Structural Dynamics in Industry

Structural Dynamics in Industry Structural Dynamics in Industry Alain Girard and Nicolas Roy First published in Fr...

Author: Alain Girard | Nicolas Roy

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Structural Dynamics in Industry

Structural Dynamics in Industry

Alain Girard and Nicolas Roy

First published in France in 2003 by Hermes Science/Lavoisier entitled “Dynamique des structures industrielles” First published in Great Britain and the United States in 2008 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 6 Fitzroy Square London W1T 5DX UK

John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA

www.iste.co.uk

www.wiley.com

© ISTE Ltd, 2008 © LAVOISIER, 2003 The rights of Alain Girard and Nicolas Roy to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Cataloging-in-Publication Data Girard, Alain. [Dynamique des structures industrielles English] Structural dynamics in industry / Alain Girard, and Nicolas Roy. p. cm. "First published in France in 2003 by Hermes Science/Lavoisier entitled "Dynamique des structures industrielles"." Includes index. ISBN: 978-1-84821-004-2 1. Structural dynamics--Mathematical models. 2. Functional analysis. 3. Industrial buildings. I. Roy, Nicolas. II. Title. TA654.G5713 2008 624.1'7--dc22 2006033667 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN: 978-1-84821-004-2 Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wiltshire.

Table of Contents

Foreword. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv

Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xvii

Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiii

Chapter 1. General Introduction to Linear Analysis . . . . . . . . . . . . . . 1.1. Introduction. . . . . . . . . . . . . . . . 1.2. Motion types . . . . . . . . . . . . . . . 1.2.1. Sine motion . . . . . . . . . . . . . 1.2.1.1. Pure sine . . . . . . . . . . . . . 1.2.1.2. Swept sine . . . . . . . . . . . . 1.2.1.3. Periodic motion . . . . . . . . . 1.2.2. Transient motion . . . . . . . . . . 1.2.3. Random motion . . . . . . . . . . . 1.2.3.1. Random process. . . . . . . . . 1.2.3.2. Time analysis . . . . . . . . . . 1.2.3.3. Statistical analysis . . . . . . . 1.2.3.4. Power spectral densities . . . . 1.3. Time domain and frequency domain . 1.3.1. Introduction . . . . . . . . . . . . . 1.3.2. The time domain . . . . . . . . . . 1.3.3. The frequency domain . . . . . . .

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1 1 2 2 2 4 5 5 7 7 8 9 12 14 14 15 16

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1.4. Frequency Response Functions . . . . . . . . . . . . 1.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . 1.4.2. Frequency Response Functions and responses . 1.4.3. The nature of excitations and responses. . . . . 1.4.4. The nature of Frequency Response Functions . 1.5. Equations of motion and solution . . . . . . . . . . . 1.5.1. Equations of motion . . . . . . . . . . . . . . . . 1.5.2. Solution using the direct frequency approach . 1.5.3. Solution using the modal approach . . . . . . . 1.5.4. Modes and 1-DOF system. . . . . . . . . . . . . 1.6. Analysis and tests . . . . . . . . . . . . . . . . . . . .

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17 17 18 19 21 24 24 26 27 28 29

Chapter 2. The Single-Degree-of-Freedom System . . . . . . . . . . . . . . .

33

2.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. The equation of motion and the solution in the frequency domain 2.2.1. Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Motion without excitation . . . . . . . . . . . . . . . . . . . . . . 2.2.2.1. The conservative system. . . . . . . . . . . . . . . . . . . . . 2.2.2.2. Dissipative system . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Solution in the frequency domain . . . . . . . . . . . . . . . . . 2.2.4. Dynamic amplifications . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. Response to a random excitation . . . . . . . . . . . . . . . . . . 2.3. Time responses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1. Response to unit impulse . . . . . . . . . . . . . . . . . . . . . . 2.3.2. Response to a general excitation . . . . . . . . . . . . . . . . . . 2.3.3. Response spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Representation of the damping . . . . . . . . . . . . . . . . . . . . . 2.4.1. Viscous damping . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2. Structural damping . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3. Other representations. . . . . . . . . . . . . . . . . . . . . . . . .

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33 35 35 35 35 37 39 43 49 51 51 55 56 61 61 62 64

Chapter 3. Multiple-Degree-of-Freedom Systems . . . . . . . . . . . . . . . .

65

3.1. Introduction. . . . . . . . . . . . . . . . . . . . . . 3.2. Determining the structural matrices . . . . . . . 3.2.1. Introduction . . . . . . . . . . . . . . . . . . . 3.2.2. Local element matrices . . . . . . . . . . . . 3.2.3. Element matrices in global reference form . 3.2.4. Assembly of element matrices . . . . . . . . 3.2.5. Linear constraints between DOF . . . . . . . 3.2.5.1. Introduction . . . . . . . . . . . . . . . . .

