore of infinite natural period ( f0 0 ), does not move in the absolute reference axes. The spectrum of the relative displacement thus has as an asymptotic value the maximum value of the absolute displacement of the base. Example 3.5.
Figure 3.8 shows the primary positive SRS z sup f 0 of a shock of half-sine shape 100 m/s2, 10 ms plotted for [ 0 between 0.01 Hz and 100 Hz.
Figure 3.8. Primary positive SRS of a half-sine (relative displacements)
The maximum displacement x m under shock calculated from the expression x t for the acceleration pulse is equal to: xm
x m W 2 S
3.18 10 3 m
The SRS tends towards this value when f0 o 0 .
Properties of Shock Response Spectra
105
For shocks of simple shape, the instant of time t p at which the first peak of the S response takes place tends towards as Z0 tends towards zero [FUN 57]. 2 Z0 The primary positive spectrum of pseudovelocities has a slope of 6 dB/octave at the low frequencies.
Example 3.6.
Figure 3.9. Primary positive SRS of a TPS pulse (four coordinate grid)
3.2.3. Shocks with 'V
In this case, for [
0 and ' D z 0 at the end of a pulse 0:
– the Fourier transform of the velocity for f V (0)
f
³f v(t) dt
'D
0 , V 0 , is equal to
[3.21]
Since acceleration is the first derivative of velocity, the residual spectrum is 2
equal to Z0 'D for low values of Z0 . The undamped residual SRS thus has a slope equal to 2 (i.e. 12 dB/octave) in this range.
106
Mechanical Shock
Example 3.7. Shock made up of one sinusoid period of amplitude 100 m/s2 and duration 10 ms.
Figure 3.10. Residual positive SRS of a “sine 1 period” shock pulse
– the primary relative displacement (positive or negative, according to the form of the shock) z sup tends towards a constant value equal to x m , absolute
displacement corresponding to the acceleration pulse x t defining the shock: zsup xm
o1
Properties of Shock Response Spectra
107
Example 3.8.
Let us consider a terminal peak sawtooth pulse of amplitude 100 m/s2 and duration 10 ms with a symmetric square pre- and post-shock of amplitude 10 m/s2. The shock has a maximum displacement given by (Chapter 7): xm
x m W 2 4
§ 2 p 1 p3 ¨ ¨ 3 2 8p 6 ©
· ¸ ¸ ¹
At the end of the shock, there is no change in velocity, but the residual displacement is equal to: x residual
x m W2 §¨ 1 p p3 ·¸ 4 ¨© 3 2 3 ¸¹
Using the numerical data of this example, we obtain: xm
4.428 mm
We find this value of x m on the primary negative spectrum of this shock (Figure 3.11). In addition: x residual
0.9576 104 mm
Figure 3.11. Primary negative SRS (displacements) of a TPS pulse with square pre- and post-shocks
108
Mechanical Shock
3.2.4. Shocks with 'V
0 and ' D
0 at the end of a pulse
For oscillatory shocks, we note the existence of the following regions [SMA 85] (Figure 3.12): – just below the principal frequency of the shock, the spectrum has, on a logarithmic scale, a slope characterized by the primary response (about 3); – when the frequency of spectrum decreases, its slope tends towards a smaller value of 2; – when the natural frequency decreases further, we observe a slope equal to 1 (6 dB/octave) (residual spectrum). In a general way, all the shocks, whatever their form, have a spectrum of slope of 1 on a logarithmic scale if the frequency is quite small. Example 3.9.
Figure 3.12. Shock response spectrum (relative displacements) of a ZERD pulse ('D = 0, 'V = 0) [FIS 77] [LAL 90] [SMA 85]
2
The primary negative SRS Z0 zsup has a slope of 12 dB/octave; the relative displacement zsup tends towards the absolute displacement x m associated with the shock movement x t .
Properties of Shock Response Spectra
109
Example 3.10.
Figure 3.13. Primary negative SRS of a half-sine pulse with half-sine pre- and post-shocks
Figure 3.14. Primary negative SRS (displacements) of a half-sine pulse with half-sine pre- and post-shocks
If the velocity change and the variation in displacement are zero at the end of the shock, but if the integral of the displacement has a non-zero value 'D , the undamped residual spectrum is given by [SMA 85]:
110
Mechanical Shock
S r ( Z0 )
3
Z0 'D
[3.22]
for small values of Z0 (slope of 18 dB/octave). Example 3.11.
Figure 3.15. Residual positive SRS of a half-sine pulse with half-sine pre- and post-shocks
3.2.5. Notes on residual spectrum Spectrum of absolute displacements
When Z0 is sufficiently small, the residual spectrum of an excitation x t is identical to the corresponding displacement spectrum in one of the following ways [FUN 61]: x( W ) z 0
a)
x ( W )
0
b)
x ( W )
x( W )
0
x ( W )
x( W )
³0 x t dt
c)
but W
0 but
³0
W
x( t ) dt z 0
³0 x t dt t
z 0 for 0 t x( W )
Properties of Shock Response Spectra
111
However, contrary to the case (c) above, if: x ( W )
x( W )
³0
W
x( t ) dt
0
but if there exists more than one value t p of time in the interval 0 t p d W for
³0
³0 x( t ) dt while t 2 the spectrum of the displacements is equal to the largest values of Z0 ³ x( t ) dt 0 which
tp
x( t ) dt
2
0 , then the residual spectrum is equal to Z0
W
p
[FUN 61]. If 'V and 'D are zero at the end of the shock, the response spectrum of the absolute displacement is equal to 2 x W where x W is the residual displacement of the base. If x W 0 , the spectrum is equal to the largest of the two quantities Z0
³0
³0
x( O ) dO is cancelled. The absolute displacement of response is not limited if the
tp
W
2
x( O ) dO and Z0
³0
tp
x( O ) O dO where t
t p is the time when the integral
input shock is such that 'V z 0 . Relative displacement
When Z0 is sufficiently small, the residual spectrum and the spectrum of the displacements are identical in the following cases: a) if x ( W ) z 0 at the end of the shock; b) if x ( W )
0 , but x t is maximum with t
W.
If not, the residual spectrum is equal to x W , while the spectrum of the displacements is equal to the largest absolute value of x t . 3.3. Properties of SRS at high frequencies
The response can be written, according to relation [2.16]: 2
Z0 z( t )
i.e., while setting u
Z0 1 [
³ x(D) e[ Z 2 0
t D:
t
0
( t D )
2
sin Z0 1 [ ( t D ) dD
112
Mechanical Shock
2
Z0 z( t )
Z0 1[
x( t u ) e[ Z ³ 2 0 t
We want to show that lim Z02 z t Z0 o f
w t
w t
Z0 [2
t
³ x(t) e 1 [2 0
Z02 x(t) [2
[ Z0 u
t
³0 e
[ Z0 u
0
u
§ 2 · sin¨ Z0 1 [ u ¸ du ¹ ©
x t . Let us set:
Z0 1 [2 u du
u du
Integrating by parts:
ª t e[ Z0 t e[ Z0 t 1 º 2 2 2 2» w(t) Z02 [2 x(t) « [ Z0 [ Z0 ¼ ¬ [ Z0
w(t) x(t) ª¬ [ Z0 t e [ Z0 t e [ Z0 t 1º¼ w t tends towards x t when Z0 tends towards infinity. Let us show that:
>
2
lim Z0 z t w t
Z0 of
@
0,
H ! 0,
i.e.,
:!0
Z02 z t w t d H . constant . Z02 z t w t
Z0 1 [
t
2
ª
³0 «¬x t Z0
1 [2 u [2
§ 2 ·º [ Z0 t du x t u sin¨ Z0 1 [ u ¸» e © ¹¼ Z02 z t w t d
Z0 1 [
t
2
³0 x t Z0
§ 2 · [ Z 0 t du x t u sin¨ Z0 1 [ u ¸ e © ¹
1 [2 u [2
such
that
Zt:,
Properties of Shock Response Spectra
113
If the function x t is continuous, the quantity
Z0
f (u)
1 [
2
x(t u) sin Z0 1 [ 2 u x(t) Z0 1 [ 2 u [ 2
tends towards zero as u tends towards zero. There consequently exists K > 0, t @ such that u > 0, K@ , f u d H and we have:
Z0
³
K
2
K
2
³
1 [ Z0
1 [
0
0
f (u) e [ Z0 u du d H
f (u) e [ Z0 u du d
Z0 1 [
³
2
K
0
e [ Z0 u du
H [ 1 [2
The function x t is continuous and therefore limited to > 0, t @ : M ! 0 when,
for all u > 0, t @ , x u d M , and we have: Z0
³K f u e [ Z t
1[
2
M [ 1 [
2
>e
[ Z0 K
0
u
Z0
du d
e
1 [
[ Z0 t
: ! 0 when, Z t : ,
³K M e[ Z t
2
2
Z0
M [ 1 [
2
>e
[ Z0 K
1 [
³ f u e [ Z 2 0 t
u
du
@ e
Thus, for Z t : : Z0 z t w t d
0
0
u
du
[ Z 0 t
@dH
114
Mechanical Shock
2
Z0 z t w t d
2
Z0 z t w t d
Z0 1 [
[ Z f u e ³ 2 0 K
H [ 1 [
2
0
u
du
Z0 1 [
[ Z f u e ³ 2 K t
0
u
du
H
2 At high frequencies, Z0 z t thus tends towards x t and, consequently, the SRS tends towards x m , a maximum x t .
3.4. Damping influence
Damping has little influence in the static region. Whatever its value, the spectrum tends towards the amplitude of the signal depending on time. This property is checked for all the shapes of shocks, except for the rectangular theoretical shock which, according to damping, tends towards a value ranging between once and twice the amplitude of the shock. In the impulse domain and especially in the intermediate domain, the spectrum has a lower amplitude when the damping is greater. This phenomenon is not great for shocks with velocity change and for normal damping (0.01 to 0.1 approximately). It is marked more for oscillatory type shocks (decaying sine for example) at frequencies close to the frequency of the signal. The peak of the spectrum here has an amplitude which is a function of the number of alternations of the signal and of the selected damping. 3.5. Choice of damping
The choice of damping should be carried out according to the structure subjected to the shock under consideration. 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 relative damping equal to 0.05 (i.e. Q = 10). No justification of this choice is given in the literature. A study of E.F. Small [SMA 66] gives the distribution function (Figure 3.16) and the probability density (Figure 3.17) of Q-factors observed on electronic equipment (500 measurements).
Properties of Shock Response Spectra
115
Figure 3.16. Distribution function of Q-factors measured on electronic equipment
Figure 3.17. Probability density of Q-factors measured on electronic equipment
The value Q = 10 appears completely acceptable here, since the values generally recorded in practice are lower than 10. Unless otherwise specified, as noted on the curve, it is the value chosen conventionally. With the spectra varying relatively little with damping (with the reservations of the preceding section), this choice is often not very important. To limit possible errors, the selected value should, however, be systematically noted on the diagram. NOTE: In practice, the most frequent range of variation of the Q factor of the structures lies between approximately 5 and 20. Larger values can be measured if the sensor is fixed on a plate or a cap [HAY 72], but measurement is not very significant since we are instead interested here in structural responses. There is no exact relation which makes it possible to obtain a SRS of a given Q factor starting from a spectrum of the same signal calculated with another Q factor.
116
Mechanical Shock
M.B. Grath and W.F. Bangs [GRA 72] proposed an empirical method deduced from an analysis of spectra of pyrotechnic shocks to carry out this transformation. It is based on curves giving, depending on Q, a correction factor, equal to the ratio of the spectrum for the quality factor Q to the value of this spectrum for Q = 10 (Figure 3.18). The first curve relates to the peak of the spectrum, the second the standard point (non-peak data). The comparison of these two curves confirms the greatest sensitivity of the peak to the choice of Q factor. These results are compatible with those of a similar study carried out by W.P. Rader and W.F. Bangs [RAD 70], which did not however distinguish between the peaks and the other values.
Figure 3.18. SRS correction factor of the SRS versus Q factor
To take account of the dispersion of the results observed during the establishment of these curves and to ensure reliability, the authors calculated the standard deviation associated with the correction factor (in a particular case, a point on the spectrum plotted for Q = 20; the distribution of the correction factor is not normal, but near to a Beta or type I Pearson law). Q 5 10 20 30 40 50
Standard points 0.085 0 0.10 0.15 0.19 0.21
Peaks 0.10 0.00 0.15 0.24 0.30 0.34
Table 3.1. Standard deviation of the correction factor
Properties of Shock Response Spectra
117
The results show that the average is conservative 65% of the time, and the average plus one standard deviation 93%. They also indicate that modifying the amplitude of the spectrum to take account of the value of Q factor is not sufficient for fatigue analysis. The correction factor being determined, they proposed to calculate the number of equivalent cycles in this transformation using the relation developed by J.D. Crum and R.L. Grant [CRU 70] (see section 4.4.2) giving the expression for the response Z 02 z t depending on the time during its establishment under a sine wave excitation as:
Z 02 z t
Q x m 1 e S N / Q cos 2 S f 0 t
[3.23]
(where N = number of cycles carried out at time t).
Figure 3.19. SRS correction factor versus Q factor
This relation, standardized by dividing it by the amount obtained for the particular case where Q 10 , is used to plot the curves of Figure 3.19 which make it possible to read N, for a given correction factor and a given Q factor. They are not reliable for Q 10 , relation [3.23] being correct only for low damping. ªZ 02 z º ¬ ¼Q ªZ 02 z º ¬ ¼ 10
10 1 e S
Q 1 eS N / Q
N / 10
[3.24]
118
Mechanical Shock
3.6. Choice of frequency range
It is customary to choose as the frequency range: – either the interval in which the resonance frequencies of the structure studied are likely to be found; – or the range including the important frequencies contained in the shock (in particular in the case of pyrotechnic shocks). 3.7. Choice of the number of points and their distribution
200 points are generally sufficient for the calculation of the SRS of simpleshaped shocks. For the SRS of shocks measured in the real environment, it may be necessary to increase this number according to the frequency content of the signal. In any case, it is preferable to choose a logarithmic distribution of points, which enables us to obtain a better distribution of the low frequency curve in logarithmic axes. 3.8. Charts
There are two spectral charts: – representation (x, y), the showing value of the spectrum versus the frequency (linear or logarithmic scales); – the four coordinate nomographic representation (four coordinate spectrum). Z0 , on the ordinates the We note here on the abscissae the frequency f0 2S pseudovelocity Z0 z m and, at two axes at 45° to the first two, the maximum relative 2
displacement z m and the pseudo-acceleration Z0 z m . This representation is interesting for it makes it possible to directly read the amplitude of the shock at the high frequencies and, at low frequencies, the velocity change associated with the shock (or if 'V 0 the displacement).
Properties of Shock Response Spectra
Figure 3.20. Four coordinate diagram
Example 3.12.
Figure 3.21. Example of SRS in the four-coordinate axes
119
120
Mechanical Shock
3.9. Relation of SRS with Fourier spectrum 3.9.1. Primary SRS and Fourier transform
The response u t of a linear undamped one-degree-of-freedom system to a generalized excitation A t is written [LAL 75] (Volume 1, Chapter 3): Z0
u( t )
³0 A(D) sin Z0 ( t D) dD t
We suppose here that t is lower than W: u t
Z0 sin Z0 t
³0 A D cos Z0D dD Z0 cos Z0t ³0 A D sin Z0D dD t
t
which is of the form: Z0 C sin Z0 t Z0 S cos Z0 t
u( t )
[3.25]
with: C S
³0 A(D) cos Z0D dD t ³ A( D ) sin Z0D dD 0 t
[3.26]
where C and S are functions of time t. u t can still be written: u( t )
Z0
2
2
C S sin Z0 t I P
[3.27]
with: tg I P
S
[3.28]
C
If the functions
2
C S
2
and sin Z 0 t I P are at a maximum for the same
value of time, the response u(t) also has a maximum for this value. sin Z 0 t I P is at a maximum when sin Z 0 t I P 2
2
C S is at a maximum when its derivative is zero:
1 . The function
Properties of Shock Response Spectra 2
2
d( C S )
1
dt
2
C 2
C S
1 2
i.e., if cos Z0 t I P absolute value of u t
> 2 C A( t ) cos Z0t 2 S A( t ) sin Z0 t@
2
C S
2
sin Z0 t
0
0 or if sin Z0 t I P
2
Z0
2
S
cos Z0 t
2
um P
C S
C S
121
0
1 . This yields the maximum
2
[3.29]
where the index P indicates that it is about the primary spectrum. However, where the Fourier transform of A t , is calculated as if the shock were non-zero only between times 0 and t with Z0 the pulsation is written as: L( Z0 )
³0 A(D) eiZ D dD t
[3.30]
0
and has as an amplitude under the following conditions: L( Z0 )
°ª ®« °¯¬
0, t
³
º2 ª A( D ) cos Z0D dD » « 0 ¼ ¬ t
³
P and
Comparison of the expressions of u m
um P
Z0 L( Z0 )
1/ 2
º2 °½ A( D ) sin Z0D dD » ¾ 0 ¼ °¿ t
L( Z0 ) shows that:
Am
L( Z0 )
0, t
[3.32]
0, t
In a system of dimensionless coordinates, with q m Z0
q m P
[3.31]
um Am
:
[3.33]
The primary spectrum of shock is thus identical to the amplitude of the reduced Fourier spectrum, calculated for t d W [CAV 64].
122
Mechanical Shock
The phase I L
0, t
of the Fourier spectrum is such that:
³ A(D) sin Z0D dD 0t ³0 A(D) cos Z0D dD t
tg I L
0,t
[3.34]
However, the phase I P is given by [3.28]
tg I P
³ A(D) sin Z0D dD 0t ³0 A(D) cos Z0D dD
[3.35]
IL
I P kS
[3.36]
t
0, y
where k is a positive integer or zero. For an undamped system, the primary positive shock spectrum and the Fourier spectrum between 0 and t are thus related in phase and amplitude. 3.9.2. Residual SRS and Fourier transform
Whatever the value of t, the response can be written as:
³0 A(D) sin Z0 t D dD t
u( t )
Z0
u( t )
Z0 sin Z0 t
For t t W , A( D ) u( t )
³0 A(D) cos Z0D dD Z0 cos Z0 t ³0 A(D) sin Z0D dD t
t
0.
Z0 sin Z0 t
³0 A(D) cos Z0D dD Z0 cos Z0 t ³0 A(D) sin Z0D dD W
W
which is of the form B1 sin Z0 t B2 cos Z0 t with B1 and B2 being constants. We also have: u( t )
C sin Z0 t I R
[3.37]
Properties of Shock Response Spectra
123
where the constant C is equal to: C
°ª Z0 ®« °¯¬
³
º2 ª A( D ) cos Z0D dD » « 0 ¼ ¬ W
³
1/ 2
º2 ½° A( D ) sin Z0D dD » ¾ 0 ¼ °¿ W
[3.38]
and the phase I R is such that:
³0 A(D) sin Z0 D dD W ³0 A(D) cos Z0 D dD W
tg I R
[3.39]
The residual spectrum, expressed in terms of displacement, is thus given by the maximum value of the response: Um
C °ª Z0 ®« °¯¬
D R ( Z0 )
³
º2 ª A( D ) cos Z0D dD » « 0 ¼ ¬ W
³
1/ 2
º2 °½ A( D ) sin Z0D dD » ¾ 0 ¼ °¿ W
[3.40]
The Fourier transform of the excitation A t is by definition equal to: L( :)
f
³f A(D) e
i : D
dD
or since outside (0, W), the function A t is zero: L( :)
³0 A(D) ei:D dD W
This expression can be written, by expressing the exponential function according to a sine and a cosine term as: L(: )
³0 A(D) cos :D dD i ³0 A(D) sin :D dD
L( :)
R > L( :)@ i I> L( :)@
W
W
[3.41]
where R : is the real part of the Fourier integral and I : the imaginary part. L( :) is a complex quantity whose module is given by:
124
Mechanical Shock
°ª ®« °¯¬
L( :)
³
º2 ª A( D ) cos :D dD » « ¼ ¬ 0 W
³
1/ 2
º2 ½° A( D ) sin :D dD » ¾ ¼ °¿ 0 W
[3.42]
Let us compare the expressions of D R : and of L : . Apart from the factor Z0 , and provided that one changes Z0 into :, these two quantities are identical. The natural frequency of the system Z0 can take an arbitrary value equal, in particular, to : since the simple mechanical system is not yet chosen. We thus obtain the relation: D R (:)
: L :
[3.43]
The phase is given by:
³ A(D) sin :D dD 0W ³0 A(D) cos :D dD W
tg I L
[3.44]
S· § S Only the values of IL ¨¨ , ¸¸ will be considered. A comparison of I R and 2¹ © 2 I L shows that: IL
: t p
2k 1 2
S
[3.45]
For an undamped system, the Fourier spectrum and the residual positive shock spectrum are related in amplitude and phase [CAV 64]. NOTE: If the excitation is an acceleration, A ( t )
x( t )
Z 02
and if, in addition, X :
is the Fourier transform of x( t ) , we have [GER 66] [NAS 65]:
DR ( : )
: L( : )
X ( : ) :
[3.46]
yielding: X(: )
: DR ( : ) VR ( : )
with V R ( Z ) being the pseudovelocity spectrum.
[3.47]
Properties of Shock Response Spectra
125
The dimension of L : is that of the variable of excitation A t multiplied by
time. The quantity : L : is thus that of A t . If the expression of A t is standardized by dividing it by its maximum value A m , it becomes, in dimensionless form: DR :
: L :
Am
Am
[3.48]
With this representation, the Fourier spectrum of the signal ( identical to its residual shock spectrum (
D R : Am
: L : Am
) is
) for zero damping [SUT 68].
3.9.3. Comparison of the relative severity of several shocks using their Fourier spectra and their shock response spectra Let us consider the Fourier spectra (amplitude) of two shocks, one being an isosceles triangle shape and the other TPS (Figure 3.22), like their positive shock response spectra, for zero damping (Figure 3.23).
Figure 3.22. Comparison of the Fourier transform amplitudes of a TPS pulse and an isosceles triangle pulse
126
Mechanical Shock
These two shocks have the same duration (1 s), same amplitude (1 m/s2) and even the same associated velocity change (0.5 m/s) (surface under the signal). They only differ in their shape.
Figure 3.23. Comparison of the positive SRS of a TPS pulse and an isosceles triangle pulse
It is noted that the Fourier spectra and shock response spectra of the two impulses have the same relative position as long as the frequency remains lower than f | 1.25 Hz, the range for which the SRS is none other than the residual spectrum, directly related to the Fourier spectrum. On the contrary, for f ! 1.25 Hz, the TPS pulse has a larger Fourier spectrum, whereas the SRS (primary spectrum) of the isosceles triangle pulse is always in the form of the envelope. The Fourier spectrum thus gives only one partial image of the severity of a shock by considering only its effects after the end of the shock (and without taking damping into account).
Properties of Shock Response Spectra
127
Example 3.13. Let us consider the two shocks in Figures 3.24 and 3.25. The amplitude of the Fourier transform of shock A is larger than that of shock B by up to about 1,000 Hz. It is smaller beyond that (Figure 3.26).
Figure 3.24. Shock A
Figure 3.25. Shock B
128
Mechanical Shock
Figure 3.26. Comparison of the amplitudes of the Fourier transforms of the shocks from Figures 3.24 and 3.25
On the other hand, the SRS of shock B is much higher than that of shock A beyond 800 Hz when the damping is equal to zero (Figure 3.27).
Figure 3.27. Comparison of SRS of shocks A and B for a zero damping
For a damping equal to 0.05, the SRS have the same amplitude over the whole frequency range (Figure 3.28). The Fourier transform does not enable us to compare the severity of the shocks.
Properties of Shock Response Spectra
129
Figure 3.28. Comparison of SRS of the shocks from Figures 3.24 and 3.25
3.10. Care to be taken in the calculation of the spectra 3.10.1. Main sources of errors Several studies carried out using the results of comparisons between laboratories have shown that errors during a calculation of a SRS can have several origins, of which the main ones are [SMI 91] [SMI 95] [SMI 96]: the algorithm used (see Volume 2, section 2.10); the presence of a continuous component and/or the technique used to suppress it; an insufficient sampling frequency (see Volume 1, Chapter 1; Volume 2, section 2.12); the presence of significant background noise. The faults observed on the spectra concern in particular, according to the case, the low or high frequencies. NOTE: Specific case of pyroshocks The dispersions observed on the SRS of pyroshocks measured in comparable conditions are often significant (3 dB to more than 8 dB with respect to the mean value, according to the authors [SMI 84] [SMI 86]). The reasons for this dispersion are generally linked to inadequate instrumentation and measurement conditions [SMI 86]:
130
Mechanical Shock
fixing the sensors onto the structure by blocks which act as mechanical filters; zero drift, due to the fact that increased accelerations make the accelerometer crystal work in a temporarily non-linear field. This drift can harm the calculation of the SRS (see section 3.9.3); amplifier saturation; sensor resonance. With the correct instrumentation, the results of measurements carried out in the same conditions are very close in reality. The spectrum does not vary with manufacturing and assembly tolerances. 3.10.2. Influence of background noise of the measuring equipment The measuring equipment is gauged according to the foreseeable amplitude of the shock to be measured. When the shock characteristics are unknown, the rule is to use a large effective range in order not to saturate the conditioning module. Even if the signal to noise ratio is acceptable, the incidence of the background noise is not always negligible and can lead to errors of the calculated spectra and the specifications which are extracted from it. Its principal effect is to increase the spectra artificially (positive and negative), increasing with the frequency and Q factor. Example 3.14.
Figure 3.29. TPS pulse with noise (rms value equal to one-tenth amplitude of the shock)
Properties of Shock Response Spectra
131
Figure 3.30 shows the positive and negative spectra of a TPS shock (100 m/s2, 25 ms) plotted in the absence of noise for a Q factor successively equal to 10 and 50, as well as the spectra (calculated in the same conditions) of a shock (Figure 3.29) composed of this TPS pulse to which is added a random noise of rms value 10 m/s2 (one-tenth of the shock amplitude).
Figure 3.30. Positive and negative SRS of the TPS pulse and with noise
Due to its random nature, it is practically impossible to remove the noise of the measured signal to extract the shock alone from it. However, techniques have been developed to try to correct the signal by cutting off the Fourier transform of the noise from that of the total signal (subtraction of the modules, conservation of the phase of the total signal) [CAI 94].
132
Mechanical Shock
3.10.3. Influence of zero shift We very often observe a continuous component superimposed on the shock signal on the recordings, the most frequent origin being the presence of a transverse high level component which disturbs the operation of the sensor. If this component is not removed from the signal before calculation of the spectra, it can lead to considerable errors [BAC 89] [BEL 88]. When this continuous component has constant amplitude, the signal treated is in fact a rectangle modulated by the true signal.
Figure 3.31. A constant zero shift is similar to a rectangular shock modulated by the signal to be analyzed in the calculation of the SRS
It is therefore not surprising to find on the spectrum of this composite signal the more or less marked characteristics of the spectra of a rectangular shock. The effect is particularly important for oscillatory shocks (with zero or very small velocity change) such as, for example, shocks of pyrotechnic origin. In this last case, the direct component consequently has a modification of the spectrum more particularly visible at low frequencies which results in [LAL 92a]: The positive and negative response spectra of this type of shock being approximately symmetric curves with respect to the frequency axis. They start from zero frequency with a very small slope at the beginning, grow with the frequency up to a maximum of several kHz (or even several dozen kHz), then tend as with all SRS towards the amplitude of the time signal. The disappearance of the quasi-symmetry of the positive and negative spectra characteristic of this type of shock is a very significant indication of a bad centering of the signal. It is recommended to consider a signal to be bad in which the positive and negative SRS are different (in absolute value) by more than 6 dB at certain frequencies (Powers-Piersol procedure) [NAS 99] [PIE 92]. According to the nature of pyroshock, the velocity change at the
Properties of Shock Response Spectra
133
the shock can be zero or not. Examining the velocity signal calculated by integration of the acceleration can also be useful for detecting a drift of varying speed of the mean value of the signal and for verifying the value of the velocity at the end of the shock. Appearance of more or less clear lobes in the negative spectrum, similar to those of a pure square shock. Example 3.15.
Figure 3.32. Pyrotechnic shock with zero shift
The example given is that of a pyrotechnic shock on which we artificially added a continuous component (Figure 3.32). Figure 3.33 shows the variation generated at low frequencies for a zero shift of about 5%. The influence of the amplitude of the shift on the shape of the spectrum (presence of lobes) is shown in Figure 3.34.
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Mechanical Shock
Figure 3.33. Positive and negative SRS of the centered and non-centered shocks
Properties of Shock Response Spectra
135
Figure 3.34. Zero shift influence on positive and negative SRS
Under certain conditions we can try to center a signal presenting a zero shift that is constant or variable according to time, by addition of a signal of the same shape as this shift and of opposite sign [SMI 85]. This correction is always a delicate operation which supposes that only the average value was affected during the disturbance of measurement. In particular one should ensure that the signal is not saturated. 3.11. Use of the SRS for pyroshocks Pyroshocks have different effects depending on their distance from the source. In the far field (section 1.1.13), the SRS is currently used to characterize their severity (comparisons, writing specifications, etc.).
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Mechanical Shock
According to the strain rate of a structure under dynamic force, the study of its behavior considers phenomena of a different nature. Table 1.1 from Volume 1, reproduced below (Table 3.2), shows the main phenomena observed according to the field of the strain rate. 10
0
Evolution of the rate of creep over Phenomenon time Type of test Test facilities
5
10
1
1
10
10
5
Constant strain rate
Structural response, resonance
Elastoplastic wave propagation
Shock wave propagation
Creep
Static
Slow dynamic
Fast dynamic (impact)
Very fast dynamic (hypervelocity)
Constant force machines
Hydraulic machines
Hydraulic cylinders Exciters
Negligible inertial forces
Metal–metal impact Explosives Shocks Gas guns pyrotechnic in origin Significant inertial forces
Table 3.2. Strain rate areas
For rates in the order of 0.1 m/s to 10 m/s, we generally consider that the structure responds globally to its natural frequencies, whereas for rates of 10 m/s to 105 m/s, the effects are instead linked to the propagation of elastoplastic waves. The bounds of these areas are given here as an example. The phenomena are not discontinuous, moving from one area to the next is not brutal (transition zone). There can be interactions. The limit between the two areas depends on the Young’s modulus and the density of the material ( c
E ), as well as the configuration specifics. U
In the first area, the stress is proportional to the relative displacement (strain), which justifies the use of the SRS of relative displacements. In the second area, H. A. Gaberson [GAB 69] [GAB 95] shows that the stress is proportional to the pseudo-velocity ( Z 0 z sup ) and that this pseudo-velocity is close to the relative velocity z of the mass of the system with one degree of freedom (model of the SRS).
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137
From this he deduces the following rule: An SRS is only considered severe if one of its components exceeds the following threshold [ENV 89] [GAB 69]: Threshold = 0.8 (g/Hz) x Natural frequency (Hz)
[3.49]
(g = 9.81 m/s2). This rule relies on an unpublished observation that military-quality equipment tends to show no faults under shock for a pseudo-velocity SRS lower than 100 inches/sec (254 cm/s). The threshold given by this relation is however equal to 49.1 in/sec (125 cm/s), which assumes that a margin of 6 dB has been taken into account. If we consider [MOE 85] [RUB 86] that the strain rate separating the two areas is of the order of 50 in/s, thus 1.25 m/s, it is necessary, in order to be outside the wave propagation area, that:
Z 0 z sup 1.25
[3.50]
2 S f 0 Z 0 z sup 2 S f 0 1.25
[3.51]
thus:
or:
Z 02 z sup g
0.8 f 0
[3.52]
This rule is used for aerospace and military-quality equipment [GRZ 08] [GRZ 08a] [GRZ 08b] [HOR 97]. It relies on the hypothesis of a stress proportional to the pseudo-velocity [CRA 62], which is only exact in the zone where there is elastoplastic wave propagation, thus in the near field: V
N EU V
[3.53]
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Mechanical Shock
where: V = stress; N = constant; E = Young’s modulus of the material; U= density; V = velocity. The application of this rule is helped by a representation of SRS in the fourcoordinate axes, which gives particular prominence to the pseudo-velocities on the y-axis.
Figure 3.35. Example of SRS of pyroshock in the four-coordinate axes
Chapter 4
Development of Shock Test Specifications
4.1. Introduction The first tests on the behavior of materials in response to shocks were carried out in 1917 by the US Navy [PUS 77] [WEL 46]. The most significant development started at the time of World War II with the development of specific free fall or pendular hammer machines. The specifications are related to the type of machine and its adjustments (drop height, material constituting the programmer, mass of the hammer). Given certain precautions, this process ensures a great uniformity of the tests. The demonstration is based on the fact that the materials, having undergone this test successfully, resist the real environment which the test claims to simulate well. It is necessary to be certain that the severity of the real shocks does not change from one project to another. It is to be feared that the material thus designed is more fashioned to resist the specified shock on the machine than the shock to which it will actually be subjected in service. Specifications appeared very quickly, contractually imposing the shape of acceleration signals, their amplitude and duration. In the mid-1950s, taking into account the development of electrodynamic exciters for vibration tests and the interest in producing mechanical shocks, the same methods were developed (it was at that time that simulation of real environment vibrations by random vibrations in the test were started). This testing on a shaker, when possible, indeed presents a certain number of advantages [COT 66]: vibration and impact tests on the same device, the possibility of carrying out shocks of very diverse shapes, etc.
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Mechanical Shock
In addition, the shock response spectrum became the tool selected for the comparison of the severity of several shocks and for the development of specifications. In this last case, the stages are as follows: – calculation of shock spectra of transient signals of the real environment; – plotting of the envelope of these spectra; – searching for a signal of simple shape (half-sine, sawtooth, etc.) of which the spectrum is close to the spectrum envelope. This operation is generally delicate and cannot be carried out without requiring an over-test or an under-test in certain frequency bands. From 1963 to 1975 the development of computers gave way to a method that consisted of giving the shock spectrum to be realized on the control system of the shaker directly. Taking into account the transfer function of the shaker (with the test item), the software then generates (on the input of the test item) a signal versus time which has the desired shock spectrum. This makes it possible to avoid the last stage of the process. The shocks measured in the real environment are, in general, complex in shape; they are difficult to describe simply and impossible to reproduce accurately on the usual shock machines. These machines can generate only simple shape shocks such as rectangle, half-sine, TPS pulses, etc. Several methods have been proposed to transform the real signal into a specification of this nature. 4.2. Simplification of the measured signal This method consists of extracting the first peak (the duration being defined by time when the signal x t is cancelled for the first time) or the highest peak.
Figure 4.1. Taking into account the largest peak
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141
The shock test specification is then described in the form of an impulse of amplitude equal to that of the chosen peak in the measured signal, of duration equal to the half-period thus defined and whose shape can vary, while approaching that of the first peak (Figure 4.1) as early as possible. The choice can be guided by the use of an abacus making it possible to check that the profile of the shock pulse remains within the tolerances of one of the standardized forms [KIR 69]. Another method consists of measuring the velocity change associated with the shock pulse by integration of the function x t during the half-cycle with greater amplitude. The shape of the shock is selected arbitrarily. The amplitude and the duration are fixed in order to preserve the velocity change [KIR 69] (Figure 4.2).