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3.2.5.2. DOF elimination . . . . . . . . . . . . . . . . . . 3.2.5.3. DOF introduction . . . . . . . . . . . . . . . . . . 3.2.6. Excitation forces . . . . . . . . . . . . . . . . . . . . 3.3. The finite element method . . . . . . . . . . . . . . . . . 3.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. The rod element . . . . . . . . . . . . . . . . . . . . . 3.3.3. Beam finite element in bending . . . . . . . . . . . 3.3.4. The complete beam finite element . . . . . . . . . . 3.3.5. Excitation forces . . . . . . . . . . . . . . . . . . . . 3.4. Industrial models . . . . . . . . . . . . . . . . . . . . . . 3.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.4.2. The element types . . . . . . . . . . . . . . . . . . . 3.4.3. Linear constraints . . . . . . . . . . . . . . . . . . . . 3.4.4. DOF management . . . . . . . . . . . . . . . . . . . 3.4.5. Rules for modeling and verification of the model . 3.4.6. Industrial examples . . . . . . . . . . . . . . . . . . . 3.5. Solution by direct integration . . . . . . . . . . . . . . . 3.5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 3.5.2. Example of explicit method . . . . . . . . . . . . . . 3.5.3. Example of implicit method. . . . . . . . . . . . . .

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73 77 79 80 80 82 83 86 88 89 89 89 91 91 93 94 95 95 96 97

Chapter 4. The Modal Approach. . . . . . . . . . . . . . . . . . . . . . . . . . .

99

4.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Normal modes . . . . . . . . . . . . . . . . . . . . . . . 4.2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 4.2.2. Free structures. . . . . . . . . . . . . . . . . . . . . 4.2.3. System static condensation . . . . . . . . . . . . . 4.2.4. Eigenvalue problem solution . . . . . . . . . . . . 4.3. Mode superposition . . . . . . . . . . . . . . . . . . . . 4.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 4.3.2. Equation of motion transformation . . . . . . . . 4.3.3. Problem caused by the damping . . . . . . . . . . 4.3.4. Frequency resolution . . . . . . . . . . . . . . . . . 4.4. From the frequency approach to the modal approach

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Chapter 5. Modal Effective Parameters . . . . . . . . . . . . . . . . . . . . . . . . . .

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5.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . 5.2. Effective modal parameters and truncation . . . . 5.2.1. Definition of the effective modal parameters 5.2.2. Summation rules . . . . . . . . . . . . . . . . .

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5.2.2.1. Direct summation . . . . . . . . . . . . . . . . . . . 5.2.2.2. Flexibilities in the presence of rigid modes. . . . 5.2.2.3. Transmissibilities and effective masses by zones 5.2.2.4. Other summation rules . . . . . . . . . . . . . . . . 5.2.3. Correction of the truncation effects . . . . . . . . . . 5.3. Particular case of a statically determined structure. . . . 5.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2. Effective mass models . . . . . . . . . . . . . . . . . . 5.4. Modal effective parameters and dynamic responses . . . 5.4.1. Frequency responses . . . . . . . . . . . . . . . . . . . 5.4.2. Random responses . . . . . . . . . . . . . . . . . . . . 5.4.3. Time responses . . . . . . . . . . . . . . . . . . . . . . 5.4.4. Time response extrema . . . . . . . . . . . . . . . . . 5.5. Industrial examples . . . . . . . . . . . . . . . . . . . . . .

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133 134 137 139 140 143 143 145 153 153 157 159 159 161

Chapter 6. Continuous Systems . . . . . . . . . . . . . . . . . . . . . . . . . . .

169

6.1. Introduction. . . . . . . . . . . . . . . . . . . . . . 6.2. The rod element . . . . . . . . . . . . . . . . . . . 6.2.1. Introduction . . . . . . . . . . . . . . . . . . . 6.2.2. Clamped-free rod . . . . . . . . . . . . . . . . 6.2.3. Free-free rod . . . . . . . . . . . . . . . . . . . 6.2.4. Clamped-clamped rod . . . . . . . . . . . . . 6.3. Bending beam element . . . . . . . . . . . . . . . 6.3.1. Introduction . . . . . . . . . . . . . . . . . . . 6.3.2. Clamped-free beam. . . . . . . . . . . . . . . 6.3.3. Free-free beam . . . . . . . . . . . . . . . . . 6.3.4. Clamped-clamped beam . . . . . . . . . . . . 6.3.5. Shear and rotary inertia effects . . . . . . . . 6.4. Plate element . . . . . . . . . . . . . . . . . . . . . 6.4.1. Introduction . . . . . . . . . . . . . . . . . . . 6.4.2. Some plate results in bending. . . . . . . . . 6.4.3. Simply supported rectangular plate . . . . . 6.5. Combined cases . . . . . . . . . . . . . . . . . . . 6.5.1. Introduction . . . . . . . . . . . . . . . . . . . 6.5.2. Combination rod + local mass or flexibility 6.5.3. Some typical results . . . . . . . . . . . . . .