Figure 4.2. Specification with same velocity change
The transformation of a complex shock environment into a simple shape shock (realizable in the laboratory) is, under these conditions, an operation which utilizes the judgment of the operator in an important way. It is rare, in practice, that the shocks observed are simple (with an easy-to-approach shape) and it is necessary to avoid falling into the trap of over-simplification. In the example in Figure 4.3, the half-sine signal can be a correct approximation of the relatively “clean” shock c. However, the real shock d, which contains several positive and negative peaks, cannot be simulated by just one unidirectional wave.
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Mechanical Shock
Figure 4.3. Difficulty of transformation of real shock pulses
It is difficult to give a general empirical rule to ensure that the quality of simulation carried out in a laboratory according to this process and the experiment of the specificator is important. It does not show that the criterion of equivalence chosen to transform the complex signal to a simple shape shock is valid. It is undoubtedly the most serious defect. This method lends itself little to statistical analysis which would be possible if we had several measurements of a particular event and which would make it possible to establish a specification covering the real environment with a given probability. In the same way, it is difficult to determine a shock enveloping various shocks measured in the life profile of the material. 4.3. Use of shock response spectra 4.3.1. Synthesis of spectra
The most complex case is where the real environment, described by curves of acceleration against time, is supposed to be composed of different events p (handling shock, inter-stage cutting shock on a satellite launcher, etc.), with each one of these events itself being characterized by ri successive measurements. These ri measurements allow a statistical description of each event. The following procedure applies for each one: – Calculate the shock response spectrum of each signal recorded with the damping factor of the principal mode of the structure if this value is known, if not, with the conventional value 0.05. In the same way, the frequency band of analysis will have to envelop the principal resonance frequencies of the structure (known or foreseeable frequencies).
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143
– If the number of measurements is sufficient, calculate the mean spectrum m (mean of the points at each frequency) and the standard deviation spectrum, then the standard deviation/mean ratio, according to the frequency; if it is insufficient, produce the envelope of the spectra. – To apply to the mean spectrum or to the mean spectrum + 3 standard deviations a statistical uncertainty coefficient k, calculated for an admissible (or contractual , if the envelope is used) probability of failure (see Volume 5). Each event thus being synthesized in only one spectrum, we proceed to an envelope of all the spectra obtained to deduce a spectrum from it covering the totality of the shocks of the life profile. After multiplication by a test factor (Volume 5), this spectrum will be used as a “real environment” reference for the determination of the specification.
Table 4.1. Process of developing a specification from real shocks measurements
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. When we envisage simulating a pyroshock in a laboratory using a shock of a different type, and particularly using a simple-shaped shock, the characteristics of
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Mechanical Shock
the simple shock can be sensitive to the choice of relative damping chosen. Pyroshock is made up of several oscillations as opposed to simple shock. This type of simulation is not advised a priori. If it is desired, however, it is advised to calculate the SRS with two damping values, for example 0.05 (Q= 10) and 0.01 (Q = 50) (or 0.005 (Q = 100)), in order to check the validity of the simulation [HIM 95] [PIE 92]. 4.3.2. Nature of the specification
According to the characteristics of the spectrum and available means, the specification can be expressed in the form of: – a simple shape signal according to time realizable on the usual shock machines (half-sine, TPS and rectangular pulse). There is an infinity of shocks which have a given response spectrum. This property is related to the very great loss of information in computing the SRS, since we only retain the largest value of the response according to time to constitute the SRS at each natural frequency. We can thus try to find a shock of simple form, to 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 (particular shape of the spectra and limitations of the facilities), the specification will be made up of two shocks, one on each half-axis. The envelope must be approaching the real environment as well as possible, ideally on all spectra in the frequency band retained for the analysis, but if not then in a frequency band surrounding the resonance frequencies of the test item (if they are known); – a shock response spectrum. In this last case, the specification is the reference SRS. 4.3.3. Choice of shape
The choice of the shape of the shock is carried out by 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 and square) (Figure 4.4).
Development of Shock Test Specifications
145
Figure 4.4. Shapes of the SRS of the realizable shocks on the usual machines
If these positive and negative spectra are nearly symmetric, we will retain a terminal peak sawtooth; it is important to remember, however, that the shock which will really be applied to the tested equipment will have a non-zero decay time so that its negative spectrum will tend towards zero at very high frequencies. This disadvantage is not necessarily onerous if, for example, a preliminary study could show that the resonance frequencies of the test item are in the frequency band where the spectrum of the specified shock envelops the real environment. If only the positive spectrum is important, we will choose any form (the selection criterion being the facility for realization) or the ratio between the amplitude of the first peak of the spectrum and the value of the spectrum at high frequencies: approximately 1.65 for the half-sine pulse ( Q 10 ), 1.18 for the terminal peak sawtooth pulse, and no peak for the square pulse.
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Mechanical Shock
4.3.4. Amplitude
The amplitude of the shock is obtained by plotting the horizontal straight line which closely envelops the positive reference SRS at high frequency.
Figure 4.5. Determination of the amplitude of the specification
This line cuts the y-axis at a point which gives the amplitude sought (here we use the property of the spectra at high frequencies which tends, in this zone, towards the amplitude of the signal in the time domain). 4.3.5. Duration
The shock duration is given by the coincidence of a particular point of the reference spectrum and the reduced spectrum of the simple shock selected above (Figure 4.6).
Figure 4.6. Determination of the shock duration
We generally consider the abscissa f01 of the first point which reaches the value of the asymptote at the high frequencies (amplitude of shock). Table 4.2 joins together some values of this abscissa for the most usual simple shocks according to the Q factor [LAL 78].
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147
f01 [
Q
Half-sine
Versed sine
TPS
Square
2
0.2500
0.413
0.542
/
0.248
3
0.1667
0.358
0.465
0.564
0.219
4
0.1250
0.333
0.431
0.499
0.205
5
0.1000
0.319
0.412
0.468
0.197
6
0.0833
0.310
0.400
0.449
0.192
7
0.0714
0.304
0.392
0.437
0.188
8
0.0625
0.293
0.385
0.427
0.185
9
0.0556
0.295
0.381
0.421
0.183
10
0.0500
0.293
0.377
0.415
0.181
15
0.0333
0.284
0.365
0.400
0.176
20
0.0250
0.280
0.360
0.392
0.174
25
0.0200
0.277
0.357
0.388
0.173
30
0.0167
0.276
0.354
0.385
0.172
35
0.0143
0.275
0.353
0.383
0.171
40
0.0125
0.274
0.352
0.382
0.170
45
0.0111
0.273
0.351
0.380
0.170
50
0.0100
0.272
0.350
0.379
0.170
f
0.0000
0.267
0.344
0.371
0.167
Table 4.2. Values of the dimensionless frequency corresponding to the first passage of the SRS by the amplitude unit
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 always remains higher than or equal to the reference spectrum. 2. It is, in general, difficult to carry out shocks of duration lower than 2 ms on standard shock machines. This difficulty can be circumvented for very light equipment with a specific assembly associated with the shock machine (dual mass shock amplifier, section 6.2). We will validate the specification by checking that the positive and negative spectra of the shock, thus determined, envelop the respective reference spectra. We will also verify if the resonance frequencies of the test item are known, that we are not excessively over-testing at these frequencies.
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Mechanical Shock
Example 4.1.
Let us consider the positive and negative spectra characterizing the real environment plotted (result of a synthesis) (Figure 4.7).
Figure 4.7. SRS of the field environment
It should be noted that the negative spectrum preserves a significant level in all of the frequency domain (the beginning of the spectrum being excluded). The most suitable simple shock shape is the terminal peak sawtooth. The amplitude of the shock is obtained by reading the ordinate of a straight line enveloping the positive spectrum at high frequencies (340 m/s2). The duration is deduced from the point of intersection of this horizontal line with the curve (point of lower frequency), which has as an abscissa equal to 49.5 Hz (Figure 4.8). We could also consider the point of intersection of this horizontal line with the tangent at the origin.
Development of Shock Test Specifications
149
Figure 4.8. Abscissa of the first passage by the unit amplitude
We read on the dimensionless spectrum of a TPS pulse (same damping ratio) the abscissa of this point: f0 W 0.415 , so that f 0 49.5 Hz, yielding:
W
0.415 49.5
0.0084 s
The duration of the shock (rounded up) will therefore be: W
9 ms
which moves the spectrum slightly towards the left and makes it possible to cover the low frequencies better. Figure 4.9 shows the spectra of the environment and those of the TPS pulse thus determined.
Figure 4.9. SRS of the specification and of the real environment
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Mechanical Shock
NOTE: In practice, it is only at this stage that the test factor can be applied to the shock amplitude. 4.3.6. Difficulties
This method leads easily to a specification when the positive spectrum of reference increases regularly from the low frequencies to a peak value not exceeding approximately 1.7 times the value of the spectrum at the highest frequencies and then decreases until it is approximately constant at high frequencies. This shape is easy to envelop since it corresponds to the shape of the spectra of normal simple shocks.
Figure 4.10. Case of a SRS presenting an important peak
In practice it can occur that the first peak of the reference spectrum is much larger, that this spectrum has several peaks and that it is almost tangential to the frequency axis at the low frequencies, etc. In the first case (Figure 4.10), a conservative method consists of enveloping the whole of the reference spectrum. After choosing the shape as previously, we note the coordinates of a particular point, e.g. the amplitude S p of the peak and its abscissa fp .
Figure 4.11. Coordinates of the peak of the dimensionless SRS of the selected shock
Development of Shock Test Specifications
151
On the dimensionless positive spectrum of the selected signal, plotted with the
same damping ratio, we read the coordinates of the first peak: I p , R
. We deduce:
– duration W
Ip fp
– amplitude x m
Sp R
Figure 4.12. Under-testing around the peak in the absence of resonance in this range
Such a shock can over-test mostly at the frequencies before and after the peak. To avoid this, if we know that the material does not have any resonance in the frequency band around the peak, a solution is to adjust the spectrum of the simple shock on the high frequency part of the reference spectrum, while cutting the principal peak (Figure 4.12). 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 over-testing at low frequencies (the oscillatory shock has a very small velocity change), the amplitude of the simple shock thus calculated is sensitive to the value of the Q factor in the intermediate frequency range. A specification using an oscillatory shock does not present this disadvantage (but presupposes that the shock is realizable on the exciter). 4.4. Other methods
Other methods were used for simulation of the shocks using their response spectrum. We will quote some of them in the following sections.
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Mechanical Shock
4.4.1. Use of a swept sine
In the past (and sometimes still today) shocks (often shocks of pyrotechnic origin, such as the separation between two stages of a satellite launcher using a flexible linear shaped charge) were simulated by a swept sine defined from the response spectrum of the shock [CUR 55] [DEC 76] [HOW 68]. The objective of this test was not the rigorous reproduction of the responses caused by the shock. This approach was used because it had proved its effectiveness as a stress-screening test (the materials thus qualified as behaving well in the presence of real pyrotechnic shocks [KEE 74]) but also because this type of test is well understood, easy to carry out and control and is reproducible. The test was defined either in a specified way (5 g between 200 and 2,000 Hz), or by looking for the characteristics of a swept sine whose extreme response spectrum envelops the spectrum of the shock considered [CUR 55] [DEC 76] [HOW 68] [KEE 74] [KER 84]. The sweeping profile is obtained, in practice, by dividing the response spectrum of the shock by the quality factor Q chosen for the calculation of the spectrum. There are many disadvantages to this process: – The result is generally very sensitive to the choice of the damping factor chosen for the calculation of the spectrum. It is therefore very important to know this factor for transformation, which also implies that if there are several resonances, the Q factor varies little with the frequencies. – A very short phenomenon, which will induce the response of few cycles, is replaced by a vibration of much larger duration which will produce a relatively significant number of cycles of stress in the system and will be able to thus damage the structures sensitive to this phenomenon in a non-representative manner [KER 84]. – The maximum responses are the same, but the acceleration signals x t are very different. In a sinusoidal test, the system reaches the maximum of its response at its resonance frequency. The input is small and it is the resonance which makes it possible to reach the necessary response. Under shock, the maximum response is obtained at a frequency more characteristic of the shock itself [CZE 67]. – The swept sine individually excites resonances, one after another, whereas a shock has a relatively broad spectrum and simultaneously excites several modal responses which will combine. The potential failure mechanisms related to the simultaneous excitation of these modes are not reproduced.
Development of Shock Test Specifications
153
4.4.2. Simulation of SRS using a fast swept sine
J.R. Fagan and A.S. Baran [FAG 67] noted in 1967 that certain shock shapes, such as the TPS pulse, excite the high frequencies of resonance of the shaker and suggested the use of a fast swept sine wave to avoid this problem. They also saw two advantages: there is neither residual velocity nor residual displacement and the specimen is tested according to two directions in the same test. The first work carried out by J.D. Crum and R.L Grant [CRU 70] [SMA 74a] [SMA 75], then by R.C. Rountree and C.R. Freberg [ROU 74] and D.H. Trepess and R.G. White [TRE 90] uses a drive signal of the form: x t
A t sin E t
[4.1]
where A t and E t are two time functions, the derivative of I t being the instantaneous pulsation of x t . The response of a linear one-degree-of-freedom mechanical system to a sine wave excitation of frequency equal to the natural frequency of the system can be written in dimensionless form (Volume 1, Chapter 6) as: q T
ª § 1 « [ ¨ cos T e [ T ¨ cos 1 [ 2 T sin 1 [ 2 « 2 ¨ 2[ 1 [ «¬ ©
·º ¸ T ¸» ¸» ¹»¼
[4.2]
If damping is weak, this expression becomes: q T |
1 2[
cos T 1 e [ T
[4.3]
Since the excitation frequency is equal to the resonance frequency, the number of cycles carried out at time t is given by: 2SN
2 S f0 t
[4.4]
For an excitation defined by an acceleration x t q T
Z20 z t xm
1 e 2[ 1
2 S N [
cos 2 S f
x m sin 2S f0 t :
0
t
[4.5]
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Mechanical Shock
Z20 z t
S N/Q Q x m 1 e cos 2 S f0 t
[4.6]
The relative displacement response z(t) is at a maximum when cos 2 S f 0 t yielding: Z20 z m
S N / Q Q x m 1 e
1,
[4.7]
The response Z20 z m depends only on the values of Q and N (for x m fixed). Given a shock measured in the real environment, J.D. Crum and R.L. Grant [CRU 70] plotted the ratio of the response spectra calculated for Q 25 and Q 5 versus frequency f0. Their study, carried out on a great number of shocks, shows that this ratio generally varies little around a value D. The specification is obtained by plotting a horizontal linear envelope of each spectrum (in the ratio D). In sinusoidal mode, the ratio
Z20 z m is, for Q, only a function of N. With a xm
swept sine excitation, we obtain a spectrum of constant amplitude if the number of cycles 'N carried out between the half-power points is independent of the natural frequency f0 , i.e. if the sweeping is hyperbolic. J.D. Crum and R.L. Grant expressed their results according to the parameter N ' Q 'N . If the sweep rate were weak, the ratio would be equal to 5 or 25 according to choice of Q (whatever the sweep mode). To obtain spectra in the ratio D (in general lower than 5), a fast sweep should thus be used. The hyperbolic swept sine is defined as follows, starting from a curve giving the Z20 z m ratio to responses for Q 25 and Q 5 versus N' and of versus N' : x m – the desired ratio D allows us to define N '
N '0 and N'0 gives
Z20 z m by using x m
the two preceding curves; – knowing the envelope spectrum Z02 zm specified for Q the necessary amplitude x m ;
5 , we deduce from it
– the authors have given, for an empirical rule, the sweep starting from a frequency f1 25% lower than the lowest frequency of the spectrum of the specified
Development of Shock Test Specifications
155
shock and finishing at a frequency f 2 25% higher than the highest frequency of the specified spectrum. The excitation is thus defined by: x t
x m sin>E t @
E t
§ f2 t · ¸ ¨ 2 S N '0 ln 1 ¸ ¨ ' N © 0¹
with:
[4.8]
if the sweep is at increasing frequencies, or by: E t
§ f2 t · ¸ 2 S N '0 ln ¨1 ¨ ¸ N '0 ¹ ©
[4.9]
for a sweep at decreasing frequencies; – the sweep duration is given by:
ts
§1 1· N'0 ¨ ¸ ¸ ¨f © 1 f2 ¹
[4.10]
The durations obtained are between a few hundreds of a millisecond and several seconds. It is possible to modulate the amplitude x m according to the frequency to satisfy a specification which would not be a horizontal line and to vary N'0 to better follow the variations of the ratio D of the spectra calculated for Q 25 and Q 5 [CRU 70] [ROU 74]. The formulation of Rountree and Freberg is more general. It is based on the relations:
156
Mechanical Shock
d > ln A t @ d > ln f t @
df t dt dE t dt
E,
A 0
a,
f 0
J R f t
2 S f t ,
E 0
0
½ f0 ° ° ° ° ¾ ° ° ° ° ¿
[4.11]
The modifiable parameters are a, E, f0 , R and J where: – a is the initial value of A t (with t
0 );
– E characterizes the variations of the amplitude A t according to time (or according to f); – f t is the instantaneous frequency, equal to f0 for t
0;
– R and J characterize the variations of t versus time. If J
0 , the law f t is linear, with a sweep rate equal to R.
If J
1 , sweep is exponential, such that f
If J §1 1 ¨ ¨f © 0 f
e
Rt
.
2 , sweep is hyperbolic (as in the assumptions of Crum and Grant) · Rt ¸ . ¸ ¹
Advantages
These methods: – produce shocks pulses well adapted for the reproduction on a shaker; – allow the simulation of a spectrum simultaneously for two values of the Q factor. Drawbacks
These methods lead to shock pulses which do not resemble the real environment at all. These techniques were developed to simulate spectra which can be represented by a straight line on log scales and they adapt badly to spectra with other shapes.
Development of Shock Test Specifications
157
4.4.3. Simulation by modulated random noise
It was recognized that the shocks measured in the domain of earthquakes have a random nature. This is why many proposals [BAR 73] [LEV 71] were made to seek a random process which, after multiplication by an adequate window, provides a shock comparable with this type of shock. The aim is to determine a waveform showing the same statistical characteristics as the signal measured [SMA 74a] [SMA 75]. This waveform is made up of a nonstationary modulated random noise having the same response spectrum as the seismic shock to be simulated. It is, however, important to note that this type of method allows reproduction of a specified shock spectrum only in one probabilistic sense. L.L. Bucciarelli and J. Askinazi [BUC 73] proposed using an excitation of this nature to simulate pyrotechnic shocks with an exponential window of the form: x t
g t n t
[4.12]
where g(t) is a deterministic function of the time, which characterizes the transitory nature of the phenomenon: g t g t
½° ¾ Et e t t 0° ¿
0 for t 0
[4.13]
and n(t) is a stationary broadband noise process with average zero and power spectral density S n : . Given a whole set of measurements of the shock, we are looking to determine S n : and the time constant E to obtain the best possible simulation. The function S n : is calculated from: : X * : @ | S : / 2E E> X n
[4.14]
: X * : @ is the mean value of the squares of the amplitudes of the where E> X Fourier spectra of the shocks measured. The constant E must be selected to be lower than the smallest interesting frequency of the shock response spectrum.
N.C. Tsai [TSA 72] was based on the following process: – choice of a sample of signal x t ;
158
Mechanical Shock
– calculation of the shock response spectrum of this sample; – being given a white noise n t , addition of energy to the signal by addition of sinusoids to n t in the ranges where the shock spectrum is small; – in the ranges where the shock spectrum is large, filtering of n t with a filter attenuating a narrow band ( ); – calculation of the shock spectrum of the modified signal n t and repetition of the process until it reaches the desired shock spectrum. Although interesting, this technique is not the subject of marketed software and is thus not used in the laboratory. NOTE: J.F. Unruth [UNR 82] suggested simulating the seisms while controlling the shock spectrum, the signal reconstituted being obtained by synthesis from the sum of pseudo-random noises into 1/6 octave. Each component of narrow band noise is the weighted sum of 20 cosine functions out of phase whose frequencies are uniformly distributed in the band considered. The relative phases have a random distribution in the interval [0, S]. 4.4.4. Simulation of a shock using random vibration 2
2
The probability that a maximum of Z0 z t is lower than Z0 z m over the
duration T is equal to 1 PP Z02 z m with PP being the distribution function of the response peaks. The number of cycles to be applied during the test is approximately equal to f0 T . If these peaks are supposed independent, the probability PT that all the
f0 T 2 2 2 maxima of Z0 z t are lower than Z0 z m is then ª1 PP Z0 z m º . The ¬ ¼
probability that a maximum of 1 ª¬1 PP Z z m º¼ 2 0
f0 T
2 Z0
z t is higher than
2 Z0
z m is thus equal to
.
Use of a narrow band random vibration
A narrow band random vibration can be applied to the material at a single frequency or several frequencies simultaneously. This process has some advantages [KER 84]: – the number of cycles exceeding a given level can be limited; – several resonances can be excited simultaneously;
Development of Shock Test Specifications
159
– amplification at resonance is reduced compared to the slow swept sine (the response varies as Q instead of Q). However, the nature of the vibration does not make it possible to ensure the reproducibility of the test. 4.4.5. Least favorable response technique Basic assumption
It is assumed that the Fourier spectrum (amplitude) is specified, which is equivalent to specifying the undamped residual shock spectrum (section 3.9.2). It is shown that if the transfer function between the input and the response of the test item (and not that of the shaker) can be characterized by: H :
H : e
i T :
[4.15]
then the peak response of the structure will be maximized by the input [SMA 74a] [SMA 75]: : X
: e i T : X e
[4.16]
: is the module of the specified Fourier transform and: where X e
x t
1
³ 2S f
f
: e i:t d: X
[4.17]
These days, calculation of the above expressions is relatively easy. The phase angle T of the transfer function is measured using a test. With this function and the : , we calculate the input x t . specified module X e The method simply assumes that the studied system is linear with a critical, welldefined response. There is no assumption on the number of degrees of freedom or on damping. It guarantees that the largest possible response peak will be reached, in practice, at about 1 to 2.5 times the response with the real shock (guarantee of a conservative test) [SMA 72] [WIT 74]. The shock spectrum techniques cannot give this insurance for systems to several degrees of freedom. The method requires significant calculations and thus numerical means.
160
Mechanical Shock
An alternative can be found in supposing that H : to be applied to the specimen so that: x t
1
1 and calculating the input
: e i:t d: X ³ e f 2S f
[4.18]
: being a real positive function and x t a real even function, an With X e input, thus defined, will resemble a SHOC waveform (Chapter 8). This input is independent of the characteristics of the test item and thus eliminates the need to define the transfer function H : . The only necessary parameter is the module of the Fourier transform (or the undamped residual shock spectrum). A series of tests showed that this approach is reasonable [SMA 72].
4.4.6. Restitution of an SRS by a series of modulated sine pulses
This method, suggested by D.L. Kern and C.D. Beam [KER 84], consists of applying a series of modulated sine wave shocks sequentially. The retained waveform resembles the response-versus-time of the mass of a one-degree-offreedom system base-excited when it is subjected to an exponentially decayed sine wave excitation; it has as an approximate equation: x t
Ate
x t
0
where :
K: t
sin : t
2 S f:
K = damping of the signal; A x m e
: e K xm ; amplitude of x t ; Neper number.
for t t 0 elsewhere
[4.19]
Development of Shock Test Specifications
161
Figure 4.13. Shock waveform (D.L. Kern and C.D. Hayes)
The choice of K must meet two criteria: – to be close to 0.05, a value characteristic of many complex structures; – to allow the maximum of x t (the largest peak) to take place at the same time as the peak of the envelope of x t .
Figure 4.14. Coincidence of the peak of the signal and its envelope
The interesting thing about this approach (which again takes a proposal of J.T. Howlett and D.J. Martin [HOW 68] containing purely sinusoidal impulses) is the facility of determination of the characteristics of each sinusoid, since each one of them is considered separately, contrary to the case of a control-per-spectrum (Chapter 8). The shocks are easy to create and realize.
162
Mechanical Shock
The adjustable parameters are the amplitude and possibly the number of cycles. The number of frequencies is selected so that the intersection point of the spectra of two adjacent signals is not more than 3 dB lower than the amplitude of the peak of the spectrum (plotted for a damping equal to 0.05). Like the slowly swept sine, this method does not make it possible to excite all resonances simultaneously. We will see in Chapter 8 how this waveform can be used to constitute a complex drive signal restoring the whole of the spectrum. 4.5. Interest behind simulation of shocks on shaker using a shock spectrum
The data of a shock specification for a response spectrum has several advantages: – the response spectrum should be more easily exploitable for dimensioning of the structure than the signal x t itself; – this spectrum can result directly from measurements of the real environment and does not require us, at the design stage, to proceed to an often-delicate equivalence with a signal of simple shape; – the spectrum can be treated in a statistical way if we have several measurements of the same phenomenon; it can be the envelope of several different transitory events and can be increased by an uncertainty coefficient; – the reference most commonly allowed to judge quality of the shock simulation is a comparison of the response spectra of the specification with the shock carried out. In a complementary way, when the shock tests can be carried out using a shaker, we can have direct control from a response spectrum: – The search for a simple form shock of a given spectrum, compatible with the usual test facilities, is not always a simple operation, according to the shape of the reference spectrum resulting from measurements of the real environment. – The shapes of the specified spectra can be very varied, contrary to those of the spectra of the usual shocks (half-sine, triangle, square pulse, etc.) carried out on the shock machines. We can therefore improve the quality of simulation and reproduce shocks which are difficult to simulate with the usual means (case of the pyroshocks for example) [GAL 73] [ROT 72]. – Taking into account the oscillatory nature of the elementary signals used, the positive and negative spectra are very close, which makes a reversal of the test item [PAI 64] useless.
Development of Shock Test Specifications
163
– In theory, simple-shaped shocks created on a shock machine are reproducible, which makes it possible to expect uniform tests from one laboratory to another. In practice, we were obliged to define tolerances on the shapes of the time history signals to take account of the distortions of the signal that are really measured and difficult to avoid. The limits are rather broad (+15 20%) and can result in accepting two shocks, included within these limits, that are likely to have very different effects (which we can evaluate with the shock spectra) [FAG 67].
Figure 4.15. Nominal half-sine and its tolerances
Figure 4.16. Shock located between the tolerances
Figure 4.17. SRS of the nominal half-sine and the tolerance limits
Figures 4.15 and 4.17 show an example of a nominal half-sine (100 m/s2, 10 ms) and its tolerance limits, as well as the shock spectra of the nominal shock and of each lower and upper limit. Figure 4.16 represents a shock made up of the sum of
164
Mechanical Shock
the nominal half-sine and of a sinusoid of amplitude 15 m/s2 and frequency 250 Hz. The spectrum of this signal is superimposed on the spectra of the tolerance limits in Figure 4.18. Although this composite signal remains within the tolerances, it is noted that it has a very different spectrum from those of the tolerance limits for small [ in a frequency band around 250 Hz and that the negative spectra of the tolerance limits intersect and thus do not delimit a well-defined domain [LAL 72].
Figure 4.18. SRS of the shock of Figure 4.16 and of the tolerance limits
NOTE: Several current standards specify the tolerance limits on the SRS with values in the order of ±6 dB [NAS 99]. With this, some practical advantages are added: – sequence of the shock and vibration tests without disassembly and with the same test fixture (saving of time and money); – maintenance of the test item with its normal orientation during the test.
Development of Shock Test Specifications
165
These last two points are not, however, specific to spectrum control, but more generally relate to the use of a shaker. Control by the spectrum, however, increases the capacities of simulation because of the possibility of the choice of the shape of the elementary waveforms and of their variety. The main control methods are described in Chapter 8.
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Chapter 5
Kinematics of Simple Shocks
5.1. Introduction The shock test is generally specified by an acceleration varying with time. This acceleration profile can be obtained with various velocity and displacement profiles depending on the initial velocity of the table supporting the specimen, leading theoretically to various types of shock programmers. All shock test facilities are, in other respects, limited in regard to force (i.e. in acceleration, taking into account the mass of the whole of the moving element made up of the table, the test fixture, the armature assembly in the case of a shaker and the specimen), velocity and displacement. It is thus useful to study the kinematics of the principal shock pulses classically carried out on the machines, namely the half-sine (or versed sine), the terminal peak sawtooth and the rectangle (or the trapezoid). 5.2. Half-sine pulse 5.2.1. General expressions of the shock motion The motion study during the application of the shock is useful for the choice of the programmer and the test facility which will make it possible to carry out the specification. We will limit ourselves, in what follows, to the most general case where the shock is defined by an acceleration pulse x t [LAL 75].
168
Mechanical Shock
With the signal of acceleration [1.1]: x t
x m sin : t
(0 d t d W )
corresponds, by integration, to the instantaneous velocity x x t v t m cos : t constant . Let us suppose that at the initial moment, : t 0 and the velocity is equal to v i : vi
xm :
constant
[5.1]
The constant is thus equal to v i v t
vi
x m :
i.e., since : W vf
:
and the velocity to:
1 cos : t
[5.2]
W of the end of the shock, velocity v f is equal to:
At the moment t v f { v W
x m
vi
x m :
1 cos : t
S:
vi
2 x m
[5.3]
:
The body subjected to this shock thus undergoes a velocity change: 'V { v f v i
2 x m
2 x m W
:
S
[5.4]
This is the area delimited by the curve x t and the time axis between 0 and W.
Kinematics of Simple Shocks
169
Figure 5.1. Velocity change of a half-sine
The displacement is calculated by a second integration; we will take the initial conditions to be t 0 , x 0 , as is practically always the case in these problems. This yields: x t
vi t
· x m § 1 ¨ t sin : t ¸ ¹ : © :
[5.5]
To further the study of this movement x t , it is preferable to specify the test conditions. Two cases arise; the velocity v i is able to be: – either zero before the beginning of the shock: the object subjected to the shock, initially at rest, acquires, under the effect of the impulse, a velocity equal to v f { 'V ; – or arbitrarily non-zero: the specimen has a velocity which varies during the shock duration W from a value v i to a value v f for t W ; it is then said that there is impact. NOTE: This refers mostly to shocks obtained on shock machines. This classification can be open to confusion insofar as the shocks can be carried out on exciters with a pre-shock and/or post-shock which communicates to the carriage (table, fixture and test item) a velocity before the application of the shock itself (we
170
Mechanical Shock
will see in Chapter 7 the need for a pre-shock and/or a post-shock to cancel the table velocity at the end of movement). 5.2.2. Impulse mode
Since v i v t
x t
'V
0: x m
1 cos : t
[5.6]
· x m § 1 ¨ t sin : t ¸ ¹ : © :
[5.7]
:
vf
2 x m W
S
[5.8]
The velocity increases without changing sign from 0 to v f . In the interval (0, W), the displacement is thus at a maximum for t W :
xm
xm
x W
· x m § 1 ¨ W sin S ¸ ¹ : © :
x m W
x m W 2
:
S
[5.9]
This is the area under the curve v t in (0, W). Equations [1.1], [5.6] and [5.7] describe the three curves x t , v t and x t in this interval.
Kinematics of Simple Shocks
Acceleration x t
x m sin
Maximum at t
S W W
2
Velocity t
171
Displacement
x m W § xm W § S · W S · ¨1 cos t ¸ x t ¨ t sin t ¸ S © W ¹ S © S W ¹ Maximum at t W for Maximum at t W : (0, W): x m W 'V v f 2 x m W 2 S xm S Zero slope at t 0 and at Zero slope at t 0 and x m W t W for t W equal to 2 S W Inflection point at t 0 . Inflection point at t 2
v t
Table 5.1. Kinematics of a half-sine shock generated by an impulse
5.2.3. Impact mode
5.2.3.1. General case The initial velocity v i is arbitrary and zero here. The body subjected to the shock arrives on the target with the velocity v i , touches the target (which has a programmer intended to shape the acceleration x t according to a half-sine) between time t 0 and t W . Several cases can arise. At time t W at the end of the shock, the velocity v f can be: – either zero (no rebound);
172
Mechanical Shock
– or arbitrary, different from zero. It is said there is rebound with velocity v R ( { v f ). We assume that the movement is carried out along only one axis, the velocity having a different direction from the velocity of impact. The velocity change is equal, in absolute terms, to 'V general case is where v R is arbitrary: vR
D v i
v R v i . The most
[5.10]
with D 0, 1 (D = coefficient of restitution). The velocity change 'V , equal to 2 x m W , makes it possible to calculate v i : S 'V
v R vi
2 S
x m W
[5.11]
i.e., in algebraic value, and by definition: 2
vR vi
vi
S
[5.12]
x m W
2 S 1 D
x m W
[5.13]
Velocity v t , given by [5.2], is thus written: v t
i.e., since : W v t
x m :
1 cos : t
2 S 1 D
x m W
S: · x m W § D 1 cos : t ¸ ¨ ¹ S ©D 1
[5.14]
and the displacement: x t
· x m W § D 1 1 t sin : t ¸ ¨ ¹ S ©D 1 :
[5.15]
Kinematics of Simple Shocks
173
To facilitate the study, we will consider some particular cases where the rebound velocity is zero, where it is equal (and opposite) to the impact velocity and finally vi where it is equal to . 2 5.2.3.2. Impact without rebound The rebound velocity is zero (D 0 ). The mobile arrives on the target with velocity v i at time t 0 , undergoes shock x t for time W and stops at t W . 'V
vi
2 x m W
[5.16]
S 'V
2 x m W S
[5.17]
and: v t
x m W § S · ¨1 cos t ¸ S © W ¹
[5.18]
x t
x m W § W S · ¨ t sin t ¸ S © S W ¹
[5.19]
The maximum displacement x m throughout the shock also takes place here for t W since v t passes from v i to 0 continuously, without a change in sign. Moreover, since v R 0 , x remains equal to x m for t t W . xm
xm
x m W § W S · ¨ W sin W ¸ S © S W ¹ x m W 2 S
[5.20]
This value of x t is equal to the area under the curve v t delimited by the curve (between 0 and W) and the two axes of coordinates.
174
Mechanical Shock
Acceleration x t
x m sin
Maximum at t
Velocity
S W W 2
Displacement
x m W § S · ¨1 cos t ¸ x t S © W ¹
t
v t
.
W (v R 0) 2 xm W vi S Zero slope when t 0 and t W
Zero at t
Inflection point at t
W 2
x m W § W S · ¨ t sin t ¸ S © S W ¹
Maximum at t xm
W x m W2
S Zero slope at t W and equal 2 x m W in t 0 to S W Inflection point at t 2
Table 5.2. Kinematics of a half-sine shock carried out by impact without rebound
5.2.3.3. Velocity of rebound equal and opposite to the velocity of impact (perfect rebound) After impact, the specimen sets out again in the opposite direction with a velocity equal to the initial velocity (D = 1 and v R v i ). It then becomes: 'V
vR vi
2 vi
2 x m W S
[5.21]
and:
vi
x m W S
[5.22]
Kinematics of Simple Shocks
175
The velocity varies from v i to v R v i when t varies from 0 to W. Let us again take the general expressions [5.14] and [5.15] for v t and x t and set D = 1: v t
and, since x x t
x m W S
cos
0 with t
x m W 2 S
2
S W
[5.23]
t
0 by assumption: sin
S W
[5.24]
t
Acceleration x t
x m sin
Maximum for t
S W W 2
Velocity t
v t
.
Zero for t vi
x m W S W
cos
S W
t
x t
x m W 2 S
2
sin
xm W
Zero slope when t t W Inflection point at t
xm
S 0 and
2
equal to and to
S
x m W
xm W S
W
t
2 x m W 2
Zero slope at t
W
S
W
Maximum for t
2
v R
Displacement
S
2
W 2
and
for t = 0
for t
W
Table 5.3. Kinematics of a half-sine shock carried out by impact with perfect rebound
176
Mechanical Shock
The displacement is maximum for t cos
S W
tm
tm
If K
xm
t m corresponding to
dx dt
0 , so that
0 § 1· ¨K ¸ W © 2¹
W
0, tm
2
[5.25]
. The maximum displacement x m thus has as a value
x m W 2 S
[5.26]
2
In the case of a perfect rebound ( v R v i ), the amplitude x m of the displacement is smaller, by a factor S, than if v R 0 . It should be noted that the amplitude x m is the area ranging between the curve v t and the two axes of W coordinates, in the time interval (0, ): 2 xm
³0
W2
v t dt
[5.27]
5.2.3.4. Velocity of rebound equal and opposed to half of the impact velocity 1
In this case, D
. The mobile arrives at the programmer with a velocity v i , 2 meets it at time t 0 , undergoes the impact for the length of time W, rebounds and vi sets out again in the opposite direction with a velocity v R : 2 vR vi
vi
vi 2
4 x m W 3
S
vi
2 x m W S
[5.28]
[5.29]
Kinematics of Simple Shocks
3
'V
[5.30]
vi
2
1
Let us set D
177
in the general expressions [5.14] and [5.15] of v t and x t ; it
2
then becomes: v t
x m W § 1 S ¨ cos S ©3 W
· t¸ ¹
x t
x m W § W S · ¨ t 3 sin t ¸ 3S © S W ¹
[5.31]
and:
The maximum displacement takes place when v t that: cos
S W
tm
tm | r
6W 10
1 3
| cos
10
[5.33] 6W 10
, yielding:
x m W 2
[5.34]
2S
¨ ©
t m such
2 K1 W
This value of x m§ D x m D 1
0, i.e. when t
6S
We will take, in (0, W), t m |
xm |
[5.32]
x m W 2 S
2
1· ¸ 2¹
lies between the two values x m D 0
x m W 2
. The hatched area under the curve v t is equal to x m .
S
and
178
Mechanical Shock
Acceleration x t
x m sin
Maximum at t
S
W W . 2
Velocity t
v t
Displacement
x m W § S · ¨1 cos t ¸ S © W ¹
Zero at t | 0.6 W 4 x m W vi 3 S v i 2 x m W vR 2 3 S Zero slope when t 0 and t W
Inflection point at t
W 2
x t
x m W § W S ¨ t sin © S S W
W
Maximum at t xm
· t¸ ¹
2 x m W 2 S
2
Zero slope at t | 0.6 W , 4 xm W equal to in 3S 2 x m W t 0 and to in 3S t W
Table 5.4. Kinematics of a half-sine shock caused by impact with 50% rebound velocity
5.2.3.5. Summary chart: remarks on the general case of an arbitrary rebound velocity All these results are brought together in Table 5.5. We can note that: – the maximum displacements required in the cases of impulse and impact without rebound are equal; – the maximum displacement in the case of a 100% rebound velocity is smaller by a factor S; the energy spent by the corresponding shock machine will thus be less [WHI 61].
Kinematics of Simple Shocks
Impact without rebound
Impulse
'V
xm
vi
0
vf
2
vi x m W
x m W 2
S 'V
S
2 x m W
0
vi
2
vi
S
vR
xm
Impact with perfect rebound
vR x m W
x m W S
2
S
'V
xm
v i
2
xm W S x m W S xm W
Impact with rebound to 50% of the initial velocity
vR
4 x m W
vi
'V
S
3
vi
2 x m W
2
3
2
S
2
xm |
S
xm W S
S x m W 2
179
2 xm W
2S
Table 5.5. Summary of the conditions for the realization of a half-sine shock
5.2.3.6. Locus of the maxima The velocity of rebound is, in the general case, a fraction of the velocity of impact:
180
Mechanical Shock
D v i
vR
However: v t
vi
x m W § S · ¨1 cos t ¸ S © W ¹
[5.35]
v t
x m W § D 1 S · cos t ¸ ¨ S ©D 1 W ¹
[5.36]
x t
x m W § D 1 W S · t sin t ¸ ¨ S ©D 1 S W ¹
[5.37]
or:
and:
x t is at a maximum when v t sin
S W
0 , i.e. when cos
S W
tm
D 1 D1
, or, since
t m is positive when 0 d t d W and since D 0, 1 , for:
sin
S W
tm
2 D D 1
Thus: xm
x tm
D 1 2 D · x m W2 § D 1 arc cos ¨ ¸ 2 ¨D 1 D 1 D 1 ¹¸ S ©
[5.38]
The locus of maxima, given by the parametric representation t m D , x m D , can
be expressed according to a relation x m t m while eliminating D between the two relations:
x tm
· S W S xm W § ¨ t m cos t m sin t m ¸ ¹ S © W S W
[5.39]
Kinematics of Simple Shocks
181
The locus of the maxima is an arc of the curve representative of this function in W the interval d t m d W . 2 5.3. Versed sine pulse
A versed sine shock pulse can be represented by [1.4]: x t x t
x m § 2S · ¨1 cos t ¸ 2 © W ¹ 0
We set here :
2S W
½ for 0 d t d W° ¾ elsewhere °¿
.
General expressions of the shock motion
By integration of [1.4], we have: v t
'V
x t
x m § sin : ¨t 2 © : vf vi
t· ¸ vi ¹
x m W 2
x m § t 2 cos : t 1 · ¨ ¸ v t i 2 ¨ ¸ : 2 ©2 ¹
(it is assumed that x 0
0 ).
[5.40]
[5.41]
[5.42]
182
Mechanical Shock
Impulse mode x m § sin : t · ¨t ¸ 2 © : ¹
v t
vf
x t
Impact mode
'V
vi
[5.43]
x m W
[5.44]
v t
[5.45]
x t
2
x m § t 2 cos : t 1 · ¨ ¸ 2 ¨ ¸ : 2 ©2 ¹
x m W 2 1 D
[5.46]
x m § sin : t W · ¨t ¸ 2 © : 1 D¹
[5.47]
2 xm § t Wt · cos : t 1 ¨ ¸ 2 ¨ ¸ : 2 ©2 1 D¹
[5.48]
Velocity
Displacement
Impact without rebound
v t
x m § sin : t · ¨t W ¸ 2 © : ¹
x t
2 xm § t cos : t 1 · ¨ W t ¸ 2 ¨ ¸ : 2 ©2 ¹
Impact with perfect rebound
v t
W sin : xm § ¨t 2 © 2 :
t· ¸ ¹
x t
2 W t cos : t 1 · xm § t ¨ ¸ 2 ¨ ¸ 2 ©2 2 : ¹
Impact with 50% rebound
v t
x m § 2 W sin : ¨t 2 © 3 :
t· ¸ ¹
x t
x m § t 2 2 W t cos : t 1 · ¸ ¨ 2 ¸ 2 ¨© 2 3 : ¹
Table 5.6. Velocity and displacement for carrying out a versed sine shock pulse
(by preserving the notation v R D v i ). Table 5.6 gives the expressions for the velocity and the displacement using the same assumptions as for the half-sine pulse.
Kinematics of Simple Shocks
Impact without rebound
Impulse
vi 'V xm
vi
0
vf x m W 4 t W
vR
x m W 2
S
with
'V xm
vi
x m W 2 0 x m W 2
x m W 2 4
Impact with perfect rebound
xm
x m W
v i
vR
4
2 xm W
16 S
with t
2
S
2
8
W 2
Impact with rebound to 50% of the initial velocity
vR
vi
x m W
2
6
xm
2 xm W
12
Table 5.7. Summary of the conditions for the realization of a haversine shock pulse
5.4. Square pulse
Analytical expression [1.7] representing this pulse shape: x t ® ¯ x t
xm
for 0 d t d W
0
elsewhere
183
184
Mechanical Shock
General expressions of the shock motion
By integration of [1.7]: v t
'V x t
x m t v i
[5.49]
x m W x m
[5.50] 2
t vi t
2
[5.51]
Impulse vf v t x t
Impact
x m W
[5.52]
x m t
[5.53]
x m 2
t
2
vi v t x t
[5.54]
x m W
§ W · x m ¨ t ¸ © 1 D¹
[5.56]
§t W · x m t ¨ ¸ ©2 1 D¹
[5.57]
Displacement
Velocity
Impact without rebound
v t
x m t W
x t
Impact with perfect rebound
v t
§ x m ¨ t ©
x t
Impact with 50% rebound
v t
[5.55]
1 D
W· ¸ 2¹
§ 2 W· x m ¨ t ¸ © 3 ¹
x t
§t · x m t ¨ W ¸ ©2 ¹ xm t 2
t W
§ t 2 W· xm t ¨ ¸ ©2 3 ¹
Table 5.8. Velocity and displacement for carrying out a rectangular shock
Kinematics of Simple Shocks
Impact without rebound
Impulse
vi
'V xm
x m W 2 2 t
vR
x m W
vf
W
x m W
vi
0
'V
vi
with xm
0 xm W x m W 2
2
Impact with perfect rebound
v i
vR
xm
with t
x m W
2 x m W 2
8 W 2
185
Impact with rebound to 50% of the initial velocity
vi
vR xm
with t x W
2 x m W 3 x m W 3 2 2 x m W 9 2W 3 3 xm 4
Table 5.9. Summary of the conditions for the realization of a rectangular shock pulse
186
Mechanical Shock
5.5. Terminal peak sawtooth pulse
Terminal peak sawtooth is represented by [1.5]: x t
xm
x t
0
½ for 0 d t d W ° ¾ elsewhere °¿
t W
General expressions of the shock motion
By integration of [1.5], we obtain: t2
v t
x m
'V
vf vi
x t
x m
v t vf x t
2W
t
vi xm W
'V
3
6W
vi t
Impact with perfect rebound Impact with 50% rebound
v t
3
x t
[5.63]
6W
Impact without rebound
vi
[5.62]
2 t
[5.60]
[5.61]
2W x m W
x m
[5.59]
2
Impulse 2 t
x m
[5.58]
Velocity x m 2 2 v t t W 2 W
v t v t
x m § 2 W 2 · ¨t ¸ 2 W ¨© 2 ¸¹ x m § 2 2 W 2 · ¨t ¸ 2 W ¨© 3 ¸¹
Impact xm W
[5.64]
2 1 D
x m § t 2 W · ¨ ¸ 2 ¨© W 1 D ¸¹ 2 xm t § t W · ¨ ¸ ¨ ¸ 2 ©3 W 1 D¹
[5.65]
[5.66]
Displacement · x m t § t 2 2¸ ¨ x t W ¨ ¸ 2 W ©3 ¹ 2 2· § x m t t W ¨ ¸ x t 2 W ¨© 3 2 ¸¹ x m t 2 2 x t t 2W 6 W
Table 5.10. Velocity and displacement to carry out a TPS shock pulse
Kinematics of Simple Shocks
Impact without rebound
Impulse
vi 'V
xm
0 x m W
vf
6
W
'V
2
vR
2
x m W t
vi
vi
xm W 2
vR
v i
0 x m W 2
with xm
Impact with perfect rebound
x m W 2 3
xm
with t
Impact with rebound to 50% of the initial velocity
x m W
x m W 2
2
x m W
vi
4
6 2 W
187
3 xm W
vR
6 2
2
9
3
with t
W
xm
x W
Table 5.11. Summary of the conditions for the realization of a TPS shock
x m W 2
2 3
x m W 2 6
188
Mechanical Shock
5.6. Initial peak sawtooth pulse Initial peak sawtooth, analytically represented by [1.6] x t x t
t· § xm¨¨ 1 ¸¸ W¹ © 0
½ for 0 d t d W ° ¾ elsewhere °¿
General expressions of the shock motion By integration of [1.6]:
v t
vf x t
v t
§ t · ¸ vi x m t ¨1 ¸ ¨ 2 W¹ ©
[5.67]
x t
x m t 2 § t · ¨1 ¸ vi t ¨ ¸ 2 © 3 W¹
[5.68]
x m
Impulse § t · ¸ t ¨¨1 ¸ 2 W¹ ©
'V
x m W
[5.70]
2
x m t 2 § t · ¸ ¨1 ¸ ¨ 2 © 3 W¹
Impact without rebound Impact with perfect rebound Impact with 50% rebound
vi
[5.69]
[5.71]
v t
[5.72]
2 1 D
2 § · W t ¸ x m ¨ t ¨ ¸ 2 W 2 1 D © ¹ 2 · § W xm t t ¨t ¸ x t ¨ ¸ 2 © 3 W 1 D¹
v t
Velocity x m t W 2 v t 2W v t
Impact x m W
x m § 2 W 2 · ¨t ¸ ¨ ¸ 2W© 2¹ x m § 2 2 W2 · ¨t ¸ ¨ ¸ 2 W© 3 ¹
[5.73]
[5.74]
Displacement · x m t § t 2 ¨ W2 ¸ x t ¨ ¸ 2W ©3 ¹ 2 2· § x m t t W ¨ ¸ x t ¨ ¸ 2 W ©3 2¹ x m t 2 2 x t t 2W 6 W
Table 5.12. Velocity and displacement needed to carry out an IPS shock pulse
Kinematics of Simple Shocks
Impact without rebound
Impulse
vi 'V xm
vi
0
2
x m W 2 3 t
vR
x m W
vf
W
'V
v i
x m W 2 0
vR
2 xm W
6 with t W
v i
x W
x m W
with xm
Impact with perfect rebound
xm
2
t
x m W 4 x m W2
Impact with rebound to 50% of the initial velocity
xm W
vR
x m W 3 x m W 6
2
v t
6 2
with § 1 · ¸¸ W ¨¨1 2¹ ©
vi
12
0.293
189
t
xm
0 for § 1 · ¸¸ W ¨¨1 3¹ ©
x m W 2 9 3
Table 5.13. Summary of the conditions for the realization of an IPS shock
Whatever the shape of the shock, perfect rebound leads to the smallest displacement (and to the lowest drop height). With traditional shock machines, this cannot really be exploited, since programmers do not allow the kinematics of the shock to be chosen.
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Chapter 6
Standard Shock Machines
6.1. Main types The first specific machines developed at the time of World War II belong to two categories: – Pendular machines equipped with a hammer which, after falling in a circular motion, strike a steel plate which is fixed to the specimen (high-impact machine) [CON 51] [CON 52] [VIG 61a]. The first of these machines was manufactured in England in 1939 to test the light equipment which was subjected, on naval ships, to shocks produced by underwater explosions (mines, torpedoes, etc.). Several models were developed in the USA and Europe to produce shocks on equipment of more substantial mass. These machines are still used today (see Figure 6.1). – Sand-drop machines are made up of a table sliding on two vertical guide columns and freefalling into a sand box. Characteristics of the shock obtained are a function of the shape and the number of wooden wedges fixed under the table, as well as the granularity of sand (see Figure 6.2) [BRO 61] [LAZ 67] [VIG 61b]. NOTE: An alternative to this machine simply comprised a wooden table supporting the specimen under which a series of wooden wedges was fixed. The table was released from a given height, without guidance, and impacted the sand in the box.
192
Mechanical Shock
Figure 6.1. Sand-drop shock testing machine
Figure 6.2. Sand-drop impact simulator
The test facilities now used are classified as follows: – freefall machines, derived from the sand-drop machines. The impact is made on a programmer adapted to the shape of the specified shock (elastomer disks, conical or cylindrical lead pellets, pneumatic programmers, etc.). To increase the
Standard Shock Machines
193
impact velocity, which is limited by the drop height, i.e. by the height of the guide columns, the fall can be accelerated by the use of bungee cords; – pneumatic machines, the velocity being derived from a pneumatic actuator; – electrodynamic exciters, the shock being specified either by the shape of a temporal signal, its amplitude and its duration, or by a shock response spectrum; – exotic machines designed to carry out non-realizable shocks by the preceding methods because their amplitude and duration characteristics are generally not compatible with the performances from these means; the desired shapes, not being normal, are not possible with the programmers delivered by the manufacturers. 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. Two principal categories are usually distinguished: – “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 velocity in the tube, is an example; – “impact” machines, which decrease the velocity of the test item throughout the shock and/or which change its direction. 6.2. Impact shock machines Most machines with free or accelerated drops belong to this last category. The machine itself allows the setting of velocity of the test item. The shock is carried out by impact on a 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. The laboratory machines of this type consist of two vertical guide rods on which the table carrying test item (Figure 6.3) slides. The impact velocity is obtained by gravity, after the dropping of the table from a certain height or using bungee cords which allow one to obtain a larger impact velocity.
194
Mechanical Shock
Figure 6.3. Elements of a shock test machine
In all cases, whatever the method for realization of the shock, it is useful to consider the complete movement of the test item between the moment when its velocity starts to take a non-zero value and when it again becomes equal to zero. One thus always observes the presence of a pre-shock and/or a post-shock. Let us consider a freefall shock machine for which the friction of the shock table on the guidance system can be neglected. The necessary drop height to obtain the desired impact velocity v i is given by:
M m g H
1 2
M m v 2i
[6.1]
if M = the mass of the moving assembly of the machine (table, fixture and programmer), m = mass of the test item and g = acceleration of gravity (9.81 m/s2),
Standard Shock Machines
195
yielding: 2
H
vi
[6.2]
2g
These machines are limited by the possible drop height, i.e. 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. We can, however, increase the impact velocity 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 [LON 63] or vertical machines [LAV 69] [MAR 65], this last configuration being less cumbersome.
Figure 6.4. Use of elastic cords
The Collins machine is an example. Its principle of operation is illustrated in Figure 6.4. The table is guided by two vertical columns in order to ensure a good position for the test item at impact. When the carriage is accelerated by elastic cords, the force applied to the table is due to gravity and to the action of these cords. If Th is the tension of the elastic cord at the instant of dropping and Ti the tension of the cord at the time of the impact, we obtain: ª Th Ti º « M m g »H ¬ 2 ¼
1 2
M m v 2i
196
Mechanical Shock
vi
ª Th Ti º » 2 g H «1 «¬ 2 M m g »¼
(neglecting the kinetic energy of the elastic cords).
Figure 6.5. Principle of operation using elastic cords
Figure 6.6. Principle behind pendular shock test machine
If a machine is of the pendular type, the impact velocity is obtained from:
M m g L 1 cos D
1 2
M m v 2i
[6.3]
Standard Shock Machines
197
i.e.: vi
2 g L 1 cos D
[6.4]
where L is the length of the arm of the pendulum and D is the angle of drop. 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 programmer. Universal shock test machine
Figure 6.7. MRL universal shock test machine (impact mode)
Figure 6.8. MRL universal shock test machine (impulse mode)
198
Mechanical Shock
The MRL (Monterey Research Laboratory) Company markets a machine able to perform shocks according to two modes: impulse and impact [BRE 66]. In the two test configurations, the test item is installed on the upper face of the table. The table is guided by two rods which are fixed at a vertical frame. To carry out a test according to the impact mode (general case), we raise the table by the height required by means of a hoist attached to the top of the frame, by the intermediary assembly for raising and dropping (Figure 6.7). 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 on the programmer, the table is again blocked to avoid a second impact. NOTE: A specific device has been developed in order to make it possible to test relatively small specimens with very short duration high acceleration pulses (up to 100,000 g, 0.05 ms) on shock machines which would otherwise not be capable of generating these pulses. This shock amplifier (“dual mass shock amplifier”, marketed by MRL) consists of a secondary shock table (receiving the specimen) and a massive base which is bolted to the top of the carriage of the shock machine. When the main table impacts and rebounds from the programmer on the base of the machine (shock duration of about 6 ms), the secondary table, initially maintained above its base by elastic shock cords, continues downward, stretching the shock cords. The secondary table impacts on a high density felt programmer placed at the base of the shock amplifier, thus generating the high acceleration shock.
The impulse mode shocks (Figure 6.8) are obtained while placing the table on the piston of the programmer (used for the realization of IPS shock pulses). The piston of this hydropneumatic 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 a second time on the programmer. Pre- and post-shocks
The realization of shocks on free or accelerated fall machines imposes de facto pre-shocks and/or post-shocks, the existence of which the user is not always aware, but which can modify the shock severity at low frequencies (section 7.6). 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 programmer. The pre-shock takes place during the fall of the table, the post-shock during its rebound.
Standard Shock Machines
199
Freefall
Let us set D as the rate of rebound (coefficient of restitution) of the programmer. If 'V is the velocity change necessary to carry out the specified shock ( 'V
³0 x t dt ), the carriage rebound velocity and the carriage impact velocity are W
related by [5.10] 'V
v R v i and v i
drop height:
'V
1 D
. We deduce from this the necessary
2
H
vi
[6.5]
2g
where g = acceleration due to gravity.
Figure 6.9. Movement of the table
The movement of the table of the machine from the moment of its release to impact is given by: z
g
[6.6]
z
g t
[6.7]
200
Mechanical Shock
z
1 2
gt
2
[6.8]
yielding, at impact, the instant of time: ti
vi
[6.9]
g
where t i is the duration of the pre-shock, which has as an amplitude g . Since the rebound velocity is equal, in absolute terms, to v R D v i , the rebound of the carriage assembly occurs until a height H R is reached so that: 2
HR
vR
[6.10]
2g
and it lasts until: tR
vR
[6.11]
g
The whole of the movement thus has the characteristics summarized in Figure 6.10.
Figure 6.10. Shock performed
Accelerated fall
Let us set m as the total impacting mass (table + fixture + test item), and k as the stiffness of the elastic cords.
Standard Shock Machines
201
Figure 6.11. Movement of the table during accelerated fall
The differential equation of the movement: 2
m
d z dt
2
m gk z
0
[6.12]
has as a solution: z
where Z
Z m cos Z t k m
g
, yielding:
z
Z m Z sin Z t
z
Z m Z cos Z t
At impact, z cos Z t i
[6.13]
2
Z
[6.14]
2
0 and t g 2
Z Zm
[6.15] t i such that: [6.16]
202
Mechanical Shock
In addition: g
zi
[6.17]
sin Z t i
1
g
2
2
[6.18]
4
Zm Z
The impact velocity is equal to:
z i
Z Z m
vi
1
g
2
4
2
Z Zm
[6.19]
yielding: 2
2
vi
g 4 2 Z Z
Zm
[6.20]
and the duration of the pre-shock is: 1
ti
Z
arc cos
g
[6.21]
Z m Z2
After the shock, the rebound is carried out with velocity v R way, we have: 2
vR
Zm R
tR
2
Z 1 Z
arc cos
g
D v i . In the same
2
[6.22]
4
Z
g
[6.23]
Zm R Z2
and: z
2
Z m R Z cos Z t R t
[6.24]
Standard Shock Machines
203
6.3. High impact shock machines 6.3.1. Lightweight high impact shock machine This machine was developed in 1939 to simulate the effects of underwater explosions (mines) on the equipment onboard military ships. Such explosions, which basically occur far away from the ships, create shocks which are propagated in all the structures. The high impact shock machine was reproduced in the USA in 1940 for use with light equipment; a third machine was built in 1942 for heavier equipment of masses ranging between 100 kg and 2,500 kg [VIG 61a] (section 6.3.2).
Figure 6.12. High impact shock machine for lightweight equipment
The procedure consisted not of specifying a shock response spectrum or a simple shape shock, but rather of the machine being used, the method of assembly, the adjustment of the machine, etc. The machine consists of a welded frame of standard steel sections, of two hammers, one sliding vertically, the other describing an arc of a circle in a vertical plane, according to a pendular motion (Figure 6.12). A target plate carrying the test item can be placed to receive one or 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.
204
Mechanical Shock
Each hammer weighs approximately 200 kg and can fall a maximum height of 1.50 m [CON 52]. The target is a plate of steel of 86 cm x 122 cm x 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 ms and the amplitudes range between 5,000 m/s2 and 10,000 m/s2. 6.3.2. Medium weight high impact shock machine This machine was designed to test equipment whose mass, including the fixture, is less than 2,500 kg (Figure 6.13). It consists of a hammer weighing 1,360 kg which swings through an arc of a circle at an angle greater than 180° and strikes an anvil at its lower face. This anvil, which is fixed under the table carrying the test item, moves vertically upwards under the impact. The movement of this unit is limited to approximately 8 cm at the top and 4 cm at the bottom ([CON 51] [LAZ 67] [VIG 47] [VIG 61b] by stops which block 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. The shocks obtained are similar to those produced with the machine for light equipment. It is difficult to accept a specification which would impose a maximum acceleration. It is easier ‘to control’ the function drop height of the hammer and total mass of the moving assembly (anvil, fixture and test item) [LAZ 67] starting from a velocity change.
Standard Shock Machines
205
Figure 6.13. High impact machine for medium weight equipment
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) [VIG 61a]. These machines can also be used to generate simple shape shocks such as halfsine or TPS pulses [VIG 63], while inserting either an elastic or plastic material between the hammer and the anvil carrying the test item. We thus obtain durations of about 10 ms at 20 ms for the half-sine pulse and 10 ms for the TPS pulse. 6.4. Pneumatic machines Pneumatic machines generally consist of a cylinder separated in two parts by a plate bored to let the rod of a piston (located lower down) pass through (Figure 6.14). The rod crosses the higher cylinder, comes out of the cylinder and supports a table receiving the test item.
206
Mechanical Shock
Figure 6.14. The principle of pneumatic machines
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 Teflon seat [THO 64]. 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 programmers (elastomers, lead cones, etc.) placed on the top of the body of the jack.
Standard Shock Machines
207
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 solid mass of reaction. The interest behind this lies in its performance and its compactness. 6.5. Specific testing facilities When the impact velocity of standard machines is insufficient, we can use other means to obtain the desired velocity: – Drop testers, equipped, for example, with two vertical (or inclined) guide cables [LAL 75] [WHI 61] [WHI 63]. The drop height can reach a few tens of meters. It is wise to make sure that the guidance is correct and, in particular, that friction is negligible. It is also desirable to measure the impact velocity (photoelectric cells or any other device). – 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 programmer fixed at the extremity of a gun on a solid reaction mass [LAZ 67] [LAL 75] [WHI 61] [WHI 63] [YAR 65]. We find the impact mode to be as above. It is necessary that the shock created at the time of setting the velocity 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 velocity setting phase 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 cable instrumentation, which must be wound or unreeled in the gun, in order to follow the movement of the projectile. – Inclined-plane impact testers [LAZ 67] [VIG 61b]. 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 running up against a wooden barrier.
Figure 6.15. Inclined plane impact tester (CONBUR tester)
208
Mechanical Shock
The shape of the shock can be modified by using elastomeric “bumpers” or springs. Tests of this type are often named “CONBUR tests”. 6.6. Programmers We will describe only the most-frequently used programmers used to carry out half-sine, TPS and trapezoid shock pulses. 6.6.1. Half-sine pulse These shocks are obtained using an elastic material interposed between the table and the solid mass reaction. Shock duration The shock duration is calculated by supposing that the table and the programmer, for this length of time, constitute a linear mass-spring system with only one degree of freedom. The differential equation of the movement can be written 2
m
d x dt
2
k x
[6.25]
0
where m = mass of the moving assembly (table + fixture + test item) and k = stiffness constant of the programmer, i.e.: x Z20 x
[6.26]
0
The solution of this equation is a sinusoid of period T
2S
. It is valid only Z0 during the elastomeric material compression and its relaxation, so long as there is contact between the table and the programmer, i.e. during a half-period. If W is the shock duration, we thus have: W
S
m k
[6.27]
This expression shows that, theoretically, the duration can be regarded as a function only dependent on the mass m and of the stiffness of the target. It is in
Standard Shock Machines
209
particular independent of the impact velocity. The mass m and the duration W being known, we deduce from it the stiffness constant k of the target: k
S2
m
[6.28]
W2
Maximum deformation of the programmer If v i is the impact velocity of the table and x m the maximum deformation of the programmer during the shock, by equalizing the kinetic loss of energy and the deformation energy during the compression of the programmer, it becomes: 1 2
2
m vi
1 2
2
[6.29]
k xm
yielding: xm
vi
m
[6.30]
k
Shock amplitude From [6.25] we have, in absolute terms, m x m
k x m , yielding x m
xm
k m
and, according to [6.30]:
x m
vi
k
[6.31]
m
where the impact velocity v i is equal to:
vi
2gH
with g = acceleration of gravity ( 1 g
[6.32] 9.81 m/s2) and H = 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.
210
Mechanical Shock
Having determined k from m and W, it is enough to act on the impact velocity, i.e. on the drop height, to obtain the required shock amplitude. Characteristics of the target
For a cylindrical programmer, we have: k
ES L
where S and L are the cross-section and the height of the programmer respectively and where E is the Young’s modulus of material in compression. Depending on the materials available, i.e. possible values of E, we choose the values of L and S which lead to a realizable programmer (by avoiding too large a height-to-diameter ratio to eliminate the risks from buckling). When the table has a large surface, it is possible to place four programmers to distribute the effort. The cross-section of each programmer is then calculated starting from the value of S determined above and divided by four. The elasticity modulus which intervenes here is the dynamic modulus, which is, in general, larger than the static modulus. This divergence is mainly a function of the type of elastomeric material used, although other factors such as the configuration, the deformation and the load can have an effect. The ratio dynamic modulus E d to static modulus Es ranges, in general, between 1 and 2. It can exceed 2 in certain cases [LAZ 67]. The greatest values of this ratio are observed with most damped materials. For materials such as rubber and Neoprene, it is close to unity.
Figure 6.16. High frequencies at impact
Figure 6.17. Impact module with conical impact face (open module)
Standard Shock Machines
211
If the surface of impact is plane, a wave created at the time of the impact is propagated in the cylinder and makes several up and down excursions. The result of this phenomena is the appearance of a high frequency oscillation at the beginning of the signal, which distorts the desired half-sine pulse, results. To avoid this phenomenon, the front face of the programmer is designed to be slightly conical in order to insert the load material gradually (open module). The shock thus created is between a half-sine and a versed sine pulse. The presence of this conical surface makes the behavior of the target non-linear. A study by A. Girard [GIR 06a] on the programmers of an IMPAC 18 x 18 machine shows that for the targets, the amplitude varies as H 0.68 (H being the height of the drop in meters) and the duration as H 0.18 . Relations [6.27] and [6.31] are modified as follows: xm
where k W
O
v1.36 i
O m
[6.33]
O x1.1 m : S §m· ¨ ¸ vi0.36 © O ¹
0.32
[6.34]
m §S· ¨ ¸ x1.1 m © W¹
4.25
[6.35]
At the height of the theoretical drop: H
W x m
2
2 S2 g
must be corrected to take into
account the energy losses: V1
H
2g H
E
E2 § 2 xm ¨ 2g ©S
where E2 | 1.25 .
2 xm W S
· W¸ ¹
[6.36]
2
[6.37]
212
Mechanical Shock
Propagation time of the shock wave
So that the target can be regarded as a simple spring and not as a system with distributed constants, it is necessary that the propagation time of the shock wave through the target is weak with respect to the duration W of the shock. If a is the velocity of the sound in material constituting the target and h its height, this condition is written: 2h a
W E
i.e., since a
2h
(E = Young’s modulus, U= density) and W
U U E
S
2h
E
m
:
k
2
2
k
m
If the mass of the target is equal to M C necessary that U
S
h US ES
2
MC k
S
h S U (S = cross-section), it is thus
m k
i.e. m MC
!!
4 S2
| 0.4
Rebound
The coefficient of restitution is a function of the material. The smallest rebounds are obtained with the elastic materials that are most strongly damped. The metal springs have small damping and thus produce significant rates of rebound, often about 75%. The elastomers vary greatly, with the rate of rebound which can be located as being between 0 and 75% of the drop height. The coefficient of restitution is also a function of the configuration and of the deformation of elastic material. The targets, which are made up of very soft material, presenting large deformations, lead in general to significant rebounds, whereas the elastomeric materials, which are stiff and thin, are calculated to become deformed
Standard Shock Machines
213
only by a few hundredths of a millimeter, and produce only very little rebound [LAZ 67]. A not very substantial rebound can mean that the material of the programmer reacts during the impact like a viscoelastic material, the table taking a rebound velocity higher than the relaxation velocity of the material [BRO 63]. To create a perfectly half-sine shock pulse with this type of programmer, we need a perfect rebound, with a rebound velocity equal to the impact velocity. It is thus necessary that damping is zero. The shock pulse obtained under these conditions is symmetrical. When the rate of rebound decreases, the return of acceleration to zero (relaxation) is faster than the rise of acceleration.
Low damping High rebound Symmetric pulse
High damping Low rebound Unsymmetric pulse
Figure 6.18. Distortion of the half-sine pulse related to the damping of the material
A good empirical rule is to limit the maximum dynamic deformation of the programmer from 10% to 15% of its initial thickness. If this limit is exceeded, the shape obtained risks non-linear tendencies.
214
Mechanical Shock
Example 6.1. Realization of a half-sine shock 300 m/s2, 10 ms. It is assumed that the mass of the moving assembly (table + fixture + test item) is equal to 600 kg.
The elastomeric programmers often have a coefficient of restitution D ( v R D vi ) of about 50%. We will consider the case where D . From [6.28]: k
m
S2 W
S2
600
2
10 2
| 5.92 107 N/m
2
The impact velocity is calculated from [6.31]: vi
xm
m k
xm
W S
300
102 S
| 0.955 m/s 2
which leads to the drop height H
vi
| 47 10
3
m. During the impact, the 2g elastomeric target will be deformed to a height equal to [6.30]:
xm
vi
m k
x m
§ W ·2 ¨ ¸ ©S¹
§ 102 ·2 ¸ | 3 103 m 300 ¨ ¨ S ¸ ¹ ©
The velocity change during the shock is equal 2 2 x m W 'V 300 10 2 | 1.91 m/s. It is checked that 'V 2 v i . S S
to
With L being the height of the target, its diameter D is calculated from ES k : L D
2
4 k S E
L
If the target is an elastomer of Young’s modulus E § 5 107 N/m2, we obtain, if L = 0.02 m, D § 0.174 m. It remains to check that the stress in material does not exceed the acceptable value.
Standard Shock Machines
215
NOTES: 1. Relations [6.27] and [6.31] were established by assuming that the material of the target is perfectly elastic and that the rebound is perfect. If this is not the case, these relations give only one approximation of x m and W (or k and v i ). In difficult cases, it is undoubtedly quicker to carry out a first test, to measure the values of x m and W obtained, then to correct k and v i using:
x m1 x m2
vi 2 vi 1
k 2 m1 k1 m2
[6.38]
i.e., according to the drop height:
x m1 x m2 W2 W1
H 2 k 2 m1 H 1 k1 m2
m2 k1 m1 k 2
[6.39]
[6.40]
Index 1 corresponds to the first shock carried out, index 2 with the required shock. These relationships remain usable as long as there is a certain rebound and as long as the shock remains symmetric. It is unfortunately difficult to maintain the same shape of the shock when we try to modify its amplitude and its duration. Thus, when a rubber target is deformed by more than approximately 30% its length at rest, its characteristic force-displacement becomes non-linear, which leads to a distortion of the profile of the shock [BRO 63] [WHI 61] [WHI 63]. 2. For a confined material (liquid for example), we have k
E dv S p
V ( E dv bulk dynamic modulus, V = volume of the liquid contained and S p effective area of the piston compressing the liquid).