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169 171 171 173 178 182 184 184 188 193 199 204 206 206 207 208 210 210 213 215

Table of Contents

Chapter 7. Complex Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Dissipative systems . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1. Complex modes . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2. Mode superposition. . . . . . . . . . . . . . . . . . . . . . . 7.2.3. Modal effective parameters and dynamic amplifications. 7.2.4. Simple example . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Gyroscopic effects. . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2. Mode superposition. . . . . . . . . . . . . . . . . . . . . . . 7.4. A more general case. . . . . . . . . . . . . . . . . . . . . . . . . 7.4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Complex modes . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3. Mode superposition. . . . . . . . . . . . . . . . . . . . . . . 7.4.4. Modal effective parameters and dynamic amplifications. 7.5. Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Simple example . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2. Industrial case . . . . . . . . . . . . . . . . . . . . . . . . . .

219 220 220 224 226 229 232 232 234 236 236 237 240 242 245 245 248

Chapter 8. Modal Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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8.1. Introduction. . . . . . . . . . . . 8.2. General approach . . . . . . . . 8.2.1. Analysis of substructures . 8.2.2. Coupling of substructures . 8.2.3. Recovery . . . . . . . . . . . 8.3. Choice of mode . . . . . . . . . 8.3.1. Introduction . . . . . . . . . 8.3.2. Boundary conditions . . . . 8.3.3. Normal modes . . . . . . . . 8.3.4. Static flexibilities . . . . . . 8.3.5. Junction modes . . . . . . . 8.3.6. Illustration . . . . . . . . . . 8.3.7. Possible combinations . . . 8.4. Some methods . . . . . . . . . . 8.4.1. Craig-Bampton method . . 8.4.2. Craig-Chang method . . . . 8.4.3. Benfield-Hruda method . . 8.4.4. Effective mass models . . . 8.4.5. Reduced models. . . . . . . 8.5. Case study . . . . . . . . . . . . 8.5.1. Benfield-Hruda truss . . . . 8.5.2. Industrial cases . . . . . . .

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ix

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249 251 251 253 255 256 256 258 259 260 262 263 265 266 266 271 276 281 283 287 287 290

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Chapter 9. Frequency Response Synthesis . . . . . . . . . . . . . . . . . . . . . 9.1. Introduction. . . . . . . . . . . . . . . . . . . . . . . 9.2. Frequency Response Functions . . . . . . . . . . . 9.2.1. FRF and other dynamic characteristics . . . . 9.2.2. Transformation of the FRF . . . . . . . . . . . 9.2.3. Simple examples . . . . . . . . . . . . . . . . . 9.3. Coupling by FRF . . . . . . . . . . . . . . . . . . . 9.3.1. FRF necessary for coupling . . . . . . . . . . . 9.3.2. Solution of the coupling . . . . . . . . . . . . . 9.3.3. Recovery . . . . . . . . . . . . . . . . . . . . . . 9.3.4. Summary . . . . . . . . . . . . . . . . . . . . . . 9.4. The basic cases. . . . . . . . . . . . . . . . . . . . . 9.4.1. Introduction . . . . . . . . . . . . . . . . . . . . 9.4.2. Free substructures at the connections . . . . . 9.4.3. Substructures constrained at the connections. 9.4.4. Mixed conditions at the connections. . . . . . 9.5. Generalization . . . . . . . . . . . . . . . . . . . . . 9.5.1. Introduction . . . . . . . . . . . . . . . . . . . . 9.5.2. Stiffness approach . . . . . . . . . . . . . . . . 9.5.3. Flexibility approach . . . . . . . . . . . . . . . 9.5.4. Comparison of the two approaches . . . . . . 9.5.5. Particular cases . . . . . . . . . . . . . . . . . . 9.6. Comparison with other substructuring techniques 9.6.1. The matrix level. . . . . . . . . . . . . . . . . . 9.6.2. The modal level . . . . . . . . . . . . . . . . . . 9.6.3. The frequency response level . . . . . . . . . . 9.6.4. Conclusion . . . . . . . . . . . . . . . . . . . . .