The manufacturers provide cylindrical modules made up of an elastomer sandwiched between two metal plates. The programmer is composed of stacked modules of various stiffnesses (Figure 6.19). It is enough for a relatively low number of different modules to cover a broad range of shock durations by combinations of these elements [BRE 67] [BRO 66a] [BRO 66b] [GRA 66].
216
Mechanical Shock
Figure 6.19. Distribution of the modules (half-sine shock pulse)
The modules are generally 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. We thus avoid exciting its bending mode at lower frequency and amplifying the vibrations due to resonance of the table. The programmers 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 programmer is composed of a cylinder of this material stuck on a plane circular plate screwed to the lower part of the table of the shock machine. 6.6.2. TPS shock pulse Programmers using crushable materials
We showed that, at the time of a shock by impact without rebound, the deflection varies according to time, according to the law:
x t
· x m t § t 2 ¨ W2 ¸ ¸ 2 W ¨© 3 ¹
which, since x t
x m
t W
and F t
m x t , can be written as:
Standard Shock Machines
x
2 2 · FW § F ¨1 ¸ 2 2 ¸ ¨ 2 m © 3 m x m ¹
217
[6.41]
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 [WHI 61]. To obtain a perfect TPS shock pulse, it is necessary that: x t
t
x m
W
and, by integration: v t
x t
t
2
xm vi
2W
3 t x m
6W
vi t
Knowing that F °S t ° ® ° x t ° ¯
m Vcr xm 6W
x t
m x t
m xm W Vcr
Vcr S t , it becomes:
t
[6.42]
t3 vi t
where S t = surface of the programmer in contact with the table at time t and V cr crush stress of material constituting the target. Law S t is thus relatively complicated. If we set S0 S t
and:
S0
t W
m x m V cr
, we can write:
218
Mechanical Shock
x t
2 º S t ª x m S t « » W v i 2 »¼ S0 «¬ 6 S0
W
O x (O = constant) and we have at time t:
It is assumed that S Vcr S
[6.43]
m x t
Vcr O x
[6.44]
yielding: x t
V cr O m
x
0
V cr O
Let us set : 2
m
x t :2 x
. This gives:
0
which has as a solution x t
x m sinh : t . Differentiating twice, we successively
have x t : x m cosh : t and x t : 2 x m sinh : t . With this assumption, the rise curve is not perfectly linear but, in practice, the approximation is sufficient. For F t or x t to present a constant slope, it is thus enough that the cross-section of the programmer increases linearly according to the distance to its top (point of impact), i.e. to define a cone. For t
0, x
0 and x
: x m sinh : W ® ¯x m : v i
vi
'V . For t
W , x W
x m , yielding:
x m
i.e.: m x 2m °O Vcr vi2 sinh : W ° ® vi2 ° sinh : W °x m x m ¯
[6.45]
Standard Shock Machines
219
These relations make it possible, in theory, to determine the characteristics of the target. However, the calculations are complex, with : being related to O. Although it is possible to determine, by calculation, the required load deformation characteristic, according to a particular law of acceleration, it is very difficult to use this information in practice. The difficulty rests in the determination of the form of the programmer and the characteristic of the dynamic crushing to produce a given shock. For each machine and each shock, it is necessary to carry out preliminary tests to check that the programmer is well calculated. The programmers are destroyed with each test. It is thus a relatively expensive method. We prefer to use, if possible, a universal programmer (section 6.6.4). The material generally used is lead or honeycomb. The cones can be calculated as follows: crushed length: xm
x m W 2
[6.46]
3
yielding the height of the cone h ! 1.2 x m (to allow material to become deformed to the necessary height); force maximum: Fm
Sm V cr
m x m
[6.47]
yielding the cross-section S m of the cone at height x m : Sm
m xm Vcr
[6.48]
When all the kinetic energy of the table is dissipated by the crushing of 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. This return time is not zero due to the inherent imperfections of the programmer. Furthermore, if the solid mass has a nonnegligible elasticity, it can become too long and unacceptable.
220
Mechanical Shock
For
lead,
the
order
of
magnitude
of
V cr
is
760 kg/cm2
( 7.6 107 N/m2 76 MPa). The range of possible durations lies between 2 and 20 ms approximately. Penetration of a steel punch in a lead block
Another method of generating a TPS shock pulse consists of using the penetration of a punch of required form in a deformable material such as lead. The punch is fixed under the table of the machine, the block of lead on the solid reaction mass. The velocity setting of the table is obtained, for example, by freefall [BOC 70] [BRO 66a] [RÖS 70]. The duration and the amplitude of the shock are functions of the impact velocity and the point angle of the cone.
Figure 6.21. Realization of a TPS shock by punching of a lead block
Figure 6.22. Penetration of the steel punch in a lead block
The force which tends to slow down the table during the penetration of the conical punch in the lead is proportional to the greatest section S x which is penetrated, at distance x from the point. If M is the point angle of the cone: 2
S x
S x tan
M
2
2
yielding, in a simplified way, if m is the total mass of the moving assembly, by equalizing the inertia and braking forces in lead: 2
m
d x dt
2
2
D S x tan
2
M 2
Standard Shock Machines
221
with D being a constant function of the crush stress of lead (by assuming that only this parameter intervenes and that the other phenomena such as steel-lead friction 2 D S tan M 2 are negligible). Let us set a : m 2
d x dt
a x
2
2
[6.49]
If v is the carriage velocity at the time t and v i the impact velocity, this relation can be written: dv
a x
dt
2
yielding: v
2
a
2 x
x
3
3
b
The constant of integration b is calculated starting from the initial conditions: for 0 , v v i yielding: v
2
2
vi
2 3
ax
3
[6.50]
Let us write [6.50] in the form: dt
1
dx
v
1 2
vi
2 3
ax
By integration it becomes: t
1 vi
dx
³0
x
1
2ax
3
2
3 vi
3
222
Mechanical Shock
If we set y
t
3
2a
3
3
3
x and T
2 vi
³0
dy
y
3
1 y
, we obtain:
T
2 a vi
Acceleration then results from [6.49]: x t
3
9
4
a vi y
4
2
3
We have, in addition, v v i 1 y . The velocity of the table is cancelled when all its kinetic energy is dissipated by the plastic deformation of lead. Then, y 1 and: 2
xm
3
3 vi
[6.51]
2a
t max
W
x m
3
Tmax 3
3 2 a vi
[6.52]
4
9 a vi
Knowing that v i W
Tmax 3
[6.53]
4 2 g H (H = drop height), 3 2a
2gH
the shock duration is not very sensitive to the drop height. From these expressions, we can establish the relations:
Standard Shock Machines
2gH
xm W
223
[6.54]
Tmax
and: 3 g H
x m x m
[6.55]
In addition:
'V
x m W 2
3 Tmax
gH
2
2
[6.56]
As an indication, this method allows us to carry out shocks of a few hundred to a few thousand grams, with durations from 4 to 10 ms approximately (for a mass m equal to 25 kg). 6.6.3. Square pulse trapezoidal pulse This test is carried out by impact. A cylindrical programmer consists of a material which is crushed with constant force (lead, honeycomb) or using the universal programmer. In the first case, the characteristics of the programmer can be calculated as follows: – the cross-section is given according to the shock amplitude to be realized using the relation: Fm
m xm
S Vcr
[6.57]
yielding: S
m xm Vcr
[6.58]
– starting from the dynamics of the impact without rebound, the length of crushing is equal to:
224
Mechanical Shock
xm
x m W 2 2
[6.59]
and that of the programmer must be equal to at least 1.4 x m , in order to allow a correct crushing of the matter with constant force. We can say that the shock amplitude is controlled by the cross-section of the programmer, the crush stress of material and the mass of the total carriage mass. The duration is affected only by the impact velocity. For this pulse shape as well, it is possible to use the penetration of a rigid punch in a crushable material such as lead. The two methods produce relatively disturbed signals because of 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 force of resistance as constant on such a stroke. The honeycombs lend themselves better to the realization of a long duration [GRA 66]. We could also use the shearing of a lead plate. 6.6.4. Universal shock programmer The so-called universal MTS Monterey programmer can be used to produce half-sine, TPS and trapezoidal shock pulses after various adjustments. This 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 6.23). 6.6.4.1. Generation of a half-sine shock pulse The chamber is put under sufficient pressure so that, during the shock, the piston cannot move (Figure 6.23). The shock pulse is thus formatted only by the compression of the stacking of elastomeric cylinders (modular programmers) placed under the piston head. We are thus brought back to the case of section 6.6.1.
Standard Shock Machines
Figure 6.23. Universal programmer MTS (half-sine and square pulse configuration)
225
Figure 6.24. Universal programmer MTS (TPS pulse configuration)
6.6.4.2. Generation of a TPS shock pulse The gas pressure (nitrogen) in the cylinder is selected so that, after compression of elastomer during duration W, the piston, assembled in the cylinder as indicated in Figure 6.24, is suddenly released for a force corresponding to the required maximum acceleration x m . The pressure 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. The rise phase is not perfectly linear, but corresponds instead to an arc of versed sine (since if the pressure were sufficiently strong, we would obtain a versed sine by compression of the elastomer alone).
226
Mechanical Shock
Figure 6.25. Realization of a TPS shock pulse
6.6.4.3. Trapezoidal shock pulse The assembly here is the same as that of the half-sine pulse (Figure 6.23). 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. 6.6.4.4. 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 limited by [IMP]: – the allowable maximum force on the table. To carry out a shock of amplitude x m , the force generated on the table, given by: F
ª m table m programmer m fixture m test item º ¬ ¼ xm
[6.60]
must be lower or equal to the acceptable maximum force Fmax . Knowing the total carriage mass, relation [6.60] allows calculation of the possible maximum acceleration under the test conditions:
Standard Shock Machines
x m max
Fmax ª¬ m table m programmer m fixture m test item º¼
This limitation is represented on the abacus by a horizontal line x m
227
[6.61] constant ;
Figure 6.26. Abacus of the limitations of a shock machine
– the maximum freefall height H or the maximum impact velocity, i.e. the velocity change 'V of the shock pulse. If v R is the rebound velocity, equal to a percentage D of the impact velocity, we have: 'V
vR vi
1 D v i
1 D
2gH
³0 x t dt W
yielding: H
'V
2
2 g 1 D 2
[6.62]
where D is a function of the shape of the shock and of the type of programmer used. In practice, there are losses of energy by friction during the fall and especially in the programmer during the realization of the shock. To take account of these losses is difficult to calculate analytically and so we can set: H
E
'V
2
2g
[6.63]
228
Mechanical Shock
where E takes losses of energy and rebound into account at the same time. As an example, the manufacturer of machine IMPAC 60 x 60 (MRL), according to the type of programmers [IMP], gives the values given in Table 6.1. Programmer
Value of E
Elastomer (half-sine pulse)
0.556
Lead (square pulse)
0.2338
Lead (TPS pulse)
1.544
Table 6.1. Loss coefficient E
Figure 6.27. Drop height necessary to obtain a given velocity change
The limitation related to the drop height can be represented by parallel straight lines on a diagram giving the velocity change 'V as a function of the drop height in logarithmic scales. The velocity change being, for all simple shocks, proportional to the product x m W , we have: 'V
O x m W
2gH E 2
yielding, while setting D
EO
2g
:
Standard Shock Machines
H
D xm W
2
[6.64]
Waveform
Half-sine TPS Square
229
Programmer
x m W max (m/s)
Elastomer
17.7
Lead cone
10.8
Universal programmer
7.0
Universal programmer
9.2
Table 6.2. Amplitude x duration limitation
On logarithmic scales ( x m ,W), the limitation relating to the velocity change is represented by parallel inclined straight lines (Figure 6.26). Limitations of programmers
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 square shapes, in association with a universal programmer. Elastomer programmers are limited by the allowable maximum force, a function of the Young’s modulus and their dimensions (Figure 6.26) [JOU 79]. This limitation is in fact related to the need to maintain the stress lower than the yield stress of material, so that the target can be regarded as a pure stiffness. The maximum stress V max developed in the target at the time of the shock can be expressed according to Young’s modulus E, to the maximum deformation x m and to the thickness h of the target according to: V max
E
xm
x m W 2
h
for an impact with perfect rebound. It is necessary that, if R e is 2 S the elastic ultimate stress:
with x m
230
Mechanical Shock 2 E x m W
hS
2
Re
i.e.: h!
2 E x m W
Re S
2
Example 6.3.
MRL IMPAC 60 x 60 shock machine Type
Colour
Hard
Maximum force (kN) Diameter 150.5 mm
Diameter 295 mm
Red
667
2,224
Mean
Blue
445
1,201
Soft
Green
111
333
Table 6.3. Examples of the characteristics of half-sine programmers
Taking into account the mass of the carriage assembly, this limitation can be transformed into maximum acceleration (Fm m x m ). Thus, without a load, with a programmer made out of a hard elastomer with diameter 295 mm and a table mass of 3,000 g, we have x m | 740 m / s 2 . With four programmers used simultaneously, maximum acceleration is naturally multiplied by four. This limitation is represented on the abacus of Figure 6.26 by the straight lines of greater slope. The universal programmer is limited [MRL]: – by the acceptable maximum force; – by the stroke of the piston: the relations established in the preceding sections, for each waveform, show that displacement during the shock is always proportional 2 to the product x m W (Figure 6.28).
Standard Shock Machines
231
Figure 6.28. Stroke limitation of universal programmers
This information 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 following conditions.
x m
constant
x m W
constant
Acceptable force on the table or on the universal programmer Drop height ( 'V )
x m W 2
constant
Piston stroke of the universal programmer
x m W 4
constant
Acceptable force for elastomers
Table 6.4. Summary of limitations on the domain of realizable shock pulses
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Chapter 7
Generation of Shocks Using Shakers
In about the mid-1950s, with the development of electrodynamic exciters for the realization of vibration tests, people soon became aware of the need for the realization of shocks on this facility. This simulation on a shaker, when possible, indeed presents a certain number of advantages [COT 66]. 7.1. Principle behind the generation of a signal with a simple shape versus time The objective is to carry out on the shaker a shock with a simple shape (half-sine, triangle, square, etc.) of given amplitude and duration similar to that carried out on the normal shock machines. This technique was mainly developed during the period 1955 to 1965 [WEL 61]. 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 control signal according to this transfer function and the signal to be realized. One of the first methods used consisted of compensating for the system using 1 analog filters gauged in order to obtain a transfer function equal to H : (if H : is the transfer function of the shaker-test item unit). The compensation must relate at the same time to the amplitude and the phase [SMA 74a]. One of the difficulties of this approach lies in the time and work needed to compensate for the system, along with the fact that a satisfactory result was not always obtained.
234
Mechanical Shock
The digital methods seemed to be much better. The process is as follows [FAV 69] [MAG 71]: – measurement of the transfer function of the installation (including the fixture and the test item) using a calibration signal; – calculation of the Fourier transform of the signal specified at the input of the test item; – division of this transform by the transfer function, calculation of the Fourier transform of the control signal; – calculation of the control signal vs time, by inverse transformation. Transfer function The measurement of the transfer function of the installation can be made using a shock, random vibration or sometimes fast swept sine-type calibration signal [FAV 74]. In all cases, the procedure consists of measuring and calculating the control signal to n dB ( 12 , 9 , 6 and/or 3 ). 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 (non-linearities). The development can be carried out using a dummy item representative of the mass of the specimen. However, and especially 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 close to it. If random vibration is used as the calibration signal, its rms value is calculated so that it is lower than the amplitude of the shock (but not too distant in order to avoid the effects of any non-linearities). This type of signal can result in applying many substantial acceleration peaks to the test item compared with the shock itself. 7.2. Main advantages of the generation of shock using shakers The realization of the shocks on shakers has very interesting advantages: – possibility of obtaining very diverse shock shapes; – use of the same means for testing with vibrations and shocks, without disassembly (reducing time costs) and using the same fixtures [HAY 63] [WEL 61]; – 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 traditional shock machines;
Generation of Shocks Using Shakers
235
– very easy realization of the test on two directions of an axis; – avoids use of a shock machine. In practice, however, we are rather quickly limited by the possibilities of exciters which therefore do not make it possible to generalize their use for shock simulation. 7.3. Limitations of electrodynamic shakers 7.3.1. Mechanical limitations Electrodynamic shakers have limited performances in the following fields [MIL 64] [MAG 72]: – the maximum stroke of the coil-table unit (according to the machines being used, 25.4 mm or 50.8 mm peak to peak). Motion study of the coil-table assembly during the usual simple form shocks (half-sine, terminal peak sawtooth, square) show that the displacement is always carried out on the same side compared with the equilibrium (rest) position of the coil. It is thus possible to improve the performances for shock generation by shifting this rest position from the central value towards one of the extreme values [CLA 66] [MIL 64] [SMA 73];
Figure 7.1. Displacement of the coil of the shaker
– the maximum velocity [YOU 64]: 1.5 to 2 m/s in sine mode (in shock, we can admit a larger velocity with non-transistorized 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, an electromotive force is produced that is opposed to the voltage supply. The velocity must thus have a value such that this emf is lower than the acceptable maximum
236
Mechanical Shock
output voltage of the amplifier. The velocity must also be zero at the end of the shock movement [GAL 73] [SMA 73]; – maximum acceleration, related to the maximum force. The limits of velocity, displacement and force are not affected by the mass of the specimen. J.M. McClanahan and J.R. Fagan [CLA 65] consider that the realizable maximum 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 fatigue of the shaker mechanical assembly. Since the number of shocks which the shaker will carry out is a lot lower than the number of cycles of vibrations than it will undergo during its lifetime, 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 this, we can observe random peaks able to reach 4.5 times this value (limitation of control system) and we can admit the same limitation in shock mode. Other values can be found in the literature, such as: – d 4 times the maximum force in sine mode, with the proviso of not exceeding 300 g on the armature assembly [HUG 72]; – more than 8 times the maximum force in sine mode in certain cases (very short shocks, 0.4 ms for example) [GAL 66]. W.B. Keegan [KEE 73] and D.J. Dinicola [DIN 64] give a factor of about 10 for the shocks with a duration lower than 5 ms. The limitation can also be due to: – the resonance of the moving element (a few thousand Hz). 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 large accelerations can involve a separation of the coil of the moving component. 7.3.2. Electronic limitations 1. Limitation of the output voltage of the amplifier [SMA 74a] which limits coil velocity.
Generation of Shocks Using Shakers
237
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 overvoltages well and thus are limited in shock mode [MIL 64]. 7.4. Remarks on the use of electrohydraulic shakers Shocks can be realized on electrohydraulic exciters [MOR 06], but with additional stresses: – contrary to the case of electrodynamic shakers, we cannot obtain shocks of amplitude larger than realizable accelerations in the steady mode in this way; – the hydraulic vibration machines are also strongly non-linear [FAV 74]. 7.5. Pre- and post-shocks 7.5.1. Requirements The velocity change 'V
³0 x t dt W
(W = shock duration) associated with
shocks with a simple shape (half-sine, square, terminal peak sawtooth 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 requirement. 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 side of positive accelerations and the side of negative accelerations. Various solutions are possible a priori: – a pre-shock alone; – a post-shock alone; – pre- and post-shocks, possibly of equal durations.
238
Mechanical Shock
Figure 7.2. Possibilities for pre- and post-shock positioning
Another parameter is the shape of these pre- and post-shocks; the most commonly used shapes being the triangle, the half-sine and the square.
Figure 7.3. Shapes of pre- and post-shock pulses
Due to discontinuities at the ends of the pulse, the square compensation is seldom satisfactory [SMA 85]. We often prefer a versed sine applied to the whole signal (Hann window) which has the advantages of being zero and smoothed at the ends (first zero derivative) and to present symmetric pre- and post-shocks. In all cases, the amplitude of pre- and post-shocks must remain small with respect to that of the principal shock (preferably lower than approximately 10%), in order not to deform the temporal signal too much and consequently, the shock spectrum. For a given pre- and post-shock shape, this choice thus imposes the duration. 7.5.2. Pre-shock or post-shock As an example, the case of a terminal peak sawtooth shock pulse (amplitude unit, duration equal to 1) with square pre- and/or post-shocks (ratio p of the absolute
Generation of Shocks Using Shakers
239
values of the pre- and post-shocks amplitude and of the principal shock amplitude equal to 0.1) is discussed below.
Figure 7.4. Terminal peak sawtooth with rectangular pre- and post-shocks
Figure 7.4 shows the signal as a function of time. The selected parameter is the duration W1 of the pre-shock.
Figure 7.5. Influence of pre-shock duration on velocity during the TPS shock pulse
It is important to check that the velocity is always zero at the beginning and the end of the shock (Figure 7.5). Between these two limits, the velocity remains positive when there is only one post-shock ( W1 0 ) and negative for a pre-shock alone ( W1 5.05 ).
240
Mechanical Shock
Figure 7.6. Influence of pre-shock duration on displacement during the TPS shock pulse
Figure 7.6 shows the displacement corresponding to this movement for the same values of the duration W1 of the pre-shock between W1 0 and W1 5.05 s. Figure 7.7 shows that the residual displacement at the end of the shock is zero for W1 | 2.4 s. The largest displacement during shock, the envelope of the residual displacement and the maximum displacement, is given according to W1 in Figure 7.8 (absolute values). This displacement has a minimum at W1 | 2 s.
Figure 7.7. Influence of the pre-shock duration on the residual displacement
Figure 7.8. Influence of the pre-shock duration on the maximum displacement
Generation of Shocks Using Shakers
241
Figure 7.9. TPS pulse with pre-shock alone (1) and post-shock alone (2)
If we compare the kinematics of the movements now corresponding to the realization of a TPS shock with only one pre-shock (1) and only one post-shock (2), we note, from Figures 7.9 to 7.11, that [YOU 64]: – the peak amplitude of the velocity is identical (in absolute value); – in (2), the acceleration peak takes place when the velocity is very large. It is thus necessary to be able to provide the maximum force when the velocity is significant [MIL 64]; – in (1) on the contrary, the velocity is at a maximum when the acceleration is zero.
Figure 7.10. Velocity curve with pre-shock alone (1) and post-shock alone (2)
Figure 7.11. Displacement curve with preshock alone (1) and post-shock alone (2)
242
Mechanical Shock
Solution (1) requires a less powerful power amplifier and thus seems preferable to (2). However, the use of symmetric pre- and post-shocks is better because of a certain number of additional advantages [MAG 72]: – the final displacement is minimal. If the specified shock is symmetric (with W respect to the vertical line ), this residual displacement is zero [YOU 64]; 2 – for the same duration W 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, i.e. when the velocity is zero (we will thus be able to have the maximum current). The solution with symmetric pre- and post-shocks requires minimal electric power.
Figure 7.12. Kinematics of the movement with pre-shock alone (1), symmetric pre- and post-shocks (2) and post-shock alone (3)
7.5.3. Kinematics of the movement for symmetric pre- and post-shock 7.5.3.1. Half-sine pulse Half-sine pulse with half-sine pre- and post-shocks Duration of pre- and post-shocks [LAL 83]:
W1
W 2p
[7.1]
Generation of Shocks Using Shakers
243
Figure 7.13. Half-sine with half-sine symmetric pre- and post-shocks
The following relations give the expressions of the acceleration, the velocity and the displacement as a function of time in each interval of definition of the signal. For 0 d t d W1 x t
p x m sin
S W1
t
[7.2]
Velocity v t
· x m W § S ¨ cos t 1¸ ¨ ¸ 2S © W1 ¹
[7.3]
Displacement x t
· x m W § W S ¨ t t¸ sin ¨ ¸ 2 S ©2 p S W1 ¹
[7.4]
For W1 d t d W1 W x t
x m sin
S W
t W1
[7.5]
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Mechanical Shock
x m W
S
t W1
v t
x t
2 º ªS x m W ° 1 1 ½° ¾ ® sin« t W1 » ¼ 4 p °¿ ¬W S °¯ S
S
cos
W
[7.6]
[7.7]
For W1 W d t d 2 W1 W S
t W W1
x t
p x m sin
v t
º x m W ª S «1 cos t W W1 » ¼ 2S ¬ W
x t
ªS x m W 2 ° 1 ® t W W1 sin« «¬ W1 2 S °¯ 2 p S
W1
[7.8]
[7.9]
½ º t 1° » 1 ¾ »¼ W p °¿
Half-sine pulse with triangular pre- and post-shocks
Figure 7.14. Half-sine with triangular symmetric pre- and post-shocks
[7.10]
Generation of Shocks Using Shakers
245
Duration of pre- and post-shocks
W1
2W
[7.11]
pS
Rise time of pre-shock W2
· W §2 ¨ p¸ ¨ ¸ S ©p ¹
[7.12]
(assuming that the slope of the segment joining the top of the triangle to the foot of the half-sine is equal to the slope of the half-sine at its origin). For 0 d t d W 2 x t
p x m
v t
x t
t W2
2 p x m t
W2
2
p x m 3 t 6 W2
[7.13]
[7.14]
[7.15]
For W 2 d t d W1 Setting T x T
v T
t W2 S x m W
T W1 W2
· · p x m W § 2 S x m T § T ¨ p¸ ¨ W 2 W1 ¸ ¨ ¸ ¹ ©2 2 S ©p W ¹
[7.16]
[7.17]
246
Mechanical Shock
x T
2 · p x W 2 · p x m W § 2 S xm T § T m 2 T ¨¨ p ¸¸ ¨ W 2 W1 ¸ ¹ 2 W ©3 2S 6 p © ¹
[7.18]
For W1 d t d W1 W If T
t W1
x T
x m sin
v T
x T
S
x m W
[7.19]
cos
S x m W 2 S
T
W
2
S
T
W
sin
S W
[7.20]
T
x m W 2 § 2· ¨p ¸ ¸ 2 ¨ 3S © p¹
[7.21]
For W1 W d t d 2 W1 W W 2 If T
t W1 W
x T
S x m
T
[7.22]
W 2
v T
x T
S x m T 2W
3
S x m T 6W
xm W
[7.23]
S
x m W T S
x m W 2 § 2· ¨p ¸ ¸ 2 ¨ p¹ 3S ©
[7.24]
Generation of Shocks Using Shakers
247
For 2 W1 W W 2 d t d 2 W1 W T
t W 2 2 W1 W W2 T
x T
p x m
v T
2 p x m T § T · x m W § p · ¨ ¸ 1 W ¸ ¨ 2 ¨ ¸ 2¹ 2 ¹ W2 © S ©
x T
2 2 2 p x m T § T · x m W T § p · x m W ¨ ¸ 1 W ¸ ¨ 2 2 ¨ ¸ 2 W2 © 3¹ 2 ¹ 3S S ©
[7.25]
W2
[7.26]
3 § 2· p ¨2 p ¸ ¨ ¸ 2 p¹ © [7.27]
Half-sine pulse with square pre- and post-shocks
Duration of pre- and post-shocks
W1
W § 2· ¨p ¸ ¨ ¸ p¹ 2S©
Figure 7.15. Half-sine with square symmetric pre- and post-shocks
[7.28]
248
Mechanical Shock
· W §2 ¨ p¸ ¨ ¸ 2 S ©p ¹
W2
[7.29]
For 0 d t d W 2
x t
p x m
[7.30]
v t
p x m t
[7.31]
x t
2 p x m t
2
[7.32]
For W 2 d t d W1
If T
t W2
x T
S x m W
T W1 W2
[7.33]
v T
S x m T § T W p · ¨ ¸ p x m W 2 ©2 W S ¹
[7.34]
x T
2 2 S x m T § T p W · p x m W 2 ¨ ¸ p x m W 2 T 2 W ©3 2 S ¹
[7.35]
Generation of Shocks Using Shakers
249
For W1 d t d W1 W If T
t W1
x T
x m sin
v T
x T
ST
[7.36]
W
x m W
cos
S
ST
[7.37]
W
2 3 xm W ª § S T· p p º 1 « sin¨ » ¸ 2 «¬ © W ¹ 2 2 p 24 »¼ S
[7.38]
For W W1 d t d 2 W1 W W 2 If T
t W W1 x T
S x m
v T
x T
T
W
[7.39]
2
S x m T
W 3
S xm T 6W
x m W
[7.40]
S
xm W T S
2 3 xm W § p p · 1 ¨ ¸ 2 ¨ 2 2 p 24 ¸ S © ¹
[7.41]
For 2 W1 W W 2 d t d 2 W1 W If T
t W2 W 2 W1
x T
p x m
[7.42]
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Mechanical Shock
v T
x T
p x m T
2 p x m T
2
x m W §¨ p 2 ·¸ 1 S ¨© 2 ¸¹
x m W S
[7.43]
2 2 § p · x m W ¨ ¸ T 1 2 ¨ ¸ S 2 ¹ ©
3 §p p · 1 ¨ ¸ ¨ ¸ p 2 2 8 © ¹
[7.44]
The expressions of the largest velocity during the movement and those of the maximum and residual displacements are brought together in Table 7.1.
Pre- and post shock pulse shape
Symmetric pre- and post-shocks Maximum velocity
Maximum displacement
Half-sines xm
x m W 2 § 1 1 · ¨ ¸ ¨ ¸ S ©S 4 p¹
xR
0
xm
x m W 2 § 2· ¨3 p ¸ ¸ 2 ¨ 3S © p¹
xR
0
xR
0
Triangles vm
r
xm W S
Residual displacement
Squares xm
3 x m W 2 § p 1 p · ¨1 ¸ 2 ¨ ¸ 2 2 p 24 ¹ S ©
Table 7.1. Half-sine – maximum velocity and displacement – residual displacement
Similar expressions can be established for the other shock shapes (TPS, square, IPS pulses). The results appear in Tables 7.2 to 7.4.
Generation of Shocks Using Shakers
251
7.5.3.2. Terminal sawtooth pulse
Symmetric pre- and post-shocks Half-sines
Pre- and post-shock pulse shape
Triangles
W1
Durations of pre- and postshocks
W1
SW
(1)
8p
Squares
W 2p
§ 1 · p ¸¸ W ¨¨ ©2 p ¹
W2
W1
· W §1 2 ¨ p ¸ ¹ 2 p ©2
W2
· W §1 2 ¨ p ¸ ¹ 2 p ©2 W3
Maximum velocity Maximum displacement Residual displacement
r
vm
xm
xR
x m W2 § 1 S · ¨ ¸ ¨ ¸ 2 © 3 2 32 p ¹
x m W 2 12
xm
xR
x m W 2 12
4p
x m W 4
x m W2 § 1 · ¨ 2 p ¸ ¨ ¸ 12 © 2 p¹
W
1 p
xm
xR
x m W 2 § p3 p 2 1 · ¨ ¸ ¨ ¸ 4 © 6 2 3 8 p¹
x m W 2 § p3 p 1 · ¨ ¸ 4 ¨© 6 2 3 ¸¹
Table 7.2. TPS pulse – maximum velocity and displacement – residual displacement
1 W1 is the total duration of the pre-shock (or post-shock if they are equal). W 2 is the duration of the first part of the pre-shock when it is composed of two straight-line segments (or of the last part of the post-shock). W3 is the total duration of the post-shock when it is different from W1 .
252
Mechanical Shock
7.5.3.3. Square pulse Symmetric pre- and post-shocks Half-sines Triangles
Squares
Pre- and post-shock pulse shape Durations of pre-and postshocks Maximum velocity Maximum displacement
W1
S 4p
W
xR
Residual displacement
xm
0
W 2p
x m W 2 2
2
xm
W1
p r
vm x m W § S · ¸ ¨1 ¨ ¸ 8 © 2 p¹
W
W1
x m W § 1 p · ¨ ¸ 2 p ©3 4 ¹
xR
xm
0
x m W 2 § 1· ¨1 ¸ ¸ ¨ 8 © p¹
xR
0
Table 7.3. Square pulse – maximum velocity and displacement – residual displacement
7.5.3.4. Initial peak sawtooth pulse Symmetric pre and post-shocks Half-sines Triangles
Squares
Pre- and postshock pulse shape W
W1
Durations of pre- and postshocks
W1
SW 8p
2p
§ 1 · p ¸¸ W ¨¨ ©2 p ¹
W2
W1 W
W2
2p W
W3
Maximum velocity Maximum displacement Residual displacement
vm xm
x m W2 § 1 1 S · ¨ ¸ ¨ ¸ 2 © 6 3 2 32 p ¹
xR
x m W 2 12
xm
xR
r
4p · §1 2 ¨ p ¸ ©2 ¹ · §1 2 ¨ p ¸ ©2 ¹
x m W 4
x m W 2 § 1 · ¨1 2 ¸ ¨ ¸ 12 © 2 p¹
x m W 2 12
2p
W
p 1
xm
xR
x m W2 § 1 2 1 · ¨ ¸ ¨ ¸ 4 © 3 3 8 p¹ 2 3 xm W § p p 1· ¨ ¸ ¨ ¸ 4 © 6 2 3¹
Table 7.4. IPS pulse – maximum velocity and displacement – residual displacement
Generation of Shocks Using Shakers
253
7.5.4. Kinematics of the movement for a pre-shock or a post-shock alone In the case of a pre-shock or a post-shock alone, the maximum velocity, equal to the velocity change 'V related to the shock, takes place at the time of transition between the compensation signal and the shock itself. The displacement starts from zero and reaches its largest value at the end of the movement (without changing sign). Like the velocity, it is negative with a pre-shock and positive with a postshock. Tables 7.5 to 7.8 bring together the expressions for this displacement according to the shape of the principal shock and that of the compensation signal, with the same notations and conventions as those in the preceding sections.
Pre-shock or post-shock only Half-sine
Pre- and post-shock pulse shape
Duration of pre-shock or post-shock
W1
SW 8p
Pre-shock: v m
xR
x m W 2 § 1· ¨1 ¸ ¨ ¸ p¹ S ©
Sp
W1
· W §4 ¨ p¸ ¨ ¸ S ©p ¹
W §p 2· ¨ ¸ ¨ ¸ S ©2 p¹
W2
W §2 p· ¨ ¸ ¨ ¸ S ©p 2¹
2 x m W
xR
S
Square
4W
W1 W2
Maximum velocity Residual displacement
Triangle
Post-shock: v m
2 xm W § 2 p 8 ¨ 1 ¨ S ©3 S 3S
· ¸ ¸ p¹
xR
2 x m W S
2 3 2· xm W § p ¨S p ¸ 2 ¨ 24 p ¸¹ S ©
Table 7.5. Half-sine with pre- or post-shock only – maximum velocity and displacement – residual displacement
254
Mechanical Shock
Pre-shock or post-shock only Triangle Half-sine
Pre- and postshock pulse shape
Square
Pre-shock: 1 · W§ W1 ¸ ¨p W1 ¸ ¨ p SW Duration of 2© 2 p¹ W § · 1 pre-shock or 1 · 4p W§ 1 W 2 W ¨¨ p ¸¸ post-shock ¸ ¨ p W 2 ©p ¹ ¸ ¨ 2 ©2 p ¹ Post-shock: Post-shock: W W W1 W1 p 2p xm W Maximum x W Pre-shock: v m m Post-shock: v m velocity 2 2 Pre-shock: Pre-shock: Pre-shock: Pre-shock:
Residual displacement
xR
x m W 2 § 2 S · ¨ ¸ xR ¨ ¸ 2 ©3 8 p¹
x m W 2 § 1· ¨ p 2 ¸P x R ¨ ¸ p¹ 6 ©
Post-shock: 2 2 § · xm W x m W 1 S ¨ xR ¨ 2 ©3 8 p¹ 6
Post-shock: xR
W
x m W 2 § 3 3· ¨p 6 p 8 ¸ ¨ ¸ p¹ 24 ©
Post-shock:
§ 1· ¨1 ¸ x R ¨ ¸ p¹ ©
x m W 2 § 1 1 · ¨ ¸ ¨ ¸ 2 ©3 4 p¹
Table 7.6. TPS with pre- or post-shock only – maximum velocity and displacement – residual displacement
Pre-shock or post-shock only Triangle Half-sine
Pre- and post-shock pulse shape Duration of e pre-shock or post-shock Maximum velocity Residual displacement
W1
SW 2p
Pre-shock: v m xR
W1
2 xm W § S · ¨1 ¸ ¨ ¸ 2 © 2 p¹
xm W xR
x m
Square
2W p
Post-shock: v m § 2 · 2 1 ¸ W ¨¨ ¸ ©2 3 p¹
W
W1
xR
p
x m W 2
x m W 2 § 1· ¨1 ¸ ¨ ¸ 2 © p¹
Table 7.7. Square pulse with pre- or post-shock only – maximum velocity and displacement – residual displacement
P
Generation of Shocks Using Shakers
255
Pre-shock or post-shock only Half-sine
Pre- and post-shock pulse shape
Triangle
Pre-shock:
Pre-shock: W1 Duration of pre-shock or post-shock
W1
SW
W1
Pre-shock: v m Pre-shock:
Residual displacement
xR
§ 1 S · 2 ¸ x m W ¨¨ ¸ © 6 16 p ¹
Post-shock: xR
§ S · 2 1 ¸ x m W ¨¨ ¸ © 3 16 p ¹
p
W p
x m W 2
W2
§1 · W ¨¨ p ¸¸ ©p ¹ 2
Pre-shock:
x m W § 1· ¨1 ¸ x R ¨ ¸ p¹ 6 ©
Post-shock: xR
x m W
2
xR
2p
Post-shock: W 2 W1 1 p 2p
Post-shock: v m
Pre-shock:
W
W1
§1 · W ¨¨ p ¸¸ ©p ¹
W2
Maximum velocity
W
Post-shock:
4p
Square
2 1 · xm W § p ¨ 1 ¸ ¨ ¸ 2 p¹ 3 ©2
x m W 2 § 1 1 · ¸ ¨ ¸ ¨ 2 ©3 4 p ¹
Post-shock: xR
§ p3 3 p 1 1 · ¸ x m W 2 ¨ ¨ 6 8 3 8 p ¸¹ ©
Table 7.8. IPS pulse with pre- or post-shock only – maximum velocity and displacement – residual displacement
7.5.5. Abacuses For a given shock and for given pre- and post-shock shapes, we can calculate, by integration of the acceleration expressions, the velocity and displacement as a function of time, as well as the maximum values of these parameters, in order to compare them with the characteristics of the facilities. This work was carried out for pre- and post-shocks – respectively half-sine, triangular and square [LAL 83] – in order to establish abacuses enabling quick evaluation of the possibility of realization of a specified shock on a given test
256
Mechanical Shock
facility (characterized by its limits of velocity and of displacement). These abacuses are made up of straight line segments on logarithmic scales (Figure 7.16): – AA', corresponding to the limit of velocity: the condition v m d v L (v L = acceptable maximum velocity on the facility considered) results in a relationship of the form x m W d constant (independent of p); – CC, DD, etc., larger slope corresponding to the limit of displacement for various values of p ( p 0.05 , 0.10, 0.25, 0.50 and 1.00). A specific shock will thus be realizable on a shaker only if the point of co-ordinates W, x m (duration and amplitude of the shock considered) is located under these lines; this useful domain increases when p increases.
Figure 7.16. Abacus of the realization domain of a shock
7.5.6. Influence of the shape of pre- and post-pulses The analysis of the velocity and the displacement varying with time associated with some simple shape shocks shows that [LAL 83]: – for all the shocks having a vertical axis of symmetry, the residual displacement is zero; – for a shock of given amplitude, duration and shape, the maximum displacement during movement is largest with half-sine pre- and post-shocks. It is smaller in the case of the triangles, followed by squares;
Generation of Shocks Using Shakers
257
– triangular pre- and post-shocks lead to the largest signal duration, the square giving the smallest duration. Under these displacement and duration criteria, it is thus preferable to use rectangular or triangular pre- and post-shocks. The square however has the disadvantage of having slope discontinuities which make its reproduction difficult and which in addition can excite resonances at high frequencies. However, it would be interesting to try to approach this form [MIL 64]; – the maximum displacement decreases, as we might expect, when p increases. It seems, however, hazardous to retain values higher than 0.10 (although possible with certain control systems), the total shock communicated to the specimen then being too deformed compared with the specification, which results in response spectra appreciably different from those of the pure shocks [FRA 77].
Example 7.1. Half-sine shock with half-sine symmetric pre- and post-shocks. Electrodynamic shaker ( v max d 1.778 m/s, d max d 1.27 cm).
Figure 7.17. Half-sine pulse with half-sine symmetric pre- and post-shocks
258
Mechanical Shock
Figure 7.18. Acceleration, velocity and displacement during the shock of Figure 7.17
Figure 7.19. Abacus for a half-sine shock with half-sine pre- and post-shocks
Generation of Shocks Using Shakers
259
Figure 7.20. Comparison between maximum displacements obtained with the typical shape of pre- and post-shocks
7.5.7. Optimized pre- and post-shocks At the time of the realization of a shock on a shaker, the displacement starts from the equilibrium position, passes through a maximum, then returns to the initial position. We in fact use only half of the available stroke. For better use of the capacities of the machine, we saw (Figure 7.1) that it is possible to shift the zero position of the table. Another method was developed [FAN 81] in order to fulfill the following objectives: – to take into account the tolerances on the shape of the signal allowed by the standards (R.T. Fandrich refers to standard MIL-STD 810C); – to use the possibilities of the shaker as well as possible. The suggested solution consists of defining the following: 1. A pre-shock 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), of the form: 1.155 sin 2 S f t 0.231 sin >2 S 3 f t @
260
Mechanical Shock
Figure 7.21. Optimized pre-shock
The rectangular shape is preferred for the reasons already mentioned, the choice of only the first two terms of the development in series being intended to avoid the disadvantages related to slope discontinuities. The pre-shock consists of one period of this signal, each half-period having a different amplitude: – positive arc: 0.24 u 0.046 x m >1.155 sin 2 S f t 0.231 sin >2 S 3 f t @@
– negative arc: 0.046 x m ^1.155 sin 2 S f t 0.231 sin >2 S 3 f t @`
where x m is the amplitude of the shock to be realized (in m/s2) and f is the fundamental frequency of the signal, estimated from the relationship f
25 0.05 x m / g
[7.45]
where g 9.81 m/s2. This expression is calculated by setting the maximum displacement during the pre-shock lower than the possible maximum displacement on the shaker (for example 1.27 cm). This maximum displacement takes place at the T is its end of the first arc, comparable at a first approximation with a square. If 2 x · 0.24 g § ¨ 0.05 m ¸ T2 , duration, the maximum displacement is equal to dmax 8 ¨© g ¸¹ 1 , yielding, if f T
Generation of Shocks Using Shakers
f2
0.24 g 0.05 xm 8 dmax
g
| 25
261
0.05 xm g
if d max 0.012 m (< 0.0127 ). The total duration of the pre-shock is thus equal to 1 W1 . The factor of 0.05 corresponds to the tolerance limit of the quoted standard f before the principal shock (5%). The constant 0.24 is the reduced amplitude of the first arc, the real amplitude for a shock of maximum value x m being equal to 0.24 0.046 xm . The second arc has a unit amplitude. The table being, before the test, in equilibrium in a median position, the objective of this pre-shock is two-fold: – to give to the velocity, just before the principal shock, 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 (Figure 7.22);
Figure 7.22. Velocity during the optimized pre-shock and the shock
– 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 all the space between the two thrusts (limitation in displacement equal, according to the machines, to 2.54 or 5.08 cm).
262
Mechanical Shock
Acceleration, velocity, displacement
Figure 7.23. Acceleration, velocity and displacement during the pre-shock
Figure 7.24. Acceleration, velocity and displacement during the pre-shock and the shock (half-sine)
2. A post-shock composed of one period of a signal of the shape y
K t sin 2 S f1 t
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.
263
Acceleration, velocity, displacement
Generation of Shocks Using Shakers
Figure 7.25. Overall movement for a half-sine shock
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 post-shock is adjusted to obtain the desired velocity change. Figure 7.25 shows the total signal obtained in the case of a principal shock halfsine 30 g, 11 ms. This methodology has been improved to provide a more general solution [LAX 01]. A. Girard [GIR 06b] recently proposed a modification to the symmetric half-sine shape pre- and post-shocks by the addition of a positive half-sine at each end in which the characteristics are calculated to minimize the maximum displacement as well as the deviation between the SRS of the nominal shock and that of the produced shock. This method, presented in the case of a main half-sine shock, can be extended to any other shape.
Figure 7.26. Half-sine with optimized pre- and post-shocks
264
Mechanical Shock
7.6. Incidence of pre- and post-shocks on the quality of simulation 7.6.1. General The specification of shock is generally expressed in the form of a signal varying with time (half-sine, triangle, etc.). We saw the need for an addition of pre- and/or post-shocks to cancel the velocity at the end of the shock when it is carried out on an electrodynamic shaker. There is no difference in principle between the realization of a shock by impact after free or accelerated fall and the realization of a shock on a shaker. On an impact-type shock machine, the test item and the table have zero velocity at the beginning of the test. The free or accelerated fall corresponds to the pre-shock phase. The rebound, if it exists, corresponds to the post-shock. The practical difference between the two methods lies in the characteristics of shape, duration and amplitude of pre- and post-shocks. In the case of impact, the duration of these signals is generally longer than in the case of the shocks on the shaker, so that the influence on the response appears for systems of lower natural frequency. 7.6.2. Influence of the pre- and post-shocks on the time history response of a onedegree-of-freedom system To highlight the problems, we will discuss the case of a specification which can be realized on a shaker or on a drop table, applied to a material protected by a suspension with a 5 Hz natural frequency and with a Q factor equal to 10. Nominal shock – half-sine pulse x m
500 m/s2, W
10 ms.
Shock on shaker – identical pre-shock and post-shock; – half-sine shape; – amplitude x p
50 m/s2 ( p
– duration such that: 'V
2 S
W x m
2
2 S
W p x p
0.1 );
Generation of Shocks Using Shakers
Wp Wp
2 x m
W
2 xp
2p
50 ms
Shock by impact – free fall 'V
2 S
x m W ; 1
– shock with rebound to 50% ( k – velocity of impact: v i ; – velocity of rebound: v R 'V vi
vi vR 'V 1 k
2
k vi
v i k v i
;
– drop height H
v i2
1
2g
2
g ti2 .
The duration of the fall is t i where ti
vi
0.216 s g Duration of the rebound tR
vR g k 'V
vR
tR
0.108 s
1 k
);
v i 1 k
265
266
Mechanical Shock
Figure 7.27. Influence of the realization mode of a half-sine shock on the response of a one-degree-of-freedom system
2
Figure 7.27 shows the response Z0 z t of a one-degree-of-freedom system ( f0 5 Hz, [ 0.05 ): – for z0
z 0
0 (conditions of the response spectrum);
– in the case of a shock with impact; – in the case of a shock on a shaker. We observed in this example the differences between the theoretical response at 5 Hz and the responses actually obtained on the shaker and the shock machine. According to the test facility used, the shock applied can under-test or over-test the material. To estimate the shock severity we must take into account the whole of the acceleration signal. 7.6.3. Incidence on the shock response spectrum In Figure 7.28, for [ (z 0
0.05 , we show the response spectrum of:
– the nominal shock, calculated under the usual conditions of the spectra z 0 0 ); – the realizable shock on shaker, with its pre- and post-shocks; – the realizable shock by impact, taking into account the fall and rebound phases.
Generation of Shocks Using Shakers
267
Figure 7.28. Influence of the realization mode of a half-sine shock on the SRS
We note in this example that for: – f0 d 10 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; – 10 Hz f0 30 Hz, the spectrum of the shock on the shaker is much overestimated; – f0 ! 30 Hz, all the spectra are superimposed. 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 which includes the resonance frequencies of the test item. This example was treated for a shock on a shaker carried out with symmetric preand post-shocks. Let us consider the case where only one pre-shock or one postshock is used. Figure 7.29 shows the response spectra of: – the nominal signal (half-sine, 500 m/s2, 10 ms); – a shock on a shaker with only one post-shock (half-sine, p velocity change; – a shock on a shaker with a pre-shock alone;
0.1 ) to cancel the
268
Mechanical Shock
– a shock on a shaker with identical pre- and post-shocks.
Figure 7.29. Influence of the distribution of pre- and post-shocks on the SRS of a half-sine shock
It should be noted that: – the variation between the spectra decreases when pre-shock or post-shock alone is used. The duration of the signal of compensation thus being larger, the spectrum is deformed at a lower frequency than in the case of symmetric pre- and post-shocks; – the pre-shock alone can be preferred with the post-shock, but the difference is weak. On the other hand, the use of symmetric pre- and post-shocks has the wellknown advantages which have already been discussed. NOTE: In the case of heavy resonant test items, or those assembled in suspension, there can be a coupling between the suspended mass m and the mass M of the coil– table-fixture unit, with a resulting modification of the natural frequency according to the rule: f 0'
f0 1
m M
[7.46]
Generation of Shocks Using Shakers
269
Figure 7.30 shows the variations of f ' 0 / f 0 according to the ratio m / M . For m close to M, the frequency f 0 can increase by a factor of about 1.4. The stress undergone by the system is therefore not as required.
Figure 7.30. Evolution of the natural frequency in the event of coupling
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Chapter 8
Control of a Shaker Using a Shock Response Spectrum
8.1. Principle of control using a shock response spectrum 8.1.1. Problems The response spectra of shocks measured in the real environment often have a complicated shape which is impossible to enclose within the spectrum of a shock with a simple shape realizable with the usual drop table-type test facilities. This problem arises in particular when the spectrum presents a significant peak [SMA 73]. The spectrum of a simple shape shock will be: – either an envelope of the peak, which will lead to significant over-testing compared with the other frequencies; – or an envelope of the spectrum except the peak with, consequently, undertesting at the frequencies close to the peak. The simulation of shocks of pyrotechnic origin leads to this kind of situation. Shock pulses with a simple shape (half-sine, terminal peak sawtooth) have, in logarithmic scales, a slope of 6 dB/octave (i.e. 45°) at low frequencies incompatible with those spectra of pyrotechnic shocks ( ! 9 dB/octave). When the levels of acceleration do not exceed the possibilities of the shakers, simulations with control using spectra are of interest.
272
Mechanical Shock
Figure 8.1. Examples of SRS which are difficult to envelop with the SRS of a simple shock
The exciters are actually always controlled by a signal which is a function of time. An acceleration-time signal gives only one shock response spectrum. However, there is an infinity of acceleration-time signals with a given spectrum. The general principle thus consists of looking for one of the signals x t with the specified spectrum. Historically, the simulation of shocks with spectrum control was first carried out using analog and then digital methods [SMA 74a] [SMA 75]. 8.1.2. Parallel filter method The analog method, suggested in 1964 by G.W. Painter and H.J. Parry ([PAI 64] [ROB 67] [SMA 74a] [SMA 75] [VAN 72]), consists of using the responses of a series of filters placed simultaneously at the output of a generator of (rectangular) impulses. The filters, distributed in the third octave, are selected to cover the frequency range of interest. Each filter output is a response impulse. If the filters are of narrow bands, each response resembles a narrow band signal which becomes established and then attenuates. If the filters are equivalent to one-degree-of-freedom systems, the response is of the decaying sinusoidal type and the reconstituted signal is oscillatory [USH 72]. Each filter is followed by an amplifier allowing us to regulate the intensity of the response. All the responses are then added together and sent to the input of the amplifier which controls the shaker. We approach the specified spectrum by modifying the gain of the amplifiers at the output of each filter. It is admitted that the output of a given filter only affects the point of the shock spectrum whose frequency is equal to
Control of a Shaker Using a Shock Response Spectrum
273
the central frequency of the filter and to which the shock spectrum is insensitive with the dephasing caused by the filters or the shaker. The complete signal corresponding to a flat spectrum resembles a swept sine of initial frequency equal to the central frequency of the highest filter, whose frequency decreases logarithmically to the central frequency of the lower filter [BAR 74] [HUG 72] [MET 67]. The disadvantage of this process is that we have practically no control over the characteristics of the total control signal (shape, amplitude and duration). According to the velocity of convergence towards the specified spectrum, the adjustment of the overall signals can also be extensive and result in applying several shocks to the test item to develop the control signal [MET 67]. This method was also used digitally [SMA 75], the essential difference being a greater number of possible shapes of shocks. Thereafter, we benefited from the development of data processing tools to create numerical control systems, which are easier to use and use elementary signals of various shapes (according to the manufacturer) to make up the control signal [BAR 74]. 8.1.3. Current numerical methods 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: – The operator must provide to the software, 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. 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 shock response spectrum presenting a frequency peak of the sinusoid whose amplitude is a function of the damping of the sinusoid.
274
Mechanical Shock
Figure 8.2. Elementary shock (a) and its SRS (b)
With an identical shock spectrum, this property makes it possible to realize on the shaker shocks what would be unrealizable with a control carried out by a temporal signal of simple shape (see Figure 8.2). For high frequencies, the spectrum of the sinusoidal signal tends roughly towards the amplitude of the signal.
Figure 8.3. SRS of the damped sinusoid defined for the first point of the SRS to be generated
Control of a Shaker Using a Shock Response Spectrum
Figure 8.4. SRS of the damped sinusoid defined for the second point of the SRS to be generated
Figure 8.5. All of the damped sinusoids and their SRS
275
276
Mechanical Shock
– All the elementary signals are added by possibly introducing a given delay (and variable) between each 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 8.6. Sum of the basic signals with delays
Figure 8.7. Sum of the basic signal without delay
– The total signal thus being made up, we calculate its SRS. Each sinusoid having an influence on the neighboring points of the SRS, the obtained SRS is different from the SRS that is being searched for.
Control of a Shaker Using a Shock Response Spectrum
277
Figure 8.8. SRS of the signal from Figure 8.7 and SRS of the specification
The software proceeds to processes correcting the amplitudes of each elementary signal using a simple rule of three or with the help of a more complicated formula (section 8.2.6) so that the spectrum of the total signal converges towards the reference spectrum after some iterations.
Figure 8.9. SRS of the signal after a first iteration and SRS of the specification
278
Mechanical Shock
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). Finally, after measurement of the transfer function of the facility, we calculate 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 time) [FAV 74]. We propose to examine below the principal shapes of elementary signals that are used or usable. 8.2. Decaying sinusoid 8.2.1. Definition 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 [BOI 81] [CRI 78] [SMA 75] [SMA 85]. Electrodynamic shakers are completely adapted to the reproduction of this type of signal. According to this, we should be able to reconstitute a given SRS from such signals, of the form: a t
a t
A e K : W sin : t 0
t t 0½° ¾ t 0 °¿
[8.1]
where: :
2Sf
f
frequency of the sinusoid
K
damping factor
NOTE: The constant A is not the amplitude of the sinusoid, which is actually equal to [CAR 74] [NEL 74] [SMA 73] [SMA 74a] [SMA 74b] [SMA 75]:
a max
§1· K arc tan¨ ¸ ©K ¹ Ae
ª § 1 ·º sin « arc tan ¨ ¸ » © K ¹¼ ¬
[8.2]
Control of a Shaker Using a Shock Response Spectrum
279
8.2.2. Response spectrum This elementary signal a t has a shock spectrum which presents a more or less significant peak to the frequency f0 f according to the value of K. This peak increases when K decreases. It can, for very weak K (about 10-3), reach an amplitude exceeding the amplitude of shock by a factor of 10 according to time [SMA 73]. It is an interesting property, since it enables, for equal SRS, a reduction in the amplitude of the acceleration signal by a significant factor and thus the ability to carry out shocks on a shaker which could not be carried out with simple shapes.
Figure 8.10. SRS of a decaying sinusoid for various values of K
Figure 8.11. SRS of a decaying sinusoid for various values of the Q factor
280
Mechanical Shock
When K o 0.5 , the SRS tends towards that of a half-sine pulse. We should not confuse the damping factor K, which characterizes the exponential decay of the acceleration signal a t , and the damping factor [, chosen for the plotting of the SRS. For given K, the SRS of the decaying sinusoid also presents a peak whose amplitude varies according to [ or Q 1 2 [ (Figure 8.11). An approximate expression of the ratio R between the peak of the spectrum and constant A of relation [8.1] is given by A.E. Galef [GAL 73] [SMA 75] for 7.103 b I b 0.5 and 0 b Y b 0.1 approximately: K
R
1 K 2
[ K
[8.3]
[
[ [ K
For this chapter, it is more interesting to consider the ratio of the peak of the spectrum to its value at the very high frequency, which can be estimated from [8.2] and [8.3]:
R
K [K
§1· K arc tan ¨ ¸ ©K¹ e
1 K [ 2 ª § 1 ·º [K sin « arc tan ¨ ¸ » [ © K ¹¼ ¬
[8.4]
This ratio is given in Figure 8.12 for various values of the damping factors K (sinusoid) and [ (SRS).
Figure 8.12. Ratio of the peak amplitude of the SRS of a decaying sinusoid and its value at the high frequency versus K and [ from [8.4]
Control of a Shaker Using a Shock Response Spectrum
281
Particular cases
1) If K
[ K
Let us set [
K H . It becomes 2 R
KH
K H
§ H· H ln 2 R H ln K H K ln¨¨1 ¸¸ © K¹
K H
, yielding
H
0
If H is small, we have ln 2 R | 1 ln K H Ho0 R
ln e ln K H , and if
1 2eK
[8.5]
From [8.4], we obtain:
Ro
1 2eK
§1· K arc tan ¨ ¸ ©K¹ e
ª § 1 ·º sin « arc tan ¨ ¸ » © K ¹¼ ¬
[8.6]
2) If [ o 0 , we have from [8.3] Ro
1 2K
[8.7]
and from [8.6]:
Ro
1 2K
§1· K arc tan ¨ ¸ ©K¹ e
ª § 1 ·º sin « arc tan ¨ ¸ » © K ¹¼ ¬
[8.8]
282
Mechanical Shock
NOTE: An approximate expression of the shock response spectrum of a damped sinusoid can be written from [8.4] as follows: a max 2 2
SRS
where h
[8.9]
(1 h ) h 2 / R 2
: / Z 0 [GAL 73].
8.2.3. Velocity and displacement
With this type of signal, the velocity and the displacement are not zero at the end of the shock. The velocity, calculated by integration of a t equal to v t
A
eK : t
2
: 1 K
K sin : t cos : t
A
: 1 K2
Ae
K : t
sin : t , is
[8.10]
If t o f v t o
A 2
: 1 K
[8.11]
The displacement is given by:
x t
Ae 2
: K t
2 2
: 1 K
K
2
sin : t 2 K cos : t sin : t
At 2
: 1 K
2KA
2
2 2
: 1 K
[8.12] If t o f , x t o f (Figure 8.13).
Control of a Shaker Using a Shock Response Spectrum
283
Figure 8.13. The velocity and the displacement are not zero at the end of the damped sine
These non-zero values of the velocity and the displacement at the end of the shock are very awkward for a test on a shaker. 8.2.4. Constitution of the total signal The total control signal is made up initially of the sum (with or without delay) of elementary signals defined separately at frequencies at each point of the SRS, added to a compensation signal of the velocity and displacement. The first stage consists of determining the constants A i and Ki of the elementary decaying sinusoids. The procedure can be as follows [SMA 74b]: – choice of a certain number of points of the spectrum of specified shock, sufficient for correctly describing the curve (couples frequency fi , value of the spectrum Si ); – choice of damping constant Ki of the sinusoids, if possible close to actual values in the real environment. This choice can be guided by examination of the shock spectra of a decaying sinusoid in reduced coordinates (plotted with the same Q factor as that of the specified spectrum), for various values of K (Figure 8.10). These curves underline the influence of K on the magnitude of the peak of the spectrum and over its width. We can also rely on the curves of Figure 8.12. However, in practice, we prefer to have a rule that is easier to introduce into the software. The value Ki | 0.1 gives good results [CRI 78]. It is, however, preferable to choose a variable damping factor according to the frequency of the sinusoid; strong at the low frequencies and weak at the high frequencies. It can, for example, decrease in a linear way from 0.3 to 0.01 between the two ends of the spectrum;
284
Mechanical Shock
NOTE: If we have acceleration signals which lead to the specified spectrum, we could use the Prony method to estimate the frequencies and damping factors [GAR 86]. – K being chosen, we can calculate, using relation [8.3], for given Q
1
2[ (damping chosen for plotting the reference shock spectrum), ratio R of the peak of the spectrum to the amplitude of the decaying sinusoid. This value of R makes it possible to determine the amplitude a max of the decaying sinusoid at the specific frequency.
Knowing that the amplitude a max of the first peak of the decaying sinusoid is related to the constant A by relation [8.2]:
a max
A
§1· K arc tan ¨¨ ¸¸ ©K¹ e
ª § 1 ·º sin «arc tan¨¨ ¸¸» © K ¹¼» ¬«
we determine the value of A for each elementary sinusoid. For small K ( 0.08 ), we have a max | A 1 1.57 K
[8.13]
8.2.5. Methods of signal compensation Compensation can be carried out in several ways: 1) By truncating the total signal until it is realizable on the shaker. This correction can, however, lead to a significant degradation of the corresponding spectrum [SMA 73]. 2) By adding to the total signal (sum of all the elementary signals) a highly damped decaying sinusoid, shifted in time, defined to compensate for the velocity and the displacement [SMA 74b] [SMA 75] [SMA 85]. 3) By adding to each component two exponential compensation functions, with a phase in the sinusoid [NEL 74] [SMA 75] ai t
^
A i : i k1 e
a t
k2 e
b t
`A
i
k3 e
c t
sin :i t T
[8.14]
Control of a Shaker Using a Shock Response Spectrum
285
Compensation using a decaying sinusoid In order to calculate the characteristics of the compensating pulse, the complete acceleration signal used to simulate the specified spectrum can be written in the form: n ª K 6 « U t Ti A i e i i 1¬
x t
U t T A c e
Kc : c t T
:i t Ti
º sin :i t Ti » ¼
sin :c t T
[8.15]
where U t Ti
for t Ti ½ ¾ for t t Ti ¿
0
U t Ti 1
[8.16]
Ti is the delay applied to the ith elementary signal. A c , Zc , Kc and T are the characteristics of the compensating signal (decaying sinusoid). The calculation of these constants is carried out by canceling the expressions of the velocity x and the displacement x obtained by integration of x . n
x t
6 U t Ti
i 1
U t T
Ai
2 :i 1 Ki
AC
2 :c 1 Kc
e ^
Ki :i t Ti ½ ® e Ki sin :i t Ti cos :i t T 1¾ ¯ ¿
>
Kc : c t T
>K
c
@
@ `
sin :c t T cos :c t T 1
[8.17] K : t Ti A ie i i ° ª K2 1 sin : t T 2 K cos : t T º 6 ® U t Ti i i i i i i ¼ 2 2 2 ¬ i 1° :i 1 Ki ¯ n
x t
K : t T ½° ° A ce c c ¾ U t T ® 2 2 2 2 2 ° ° :2 1 K2 ¿ :i 1 Ki :i 1 Ki c ¯ c
A i t Ti
2Ki A i
½° ª K2 1 sin : t T 2K cos : t T º A c t T 2Kc A c ¾ c c c 2 ¬ c ¼ 2 2 2 ° ¿ :c 1 Kc :c 1 Kc
[8.18]
286
Mechanical Shock
The cancelation of the velocity and displacement for t equal to infinity leads to:
Ac 2
:c 1 Kc
i 1
Ac T
Ai
n
6
§ 22 · :i ¨1 Ki i ¸ ¹ ©
2 Kc A c
:c 1 Kc
2 2
:c 1 Kc
2
[8.19]
ª « 2 Ki A i 6« i 1« :2 1 K2 i ¬ i
2 2
º » 2 » :i 1 Ki » ¼ A i Ti
n
2
[8.20]
yielding Ac
T
T
n
Ai
: c 1 Kc2 6
i 1
:i 1 Ki2
2 :c 1 Kc ° 2 Kc A c ® ° :2 1 K2 Ac c ¯ c
1 K :
c
2 Kc
:c 1
Kc2
2 c
Ac
[8.21]
ª 2 Ki A i « A i Ti 6 « 2 2 2 i 1« :i 1 Ki :i 1 Ki ¬ n
2
n
6
i 1
Ai
1
Figure 8.14. Acceleration pulse compensated by a decaying sinusoid
Ki2
º½ »° ¾ 2» »¼° ¿
ª º 2 Ki « Ti » 2 :i ¬« :i 1 Ki »¼
[8.22]
[8.23]
Figure 8.15. Velocity associated with the signal compensated by a decaying sinusoid
Control of a Shaker Using a Shock Response Spectrum
287
Figure 8.16. Displacement associated with the signal compensated by a decaying sinusoid
Constants A c and T characterizing the compensating sinusoid are thus a function of the other parameters ( :c , Kc ). The frequency of the compensating waveform § : · c¸ fc ¨¨ ¸ should be about a half or third of the smallest frequency of the points © 2 K¹ selected to represent the specified spectrum. Damping Kc is selected to be between 0.5 and 1 [SMA 74b]. Constants A c and T can then be determined. Compensation using two exponential signals This method, suggested by Nelson and Prasthofer [NEL 74] [SMA 85], consists of adding to the decaying sinusoid two exponential signals and a phase shift T. Each elementary waveform is of the form: ai t
^
`
Ai :i k1 ea t k2 eb t Ai k3 ec t sin :i t T
[8.24]
The exponential terms are defined in order to compensate the velocity and the displacement (which must be zero at the beginning and the end of the shock). The phase T is given to cancel acceleration at t 0 . With this method, each individual component is thus compensated. The choice of parameters a, b and c is somewhat arbitrary. The idea being to create a signal resembling a damped decaying sinusoid, we choose: c
Ki :n
[8.25]
288
Mechanical Shock
where :i
:n
a
b
2
1 Ki
[8.26]
:n 2S
[8.27]
2 Ki :n
[8.28]
The acceleration, velocity and displacement are zero at the beginning and the end of the shock if: k1
k2
k3
T
a2
[8.29]
b a ª«¬ c a 2 : i2 º»¼ b2
[8.30]
a b ª«¬ c b 2 : i2 º»¼
c
2
: i2
2
4 c 2 : i2
[8.31]
ª b c 2 : 2 º ª a c 2 : 2 º i »« i » «¬ ¼¬ ¼
arc tan
2 c :i 2
c
:i2
arc tan
:i ac
arc tan
:i
[8.32]
bc
With these notations, the velocity and the displacement are given by: v t
ª k1 a t k 2 b t º A i k 3 e c t A i :i « e e c sin :i t T :i cos :i t T » 2 2 ¬ a ¼ b c :i
>
@
[8.33]
Control of a Shaker Using a Shock Response Spectrum
d t
ª k1 a t k 2 b t º A i k3 A i :i « 2 e 2e » ¬a ¼ c 2 :2 b i
2
e
289
c t
ª c 2 : 2 sin : t T 2 c : cos : t T º i i i i «¬ »¼
[8.34]
Figure 8.17 shows as an example a signal compensated according to this method. It is noted that the first negative peak is larger than the first positive peak. The waveform resembles overall a decaying sinusoid. :i
2 S fi
k1
0.067 43
a
1.001 25
fi
1
k2
0.026 87
b
0.629 1
Ki
0.05
k3
0.995 3
c
0.314 55
Ai
1
T
2.882 67
Figure 8.17. Waveform compensated by two exponential signals
Figures 8.18 and 8.19 give the corresponding velocity and displacement.
290
Mechanical Shock
Figure 8.19. Displacement after compensation by two exponential signals
Figure 8.18. Velocity after compensation by two exponential signals
NOTE: We also find the equivalent form ai t
A : i ª k 1 e a t k 2 e b t e c t k 3 cos : i t k 4 sin : i t º ¬ ¼
with k3
k4
k 1 k 2
a b c k 1 k 2 c 2 : i2 a b :i
k
1
b k2 a
where k1 and k2 have the same definition as previously. 8.2.6. Iterations
Once all these coefficients are determined, we calculate the spectrum of the signal thus made up. All this work has been carried out up to now by assuming the influence of each decaying sine of frequency fi on the other points of the spectrum to be negligible. This assumption is actually too simplistic, and the spectrum obtained does not match the specified spectrum. It is thus necessary to proceed to successive iterations to refine the values of the amplitudes A i of the components of x t : x t
¦ 1 i A i e K : t sin :i t i
i
i
[8.35]
Control of a Shaker Using a Shock Response Spectrum
291
The iterations can be carried out by correcting the amplitudes of the elementary waveforms by a simple rule of three. More elaborate relations have been proposed, such as [BOI 81] [CRI 78]: A i n 1
c i
. i
S fi S c n f i n Ai p S f S n f c
'A i
[8.36]
where:
– Sc n fi is the amplitude of the spectrum calculated using the values A i of the n iteration at the frequency fi ; th
– S fi is the value of the reference spectrum to the frequency fi ; – Sˆc f i is the amplitude of the spectrum calculated with the values A i of the nth iteration at the frequency fi , except for the coefficient A i which is added with an increment 'A i (0.05 gives satisfactory results); – p is a weighting factor (0.5 gives an acceptable velocity of convergence). If the procedure converges (and this is fortunately the case in general), the reconstituted spectrum and the specified spectrum are very close at the selected frequencies fi . The agreement can be worse between the frequencies fi . These intermediate values can be modified by changing the sign of the constants A i (taken to be initially positive) [NEL 74]. We often retain the empirical rule which consists of alternating the signs of the components. If the levels are too high between the frequencies fi , we must decrease the values of Ki . If, on the contrary, they are too weak, it is necessary to increase the values of Ki or to add components between the frequencies fi . The calculated spectrum must match the specification as well as possible, as close to the peaks of the spectrum as to the troughs. We sometimes act on the frequency fc to readjust the residual velocity v R and displacement d R to zero (to increase fc led to a reduction in v R and d R in general [SMA 74a]).