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295 296 296 298 299 301 301 303 304 305 306 306 306 308 309 310 310 311 312 314 317 318 318 319 320 321

Chapter 10. Introduction to Non-linear Analysis. . . . . . . . . . . . . . . . .

323

10.1. Introduction . . . . . . . . . . . . . . . . . . . . 10.2. Non-linear systems. . . . . . . . . . . . . . . . 10.2.1. Introduction . . . . . . . . . . . . . . . . . 10.2.2. Simple examples of large displacements 10.2.3. Simple example of variable link . . . . . 10.2.4. Simple example of dry friction . . . . . . 10.2.5. Material non-linearities. . . . . . . . . . . 10.3. Non-linear 1-DOF system . . . . . . . . . . . 10.3.1. Introduction . . . . . . . . . . . . . . . . . 10.3.2. Undamped motion without excitation . . 10.3.3. Case of a stiffness of form k (1 + µ x 2 ) . 10.3.4. Undamped motion with excitation . . . .

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295

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323 324 324 326 328 328 329 329 329 331 332 336

Table of Contents

10.3.5. Damped motion with excitation. . . . 10.4. Non-linear N-DOF systems . . . . . . . . 10.4.1. Introduction . . . . . . . . . . . . . . . 10.4.2. Non-linear link with periodic motion 10.4.3. Direct integration of equations . . . .

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340 343 343 344 346

Chapter 11. Testing Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . .

349

11.1. Introduction . . . . . . . . . . . . . . . . . . . . . . 11.2. Dynamic tests . . . . . . . . . . . . . . . . . . . . . 11.2.1. Development plan of a structure . . . . . . . 11.2.2. Types of tests . . . . . . . . . . . . . . . . . . 11.2.3. Test hardware . . . . . . . . . . . . . . . . . . 11.3. The identification tests . . . . . . . . . . . . . . . 11.3.1. Introduction . . . . . . . . . . . . . . . . . . . 11.3.2. Modal parameters to be identified . . . . . . 11.3.3. Phase resonance modal tests. . . . . . . . . . 11.3.4. Phase separation modal tests . . . . . . . . . 11.3.5. Extraction of modal parameters. . . . . . . . 11.3.6. Single DOF (SDOF) methods . . . . . . . . . 11.3.7. Multi-DOF (MDOF) methods. . . . . . . . . 11.4. Simulation tests. . . . . . . . . . . . . . . . . . . . 11.4.1. Introduction . . . . . . . . . . . . . . . . . . . 11.4.2. Tests with shakers. . . . . . . . . . . . . . . . 11.4.3. Shock device tests. . . . . . . . . . . . . . . . 11.4.4. The tests in a reverberant acoustic chamber 11.4.5. Elaboration of specifications . . . . . . . . . 11.4.6. Impact of a structure on its environment . .

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385

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Chapter 12. Model Updating and Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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349 350 350 352 353 358 358 359 362 364 366 368 370 372 372 373 375 376 377 379

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12.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . 12.2. Sensitivity analysis. . . . . . . . . . . . . . . . . . . 12.2.1. Introduction . . . . . . . . . . . . . . . . . . . . 12.2.2. Sensitivity of the natural frequencies . . . . . 12.2.3. Sensitivity of the eigenvectors . . . . . . . . . 12.2.4. Sensitivity of the modal effective parameters 12.2.5. Simple example . . . . . . . . . . . . . . . . . . 12.3. Ritz reanalysis . . . . . . . . . . . . . . . . . . . . . 12.3.1. Introduction . . . . . . . . . . . . . . . . . . . . 12.3.2. Utilization of the normal modes . . . . . . . . 12.3.3. Utilization of additional modes . . . . . . . . .

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xi

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385 387 387 388 388 389 390 392 392 392 393

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12.3.4. Simple example . . . . . . . . . . . 12.4. Model updating. . . . . . . . . . . . . . 12.4.1. Physical parameters . . . . . . . . . 12.4.2. Test/analysis correlation . . . . . . 12.4.3. Updating procedure . . . . . . . . . 12.5. Optimization processes . . . . . . . . . 12.5.1. Introduction . . . . . . . . . . . . . 12.5.2. Non-linear optimization methods. 12.5.3. Non-linear simplex method . . . . 12.6. Applications. . . . . . . . . . . . . . . . 12.6.1. Optimization of a simple system . 12.6.2. Updating a simple system . . . . . 12.6.3. Industrial case . . . . . . . . . . . .

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393 395 395 398 400 401 401 402 403 404 404 405 407

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

411

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

417

Foreword

When Alain Girard and Nicolas Roy told me of their intention of writing a book on structural dynamics in industry, I was instantly won over by their initiative. In fact, experience has shown me that the essential theory-practice connection is the subject of so few works that I could only support the initiative of two experts wanting to impart their knowledge in a pragmatic way. Having had a chance to appreciate the relevance of their analyses and expertise, I am convinced that the reader will find in this book concrete answers to multiple problems, and at the very least the elements that will enable them to make progress notably in their methodology. I wish a long life to this book which illustrates so well our INTESPACE slogan “Environmental intelligence” with “The Intelligence of Structural Dynamics in Industry” that it brings.

Jean-Louis MARCÉ President General Manager

Preface

Structural dynamics has become increasingly important in different fields such as automotive, aeronautics or space, where the need for performance is everincreasing. This expansion was encouraged by the capability of computers to simulate phenomena through increasingly large models. However, to control these phenomena, we must first understand them. This book sets out to explain basic notions with a methodic approach to help improve this comprehension and to deal with industrial structures without becoming too specific. The developments are outlined with minimum mathematics and are often illustrated by simple examples before moving on to cases taken from industrial reality. This book is addressed to the student or the researcher wanting to organize and clarify their knowledge in this field, as well as to the engineer concerned with bringing a practical response to problems they have to solve. This field is vast; consequently, certain related aspects are limited to a relatively quick presentation and readers are referred to more detailed works, in order to refocus on the most fundamental principles. This is also the case, to differing degrees, with numerical analysis, finite elements, fluid-structure coupling, high frequency, non-linear, fatigue or experimental techniques. The content is the result of the authors’ experience acquired through their involvement with research and industrial activities mainly at the Centre National d'Etudes Spatiales and INTESPACE Test Center, as well as from teaching, specifically at the engineering schools SUPAÉRO, ENSICA and internships.

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The authors wish to thank all who have contributed in one way or another to the achievement of this work, and in particular: – Jean-Louis Marcé, President General Manager of INTESPACE, and LouisPatrice Bugeat, Head of Systems and Expertise Department , for providing a favorable environment for their work; – all the colleagues who, through discussions, developments or implementations, have helped the subject move forward, especially Jacky Chatelain and Paul-Eric Dupuis; – Jean-François Imbert, former Structures Department Manager at CNES then Assistant Director for Studies and Engineering of INTESPACE, who has provided the impetus and the guidance for these activities with proper judgment; – Jean-Noël Bricout, from CNES, who gave the manuscript a meticulous rereading; – and finally, all departments from the different companies that have encouraged research in this field pushing back the limits: in particular, ALCATEL/Cannes, ASTRIUM/Toulouse, BMW AG Dept EK-213, CNES/Toulouse, DGA/DCE/Toulon, EDF/DER/Clamart, ESA/ESTEC/Noordwijk, PSA PEUGEOT CITROEN, SNECMA PROPULSION SOLIDE/Le Haillan.

Introduction

The title of this book is derived from three key words that should be explained in order to clarify the context: – Structures: structures are referred to here as supports for objects of any nature, for example a mast, bridge, building, car body, a plane’s fuselage, etc. We encounter them in all fields and at all levels since the supported objects can in turn serve as support for smaller objects. A launcher then has a structure supporting a payload which in turn has a structure supporting equipment with a structure supporting circuit boards, etc. In all these structures, the main mission is to resist their environment. – Dynamic: structural dynamics is the study of structures subjected to a mechanical environment which depends on time and leading to a movement. We can compare dynamics to statics where the environment does not depend on time, such as gravity or constant pressure. We can also consider statics as a particular aspect of dynamics where the frequency of movement is zero. In this regard, these two disciplines connect in the study of structures. In statics, the idea is mainly to verify that movements or stresses remain within acceptable limits. In dynamics, we must first study the movement which, in certain conditions, can lead to much higher levels than those from a static behavior. Prediction of these phenomena must be controlled in order to understand, remedy and optimize. – Industrial: structures considered will be industrial in the sense that they will not be limited to academic cases. The idea is of course not to oppose these two domains, but to make them complementary. A simple example will enable us to understand and make basic calculations which could be sufficient in certain cases. The complex application processed by a computer will illustrate possibilities in concrete cases. Another aspect of industrial structures is that prediction can rest on two types of activities: analysis and test. To ignore one in favor of the other is dangerous: only relying on analysis is not reliable enough, and relying only on