292
Mechanical Shock
8.3. D.L. Kern and C.D. Hayes’ function 8.3.1. Definition
To reproduce a given spectrum one frequency after another (and not a whole spectrum simultaneously), D.L. Kern and C.D. Hayes [KER 84] proposed using a shock of the form: a t a t
t t 0 ½° ¾ t 0 °¿
A t e K : t sin : t 0
[8.37]
where: f = frequency; K = damping; A = amplitude of the shock; e = Neper number. The signal resembles the response of a one-degree-of-freedom system to a Dirac impulse function. It can seem interesting a priori to examine the potential use of this function for the synthesis of the spectra. The first parameters to be considered are residual velocity and displacement. 8.3.2. Velocity and displacement Velocity
By integration we obtain: v t
A
A
e :
2
te
K : t
2
: 1 K
K : t
2 2
1 K
K sin : t cos : t
ª 1 K2 sin : t 2 K cos : t º ¬ ¼
[8.38]
The velocity, different from zero for t 0 , can be cancelled by adding the term 2KA , but then, the residual velocity is no longer zero. 2 2 2 : 1 K
Control of a Shaker Using a Shock Response Spectrum
293
Displacement
The integration of v t gives: Ate
d t
:
Ae 3
:
2
K : t
1 K
K : t
ª 2 K K2 3 sin : t 2 3 K2 1 cos : t º ¼
2 3¬
1 K
ª K2 1 sin : t 2 K cos : t º ¬ ¼
2 2
[8.39]
We cannot cancel the velocity and the displacement at the same time for t and large t.
0
As an example, the curves of Figures 8.20, 8.21 and 8.22 show the acceleration, the velocity and the displacement for f 1 and K 0.05 .
Figure 8.20. Example of D.L. Kern and C.D. Hayes’ waveform
Figure 8.21. Velocity
Figure 8.22. Displacement
294
Mechanical Shock
8.4. ZERD function 8.4.1. Definition
8.4.1.1. D.K. Fisher and M.R. Posehn expression The use of a decaying sinusoid to compose a shock of a given SRS has the disadvantage of requiring the addition of a compensation waveform intended to reduce the velocity and the displacement at the end of the shock to zero. This signal modifies the response spectrum at the low frequencies and, in certain cases, can harm the simulation quality. D.K. Fisher and M.R. Posehn [FIS 77] proposed using a waveform which they named ZERD (ZEro Residual Displacement) defined by the expression: a t
ª1 º A e K : t « sin : t t cos : t I » ¬: ¼
[8.40]
where I
§ 2K · ¸ arc tan ¨ ¨ 1 K2 ¸ ¹ ©
This function resembles a damped sinusoid and has the advantage of leading to zero velocity and displacement at the end of the shock.
Figure 8.23. ZERD waveform of D.K. Fisher and M.R. Posehn (example)
Control of a Shaker Using a Shock Response Spectrum
295
Gradually, the peaks of maximum amplitude arrive before it decreases at a regular rate. The positive and negative peaks are almost symmetric. Therefore, it is well adapted to a reproduction on a shaker. 8.4.1.2. D.O. Smallwood expression D.O. Smallwood [SMA 85] defined the ZERD function using the relation: a t
A eKe
K : t
>sin : t : t cos : t \ @
[8.41]
where § 2K · ¸ arc tan¨ ¨ 1 K2 ¸ ¹ ©
\
[8.42]
It is this definition which we will use hereafter. 8.4.2. Velocity and displacement
By integration we obtain the velocity: ½ ° t e AK e® > K cos : t \ sin : t \ @°¾ 2 °¿ °¯ 1 K K : t
v t
e
K : t
2
: 1 K e
K sin : t cos : t
K : t
2 2
: 1 K
ª 1 K2 ¬
º½ »° cos : t \ 2K sin : t \ »¾ »¼° ¿
[8.43]
and the displacement d t
AKe t
2 2
: 1 K
e
K : t ª
¬
1 K cos : t \ 2 K sin : t \ º¼ 2
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Mechanical Shock
A K e e K: t ª 2 K K 2 3 cos : t \ 2 1 3 K 2 sin : t \ 3 «¬ 2 2 : 1 K
2 K 1 sin : t 2K cos : t º ¼
[8.44]
It can be interesting to consider the envelope of this signal. If we pose: M t
sin : t : t cos : t \
[8.45]
M t
C t cos > : t T t @
[8.46]
C t
t
2
2
1 T t
2
: cos \ sin \
arc tan
:t
[8.47]
sin \
[8.48]
cos \
If K is small, < is also small, so that C t | : t and M t
: t cos > : t T t @ .
Figure 8.24. ZERD waveform of D.O. Smallwood (example)
Control of a Shaker Using a Shock Response Spectrum
297
Acceleration becomes: a t
Ke A
2
2
2
: cos \ sin \ t e
K : t
cos > : t T t @
[8.49]
and for small K a t | K e A : t e
K : t
cos > : t T t @
The maximum of the envelope t e
K: t
takes place for t
[8.50] 1
. Since
1
is K: K: the time-constant, the function reaches its maximum in time 1 / K : . The value of 1 this maximum is , the maximum of cos > : t T t @ being equal to 1 :Ke a max
Ke A : :Ke
A
[8.51]
A is thus the amplitude of a t . 8.4.3. Comparison of ZERD waveform with standard decaying sinusoid
This comparison can be carried out through the envelopes, i.e. of e D t e e
Dt
(where D
Dt
K : ):
Figure 8.25. Comparison of the ZERD and decaying sinusoid waveform envelopes
and
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Mechanical Shock
The plotting of these two curves shows that they have: – the same amplitude A; – the same slope (-1) in a semi-logarithmic scale for large D t ; – a different decrement with e D t e
Dt
decreasing less quickly.
8.4.4. Reduced response spectra
8.4.4.1. Influence of the damping K of the signal The response spectra of this shock are plotted in Figure 8.26 with a Q factor equal to 10; they correspond to signals according to the times defined for K 0.01 , 0.05 and 0.1 respectively.
Figure 8.26. ZERD waveform – influence of damping K on the SRS
The peak of these spectra becomes increasingly narrow as K decreases. 8.4.4.2. Influence of the Q factor (K
The response spectra of Figure 8.27 are plotted from a ZERD waveform 0.05 ) for Q 50 , 10 and 5, respectively.
Control of a Shaker Using a Shock Response Spectrum
299
Figure 8.27. ZERD waveform – influence of the Q factor on the SRS
8.5. WAVSIN waveform 8.5.1. Definition
R.C. Yang [SMA 74a] [SMA 75] [SMA 85] [YAN 70] [YAN 72] proposed (initially for the simulation of the earthquakes) a signal of the form: a t
a m sin 2 S b t sin 2 S f t
a t
0
0 d t d W ½° ¾ elsewhere °¿
[8.52]
where f
W
Nb
[8.53]
1 2b
[8.54]
where N is an integer (which, as we will see, must be odd and higher than 1). The first term of a t is a window of half-sine form of half-period W. The second describes N half-cycles of a sinusoid of greater frequency (f), modulated by the preceding window.
300
Mechanical Shock
Example 8.1.
Figure 8.28 shows an acceleration signal WAVSIN plotted for f
1 Hz,
N
5,
am
1.
Figure 8.28. Example of WAVSIN waveform
8.5.2. Velocity and displacement
The function a t can also be written: a t
am 2
> cos 2 S b 1 N t cos 2 S b 1 N t@
[8.55]
Velocity v t
am
4 S b 1 N
2
> 1 N sin 2 S b t 1 N 1 N sin 2 S b t 1 N @ [8.56]
( N z 1). At the end of the shock
Control of a Shaker Using a Shock Response Spectrum
1
W
t
301
2b am
v W
4 S b 1 N
2
> 1 N sin S 1 N 1 N sin S 1 N @
Whatever the value of N v W
0
Displacement am
d t
2
8S b
2
1 N
2 2
> 1 N 2 cos 2 S b t 1 N
1 N 2 cos 2 S b t 1 N 4 N
For t d W
@
[8.57]
W
am 2
8S b
2
1 N
2 2
> 1 N 2 cos S 1 N 1 N 2 cos S 1 N 4 N@
Figure 8.29. Velocity corresponding to the waveform in Figure 8.28
Figure 8.30. Displacement corresponding to the waveform in Figure 8.28
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Mechanical Shock
d W
am
8 1 N
If N is even ( N d W
2
S b
2
> 1 N 2 cos S 1 N 1 N 2 cos S 1 N 4 N@
2 n ), cos 1 2 n S
cos 1 2 n S
1 and
N
1 N
2 2
If N is odd ( N d W
2 2
2
S b
2
2 n 1 ): am
8 1 N
2 2
2
b S
2
> 1 N 2 1 N 2 4 N @
0
For the displacement to be zero at the end of the shock, it is thus necessary that N is an odd integer. The advantages of this waveform are: – the residual velocity and displacement associated with each elementary signal a t are zero;
t
– with N being odd, the two sine functions intervening in a t are maximum for W . The maximum of a t is thus a m ; 2
– the components have a finite duration, which makes it possible to avoid the problems involved in a possible truncation of the signals (the case of decaying sinusoids for example). 8.5.3. Response of a one-degree-of-freedom system
Let us set f0 as the natural frequency of this system and Q as its quality factor. To simplify the writing, let us express a t in dimensionless coordinates, in the form: O T
where
1 2
> cos D T cos E T@
[8.58]
Control of a Shaker Using a Shock Response Spectrum
D
2 S b 1 N
T
2 S f0 t
T0
E
Z0 t
E
303
2 S b 1 N S 1 N
S 1 N
W
T0
Z0
Z0 W
yielding 2 S b 1 N
D
Z0 S 1 N
D
Z0 W
Dt
Z0
Z0
S 1 N T0
S 1 N T0
Et
S 1 N T0
T
Z0
T
a t
O
am
8.5.3.1. Relative response displacement 0 d T d T0 q T
1 2
^ 2 [ E P sin E T D M sin D T D2 1 M cos D T E 2 1 P cos E T
º 2 2 sin 1 [ T cos 1 [ T» « 1 [2 » ¬ ¼ ª [ 2 2 [ T 2 2 E P D M e « cos 1 [ T sin 1 [ 2 « 1[ ¬ e
where:
[ T
ª
M P «
[
º½ ° T»¾ »° ¼¿
[8.59]
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Mechanical Shock
M
1 D
1
2 2
P
4[2 D2
1 E
1
2 2
2
4[ E
2
Particular cases [
0 and E
T sin T ½ 1 2 ® M D 1 cos D T cos T ¾ 2 ¯ 2 ¿
q T
[8.60]
1
M
[
1
1 D
2 2
1
1
q T
2
^ 2 E P sin E T D M sin D T D 1 M cos D T 2
2 T E 1 P cos E T M P T 1 e
e
T
1 T P E 2 M D2
`
[8.61]
where: M
1
P
1 D
2 2
1
1 E
2 2
Let us set, for 0 d T d T0 : A T
For T t T0 q T
q T
[8.62]
A T A T T0
[8.63]
Control of a Shaker Using a Shock Response Spectrum
305
8.5.3.2. Absolute response acceleration 0 d T d T0
^
q T
2
e
cos 1 [ T E P D M ½ ° 2 ª 2 2 2 4 4 º 1 [ T 1 2 [ D M E P E P D M ¾ ¬ ¼°
2
2
2 [ D M cos D T E P cos E T 2 [ e
2
A T [ D M D 1 sin D T E P E 1 sin E T
[ T
1[
2
sin
[ T
2
2
2
¿ [8.64]
Particular cases [
0 and E
[
1
1: the same relations as for the relative displacement
^
q T
2
2
2
A T 2 M D cos D T 2 P E cos E T M D D 1 sin D T
2
P E E 1 sin E T e
T
ª 2 T P E 2 M D2 T P E 4 M D4 º½¾ ¬ ¼¿
[8.65] For all the cases where 0 d T d T0 , let us set: q T
A T B T
a T
[8.66]
[8.67]
This becomes, for T ! T0 : q T
a T a T T0
8.5.4. Response spectrum
The SRS of this waveform presents a peak whose amplitude varies with N with its frequency close to f. Figure 8.31 shows the spectra plotted in reduced coordinates for N 9 ( Q 10 ).
3 , 5, 7 and
306
Mechanical Shock
Figure 8.31. WAVSIN – influence of the number of half-cycles N on the SRS
Figure 8.32. WAVSIN – amplitude of the peak of the SRS versus N and Q
Figure 8.32 gives the value of the peak of the shock spectrum R Q, N standardized by the peak R 10, N according to the number of half-cycles N, for various values of Q [PET 81]. 8.5.5. Time history synthesis from shock spectrum
The process consists here of choosing a certain number n of points of the spectrum of reference and, at the frequency of each one of these points, choosing the parameters b, N and a m to correspond to the peak of the spectrum of the elementary waveform with the point of the reference spectrum. This operation being carried out for n points of the spectrum of reference, the total signal is obtained by making the sum:
Control of a Shaker Using a Shock Response Spectrum
x t
307
n
¦ a i t Ti
[8.68]
i 1
with Ti being a delay intended to constitute a signal x t resembling the signal of the field environment to be simulated (the amplitude and the duration being preserved if possible) as well as possible. The delay has little influence on the shock spectrum of x t . Choice of components
The frequency range can correspond to the interval of definition of the shock spectrum (1/3 or 1/2 octave). Convergence is faster for the 1/2 octave. With 1/12 octave, the spectrum is smoother, without troughs or peaks. The amplitude of each component can be evaluated from the ratio of the value of the shock spectrum at the frequency considered and the number of half-cycles chosen for the signal [BAR 74]. a m i enables a change of amplitude at all the points of the spectrum. N i enables the modification of the shape and the amplitude of the peak of the spectrum of the elementary waveform at the frequency fi .
The errors between the specified spectrum and the realized spectrum are calculated from an average on all the points to arrive at a value of the “total” error. If the error is unacceptable, we proceed to other iterations. Four iterations are generally sufficient to reach an average error lower than 11% [FAV 74]. With the ZERD waveform, the WAVSIN pulse is that which gives the best results. It is finally necessary to check before the test that the maximum velocity and displacement corresponding to the drive acceleration signal remain within the limits of the test facility (by integration of x t ). 8.6. SHOC waveform 8.6.1. Definition
The SHOC (SHaker Optimized Cosines) method suggested by D.O. Smallwood [SMA 73] [SMA 74a] [SMA 75] is based on the elementary waveform defined by:
308
Mechanical Shock
a t
St a m e K : t cos : t G cos 2 W
a t
0
a t
a t
W½ ° 2° W ° if t ! ¾ 2 ° for t 0 ° ° ¿ 0dtd
[8.69]
The signal is oscillatory, of increasing amplitude according to time, and thus decreasing (symmetry with respect to the ordinates). The duration W of the signal is selected to be quite long so that the signal can be W W regarded as zero when t ! and t . The waveform is made up of a decaying 2 2 cosine and a function of the “haversine” type, the latter being added only to be able to cancel the velocity and the displacement at the end of the shock. In theory, the added signal should modify the initial signal as little as possible.
Figure 8.33. Composition of a SHOC waveform
Control of a Shaker Using a Shock Response Spectrum
309
The characteristics of this compensation function are given while equalizing, except for the sign, the area under the curve and the area under the decaying cosine. The expression of the haversine used by D.O. Smallwood can be written in the form: s t
B cos 2
s t
0
W ·½ § W for ¨¨ d t d ¸¸° 2 ¹¾ © 2 ° elsewhere ¿
St W
[8.70]
This relation has two independent variables with which we can cancel the residual conditions [SMA 73]. The velocity at the end of the shock is equal to 2 'V , if 'V is the change of velocity created by the positive part (t t 0) of the shock.
³0
'V
½ K : t ªt W 2 S t º° 2 ° am e »¾ ® > K cos : t sin : t@ G « sin 2 «¬ 2 4 S ° : 1 K W »¼° ¿0 ¯
W 2
K : t
am e
cos :W dt G
³0
St
'V
W 2
cos
2
W
dt W
being sufficiently large 'V |
am K
2
: 1 K
G
W 4
'V is zero at the end of the shock if: G
4 K am
2
: W 1 K
[8.71]
The largest value of a t occurs for t a 0
a 0
0:
am G
am
ª 4K «1 2 « : W 1 K ¬
º » » ¼
[8.72]
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Mechanical Shock
8.6.2. Velocity and displacement
By integration of the acceleration: a t
am e
K : t
cos : t G cos
2
S W
t
we obtain the velocity v t
am
e
K : t
2
: 1 K
> K cos : t sin : t@
[8.73]
and the displacement d t
2
am
K 1 :
2
2 2
1 K
e
K : t
cos : t
Example 8.2.
SHOC waveform, f
0.8 Hz and K
0.065
Figure 8.34. Example of SHOC waveform
[8.74]
Control of a Shaker Using a Shock Response Spectrum
311
8.6.3. Response spectrum
8.6.3.1. Influence of damping K of the signal Figure 8.35 shows the response spectra of a SHOC waveform of frequency 1 Hz with damping factors K successively equal to 0.01, 0.02, 0.05 and 0.1. These spectra are plotted for Q 10 . We observe the presence of an important peak centered on : whose amplitude varies with K. the frequency f 2S
Figure 8.35. SHOC – influence of K on the SRS
8.6.3.2. Influence of Q factor on the spectrum The Q factor has significance only in the range centered on the frequency of the signal. The spectrum presents a peak that is all the more significant since the Q factor is larger.
312
Mechanical Shock
Figure 8.36. SHOC – influence of the Q factor on the SRS
8.6.4. Time history synthesis from shock spectrum
To approach a point of the shock spectrum to simulate, we have the following parameters: – damping K for the shape of the curve; – the frequency f at the point of the spectrum to be reproduced; – the amplitude a m related to the amplitude of the spectrum (scale factor on the whole of the curve); – duration W, selected in order to limit the maximum displacement during the shock according to the possibilities of the test facility. In fact, K and W are dependent since we also require that at the moment W / 2 the decaying cosine is close to zero. Considering the envelope, we can for example ask that with W / 2 , the amplitude of the signal is lower than p% of the value with t 0 yielding: K :
e
W 2
p 100
For given f, it is thus necessary that
Control of a Shaker Using a Shock Response Spectrum
ln
KWt
313
100 p
Sf
The curve of Figure 8.33 is plotted, as an example, for p 2 which leads to the relation W t . K
[8.75] 100 / e 2S | 0.187 ,
Examination of the dimensionless SRS shows that the advantages of the decaying sinusoid are preserved. If W decreases, the necessary displacement decreases and, as the low frequency energy decreases, the spectrum is modified at frequencies lower than approximately 2 / W . Each time that a correction proves to be necessary, a compromise must thus be carried out between the smallest frequency to which the shock spectrum must be correctly reproduced and the displacement available. If 1/W is small compared to the frequency of the lowest resonance of the system, the effect of the correction on the response of the structure is weak [SMA 73]. Due to symmetry around the y-axis t 0 , the shocks are added in the frequency domain (i.e. of the shock response spectra) as well as in the time domain. This simplifies the construction of complex spectra. Variations can, however, be observed between the specified and carried out shock spectra, which had with non-linearities of the assembly, with the noise. In general, these variations do not exceed 30%. The peaks of an acceleration signal built from SHOC functions are positive in a dominating way. For certain tests, we can carry out a shock which has roughly an equal number of positive peaks and negative peaks with comparable amplitude. This can be accomplished by alternating the signs of the various components. This alternation can lead, in certain cases, to a reduction in the displacement necessary to carry out the specified spectrum. 8.7. Comparison of WAVSIN, SHOC waveforms and decaying sinusoid
The cases treated by D.O. Smallwood [SMA 74a] 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.
314
Mechanical Shock
8.8. Use of a fast swept sine
The specified shock response spectrum can also be restored by generation of a fast swept sine. It is pointed out that a swept sine can be described by a relation of the form (Volume 1, Chapter 8) x t
x m sin> E t @
where for a linear sweep ( f E t
b t f1)
§b t · 2St¨ f1 ¸ © 2 ¹
[8.76]
f1 is the initial sweep frequency and b the sweep rate. The number of cycles f1 f2 N b carried out between f1 and f2 for the duration T is given by N b T. 2 The signal describing this sweep presents the property of a Fourier transform of roughly constant amplitude in the swept frequency band, being represented by [REE 60]:
X
x m
[8.77]
b
The first part of the response spectrum consists of the residual spectrum (low frequencies). Knowing that, for zero damping, the residual shock response spectrum S R is related to the amplitude of the Fourier transform by SR
2 S f0 X
[8.78]
we have, by combining [8.77] and [8.78] SRS
2 S f0
x m b
[8.79]
From these results, for the method which makes it possible to determine the characteristics of a fast swept sine from a response spectrum [LAL 92b]: – we fix the number of points N of definition of the swept sine signal; – we are given a priori, to initialize calculation, a number of cycles ( N b example), from which we deduce the sweep duration
12 for
Control of a Shaker Using a Shock Response Spectrum
T
315
2 Nb f1 f2
and the sweep rate b
f2 f1 T
– between two successive points ( fi , SRSi ) and ( fi1, SRSi 1 ) from the specified spectrum, the frequency of the signal is obtained from the sweeping law f b t f1; – the amplitude of the sinusoid at time t corresponding to the frequency f included between fi and fi1 is calculated by linear interpolation according to amp
1.3
b SRSi 1 fi 1 SRSi fi SRSi f fi S fi 1 fi S fi
[8.80]
(the constant 1.3 makes it possible to take into account the fact that relation [8.78] is valid only for one zero damping whereas the spectra are generally plotted for a value equal to 0.05. This constant is not essential, but makes it possible to have a better result for the first calculation); – we deduce, starting from [8.76], the expression of the signal: x t
ª §b t ·º amp sin« 2 S t ¨ f1 ¸» «¬ © 2 ¹»¼
– by integration of this signal of acceleration, we calculate the associated velocity change 'V (by supposing the initial velocity equal to zero). By comparison with the velocity change 'V0 read on the specified response spectrum (given by the slope at the origin of this spectrum, calculated from the first two points of the spectrum and divided by 2 S ), we determine the duration and the number of cycles N b (up to now selected a priori) necessary to guarantee the same change velocity from bc
Tc
b 'V0 1.2 'V f2 f1 bc
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Mechanical Shock
Ncb
f1 f2 2
Tc
– with the same procedure as previously, we realize a re-sampling of the signal x t ; – the response spectrum of this waveform is calculated and compared with the specified spectrum. From the noted variations of each N points, we readjust the amplitudes by performing rules of three. One to two iterations are generally enough to obtain a signal of which: – the spectrum is very close to the specified spectrum; – the amplitude and the velocity change are of the same order of magnitude as those of the signal having been used to calculate the specified spectrum. This signal, to be realizable on the shaker, must be modified by the addition of a pre-shock and/or a post-shock ensuring an overall zero velocity change. Example 8.3.
Let us assume that the specified spectrum is the shock spectrum of a half-sine waveform, of amplitude 500 m/s2 and of duration 10 ms (associated velocity change: 3.18 m/s). Figure 8.37 shows the signal obtained after three iterations carried out to readjust the amplitudes (without pre- or post-shock) and Figure 8.38 shows the corresponding response spectrum, superimposed on the specified spectrum.
Figure 8.37. Example of fast swept sine
Figure 8.38. SRS of the equivalent swept sine to the SRS of a half-sine shock
Control of a Shaker Using a Shock Response Spectrum
317
The velocity change is equal to 3 m/s, the amplitude is very close to 500 m/s2 and the duration of the first peak of the signal, dominating, is close to 10 ms. This method is thus of interest to roughly represent the characteristics of amplitude, duration and velocity change of the signal at the origin of the specified spectrum. It has the disadvantage of not always converging according to the shape of the specified spectrum. 8.9. Problems encountered during the synthesis of the waveforms
The principal problems encountered are as follows [SMA 85]. Problem
Possible remedy
The step assumes that the elementary waveforms which constitute the shock of the specified spectrum are not too dependent on one another, i.e. the modification of the amplitude of the one of them only slightly modifies the other points of the spectrum. If the points chosen on the specified spectrum are too close to one another, if the damping is too large, it can be impossible to converge. The search for a solution can be based on the following: – the amplitude of a component cannot be reduced if the SRS is too high at this frequency: there is thus a limit with the possibilities of compensation with respect to the contribution of the near components; The iterations do not converge.
– a small increase in the amplitude of one component can sometimes reduce the shock spectrum to this frequency because of the interaction of the near components; – to change the sign of the amplitude of one component will not lower the SRS in general. It should be noted that convergence is better if the signs of the components are alternated.
318
Mechanical Shock
If the SRS is definitely smaller at the high frequencies than in certain ranges of intermediate frequencies, there cannot be a solution. It is known that any SRS tends at high frequencies towards the amplitude of the signal. The SRS limit at high frequencies of the components designed to reproduce a very large peak can sometimes be higher than the values of the reference SRS at high frequency. The SRS of the total signal is then always too large in this range. The solutions in the event of a convergence problem can be the following: – to give a high damping to the low frequency components and decreasing it in a continuous way when the component frequency increases; – to change the frequency of the components; The iterations do not converge (continuation).
– to lower damping (each component); – to reduce the number of component; – to change the sign of certain components. This problem can be corrected by:
The spectrum is – increasing the number of components, while placing the well simulated at new ones close to the “valleys” of the spectrum; the frequencies – increasing the damping of the components; chosen, but is too small between – changing the sign of the components; it should, however, these be known that the components interact in an way that cannot be frequencies. envisaged when the sign of one of the two near components is changed. The spectrum is well simulated at the frequencies chosen, but is too large between certain frequencies.
We can try to correct this defect: – by removing a component; – by reducing the damping of the near components; – by changing the sign of one of the near components.
Control of a Shaker Using a Shock Response Spectrum
319
If acceleration is too high, we can: The resulting – lower the damping to increase the ratio peak of the x t signal is not spectrum/amplitude of the signal; realizable (going – increase the delays between the components; beyond the – change the sign of certain components; limiting performances of – use another form of elementary waveform at each the shaker). frequency; – as a last resort, reduce the frequency range on which the specification is defined. If the velocity is too large: – The low frequency components are usually at the origin of this problem and a compromise must be found at these The resulting frequencies, which can result in removing the first points of the signal x t is not reference spectrum (these components also produce a large realizable (going displacement). This is also a means of reducing the duration of beyond of the the signal when the specification imposes one duration limiting maximum. This modification can be justified by showing that performances of the test item does not have any resonance frequency in this the shaker). domain. – If possible, we can try to change the elementary waveform; the displacement and the velocity associated with the new waveform being different. The ZERD waveform often gives better results. – If no compromise is satisfactory, it is necessary to change the shock test facility, with no certainty of obtaining better results.
8.10. Criticism of control by SRS
Whatever the method adopted, simulation on a test shock facility measured in the field requires the calculation of their response spectra 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 shape, duration and amplitude characteristics of the signal, with the already discussed difficulties.
320
Mechanical Shock
If the specification is given in the form of a SRS, the operator inputs the given spectrum in the control system, 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 response spectra. This phenomenon is related to the loss of most of the information initially contained in the signal x t during the calculation of the spectrum [MET 67]. It was also seen that the oscillatory shock pulses have a spectrum which presents an important peak to the frequency of the signal. This peak can, according to the choice of parameters, exceed the amplitude of the same spectrum by a factor of 5 at the high frequencies, i.e. 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 versus time of amplitude S / 5 to reproduce the point. For a simple-shaped shock, this factor does not exceed 2 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 we can call into question. This experiment makes it possible to note that, if any particular precaution is not taken, the signals created by these methods generally have one duration that is much larger and an amplitude much smaller than the shocks which were used to calculate the reference SRS (factor of about 10 in both cases). We saw in Chapter 4 that we can use a slowly swept sine to which the response spectrum is close to the specified shock spectrum [CUR 55] [DEC 76] [HOW 68]. In the face of such differences between the excitations, we 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 which we choose a priori the Q factor. We must be aware that: – the behavior of the structure is in practice far from linear and that the equivalence of the spectra does not lead to stresses of the same amplitude. Another effect of these non-linearities sometimes appears due to the inability of the system to correct the drive waveform to take into account the transfer function of the installation;
Control of a Shaker Using a Shock Response Spectrum
321
Figure 8.39. Example of shocks having spectra near the SRS
– 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 quite different when the number of shocks to be applied is significant; – the tests carried out by various laboratories do not have the same severity. R.T. Fandrich [FAN 69] and K.J. Metzgar [MET 67] confronted this problem, based on the example of the signals of Figure 8.39. These signals A and B, with very different characteristics, have similar response spectra, within acceptable tolerances (Figure 8.40), but are they really equivalent? Does the response spectrum constitute a sufficient specification [SMA 74a] [SMA 75]? These questions did not receive a particularly 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 ( 'V , duration of the shock).
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Figure 8.40. SRS of the shocks shown in Figure 8.39 and their tolerances
Example 8.4.
Figure 8.41. Shocks A and B with very different durations and amplitudes,but having very close SRS
Control of a Shaker Using a Shock Response Spectrum
323
Figure 8.42. SRS of the shocks shown in Figure 8.41
8.11. Possible improvements
To obtain a better specification we can, for example: – consider the acceleration signals at the origin of the specified spectrum and specify if they are shocks with a velocity change or are oscillatory. The shock spectrum can, if it is sufficiently precise, give this information by its slope at very low frequencies. The choice of the type of simulation should be based on this information; – specify in addition to the spectrum other complementary data such as the duration of the signal time or the number of cycles (less easy) or one of the pre-set parameters in the following sections, in order to deal with the spectrum and the couple amplitude/duration of the signal at the same time. 8.11.1. IES proposal
To solve this problem, a commission of the IES (Institute of Environmental Sciences) proposed in 1973 four solutions consisting of specifying additional parameters [FAV 74] [SMA 74a] [SMA 75] [SMA 85] as follows. 1. Limit the transient duration This is a question of imposing minimum and maximum limits over the duration of the shock by considering that if the shock response spectrum is respected and if
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Mechanical Shock
the duration is comparable, the damage should be roughly the same [FAN 69]. For complex shapes, we should pay a lot of attention to how the duration is defined. 2. Require SRS at two different values of damping Damping is generally poorly known and has values different at each natural frequency of the structure. It can be assumed that if the SRS is respected for two different damping values, for example [ 0 .1 (Q = 5) and [ 0.02 (Q = 25), the corresponding shock should be a reasonable simulation for any value of [. This approach also results in limiting the duration of the acceptable shocks. It is not certain in particular that a solution always exists, when the reference spectra come from smoothed spectra or an envelope of spectra of several different events. This approach intuitively remains attractive however; it is not used often except in the case of fast sweep sines. It deserves to be paid some attention to evaluate its consequences over the duration of the drive waveform thus defined with shapes such as WAVSIN, SHOC, and a decaying sinusoid. 3. Specification of the allowable ratios between the peak of the SRS of each elementary waveform and the amplitude of the corresponding signal versus time The goal is here to prevent or encourage the use of an oscillatory-type shock or a simple shape shock (with velocity change). It should, however, be recalled that if the shock spectrum is plotted at a sufficiently high frequency, the value of the spectrum reflects the amplitude of the shock in the time domain. This specification is thus redundant. It is, however, interesting, as it can be effective over the duration of the shock and thus leads us to be better able to take into account the couple shock duration/amplitude. 4. To specifically exclude certain methods The test requester can give an opinion on the way of proceeding so that the test is correct. He/she can also exclude certain testing methods a priori when he/she knows that they cannot give good results. He/she can even remove the choice from the test laboratory and specify the method to be used, as well as all of the parameters that define the shock (for example, for components of damped sine type, frequencies, damping factors and amplitudes, etc.). Only laboratories equipped to use this method will be able to carry out this test, however [SMA 75] [SMA 85]. 8.11.2. Specification of a complementary parameter
Several proposals have been made. The simplest suggests arbitrarily limiting the duration of the synthesized shock to 20 ms (with equal shock spectrum) [RUB 86]. Others introduce an additional parameter to better attempt to respect the amplitude shock duration.
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325
8.11.2.1. Rms duration of the shock : . The rms duration W Let us set x t as a shock of Fourier transform of X rms is defined by [SMA 75]:
1 f 2 2 t x t dt f E
³
W2rms
[8.81]
where E
³f > x t 2 @ dt f
[8.82]
E is referred as the “energy of the shock”. It is necessary that E is finite, i.e. that x t approaches zero more quickly than 12 , when t tends towards f or f . t
In general, the rms duration of a transient is a function of the origin of the times selected. To avoid this difficulty, we choose the origin in order to minimize the rms duration. If another origin is considered, the rms duration can be minimized by introducing a time-constant T (shift) equal to [SMA 75]: T
1
³ E f f
t x t
2
dt
[8.83]
The rms duration is a measure of the central tendency of a shock. Let us consider, for example a transient of a certain finite energy, composed of all the frequencies equal in quantity. An impulse (function G) represents a shock of this type with one minimum duration. A low level random signal, long duration represents it with one maximum duration. The rms duration of most current transients is given in Table 8.1. This duration can also be calculated starting from [PAP 62]: W 2rms
2 ª§ dX · 2 § dI · º «¨¨ m ¸¸ A 2 ¨¨ ¸¸ » d: 2 S E f «¬© d: ¹ © d: ¹ »¼
[8.84]
X m : ei I :
[8.85]
1
³
f
where X :
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Mechanical Shock
: of the Fourier transform is specified, the rms duration If the amplitude X m dI 0 , i.e. by I : constant. The constant can be zero. minimum is given by d: Equation [8.84] implies that the rms duration is related to the smoothness of the Fourier spectrum (amplitude and phase at the same time). The smoother the spectrum, the shorter the rms duration.
Function
Rectangle Half-sine TPS Triangle
x t x t
x t x t x t x t x t x t
Haversine
x t x t
Decaying sinusoid
x t x t
Equation 1 for 0 t W 0 elsewhere St sin for 0 t W W 0 elsewhere t / W for 0 t W 0 elsewhere W W 2 t 1 for t W 2 2 0 elsewhere 1§ 2 S t· ¨1 cos ¸ for 0 t W 2© W ¹ 0 elsewhere
Rms duration 0.29 W
0.23 W 0.19 W 0.16 W
0.14 W
a : t
e sin : t for t ! 0 0 elsewhere for a << 1
1 2a:
Table 8.1. Rms duration of some often-used shocks [PAP 62] [SMA 75]
8.11.2.2. Rms value of the signal T.J. Baca ([BAC 82] [BAC 83] [BAC 84] [BAC 86]) proposed characterizing the shock by its rms value, given by: x rms
1 W 2 x t dt W 0
³
[8.86]
where W is the shock duration. The rms value is an “average” of the energy of the signal over the shock duration.
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327
NOTE: This method can be used to compare shocks by observing the variations of their rms value with time [CAN 80]. 1 t
x rms t
t
³0 x
2
O
dO
0 d t dW
[8.87]
j
[8.88]
i.e., numerically 1 j 1
x rms j
j
¦xi2
2, 3,..., N
i 2
where N = total number of points defining the signal.
8.11.2.3. Rms value in the frequency domain The rms value is a means of highlighting the contents of the frequency of a shock. It can also be calculated in the domain of the frequencies [BAC 82] [BAC 83] [BAC 84], by application of Parseval’s theorem:
³f x2 t dt ³f f
f
f 2 df X
[8.89]
i.e.
³0 x2 t dt W
2
³0
FC
f 2 df X
[8.90]
f of x t is such that: where the Fourier transform X f X
f
³f x t e
2 S i f t
dt
[8.91]
and where FC is the cut-off frequency which limits the useful frequency range, f 2 , yielding: taking into account the symmetry of X
x rms
ª 2 F 2 º «¬ W ³0 C X f df »¼
1/ 2
[8.92]
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Mechanical Shock
NOTE: Also, according to the frequency: ª2 «W ¬
x rms f
FC
³0
2 º ª¬ X O º¼ d O » ¼
1/ 2
[8.93]
where f is the current frequency at which the rms value is calculated. This expression thus makes it possible to highlight the contribution of all the frequencies (lower than Fc ) for a shock of duration W.
8.11.2.4. Histogram of the peaks of the signal The shock spectrum does not give any direct information concerning the number of peaks of the signal x t . The histogram of the peaks (a peak being the maximum or minimum between two zero passages) could constitute complementary data (by possibly standardizing the ordinate of the curve by division by the amplitude of the largest peak). If this technique is used, it is important to specify the type of filtering which the signal underwent before establishment of the histogram; the comparison of two shocks making sense only if they were filtered under the same conditions (same frequency limits). 8.11.2.5. Use of the fatigue damage spectrum
Given a shock x t , the fatigue damage spectrum D f0 (see Volume 5) is calculated starting from the response relative displacement z t of a linear onedegree-of-freedom mechanical system of natural frequency f0 and of quality factor Q:
¦
D f0
i
ni Ni
[8.94]
The number N of cycles to the rupture is extracted from the Basquin law b
NV C (b and C are constant functions of the material constituting the part). This yields, since V K z ,
¦
D f0
i
b
ni
b
Vi
K
C
C
¦ n i zib i
[8.95]
Control of a Shaker Using a Shock Response Spectrum
329
The damage at frequency f0 takes into account the number of peaks n i of amplitude zi (histogram of the peaks of the response). The fatigue damage spectrum, which includes characteristics of the signal such as the duration and the histogram, is very complementary to the response spectrum and could thus be specified jointly with the shock response spectrum to define a test. 8.11.3. Remarks on the properties of the response spectrum
The need to specify a complementary parameter is related to the fact that the specified spectrum is defined only for one limited frequency range. When it is calculated in a sufficiently large frequency interval, the response spectrum makes it possible to read very useful information, such as: – at low frequencies, the velocity change 'V associated with the shock (slope of the spectrum in the origin). This is not strictly exact if the damping is zero. However, for the usual values of [, the approximation is sufficient to determine if the shock is associated with a velocity change or not; – at high frequencies, the magnitude of the signal varies with time. Consideration of these parameters should make it possible to obtain a simulation much more correctly akin to the real shock. It is thus desirable to specify the shocks with a spectrum calculated in a sufficiently broad frequency range to make it possible to read these values on the curve. A signal which has a SRS very close to the SRS specified across the whole frequency band necessarily has the same amplitude and the same velocity change as the origin of shock. 8.12. Estimate of the feasibility of a shock specified by its SRS
Several methods have been proposed to evaluate the feasibility of a shock specified using a shock response spectrum. 8.12.1. C.D. Robbins and E.P. Vaughan’s method
The shakers are limited with respect to force. Possible rms acceleration x rms is a function of the total mass M of the test package including the mounting fixture and attachments:
x rms
Frms M
[8.96]
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Mechanical Shock
where Frms maximum rms random force realizable on the shaker. The shaker can accept peak values equal to three times the rms force: x m
3 Frms
[8.97]
M
Each point of the SRS is simulated by an oscillatory signal having the frequency of the SRS at this point (Figure 8.43).
Figure 8.43. Simulation of an SRS point of reference
The spectrum of this elementary waveform has a peak at this frequency whose amplitude is R times its value at high frequencies, i.e. R times the amplitude of the signal in the time domain (R being a function of the type of signal used and of the number of oscillations). The possible maximum value SRS max of the shock response spectrum is thus: SRS max
R x m
SRSmax
R
[8.98]
3 Frms
[8.99]
M
where R is equal to or higher than 2. If the penalizing value R SRSmax
6
Frms M
2 is taken, we obtain
[8.100]
Control of a Shaker Using a Shock Response Spectrum
With the more realistic value R SRSmax
12
331
4 , we have
Frms M
[8.101]
K.J. Metzgar [MET 67], C.D. Robbins and P.E. Vaughan [ROB 67] checked by experiment that it was possible to reach spectrum values higher than 1,000 g (they specified neither the mass nor the type of shaker). Tests confirming this value in addition were carried out with one 135 kN shaker and a test item mass of 200 kg. Problems of non-linearity
After input of the specified spectrum on the control system, the calculation of the drive signal is normally carried out at low amplitude, for example 10% of the specification, to avoid damaging the test item while making it undergo all the shocks necessary to the development procedure. Once the spectrum obtained is considered to be satisfactory, we apply the shock to the test item. For small test items or larger dead mass-type test items, it can be agreed that the passage from level 1/10th to level 1 is effected linearly except for 10%. For heavier test resonant items, it is preferable if possible to use a dummy item which is representative for the development of the test in order to guard against possible significant non-linearity, and to carry out intermediate level shocks. 8.12.2. Evaluation of the necessary force, power and stroke
The dynamic force necessary to carry out a shock is related to acceleration on the table by the relation [DES 83]: F :
: m H : X
[8.102]
where : are the Fourier transforms of the force F t and acceleration – F : and X x t respectively;
– m is the total mass of the test package (armature, table, fixture and test item); – H : is the transfer function of the test item. It is necessary to take into account the weight of the moving unit if the shaker, for a vertical configuration, supports this load directly.
332
Mechanical Shock
In most practical cases, the test item can be represented, approximately, by only a one-degree-of-freedom system, of natural frequency Zsp and damping factor [sp ; this transfer function is then written: H :
where h
: Zsp
1 2 i [sp h 1 h 2 2 i [sp h 1 .
and i
Figure 8.44. One-degree-of-freedom model of an equipment tested on a shaker
Let us set SRS f as the SRS specified calculated for a damping equal to [. By definition, each point of this spectrum gives the largest response of a linear onedegree-of-freedom transfer function system: H f
1 2
§f · f 1 ¨ ¸ 2i[ ¨f ¸ f0 © 0¹
[8.103]
For the usual values of damping factor [, we have, at the resonance ( f f0 ), 1 H f | . Knowing in addition that the response of a mechanical system is 2i[ f , the maximum of the related to the excitation by the relation R f H f X
Control of a Shaker Using a Shock Response Spectrum
333
response being the point of the SRS, we obtain, by noting V : a function whose module is the SRS ( SRS : V : ): V : |
1 2i[
: X
[8.104]
yielding F : | m H : 2 i [ V :
[8.105]
The necessary maximum force is obtained in a conservative way for the maximum SRS value. Fm | m
Fm | m
1 2 i [sp [ [ sp
2 i [ V : max
[8.106]
SRS max
Let us set V : as the Fourier spectrum of the velocity of the table during the shock movement. The power necessary is given by the real part of F V : :
p
ª¬ F V : º¼
: i : V : , the necessary maximum power is estimated Knowing that X conservatively from [8.102] and [8.104] by : ½° ° X ®m H : ª¬ 2 j [ V : º¼ ¾ j : °¿ ¯°
p
V º ª¬ m H : X ¼
p
2 ª V : º » «m H : 4 [ 2 « j: » ¬ ¼
p
4 m [2 ª 2 j H : V : º ¬« ¼» :
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Mechanical Shock
Conservatively, H
1 2 i [sp
, yielding
2 m [ 2 §¨ SRS 2 ·¸ [ sp ¨© : ¸¹ max
p max
[8.107]
Taking into account [8.104], the Fourier transform of the displacement during the shock can be written: : X
X
i :
2
2i[
V :
i : 2
This yields an estimate of the maximum stroke: xm
ª SRS º 2[« » ¬ : 2 ¼ max
[8.108]
Relations [8.106], [8.107] and [8.108] show that, if n is the slope of the SRS, the power, the stroke and the force are respectively of the form p max 1 1 is a function increasing for n ! and decreasing for n . 2 2 xm
k2 :
Fm
k3
k1 :
2 n 1
. This
n 2
(where k1, k 2 and k 3 are constants). Thus, in logarithmic scales: – the necessary power is given by the highest point of contact between the SRS and the line of slope 1/2; – the maximum stroke corresponds to the highest point of contact of the SRS with a line of slope 2; – the force at the highest point of contact between the SRS and a line of zero slope.
Control of a Shaker Using a Shock Response Spectrum
335
Figure 8.45. Calculation of the force, the power and the stroke necessary to carry out a shock of a given SRS
The coordinates of these contact points are used in relations [8.106], [8.107] and [8.108] to calculate the maximum force, the necessary power and the stroke. If the damping [sp of the test equipment is not known, we can use the approximate value of 0.05. NOTES: 1. In the case of a specification defined by a velocity SRS, we will consider in the same way the points of intersection of the spectrum with [DES 83]: – a line of slope 1 / 2 for maximum power; – a line of slope 1 for the stroke; – a line of slope –1 for the force. Slope Quantity
Power
Relative displacement SRS (or absolute acceleration) 1 2
Velocity SRS
1 2
Stroke
2
1
Force
0
-1
Table 8.2. Slopes of the straight lines allowing evaluation of the possibility of carrying out a shock of a given SRS on a shaker
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Mechanical Shock
2. If the dynamic behavior of the material is known and if it is known that, in the frequency domain studied, the material does not have any resonance, we will be able, to avoid too great a conservatism, to take [ sp 0.5 for transfer function H : equal to 1.
3. This method provides only one order of magnitude of the force, the power and the stroke necessary to carry out a shock specified from an SRS on shaker because: – the relations used are approximations; – according to the shape of the elementary waveform used to build the drive waveform, the actually generated waveform will require conditions slightly different from those estimated by this test.
Chapter 9
Simulation of Pyroshocks
Many works have been published on the characterization, measurement and simulation of shocks of pyrotechnic origin (generated by bolt cutters, explosive valves, separation nuts, etc.) [ZIM 93]. There are many test facilities suggested, ranging from traditional machines to very exotic means. The tendency today is to consider that the best simulation of shocks measured in the near field (see section 1.1.13) can be obtained only by subjecting the material to the shock produced by the real device (which poses the problem of the application of an uncertainty factor to cover the variability of this shock). In certain cases, simulation using a metal-metal impact may be appropriate. For shocks in the mid-field, simulation can be carried out either using the real pyrotechnic source and a specific mechanical assembly or using specific equipment with explosives, or by impacting metal to metal or using a shaker if the structural response is more important [BAT 08] [NAS 99]. In the far field, when the real shock is practically made up only of the response of the structures, a simulation on a shaker is possible (when use of this method allows). 9.1. Simulations using pyrotechnic facilities If we are looking to carry out shocks close to those experienced in the real environment, the best simulation should be the generation of shocks of a comparable nature on the material concerned. The simplest solution consists of making
338
Mechanical Shock
functional, real pyrotechnic devices on real structures. Simulation is perfect but [CON 76] [LUH 76]: – it can be expensive and destructive; – we cannot apply an uncertainty factor without being likely to create unrealistic local damage (a larger load, which requires an often expensive modification of the devices can be much more destructive). To avoid this problem, an expensive solution consists of carrying out several tests in a statistical matter. We often prefer 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 described below: 1) A test facility made up of a cylindrical structure [IKO 64] which comprises a “consumable” sleeve cut out for the test by an explosive cord (Figure 9.1). 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 equipment to be tested (fixed on the structure as in the real case if possible) and the explosive cord.
Figure 9.1. Barrel tester for pyroshock simulation
2) For a large-sized structure subjected to this type of shock, we generally prefer to make the real pyrotechnical systems placed on the structure as they would be under operating conditions. The problem of the absence of the uncertainty factor for the qualification tests remains. 3) D.E. White, R.L. Shipman and W.L. Harlvey proposed placing a greater number of small explosive charges near the equipment to be tested on the structure, in “flower pots”. The number of pots to be used per axis depends on the amplitude
Simulation of Pyroshocks
339
of the shock, the size of the equipment and the local geometry of the structure. They are manufactured in a stainless steel pipe which is 10 cm in height, 5 cm in interior diameter, 15 cm in external diameter and welded to approximately 13 mm steel base plates [CAR 77] [WHI 65].
Figure 9.2. “Flower pot” provided with an explosive charge
A number of preliminary shots, reduced as a result of the experience we acquire from experimenting, are necessary to obtain the desired shock. The shape of the shock can be modified within certain limits by use of damping devices, placing the pot closer to or further away from the equipment, or by putting suitable padding in the pot. If, for example, we put sand on the charge in the pot, we transmit more low frequency energy to the structure and the shape of the spectrum is more regular and smoother. We can also place a crushable material between the flower pot and the structure in order to absorb the high frequencies. When the necessary explosive charge is substantial, this process can lead to notable permanent deformations of the structure. The transmitted shock then has an amplitude lower than that sought and, to compensate, we can be tempted to use a larger charge with the next shooting. To avoid entering this vicious circle, it is preferable, with the next shooting, either to change the position of the pots, or to increase the number by using weaker charges. The advantages of this method are the following: – the equipment can be tested in its actual assembly configuration; – high intensity shocks can be obtained simultaneously along the three principal axes of the equipment.
340
Mechanical Shock
There are also some drawbacks: – no analytical method of determination a priori of the charge necessary to obtain a given shock exists; – the use of explosive requires testing under specific conditions to ensure safety; – the shocks obtained are not very reproducible, with many influential parameters; – the tests can be expensive if the structure is deformed each time [AER 66]. 4) A test facility made up of a basic rectangular steel plate (Figure 9.3) suspended horizontally. This plate receives an explosive load (chalk line, explosive in plate or bread) on its lower part, directly or by the intermediary of an “expendable” item.
Figure 9.3. Plate with resonant system subjected to detonation
A second plate supporting the test item rests on the base plate via four elastic supports. Tests carried out in this way showed that the shock spectrum generated at the input of the test item depends on: – the explosive charge; – the nature and thickness of the plate carrying the test item; – the nature of the elastic supports and their prestressing; – the nature of material of the base plate and its dimensions; constituting – the mass of the test item [THO 73].
Simulation of Pyroshocks
341
The reproducibility of the shocks is better if the explosive charge is not in direct contact with the base plate. 9.2. 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: large amplitude, short duration, high frequency content, shock response spectrum comparable with a low frequency slope of 12 dB per octave, etc. Simulation is in general satisfactory up to approximately 10 kHz.
Figure 9.4. Simulation by metal to metal impact (Hopkinson bar)
The shock can be created by the impact of a hammer on the structure itself, a Hopkinson bar or a resonant plate [BAI 79] [DAV 85] [DAV 92] [LUH 81].
Figure 9.5. Simulation by the impact of a ball on a steel beam
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 the system (length of the bar between two fixing points, the addition or removal of runners, etc.) or by the addition of a deformable material between the hammer and the anvil. To generate shocks of great amplitude, the hammer can be replaced by a ball or a projectile with a plane front face made out of
342
Mechanical Shock
steel or aluminum, and launched by a pneumatic gun (air or nitrogen) [DAV 92]. The impact can be carried out directly on the resonant beam or on a surmounted plate of a resonant mechanical system composed of a plate supporting the test item connecting it to the impact plate. 9.3. Simulation using electrodynamic shakers The possibilities of creating shocks using an electrodynamic shaker are limited by the maximum stroke of the table and more particularly by the acceptable maximum force. The limitation relating to the stroke is not very constraining for the pyrotechnical shocks, since they are at high frequencies. There remains a limitation on the maximum acceleration of the shock [CAR 77] [CON 76] [LUH 76] [POW 76]. If, with the reservations of section 4.3.6, we agree to cover only part of the spectrum, then when we perform a possible simulation on the shaker, this 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 [DIN 64] [GAL 66], but also random noise or a combination of simple elementary signals with the characteristics to reproduce a specified response spectrum (direct control from a shock spectrum; see Chapter 8). The problem of over-testing at low frequencies as previously discussed is eliminated and it is possible, in certain cases, to reproduce the real spectrum up to 1,000 Hz. If we are 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 2,000 Hz. Certain facilities of this type were modified in the USA to make it possible to simulate the effects of pyrotechnical shocks up to 4,000 Hz. We can thus manage to simulate shocks whose spectrum can reach 7,000 g [MOE 86]. We have already seen, however, in Chapter 8 the limits and disadvantages of this method. 9.4. 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, by a simple shape shock, such as half-sine, triangle, trapezoid, etc., starting from the “shock response spectrum” equivalence criterion (with the application of a given or calculated uncertainty factor1 to the shock amplitude) [LUH 76].
1 See Volume 5.
Simulation of Pyroshocks
343
With the examination of the shapes of the response spectra of standard simple shocks, it seems that the best adapted signal is the terminal peak saw tooth pulse, whose spectra are also appreciably symmetric. The search for the characteristics of such a triangular impulse (amplitude, duration) with a spectrum envelope of that of a pyrotechnical shock often leads to a duration of about 1 ms and to an amplitude being able to reach several tens of thousands of ms-2. Except in the case of very small test items, it is generally not possible to carry out such shocks on the usual drop tables: – limitation in amplitude (acceptable maximum force on the table); – duration limit: the pneumatic programmers do not allow it to go below 3 ms to 4 ms. Even with the lead programmers, it is difficult to obtain a duration of less than 2 ms. However, spectra of the pyrotechnical shocks with, in general, averages close to zero have a very weak slope at low frequencies, which leads to a very small duration of simple shock, of about one millisecond (or less); – the spectra of the pyrotechnical shocks are much more sensitive to the choice of damping than simple shocks carried out on shock machines. To escape the first limitation, we accept, in certain cases, simulation of the effects of the shock only at low frequencies, as indicated in Figure 9.6. The “equivalent” shock has in this case a larger amplitude since fa ; the last covered frequency is higher.
Figure 9.6. Need for a TPS shock pulse of very short duration
Figure 9.7. Realizable durations lead to over-testing
344
Mechanical Shock
With this approximation, the shape of the shock has little importance, all the shocks of simple shape having in the zone which interests us (impulse zone) symmetric spectra. However, we often choose the terminal peak saw tooth to be able to reach, with lead programmers, levels of acceleration difficult to obtain with other types of programmers. This procedure, one of the first used, is open to criticism for several reasons: – if the tested item has only one frequency fa , simulation can be regarded as correct (insofar as the test facilities are able to carry out the specified shock perfectly). However, very often, in addition to a fundamental frequency ff of rather low resonance such as we can realize easily for fa ! ff , the specimen has other resonances at higher frequencies with substantial Q factors. In this case, all resonances are excited by shock and because of the frequency content particular to this kind of shock, the responses of the modes at high frequencies can be dominating. This process can thus lead to important undertesting; – by covering only the low frequencies, we can define an “equivalent” shock of sufficiently low amplitude to be realizable on the drop testers. However, nothing is solved from the point of view of shock duration. The limitation of 2 ms on the crusher programmers or 4 ms approximately on the pneumatic programmers will not make it possible to carry out a sufficiently short shock. Its spectrum will in general envelop much too much of the pyrotechnical shock at low frequencies (Figure 9.7). Except for the intersection point of the spectra ( f fa ), simulation will then be incorrect over all the frequency band. Over-testing is sometimes acceptable for f fa , and under-testing beyond. We tried to show in this chapter 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 [CON 76] [NEL 74] [POW 74] [POW 76].
Appendix
Similitude in Mechanics
In certain cases where the material is too heavy or bulky to be tested using the usual test facilities, it is possible to carry out tests on a specimen on a reduced scale. This method is often used for example in certification tests applied to containers for the transportation of nuclear matter which consist, for example, of a freefall test of 9 m onto a concrete flagstone (type B containers). The determination of the model can be made according to two different assumptions. A1. Conservation of materials With this assumption, which is most frequently used, the materials of the object on scale 1 and the model on a reduced scale are the same. The stresses and the velocities (in particular the sound velocity in each material) are retained. Let us set L as a length on scale 1 and A the length corresponding to the reduced scale. The scale ratio [BAK 73] [BRI 31] [FOC 53] [LAN 56] [MUR 50] [PAS 67] [SED 72] is: O
A L
[A.1]
346
Mechanical Shock
The velocity being retained, i.e. V
L T
v
A t
, the changing duration is given
by t
A
OL
v
V
Acceleration *
J
OT V
[A.2]
becomes
T
v
V
*
t
OT
O
[A.3]
In a similar way, analysis of the dimensions leads to the following relations. 2
Surface
s
O S
Volume
w
O W
Density
Unchanged
Mass
m M
Frequency
3
3
O M ) O 2
Force
f
O F
Energy
e
O E
Power
p
O P
Pressure (stress)
3
2
Unchanged
Table A.1. Scale factor of various parameters
Several requirements exist at the time of the definition of the model on a reduced scale, such as: – the clearance limits on the scale, the manufacturing tolerances, the states of surface;
Appendix
347
– for the screwed parts, the number of screws (dimensions on the scale) is taken into account, if possible. If the ratio of the scale leads to too small screws, we can replace them with screws of a bigger size and of a lower number, taking into account the total area of the cross-section of all of the screws; – in the case of stuck parts, taking into account the bonding strength in similitude; – in the case of measurements taken on the model by similarity, taking into account the similitude of the frequency bandwidth of the measuring equipment. This is not always easy to do, in particular in the case of impact tests where the shock to be measured already has high frequency contents on scale 1; – the effects of gravity (when they are not negligible); – as far as possible the choice of sensors in similitude (dimensions, mass, etc.). It is advisable to also consider the scale factor with respect to their functional characteristics (acceleration and the frequency are 1/O times on a reduced scale). Difficulties can arise and their importance needs to be evaluated according to the case studied. For example, the gradient of the stresses in the part is not taken into account. If G is this gradient on scale 1 and if 6 is the stress, we have on a reduced scale g
V2 V1
6 2 61
G
A
OL
O
[A.4]
i.e. g
G
[A.5]
O
A2. Conservation of acceleration and stress
We always have A
OL
Acceleration being preserved, it becomes, since J
[A.6] v
A
t
t
2
*
348
Mechanical Shock
t
2
A
OL
J
*
OT
2
[A.7]
yielding t
O T
[A.8]
Table A.2 summarizes the main relations for this. Velocity
32
O
v
V
2
Surface
s
O S
Volume
w
O W D
d
Density Mass
3
m
O 2
O M )
Frequency
M
Force
f
O F
Energy
e
O E
O 2
3
52
O
Power
p
Pressure (stress)
Unchanged Table A.2. Scale factors
P
Mechanical Shock Tests: A Brief Historical Background
The very first tests were performed by the US Navy in around 1917 [PUS 77] [WEL 46]. Progress, which was slow up to World War II, accelerated in the 1940s. The following list gives a few of the major dates. 1932
First publication on the shock response spectrum for the study of earthquakes.
1939
First high impact shock machine (pendular drop-hammer) for the simulation of the effects of submarine explosions on on-board equipment [CLE 72] [OLI 47].
1941
Development of a 10-foot freefall test method [DEV 47].
1945
Drafting of the first specifications for aircraft equipment [KEN 51] under various environmental conditions (A.F. Specification 410065).
1947
Environmental measurements on land vehicles for drafting specifications [PRI 47].
1947
Use of an air gun to simulate shocks on electronic components (Naval Ordnance Laboratory) [DEV 47].
1948
Freefall machine on sand, with monitoring of the amplitude and duration of the shock (US Air Force) [BRO 61].
1955
Use of exciters for shock simulation (reproduction of simple shape shocks) [WEL 61].
350
Mechanical Shock
1964
Taking into account shocks of pyrotechnic origin, demonstrating the difficulty of simulating them with classic facilities. Development of special facilities [BLA 64].
1966
Initial research into shock simulation on exciters driven from a shock response spectrum [GAL 66]. These methods were only fully developed in the mid-1970s.
1984
The shock response spectrum becomes the benchmark in the MIL-STD 810 D standard for the definition of specifications.
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360
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[PAI 64] PAINTER G.W. and PARRY H.J., “Simulating flight environment shock on an electrodynamic shaker”, Shock, Vibration and Associated Environments Bulletin, no. 33, Part 3, 1964, p. 85/96. [PAP 62] PAPOULIS A., The Fourier Integral and its Applications, McGraw-Hill, 1962, p. 62. [PAS 67] PASCOUET A., Similitude et résistance dynamique des matériaux. Application aux ondes de pression dans l'eau, Mémorial de l'Artillerie Française, Fascicule 1, 1967. [PIE 92] PIERSOL A.G., “Guidelines for Dynamic Data Acquisition and Analysis”, Journal of the IES, Volume 35, Number 5/September-October 1992, p. 21/26. [PET 81] PETERSEN B.B., Applications of Mechanical Shock Spectra, Elektronikcentralen, Danish Research Center for Applied Electronics, Denmark, ECR 106, January 1981. [POW 74] POWERS D.R., “Development of a pyrotechnic shock test facility”, The Shock and Vibration Bulletin, no. 44, Part 3, August 1974, p. 73/82. [POW 76] POWERS D.R., “Simulation of pyrotechnic shock in a test laboratory”, IES Proceedings, 1976, p. 5/9. [PRI 47] PRIEBE F.K., “Vehicular shock and vibration instrumentation and measurements with special consideration of military vehicles”, The Shock and Vibration Bulletin, no. 6, November 1947. [PUS 77] PUSEY H.C., “An historical view of dynamic testing”, The Journal of Environmental Sciences, September/October 1977, p. 9/14. [RAD 70] RADER W.P. and BANGS W.F., “A summary of pyrotechnic shock in the aerospace industry”, The Shock and Vibration Bulletin, no. 41, Part 5, December 1970, p. 9/15. [REE 60] REED W.H., HALL A.W. and BARKER L.E., “Analog techniques for measuring the frequency response of linear physical systems excited by frequency sweep inputs”, NASA TN D 508, 1960. [RID 69] RIDLER K.D. and BLADER F.B., “Errors in the use of shock spectra”, Environmental Engineering, July 1969, p. 7/16. [ROB 67] ROBBINS C.D. and VAUGHAN P.E., “Laboratory techniques for utilization of a shock synthetizer/analyzer”, IES Proceedings, 1967, p. 211/214. [RÖS 70] RÖSSLER F. and DAVID E., Fallversuche auf Blei, ISL Notiz N 32/70, 09/07/1970. [ROT 72] ROTHAUG R.J., “Practical aspects of shock spectrum testing”, IES Proceedings, 1972, p. 303/308. [ROU 74] ROUNTREE R.C. and FREBERG C.R., “Identification of an optimum set of transient sweep parameters for generating specified response spectra”, The Shock and Vibration Bulletin, no. 44, Part 3, 1974, p. 177/192. [RUB 58] RUBIN S., “Response of complex structures from reed-gage data”, Journal of Applied Mechanics, Vol. 35, December 1958. [RUB 86] RUBIN S., “Pyrotechnic shock”, The Shock and Vibration Bulletin, no. 56, Part 3, August 1986, p. 1/2.