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testing only shows a partial view of the situation. These two worlds must be perfectly interlinked in the development of a structure, with correlation and mathematical model updating phases in relation to tests. Structural dynamics in industry will be addressed as methodically as possible starting with basic notions, in order to pave the way for handling more complex cases. The starting point is a dynamic environment imposed on the structure. This excitation may take on very different forms that we must categorize in order to complete an analysis adapted to the situation. We can make the following distinctions: – concerning the excitation transmission type: this transmission can be: - mechanical, i.e. using the solid path: the structure is excited by contact with a moving adjacent structure, leading to localized or distributed forces, such as a car body excited by motor vibrations; - acoustic, using the airborne path or, more generally, that of any fluid: the structure is excited by a pressure field exerted on all exposed surfaces, for example, the same body excited by aerodynamic turbulence. This case involves two completely different environments, structure and fluid, each requiring specific techniques. These two transmissions often coexist: besides the previous example, a typical case is the satellite on its launcher, excited by its mechanical interface and its surfaces exposed to acoustic field under the fairing. This environment is called vibroacoustics; – concerning the type of time functions: it may be: - sinusoidal: this particular case is very important because of the movement amplifications that it can generate. It also represents a basic case which helps to solve the general case that we will see later; - transient: actually, any real excitation can be qualified as transient, because everything has a beginning and an end. However, we generally reserve the word “transient” for a relatively short excitation. This can be from a few milliseconds for pyrotechnic events, for example, to several seconds for seismic events. For very short durations, we talk about shock. For a relatively longer duration, the deterministic analysis can become awkward, in which case a possible solution is illustrated by the following characteristic; - random (or stochastic): this is the majority of actual excitations resulting from various and generally independent causes. It will be advantageous to work with them in a probabilistic manner by statistical analysis, in order to bring out the movement’s major properties;

Introduction

xix

– Concerning the type of frequencies involved: this aspect is particularly important. Any function of time, under certain conditions, may be decomposed into a sum of sinusoids: the decomposition given by its Fourier transform. We then have a description of the excitation in the field of frequencies and its content can be: - low frequency; or - broadband (low frequency + high frequency); in relation to the structure involved. A low frequency excitation will generate relatively simple shaped responses on the structure, characterized by wavelengths similar to its size, making analysis of movement easier, contrary to a higher frequency. To be more precise, low frequency is the domain which only involves a limited number of normal modes, stationary shapes related to frequencies that we will explain in more detail later and which enable efficient analysis. At high frequency, these normal modes are numerous, complex and less significant individually, which requires the use of appropriate analysis techniques. This is the case with vibroacoustic environments frequently generating broadband random vibrations, and shock environments where phenomena are often more propagative than stationary. Concerning analysis, we mainly focus here on low frequency. High frequency techniques come from a slightly different world, which is more difficult to master and which will not be explained here. However, the different functions of time will all be addressed since they all end up using the same solution approach. Finally, concerning types of transmission, acoustic excitation will be considered as long as it can be represented by a mechanical excitation by transforming pressures into known forces: fluid-structure coupling will therefore not be discussed as such. Next, excited by its dynamic environment, a structure responds. There again, analysis will depend on several factors: – concerning the nature of structural behavior: it can be linear or proportional to excitation, or non-linear for different reasons. In this last case, the analysis is much more difficult, unless we can linearize it. In the linear case, equations of motion can be easily integrated but in a rather inefficient way; – at low frequency, as indicated above, the notion of normal mode enables an efficient analysis technique called mode superposition. Each mode, in fact, behaves as a “spring-mass” system, or single-degree-of-freedom system, i.e. the simplest dynamic system, whose state only depends on one parameter. It is therefore doubly interesting to start by analyzing this system: its simplicity makes it possible to understand basic phenomena well and it prepares for more complex systems, at N degrees-of-freedom, with the help of their normal modes. In addition, it can be used as a structure of reference for environmental characterization;

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– concerning the domain of analysis: we can work by remaining in the time domain, or we can move to the frequency domain. These two worlds communicate with the help of Fourier transform and its inverse. In the frequency domain, relations between excitations and responses are commonly called frequency response functions (FRF) and are easily handled, which has many advantages; – concerning structure representation with a mathematical model: this can be continuous or discrete. Any real structure is continuous (at the macroscopic level) and the equations controlling its movement are initially partial differential equations. The difficulty in resolving them analytically in the case of complex structures might lead us to discretize them, or in other words to describe their state by a finite number of parameters, or degrees of freedom, for example by the method of finite elements. In this case, equations are expressed using matrices, and the size of matrices directly depends on the discretization involved. The continuous approach applies to simple cases which can be used as a reference when needed, for example, to evaluate discretization errors; – if the structure is modular, or in other words if it is made up of distinct parts connected together by simple interfaces, we could be tempted to first analyze each part before assembling them: this technique, called substructuring, has several advantages from a time of calculation point of view but also with organization. It can be executed at different levels by matrix assembly, in particular with FRF for frequency response synthesis and for normal modes for modal synthesis; – if there is room for maneuver in the definition of structure, we may want to optimize it on dynamic criteria, for example, maximizing natural frequencies or minimizing certain responses to a given excitation. In addition, if test results are available, they must be compared to results from the model and the model must be modified by an operation called updating to make it more representative by minimizing the gap between results. These two problems use the same optimization techniques. We mainly focus here on linear analysis; non-linear analysis being presented in an introductory chapter. The modal approach to discrete systems will be discussed in more detail to better use its advantages for low frequency, once we have examined the single-degree-of-freedom system, which is the keystone of this analysis. Continuous systems will follow to provide some references, and then complex modes to take certain phenomena into consideration in a more effective manner. In light of these developments, substructuring, optimization and updating in relation to tests will be addressed, bringing responses adapted to generally expressed needs in the industry. We will complete these analysis techniques with a description of the different test techniques, with the goal of identifying dynamic properties or of simulating an environment.