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Index
50 % rebound, 176
D, E
A, B, C
damping constant of the sinusoids, 283 influence, 114 decaying sinusoid, 273, 278, 279, 297, 313 sinusoidal shock, 8 development of shock test specifications, 139 displacement, 169, 172, 176 drop height, 195, 199, 207, 227 tester, 207 duration of the pre-shock, 240 dynamic modulus, 210 earthquake, 2, 52, 93, 158, 299 elasticity modulus, 210 elastomer, 212, 228, 229 electrodynamic shaker, 342 electrohydraulic exciter, 237 electronic limitations, 236 energy of the pulse, 325 energy spectrum, 13 equivalent static acceleration, 65 explosive charge, 338 cord, 338 extreme response spectrum, 82
abacuses, 255 accelerated fall, 200 algorithms, 82 aliasing, 31 allowable maximum force, 226 time-frequency analysis, 36 arbitrary triangular pulse, 20, 63 background noise, 130 bump test, 9 bungee cord, 195 choice of damping, 114 of the shock shape, 144 coefficient of restitution, 172, 199, 212 compensating sinusoid, 287 compensation of the signal, 284 using two exponentials, 287 CONBUR test, 208 control by a shock response spectrum, 271 correction factor, 116 crushing, 223
366
Mechanical Shock
F, G fast swept sine, 153, 314 fatigue damage, 321 spectrum, 328 final peak sawtooth pulse, 5 four-coordinate representation, 118 Fourier reciprocity formula, 13 short-term, 39 spectrum, 13, 125, 326 transform, 12, 27, 36, 120, 159, 234, 314, 327, 331 frequency range, 118 Gabor window, 41 GAM EG 13, 9 gas gun, 207 Gauss wavelet, 45 H, I half-sine, 3, 205, 238, 242 pulse, 14, 61, 71, 109, 167, 208, 224 with half-sine pre- and postshocks, 242 shock pulse, 98 with post-shock alone, 253 with square pre and post-shock, 247 with triangular pre and postshock, 244 Hamming window, 42 haversine shock, 4 high impact shock machine, 203 honeycomb, 219, 223 Hopkinson bar, 341 hyperbolic swept sine, 154 impact, 169, 193, 198 machine, 193 mode, 182 shock machine, 193 velocity, 194, 209, 221, 227 without rebound, 173, 216, 223
impulse, 169 domain, 95, 114 machine, 193 mode, 170, 182, 198 zone, 344 inclined plane impact tester, 207 initial negative shock response spectrum, 66 initial peak sawtooth (IPS) pulse, 6, 62, 75, 188, 252 with post-shock alone, 255 initial positive shock response spectrum, 66 intermediate domain, 95 inversion formula, 13 isosceles triangle pulse, 125 iteration, 290, 307, 316 J, K, L, M jerk, 3 kinematics, 167 lead programmers, 343 lead, 219, 223, 228 least favourable response, 159 limitations of programmers, 229 of the shock machines, 226 logarithmic four coordinate spectrum, 91 loss coefficient, 228 Collins machine, 195 maximax shock response spectrum, 68 maximum acceleration, 236 displacement, 240, 257 force, 236 stroke, 235, 334 velocity, 235 metal to metal impact, 341 mode, 92 modulated random noise, 157 sine wave, 160
Index
module, 215 Morlet wavelet, 45 motion of the coil-table assembly, 235 N, O narrow band random vibration, 158 negative shock response spectrum, 67 noise, 131 nominal shock, 163, 266 non-linearity, 320, 331 one period of sinusoid, 106 optimized pre- and post-shocks, 259 oscillatory shock, 108, 132 over-testing, 140, 151 package, 58, 90 peak histogram, 328 pendular machine, 196 perfect rebound, 174, 229 phase angle, 13 pneumatic gun, 342 machine, 205 positive shock response spectrum, 67 post-shock, 198, 237 power, 334 pre-shock, 198, 202, 237 primary negative shock response spectrum, 66, 108 primary positive shock response spectrum, 66 primary positive spectrum, 102, 104, 105 primary shock response spectrum, 120 programmer, 197, 208, 228 deformation, 209 using crushable material, 216 Prony method, 284 pseudo-acceleration, 66 pseudo-velocity, 66, 100 punch, 220, 224 pyroshock, 9, 152, 157
367
pyrotechnic facility, 337 shock, 118, 132, 271 Q, R Q factor, 115 random vibration, 158 rate of rebound, 199, 212 reaction mass, 207 rebound, 171, 212 velocity, 227 rectangular pulse, 183 relative displacement, 240 shock spectrum, 65 residual positive, 100 shock response spectrum, 66, 122 spectrum, 105, 110 resonance of the table, 216 spectrum, 65 rms duration, 325 value, 326, 327 rubber, 210 S sampling frequency, 86 secondary shock response spectrum, 66 shaker, 233, 272 control from a SRS, 162 Shannon theorem, 31, 86 SHOC, 307, 313 shock amplifier, 198 amplitude, 146 by impact, 264 defined by a force, 55 defined by an acceleration, 56 duration, 146, 323, 344 machine, 191 on shaker, 233, 264
368
Mechanical Shock
response spectrum, 51, 54 simple, 3 wave, 212 of pyrotechnic origin, 337 sinus-verse, 72 slope at the origin, 101 solid mass of reaction, 207, 219 solid reaction mass, 208 specification of shock, 342 specification, 139, 319 square pulse, 22, 62, 77, 223, 252 square shock, 7 square with post-shock alone, 254 square, 238 SRS slope, 108 SRS tolerance, 163 standardized response spectrum, 69 static domain, 95 subroutine for the calculation, 83 swept sine, 152, 320 synthesis of the spectra, 142 synthesis, 317 system with several degrees of freedom, 92 T temporal step, 31 terminal peak sawtooth (TPS) pulse, 5, 16, 63, 74, 99, 101, 107, 125, 186, 205, 216, 220, 225, 241, 251, 343
test factor, 143, 150 time history synthesis, 306, 312 TPS with post-shock alone, 254 transfer function, 233, 278, 332 transient vibration, 2 trapezoidal pulse, 8, 23, 64, 77, 223, 226 triangle, 238 U uncertainty coefficient, 143 uncertainty factor, 338 universal programmer, 223, 230 universal shock programmer, 224 universal shock test machine, 197 velocity 168, 172, 175 change, 12, 97, 109, 141, 168, 172, 199, 227, 267, 315 versed sine, 238 pulse, 15, 61, 72, 181 shock, 4 WAVSIN, 300, 313 waveform, 299 ZERD, 294, 307, 313 function, 294 pulse, 108 zero derivative, 130 shift, 132
Summary of Other Volumes in the Series
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Summary of Volume 1 Sinusoidal Vibration
Chapter 1. The Need 1.1. The need to carry out studies into vibrations and mechanical shocks 1.2. Some real environments 1.2.1. Sea transport 1.2.2. Earthquakes 1.2.3. Road vibratory environment 1.2.4. Rail vibratory environment 1.2.5. Propeller airplanes 1.2.6. Vibrations caused by jet propulsion airplanes 1.2.7. Vibrations caused by turbofan aircraft 1.2.8. Helicopters 1.3. Measuring vibrations 1.4. Filtering 1.4.1. Definitions 1.4.2. Digital filters 1.5. The frequency of a digitized signal 1.6. Reconstructing the sampled signal 1.7. Characterization in the frequency domain 1.8. Elaboration of the specifications 1.9. Vibration test facilities 1.9.1. Electro-dynamic exciters 1.9.2. Hydraulic actuators Chapter 2. Basic Mechanics 2.1. Basic principles of mechanics 2.1.1. Principle of causality
Mechanical Vibration and Shock Analysis
2.1.2. Concept of force 2.1.3. Newton’s First law (inertia principle) 2.1.4. Moment of a force around a point 2.1.5. Fundamental principle of dynamics (Newton’s second law) 2.1.6. Equality of action and reaction (Newton’s third law) 2.2. Static effects/dynamic effects 2.3. Behavior under dynamic load (impact) 2.4. Elements of a mechanical system 2.4.1. Mass 2.4.2. Stiffness 2.4.3. Damping 2.4.4. Static modulus of elasticity 2.4.5. Dynamic modulus of elasticity 2.5. Mathematical models 2.5.1. Mechanical systems 2.5.2. Lumped parameter systems 2.5.3. Degrees of freedom 2.5.4. Mode 2.5.5. Linear systems 2.5.6. Linear one-degree-of-freedom mechanical systems 2.6. Setting an equation for n degrees-of-freedom lumped parameter mechanical system 2.6.1. Lagrange equations 2.6.2. D’Alembert’s principle 2.6.3. Free-body diagram Chapter 3. Response of a Linear One-Degree-of-Freedom Mechanical System to an Arbitrary Excitation 3.1. Definitions and notation 3.2. Excitation defined by force versus time 3.3. Excitation defined by acceleration 3.4. Reduced form 3.4.1. Excitation defined by a force on a mass or by an acceleration of support 3.4.2. Excitation defined by velocity or displacement imposed on support 3.5. Solution of the differential equation of movement 3.5.1. Methods 3.5.2. Relative response 3.5.3. Absolute response 3.5.4. Summary of main results 3.6. Natural oscillations of a linear one-degree-of-freedom system
Summary of Volume 1
3.6.1. Damped aperiodic mode 3.6.2. Critical aperiodic mode 3.6.3. Damped oscillatory mode Chapter 4. Impulse and Step Responses 4.1. Response of a mass–spring system to a unit step function (step or indicial response) 4.1.1. Response defined by relative displacement 4.1.2. Response defined by absolute displacement, velocity or acceleration 4.2. Response of a mass–spring system to a unit impulse excitation 4.2.1. Response defined by relative displacement 4.2.2. Response defined by absolute parameter 4.3. Use of step and impulse responses 4.4. Transfer function of a linear one-degree-of-freedom system 4.4.1. Definition 4.4.2. Calculation of H h for relative response 4.4.3. Calculation of H(h) for absolute response 4.4.4. Other definitions of the transfer function Chapter 5. Sinusoidal Vibration 5.1. Definitions 5.1.1. Sinusoidal vibration 5.1.2. Mean value 5.1.3. Mean square value – rms value 5.1.4. Periodic vibrations 5.1.5. Quasi-periodic signals 5.2. Periodic and sinusoidal vibrations in the real environment 5.3. Sinusoidal vibration tests Chapter 6. Response of a Linear One-Degree-of-Freedom Mechanical System to a Sinusoidal Excitation 6.1. General equations of motion 6.1.1. Relative response 6.1.2. Absolute response 6.1.3. Summary 6.1.4. Discussion 6.1.5. Response to periodic excitation 6.1.6. Application to calculation for vehicle suspension response 6.2. Transient response 6.2.1. Relative response
Mechanical Vibration and Shock Analysis
6.2.2.Absolute response 6.3. Steady state response 6.3.1. Relative response 6.3.2. Absolute response Z z Z z 6.4. Responses 0 , 0 and xm x m
k m z Fm
6.4.1. Amplitude and phase 6.4.2. Variations of velocity amplitude 6.4.3. Variations in velocity phase kz Z2 z and 0 Fm xm 6.5.1. Expression for response 6.5.2. Variation in response amplitude 6.5.3. Variations in phase y y y F , , and T 6.6.Responses xm xm xm Fm 6.6.1. Movement transmissibility 6.6.2. Variations in amplitude 6.6.3. Variations in phase 6.7. Graphical representation of transfer functions
6.5.Responses
Chapter 7. Non-Viscous Damping 7.1. Damping observed in real structures 7.2. Linearization of non-linear hysteresis loops – equivalent viscous damping 7.3. Main types of damping 7.3.1. Damping force proportional to the power b of the relative velocity 7.3.2. Constant damping force 7.3.3. Damping force proportional to the square of velocity 7.3.4. Damping force proportional to the square of displacement 7.3.5. Structural or hysteretic damping 7.3.6. Combination of several types of damping 7.3.7. Validity of simplification by equivalent viscous damping 7.4. Measurement of damping of a system 7.4.1. Measurement of amplification factor at resonance 7.4.2. Bandwidth or 2 method 7.4.3. Decreased rate method (logarithmic decrement) 7.4.4. Evaluation of energy dissipation under permanent sinusoidal vibration
Summary of Volume 1
7.4.5. Other methods 7.5. Non-linear stiffness Chapter 8. Swept Sine
8.1. Definitions 8.1.1. Swept sine 8.1.2. Octave – number of octaves in frequency interval ( f 1 , f 2 ) 8.1.3. Decade 8.2. ‘Swept sine’ vibration in the real environment 8.3. ‘Swept sine’ vibration in tests 8.4. Origin and properties of main types of sweepings 8.4.1. The problem 8.4.2. Case 1: sweep where time 't spent in each interval ' f is constant for all values of f 0 8.4.3. Case 2: sweep with constant rate 8.4.4. Case 3: sweep ensuring a number of identical cycles ' N in all intervals ' f (delimited by the half-power points) for all values of f 0 Chapter 9. Response of a One-Degree-of-Freedom Linear System to a Swept Sine Vibration
9.1. Influence of sweep rate 9.2. Response of a linear one-degree-of-freedom system to a swept sine excitation 9.2.1. Methods used for obtaining response 9.2.2. Convolution integral (or Duhamel’s integral) 9.2.3. Response of a linear one-degree-of freedom system to a linear swept sine excitation 9.2.4. Response of a linear one-degree-of-freedom system to a logarithmic swept sine 9.3. Choice of duration of swept sine test 9.4. Choice of amplitude 9.5. Choice of sweep mode Appendix: Laplace Transformations Vibration Tests: a Brief Historical Background Bibliography Index
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Summary of Volume 3 Random Vibration
Chapter 1. Statistical Properties of a Random Process
1.1. Definitions 1.1.1. Random variable 1.1.2. Random process 1.2. Random vibration in real environments 1.3. Random vibration in laboratory tests 1.4. Methods of random vibration analysis 1.5. Distribution of instantaneous values 1.5.1. Probability density 1.5.2. Distribution function 1.6. Gaussian random process 1.7. Rayleigh distribution 1.8. Ensemble averages: through the process 1.8.1. n order average 1.8.2. Centered moments 1.8.3. Variance 1.8.4. Standard deviation 1.8.5. Autocorrelation function 1.8.6. Cross-correlation function 1.8.7. Autocovariance 1.8.8. Covariance 1.8.9. Stationarity 1.9. Temporal averages: along the process 1.9.1. Mean 1.9.2. Quadratic mean – rms value 1.9.3. Moments of order n 1.9.4. Variance – standard deviation 1.9.5. Skewness
Mechanical Vibration and Shock Analysis
1.9.6. Kurtosis 1.9.7. Temporal autocorrelation function 1.9.8. Properties of the autocorrelation function 1.9.9. Correlation duration 1.9.10. Cross-correlation 1.9.11. Cross-correlation coefficient 1.9.12. Ergodicity 1.10. Significance of the statistical analysis (ensemble or temporal) 1.11. Stationary and pseudo-stationary signals 1.12. Summary chart of main definitions 1.13. Sliding mean 1.14. Identification of shocks and/or signal problems 1.15. Breakdown of vibratory signal into “events”: choice of signal samples 1.16. Interpretation and taking into account of environment variation Chapter 2. Random Vibration Properties in the Frequency Domain
2.1. Fourier transform 2.2. Power spectral density 2.2.1. Need 2.2.2. Definition 2.3. Cross-power spectral density 2.4. Power spectral density of a random process 2.5. Cross-power spectral density of two processes 2.6. Relationship between the PSD and correlation function of a process 2.7. Quadspectrum – cospectrum 2.8. Definitions 2.8.1. Broad band process 2.8.2. White noise 2.8.3. Band-limited white noise 2.8.4. Narrow band process 2.8.5. Pink noise 2.9. Autocorrelation function of white noise 2.10. Autocorrelation function of band-limited white noise 2.11. Peak factor 2.12. Effects of truncation of peaks of acceleration signal on the PSD 2.13. Standardized PSD/density of probability analogy 2.14. Spectral density as a function of time 2.15. Relationship between the PSD of the excitation and the response of a linear system 2.16. Relationship between the PSD of the excitation and the crosspower spectral density of the response of a linear system 2.17. Coherence function
Summary of Volume 3
2.18. Transfer function calculation from random vibration measurements 2.18.1. Theoretical relations 2.18.2. Presence of noise on the input 2.18.3. Presence of noise on the response 2.18.4. Presence of noise on the input and response 2.18.5. Choice of transfer function Chapter 3. Rms Value of Random Vibration
3.1. Rms value of a signal as a function of its PSD 3.2. Relationships between the PSD of acceleration, velocity and displacement 3.3. Graphical representation of the PSD 3.4. Practical calculation of acceleration, velocity and displacement rms values 3.4.1. General expressions 3.4.2. Constant PSD in frequency interval 3.4.3. PSD comprising several horizontal straight line segments 3.4.4. PSD defined by a linear segment of arbitrary slope 3.4.5. PSD comprising several segments of arbitrary slopes 3.5. Rms value according to the frequency 3.6. Case of periodic signals 3.7. Case of a periodic signal superimposed onto random noise Chapter 4. Practical Calculation of the Power Spectral Density
4.1. Sampling of signal 4.2. PSD calculation methods 4.2.1. Use of the autocorrelation function 4.2.2. Calculation of the PSD from the rms value of a filtered signal 4.2.3. Calculation of the PSD starting from a Fourier transform 4.3. PSD calculation steps 4.3.1. Maximum frequency 4.3.2. Extraction of sample of duration T 4.3.3. Averaging 4.3.4. Addition of zeros 4.4. FFT 4.5. Particular case of a periodic excitation 4.6. Statistical error 4.6.1. Origin 4.6.2. Definition 4.7. Statistical error calculation 4.7.1. Distribution of the measured PSD 4.7.2. Variance of the measured PSD 4.7.3. Statistical error
Mechanical Vibration and Shock Analysis
4.7.4. Relationship between number of degrees of freedom, duration and bandwidth of analysis 4.7.5. Confidence interval 4.7.6. Expression for statistical error in decibels 4.7.7. Statistical error calculation from digitized signal 4.8. Influence of duration and frequency step on the PSD 4.8.1. Influence of duration 4.8.2. Influence of the frequency step 4.8.3. Influence of duration and of constant statistical error frequency step 4.9. Overlapping 4.9.1. Utility 4.9.2. Influence on the number of dofs 4.9.3. Influence on statistical error 4.9.4. Choice of overlapping rate 4.10. Information to provide with a PSD 4.11. Difference between rms values calculated from a signal according to time and from its PSD 4.12. Calculation of a PSD from a Fourier transform 4.13. Amplitude based on frequency: relationship with the PSD 4.14. Calculation of the PSD for given statistical error 4.14.1. Case study: digitization of a signal is to be carried out 4.14.2. Case study: only one sample of an already digitized signal is available 4.15. Choice of filter bandwidth 4.15.1. Rules 4.15.2. Bias error 4.15.3. Maximum statistical error 4.15.4. Optimum bandwidth 4.16. Probability that the measured PSD lies between r one standard deviation 4.17. Statistical error: other quantities 4.18. Peak hold spectrum 4.19. Generation of random signal of given PSD 4.19.1. Random phase sinusoid sum method 4.19.2. Inverse Fourier transform method 4.20. Using a window during the creation of a random signal from a PSD Chapter 5. Statistical Properties of Random Vibration in the Time Domain
5.1. Distribution of instantaneous values 5.2. Properties of derivative process 5.3. Number of threshold crossings per unit time
Summary of Volume 3
5.4. Average frequency 5.5. Threshold level crossing curves 5.6. Moments 5.7. Average frequency of PSD defined by straight line segments 5.7.1. Linear-linear scales 5.7.2. Linear-logarithmic scales 5.7.3. Logarithmic-linear scales 5.7.4. Logarithmic-logarithmic scales 5.8. Fourth moment of PSD defined by straight line segments 5.8.1. Linear-linear scales 5.8.2. Linear-logarithmic scales 5.8.3. Logarithmic-linear scales 5.8.4. Logarithmic-logarithmic scales 5.9. Generalization: moment of order n 5.9.1. Linear-linear scales 5.9.2. Linear-logarithmic scales 5.9.3. Logarithmic-linear scales 5.9.4. Logarithmic-logarithmic scales Chapter 6. Probability Distribution of Maxima of Random Vibration
6.1. Probability density of maxima 6.2. Expected number of maxima per unit time 6.3. Average time interval between two successive maxima 6.4. Average correlation between two successive maxima 6.5. Properties of the irregularity factor 6.5.1. Variation interval 6.5.2. Calculation of irregularity factor for band-limited white noise 6.5.3. Calculation of irregularity factor for noise of form G = Const. fb 6.5.4. Case study: variations of irregularity factor for two narrow band signals 6.6. Error related to the use of Rayleigh’s law instead of a complete probability density function 6.7. Peak distribution function 6.7.1. General case 6.7.2. Particular case of a narrow band Gaussian process 6.8. Mean number of maxima greater than the given threshold (by unit time) 6.9. Mean number of maxima above given threshold between two times 6.10. Mean time interval between two successive maxima 6.11. Mean number of maxima above given level reached by signal excursion above this threshold 6.12. Time during which the signal is above a given value
Mechanical Vibration and Shock Analysis
6.13. Probability that a maximum is positive or negative 6.14. Probability density of the positive maxima 6.15. Probability that the positive maxima is lower than a given threshold 6.16. Average number of positive maxima per unit of time 6.17. Average amplitude jump between two successive extrema Chapter 7. Statistics of Extreme Values
7.1. Probability density of maxima greater than a given value 7.2. Return period 7.3. Peak A p expected among N p peaks 7.4. Logarithmic rise 7.5. Average maximum of N p peaks 7.6. Variance of maximum 7.7. Mode (most probable maximum value) 7.8. Maximum value exceeded with risk D 7.9. Application to the case of a centered narrow band normal process 7.9.1. Distribution function of largest peaks over duration T 7.9.2. Probability that one peak at least exceeds a given threshold 7.9.3. Probability density of the largest maxima over duration T 7.9.4. Average of highest peaks 7.9.5. Mean value probability 7.9.6. Standard deviation of highest peaks 7.9.7. Variation coefficient 7.9.8. Most probable value 7.9.9. Median 7.9.10. Value of density at mode 7.9.11. Expected maximum 7.9.12. Average maximum 7.9.13. Maximum exceeded with given risk D 7.10. Wide band centered normal process 7.10.1. Average of largest peaks 7.10.2. Variance of the largest peaks 7.10.3. Variation coefficient 7.11. Asymptotic laws 7.11.1. Gumbel asymptote 7.11.2. Case study: Rayleigh peak distribution 7.11.3. Expressions for large values of N p 7.12. Choice of type of analysis 7.13. Study of the envelope of a narrow band process 7.13.1. Probability density of the maxima of the envelope 7.13.2. Distribution of maxima of envelope 7.13.3. Average frequency of envelope of narrow band noise
Summary of Volume 3
Chapter 8. Response of a One-Degree-of-Freedom Linear System to Random Vibration
8.1. Average value of the response of a linear system 8.2. Response of perfect bandpass filter to random vibration 8.3. The PSD of the response of a one-dof linear system 8.4. Rms value of response to white noise 8.5. Rms value of response of a linear one-dof system subjected to bands of random noise 8.5.1. Case where the excitation is a PSD defined by a straight line segment in logarithmic scales 8.5.2. Case where the vibration has a PSD defined by a straight line segment of arbitrary slope in linear scales 8.5.3. Case where the vibration has a constant PSD between two frequencies 8.5.4. Excitation defined by an absolute displacement 8.5.5. Case where the excitation is defined by PSD comprising n straight line segments 8.6. Rms value of the absolute acceleration of the response 8.7. Transitory response of a dynamic system under stationary random excitation 8.8. Transitory response of a dynamic system under amplitude modulated white noise excitation Chapter 9. Characteristics of the Response of a One-Degree-of-Freedom Linear System to Random Vibration
9.1. Moments of response of a one-degree-of-freedom linear system: irregularity factor of response 9.1.1. Moments 9.1.2. Irregularity factor of response to noise of a constant PSD 9.1.3. Characteristics of irregularity factor of response 9.1.4. Case of a band-limited noise 9.2. Autocorrelation function of response displacement 9.3. Average numbers of maxima and minima per second 9.4. Equivalence between the transfer functions of a bandpass filter and a one-dof linear system 9.4.1. Equivalence suggested by D.M. Aspinwall 9.4.2. Equivalence suggested by K.W. Smith 9.4.3. Rms value of signal filtered by the equivalent bandpass filter Chapter 10. First Passage at a Given Level of Response of a One-Degree-of-Freedom Linear System to a Random Vibration
10.1. Assumptions 10.2. Definitions
Mechanical Vibration and Shock Analysis
10.3. Statistically independent threshold crossings 10.4. Statistically independent response maxima 10.5. Independent threshold crossings by the envelope of maxima 10.6. Independent envelope peaks 10.6.1. S.H. Crandall method 10.6.2. D.M. Aspinwall method 10.7. Markov process assumption 10.7.1. W.D. Mark assumption 10.7.2. J.N. Yang and M. Shinozuka approximation 10.8. E.H. Vanmarcke model 10.8.1. Assumption of a two state Markov process 10.8.2. Approximation based on the mean clump size Appendices Bibliography Index
Summary of Volume 4 Fatigue Damage
Chapter 1. Concepts of Material Fatigue
1.1. Introduction 1.2. Types of dynamic loads (or stresses) 1.2.1. Cyclic stress 1.2.2. Alternating stress 1.2.3. Repeated stress 1.2.4. Combined steady and cyclic stress 1.2.5. Skewed alternating stress 1.2.6. Random and transitory stresses 1.3. Damage arising from fatigue 1.4. Characterization of endurance of materials 1.4.1. S-N curve 1.4.2. Statistical aspect 1.4.3. Distribution laws of endurance 1.4.4. Distribution laws of fatigue strength 1.4.5. Relation between fatigue limit and static properties of materials 1.4.6. Analytical representations of S-N curve 1.5. Factors of influence 1.5.1. General 1.5.2. Scale 1.5.3. Overloads 1.5.4. Frequency of stresses 1.5.5. Types of stresses 1.5.6. Non-zero mean stress 1.6. Other representations of S-N curves 1.6.1. Haigh diagram 1.6.2. Statistical representation of Haigh diagram
Mechanical Vibration and Shock Analysis
1.7. Prediction of fatigue life of complex structures 1.8. Fatigue in composite materials Chapter 2. Accumulation of Fatigue Damage
2.1. Evolution of fatigue damage 2.2. Classification of various laws of accumulation 2.3. Miner’s method 2.3.1. Miner’s rule 2.3.2. Scatter of damage to failure as evaluated by Miner 2.3.3. Validity of Miner’s law of accumulation of damage in case of random stress 2.4. Modified Miner’s theory 2.4.1. Principle 2.4.2. Accumulation of damage using modified Miner’s rule 2.5. Henry’s method 2.6. Modified Henry’s method 2.7. Corten and Dolan’s method 2.8. Other theories Chapter 3. Counting Methods for Analyzing Random Time History
3.1. General 3.2. Peak count method 3.2.1. Presentation of method 3.2.2. Derived methods 3.2.3. Range-restricted peak count method 3.2.4. Level-restricted peak count method 3.3. Peak between mean-crossing count method 3.3.1. Presentation of method 3.3.2. Elimination of small variations 3.4. Range count method 3.4.1. Presentation of method 3.4.2. Elimination of small variations 3.5. Range-mean count method 3.5.1. Presentation of method 3.5.2. Elimination of small variations 3.6. Range-pair count method 3.7. Hayes’ counting method 3.8. Ordered overall range counting method 3.9. Level-crossing count method 3.10. Peak valley peak counting method 3.11. Fatigue-meter counting method 3.12. Rainflow counting method
Summary of Volume 4
3.12.1. Principle of method 3.12.2. Subroutine for rainflow counting 3.13. NRL (National Luchtvaart Laboratorium) counting method 3.14. Evaluation of time spent at a given level 3.15. Influence of levels of load below fatigue limit on fatigue life 3.16. Test acceleration 3.17. Presentation of fatigue curves determined by random vibration tests Chapter 4. Fatigue Damage by One-degree-of-freedom Mechanical System
4.1. Introduction 4.2. Calculation of fatigue damage due to signal versus time 4.3. Calculation of fatigue damage due to acceleration spectral density 4.3.1. General case 4.3.2. Approximate expression of the probability density of peaks 4.3.3. Particular case of a wide-band response, e.g. at the limit r = 0 4.3.4. Particular case of narrow band response 4.3.5. Rms response to narrow band noise G0 of width 'f when G0 'f = constant 4.4. Equivalent narrow band noise 4.4.1. Use of relation established for narrow band response 4.4.2. Alternative: use of mean number of maxima per second 4.4.3. Approximation to real maxima distribution using a modified Rayleigh distribution 4.5. Calculation of fatigue damage from the probability density of domains 4.5.1. Differences between the probability of peaks and of ranges 4.5.2. Wirsching’s approach 4.5.3. Chaudhury and Dover’s approach 4.5.4. Dirlik’s probability density 4.5.5. Expression of the fatigue damage from the Dirlik probability density 4.6. Comparison of S-N curves established under sinusoidal and random loads 4.7. Comparison of theory and experiment 4.8. Influence of shape of power spectral density and value of irregularity factor 4.9. Effects of peak truncation 4.10. Truncation of stress peaks 4.10.1. Particular case of a narrow band noise 4.10.2. Layout of the S-N curve for a truncated distribution
Mechanical Vibration and Shock Analysis
Chapter 5. Standard Deviation of Fatigue Damage
5.1. Calculation of standard deviation of damage: Bendat’s method 5.2. Calculation of standard deviation of damage: method of Crandall et al. 5.3. Comparison of Mark and Bendat’s results 5.4. Statistical S-N curves 5.4.1. Definition of statistical curves 5.4.2. Bendat’s formulation 5.4.3. Mark’s formulation Chapter 6. Fatigue Damage using other Calculation Assumptions
6.1. S-N curve represented by two segments of a straight line on logarithmic scales (taking into account fatigue limit) 6.2. S-N curve defined by two segments of straight line on log-lin scales 6.3. Hypothesis of non-linear accumulation of damage 6.3.1. Corten-Dolan’s accumulation law 6.3.2. Morrow’s accumulation model 6.4. Random vibration with non-zero mean: use of modified Goodman diagram 6.5. Non-Gaussian distribution of instantaneous values of signal 6.5.1. Influence of distribution law of instantaneous values 6.5.2. Influence of peak distribution 6.5.3. Calculation of damage using Weibull distribution 6.5.4. Comparison of Rayleigh assumption/peak counting 6.6. Non-linear mechanical system Chapter 7. Low Fatigue Cycle
7.1. Overview 7.2. Definitions 7.2.1. Baushinger effect 7.2.2. Cyclic strain hardening 7.2.3. Properties of cyclic stress-strain curves 7.2.4. Stress-strain curve 7.2.5. Hysteresis and fracture by fatigue 7.2.6. Significant factors influencing hysteresis and fracture by fatigue 7.2.7. Cyclic stress-stress curve (or cyclic consolidation curve) 7.3. Behavior of materials experiencing strains in the oligocyclic domain 7.3.1. Types of behaviors
Summary of Volume 4
7.3.2. Cyclic strain hardening 7.3.3. Cyclic strain softening 7.3.4. Cyclically stable metals 7.3.5. Mixed behavior 7.4. Influence of the level application sequence 7.5. Development of the cyclic stress-strain curve 7.6. Total strain 7.7. Fatigue strength curve 7.7.1. Basquin curve 7.8. Relation between plastic strain and number of cycles to fracture 7.8.1. Orowan relation 7.8.2. Manson relation 7.8.3. Coffin relation 7.8.4. Shanley relation 7.8.5. Gerberich relation 7.8.6. Sachs, Gerberich, Weiss and Latorre relation 7.8.7. Martin relation 7.8.8. Tavernelli and Coffin relation 7.8.9. Manson relation 7.8.10. Ohji et al. relation 7.8.11. Bui-Quoc et al. relation 7.9. Influence of the frequency and temperature in the plastic field 7.9.1. Overview 7.9.2. Influence of frequency 7.9.3. Influence of temperature and frequency 7.9.4. Effect of frequency on plastic strain range 7.9.5. Equation of generalized fatigue 7.10. Laws of damage accumulation 7.10.1. Miner rule 7.10.2. Yao and Munse relation 7.10.3. Use of the Manson–Coffin relation 7.11. Influence of an average strain or stress 7.11.1. Other approaches 7.12. Low cycle fatigue of composite material Chapter 8. Fracture Mechanics
8.1. Overview 8.1.1. Definition: stress gradient 8.2. Fracture mechanism 8.2.1. Major phases 8.2.2. Initiation of cracks 8.2.3. Slow propagation of cracks
Mechanical Vibration and Shock Analysis
8.3. Critical size: strength to fracture 8.4. Modes of stress application 8.5. Stress intensity factor 8.5.1. Stress in crack root 8.5.2. Mode I 8.5.3. Mode II 8.5.4. Mode III 8.5.5. Field of equation use 8.5.6. Plastic zone 8.5.7. Other form of stress expressions 8.5.8. General form 8.5.9. Widening of crack opening 8.6. Fracture toughness: critical K value 8.6.1. Units 8.7. Calculation of the stress intensity factor 8.8. Stress ratio 8.9. Expansion of cracks: Griffith criterion 8.10. Factors affecting the initiation of cracks 8.11. Factors affecting the propagation of cracks 8.11.1. Mechanical factors 8.11.2. Geometric factors 8.11.3. Metallurgical factors 8.11.4. Factors linked to the environment 8.12. Speed of propagation of cracks 8.13. Effect of a non-zero mean stress 8.14. Laws of crack propagation 8.14.1. Head 8.14.2. Modified Head law 8.14.3. Frost and Dugsdale 8.14.4. McEvily and Illg 8.14.5. Paris and Erdogan 8.15. Stress intensity factor 8.16. Dispersion of results 8.17. Sample tests: extrapolation to a structure 8.18. Determination of the propagation threshold Ks 8.19. Propagation of cracks in the domain of low cycle fatigue 8.20. Integral J 8.21. Overload effect: fatigue crack retardation 8.22. Fatigue crack closure 8.23. Rules of similarity 8.24. Calculation of a useful lifetime
Summary of Volume 4
8.25. Propagation of cracks under random load 8.25.1. Rms approach 8.25.2. Narrow band random loads 8.25.3. Calculation from a load collective Appendix Bibliography Index
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Summary of Volume 5 Specification Development
Chapter 1. Extreme Response Spectrum of a Sinusoidal Vibration 1.1. The effects of vibration 1.2. Extreme response spectrum of a sinusoidal vibration 1.2.1. Definition 1.2.2. Case of a single sinusoid 1.2.3. Case of a periodic signal 1.2.4. General case 1.3. Extreme response spectrum of a swept sine vibration 1.3.1. Sinusoid of constant amplitude throughout the sweeping process 1.3.2. Swept sine composed of several constant levels Chapter 2. Extreme Response Spectrum of a Random Vibration 2.1. Unspecified vibratory signal 2.2. Gaussian stationary random signal 2.2.1. Calculation from peak distribution 2.2.2. Use of the largest peak distribution law 2.2.3. Response spectrum defined by k times the rms response 2.2.4. Other ERS calculation methods 2.3. Limit of the ERS at the high frequencies 2.4. Response spectrum with up-crossing risk 2.4.1. Complete expression 2.4.2. Approximate relation 2.4.3. Calculation in a hypothesis of independence of threshold overshoot 2.4.4. Use of URS 2.5. Comparison of the various formulae 2.6. Effects of peak truncation on the acceleration time history
Mechanical Vibration and Shock Analysis
2.6.1. Extreme response spectra calculated from the time history signal 2.6.2. Extreme response spectra calculated from the power spectral densities 2.6.3. Comparison of extreme response spectra calculated from time history signals and power spectral densities 2.7. Sinusoidal vibration superimposed on a broad band random vibration 2.7.1. Real environment 2.7.2. Case of a single sinusoid superimposed to a wide band noise 2.7.3. Case of several sinusoidal lines superimposed on a broad band random vibration 2.8. Swept sine superimposed on a broad band random vibration 2.8.1. Real environment 2.8.2. Case of a single swept sine superimposed to a wide band noise 2.8.3. Case of several swept sines superimposed on a broad band random vibration 2.9. Swept narrow bands on a wide band random vibration 2.9.1. Real environment 2.9.2. Extreme response spectrum Chapter 3. Fatigue Damage Spectrum of a Sinusoidal Vibration 3.1. Fatigue damage spectrum definition 3.2. Fatigue damage spectrum of a single sinusoid 3.3. Fatigue damage spectrum of a periodic signal 3.4. General expression for the damage 3.5. Fatigue damage with other assumptions on the S–N curve 3.5.1. Taking account of fatigue limit 3.5.2. Cases where the S–N curve is approximated by a straight line in log-lin scales 3.5.3. Comparison of the damage when the S–N curves are linear in either log-log or log-lin scales 3.6. Fatigue damage generated by a swept sine vibration on a single degree-of-freedom linear system 3.6.1. General case 3.6.2. Linear sweep 3.6.3. Logarithmic sweep 3.6.4. Hyperbolic sweep 3.6.5. General expressions for fatigue damage 3.7. Reduction of test time 3.7.1. Fatigue damage equivalence in the case of a linear system 3.7.2. Method based on fatigue damage equivalence according to Basquin’s relationship 3.8. Remarks on the design assumptions of the ERS and FDS
Summary of Volume 5
Chapter 4. Fatigue Damage Spectrum of a Random Vibration 4.1. Fatigue damage spectrum from the signal as function of time 4.2. Fatigue damage spectrum derived from a power spectral density 4.3. Simplified hypothesis of Rayleigh’s law 4.4. Calculation of the fatigue damage spectrum with Dirlik’s probability density 4.5. Reduction of test time 4.5.1. Fatigue damage equivalence in the case of a linear system 4.5.2. Method based on a fatigue damage equivalence according to Basquin’s relationship taking account of variation of natural damping as a function of stress level 4.6. Truncation of the peaks of the ‘input’ acceleration signal 4.6.1. Fatigue damage spectra calculated from a signal as a function of time 4.6.2. Fatigue damage spectra calculated from power spectral densities 4.6.3. Comparison of fatigue damage spectra calculated from signals as a function of time and power spectral densities 4.7. Sinusoidal vibration superimposed on a broad band random vibration 4.7.1. Case of a single sinusoidal vibration superimposed on broad band random vibration 4.7.2. Case of several sinusoidal vibrations superimposed on a broad band random vibration 4.8. Swept sine superimposed on a broad band random vibration 4.8.1. Case of one swept sine superimposed on a broad band random vibration 4.8.2. Case of several swept sines superimposed on a broad band random vibration 4.9. Swept narrow bands on a broad band random vibration Chapter 5. Fatigue Damage Spectrum of a Shock 5.1. General relationship of fatigue damage 5.2. Use of shock response spectrum in the impulse zone 5.3. Damage created by simple shocks in static zone of the response spectrum Chapter 6. Influence of Calculation Conditions of ERSs and FDSs 6.1. Variation of the ERS with amplitude and vibration duration 6.2. Variation of the FDS with amplitude and duration of vibration 6.3. Should ERSs and FDSs be drawn with a linear or logarithmic frequency step? 6.4. With how many points must ERSs and FDSs be calculated? 6.5. Difference between ERSs and FDSs calculated from a vibratory signal according to time and from its PSD
Mechanical Vibration and Shock Analysis
6.6. Influence of the number of PSD calculation points on ERS and FDS 6.7. Influence of the PSD statistical error on ERS and FDS 6.8. Influence of the sampling frequency during ERS and FDS calculation from a signal based on time 6.9. Influence of the peak counting method 6.10. Influence of a non-zero mean stress on FDS Chapter 7. Tests and Standards 7.1. Definitions 7.1.1. Standard 7.1.2. Specification 7.2. Types of tests 7.2.1. Characterization test 7.2.2. Identification test 7.2.3. Evaluation test 7.2.4. Final adjustment/development test 7.2.5. Prototype test 7.2.6. Pre-qualification (or evaluation) test 7.2.7. Qualification 7.2.8. Qualification test 7.2.9. Certification 7.2.10. Certification test 7.2.11. Stress screening test 7.2.12. Acceptance or reception 7.2.13. Reception test 7.2.14. Qualification/acceptance test 7.2.15. Series test 7.2.16. Sampling test 7.2.17. Reliability test 7.3. What can be expected from a test specification? 7. 4. Specification types 7.4.1. Specification requiring in situ testing 7.4.2. Specifications derived from standards 7.4.3. Current trend 7.4.4. Specifications based on real environment data 7.5. Standards specifying test tailoring 7.5.1. The MIL–STD 810 standard 7.5.2. The GAM.EG 13 standard 7.5.3. STANAG 4370 7.5.4. The AFNOR X50–410 standard
Summary of Volume 5
Chapter 8. Uncertainty Factor 8.1. Need – definitions 8.2. Sources of uncertainty 8.3. Statistical aspect of the real environment and of material strength 8.3.1. Real environment 8.3.2. Material strength 8.4. Statistical uncertainty factor 8.4.1. Definitions 8.4.2. Calculation of uncertainty factor 8.4.3. Calculation of an uncertainty coefficient when the real environment is only characterized by a single value Chapter 9. Aging Factor 9.1. Purpose of the aging factor 9.2. Aging functions used in reliability 9.3. Method for calculating aging factor 9.4. Influence of standard deviation of the aging law 9.5. Influence of the aging law mean Chapter 10. Test Factor 10.1. Philosophy 10.2. Calculation of test factor 10.2.1. Normal distributions 10.2.2. Log-normal distributions 10.2.3. Weibull distributions 10.3. Choice of confidence level 10.4. Influence of the number of tests n Chapter 11. Specification Development 11.1. Test tailoring 11.2. Step 1: Analysis of the life cycle profile. Review of the situations 11.3. Step 2: Determination of the real environmental data associated with each situation 11.4. Step 3: Determination of the environment to be simulated 11.4.1. Need 11.4.2. Synopsis methods 11.4.3. The need for a reliable method 11.4.4. Synopsis method using power spectrum density envelope 11.4.5. Equivalence method of extreme response and fatigue damage 11.4.6. Synopsis of the real environment associated with an event (or sub-situation)
Mechanical Vibration and Shock Analysis
11.4.7. Synopsis of a situation 11.4.8. Synopsis of all life profile situations 11.4.9. Search for a random vibration of equal severity 11.4.10. Validation of duration reduction 11.5. Step 4: Establishment of the test program 11.5.1. Application of a test factor 11.5.2. Choice of the test chronology 11.6. Applying this method to the example of the ‘round robin’ comparative study 11.7. Taking environment into account in project management Chapter 12. Influence of Calculation Conditions of Specification 12.1. Choice of the number of points in the specification (PSD) 12.2. Influence of Q factor on specification (outside of time reduction) 12.3. Influence of Q factor on specification when duration is reduced 12.4. Validity of a specification established for Q factor equal to 10 when the real structure has another value 12.5. Advantage in the consideration of a variable Q factor for the calculation of ERSs and FDSs 12.6. Influence of the value of parameter b on the specification 12.6.1. Case where test duration is equal to real environment duration 12.6.2. Case where duration is reduced 12.7. Choice of the value of parameter b in the case of material made up of several components. 12.8. Influence of temperature on parameter b and constant C 12.9. Importance of a factor of 10 between the specification FDS and the reference FDS (real environment) in a small frequency band 12.10. Validity of a specification established by reference to a 1 dof system when real structures are multi dof systems Chapter 13. Other Uses of Extreme Response Up-Crossing Risk and Fatigue Damage Spectra 13.1. Comparisons of the severity of different vibrations 13.1.1. Comparisons of the relative severity of several real environments 13.1.2. Comparison of the severity of two standards 13.1.3. Comparison of seism severity 13.2. Swept sine excitation – random vibration transformation 13.3. Definition of a random vibration with the same severity as a series of shocks
Summary of Volume 5
13.4. Writing a specification only from an ERS (or a URS) 13.4.1. Matrix inversion method 13.4.2. Method by iteration 13.5. Establishment of a swept sine vibration specification Appendices Formulae Index