Introduction

xxi

The different types of analysis that we have identified are summarized in Figure 1 and will be developed in the following chapters: 1) General points on linear analysis techniques, explaining the previous comments, which should be read first to provide an overview of the subject. 2) The single-degree-of-freedom system, the keystone of this analysis: equations on movement and resolution. Responses, response spectrums for environment characterization. 3) N-degree-of-freedom systems: equations of motion. Determination of structural matrices. The finite element method. Resolution by direct integration. 4) Modal approach to discrete systems: real and complex modes. Mode superposition techniques with real modes for efficient calculation of low frequency responses. 5) Modal effective parameters: introduced in the previous chapter, representing essential information for understanding the phenomena and mastering the behavior. 6) Continuous systems, the limiting case of discrete systems: analysis techniques. Processing simple cases which can serve as references. 7) Complex modes approach, for a better consideration of certain phenomena: dissipative systems, gyroscopic effects, general cases which can be handled by mode superposition. 8) Substructuring by modal synthesis: representation of each substructure by its normal modes in a given configuration. Formulation based on selected configurations. 9) Substructuring by frequency response synthesis: representation of each substructure by its FRF. Complementarity with modal synthesis. 10) An introduction to nonlinear analysis: sources of nonlinearities and simple examples. The case of the single-degree-of-freedom system. Some analysis approaches for multiple-degree-of-freedom systems. 11) Testing techniques: tests in structure development. Identification of dynamic properties. Environment simulation. 12) Model updating and optimization: test/analysis comparison, model reanalysis and updating. Optimization techniques.

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linearization

LINEAR PHENOMENON

DETERMINISTIC

RANDOM

STANDARD ANALYSIS

STOCHASTIC ANALYSIS

TIME ANALYSIS

FREQUENCY ANALYSIS

LOW FREQUENCY

BROADBAND

MODAL APPROACH

DIRECT RESOLUTION

SUBSTRUCTURING

RESPONSES

NON-LINEAR PHENOMENON

TEST RESULTS

CORRELATION UPDATING

OPTIMIZATION

Figure 1. Major types of analyses

Glossary

Abbreviations DOF FRF PSD

Degree-of-freedom Frequency response function Power spectral density

Convention for matrices In order to efficiently develop a formulation implying a certain formalism with matrices, we will use notations where subscripts play an important role: first mnemonics (whenever possible), second, avoiding multiplicity of matrix notations, and finally coherence in matrix products by subscript ordering. Matrices introduced in the different chapters represent properties relative to DOF, or in a broad sense, parameters defining the state of the structure involved. By general convention, X ij designates a matrix of dimension (n, p) where rows are relative to n DOF i and columns to p DOF j. We should note that this convention implies the relation X ji = X ij T (transposed matrix) since it comes down to permuting rows and columns. In particular, matrix Xii (which is not necessarily diagonal) is necessarily symmetric. These properties are verified here as long as the principle of reciprocity applies. Otherwise, in the presence of rotating parts destroying certain symmetries for example, precautions will be taken to remove ambiguities without losing the advantages of this notation.

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A subscript related to all DOF i and j will logically be named (i+j): for example, we will write X (i + j ) k or X (i + j )(i + j ) . The previous convention must be completed by another convention to be able to distinguish rows, columns or terms of a matrix X ij . By convention, an underlined subscript has a fixed value. Therefore:

Xi j

represents row i in matrix Xij ,

Xi j

represents column j in matrix X ij ,

X ij

represents term ij in matrix X ij .

In certain cases, matrix subscripts are followed by other information after a comma to explain affiliation: Therefore:

Xij , k X ij , res

represents a matrix Xij relative to mode k , represents a matrix X ij “res” (of residual value: term explained when appropriate).

These conventions are not customary, particularly when they replace the notation T for a transpose of a matrix. Product ( Xij ) T Yi is simply written as X ji Yi where the sequencing of subscripts is clearly visible. Experience has shown that any reading problems disappear after an adaptation period.

Notations Below is a list of the main notations used in this book. Some have more than one meaning, but have different symbols or are used in different contexts. Other notations can appear in certain chapters for particular points: they are explained locally to remove any ambiguity. Scalars or matrices A A, B a, b C, c

Accelerance (acceleration/force), dynamic amplification H or T, interpolation function Matrices in space 2N Generalized parameters in space 2N Linear viscous damping, coefficient

Glossary

C c D d E F f G g H h I i J K, k k L M, m P p Q q R S T

t U u, v, w V W x x, y, z Y, y Z

α, β γ ε ζ η

xxv

Linear combination Propagation speed Power dissipation, plate bending stiffness, differential operator Distance Young’s modulus (stress/strain) Force, objective function Frequency Flexibility (displacement/force), shear modulus Gravity acceleration Frequency response function (FRF), dynamic amplification factor Impulse response, plate thickness Identity, inertia (of section or of mass) -1 Torsional inertia (of section or of mass) Stiffness (force/displacement) Form factor of a cross-section Modal participation factor, length Mass (force/acceleration) Probability Probability density Amplification at resonance: Q = 1 / (2ζ )= 1 / η Modal displacement Auto or cross-correlation, radius, gyroscopic effect Power spectral density (PSD), cross-section area Transmissibility (displacement/displacement ...), dynamic transmissibility factor, period or duration, kinetic energy, transformation Time, transmissibility impulse response, thickness Elastic energy Physical displacement Total potential energy, volume Power spectral density (PSD) “physical”, work of external forces Relative displacement Coordinates (position) Admittance (velocity/force), left eigenvector Impedance (force/velocity) Parameters Coherence Strain Viscous damping factor Structural damping

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Structural Dynamics in Industry

θ λ µ ν ρ σ τ Φ, φ φ ϕ Ψ Ω ω

Rotation Lagrange multiplier, dimensionless circular frequency Mass per unit length or surface density Poisson’s ratio Mass density Stress, standard deviation Delay Eigenvector, mode shape, right eigenvector Shear factor Phase, restoring force Junction mode Rotating speed Circular frequency: ω = 2π f

Subscripts c e g i j k l m n o p r res s x, y z

Connection Element, excitation General: g = i+j Internal Junction (support) Normal or complex mode Complement of r to i: l = i-r Linear constraint (multi-DOF) Complement of m to g: n = g-m Omit (complement of s to i: o = i-s), observation Physical parameter Rigid (statically determinate junction) Residual Selection Excitation, response Zone

Other notations ∆x ℜ(x) , ℑ(x) X(ω) ∂X X,p = ∂x p / x p

Difference between two values of x Real part of x, imaginary part of x Fourier transform of x(t) Relative derivative of X with respect to x p

Glossary

~ X

x

*

Effective parameter Conjugate of x

σx

dx / dt , d 2 x / dt 2 Average of x, x made complex Standard deviation of x

XT X

Transposed X matrix Condensed X matrix

x , x x

xxvii

Chapter 1

General Introduction to Linear Analysis

1.1. Introduction The essence of this work is dedicated to linear analysis, i.e. relying on the following two properties: – the structure response to a sum of excitations is the sum of the responses to each excitation. We can thus analyze each excitation separately and then superpose them; – the response is proportional to the excitation. The ratio between excitation and response is thus a characteristic of the structure that can be determined independently of the excitation. Under these conditions, the processing is considerably simplified and makes it possible to consider large size models without too many difficulties. The appearance of non-linearities will be discussed in Chapter 10. In section 1.2 we will first take note of the various motion types that can be analyzed in a traditional way with certain reservations. We will describe here only the properties that are useful for what comes next (full developments on the characterization of the environments can be found in [LAL 02]). This description will reveal two worlds that communicate due to the Fourier transform in the time and frequency domains, whose impact on analysis will be examined in section 1.3, logically leading to the description of the dynamic properties of the structure in the form of frequency response functions in section 1.4. After these basic considerations, the equations of motion and their solution will be discussed in section 1.5, before concluding with section 1.6 on a parallel between analysis and tests enabling the establishment of a first link between these two complementary techniques.

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Structural Dynamics in Industry

1.2. Motion types 1.2.1. Sine motion 1.2.1.1. Pure sine The response of a linear structure to a sinusoidal excitation, sine in abbreviated form, of frequency f is a sine of the same frequency. This frequency f or number of cycles per second, expressed in hertz (Hz), is the inverse of period T, expressed in seconds (s). Although it constitutes the most physically interpretable parameter, it is often preferred to use the circular frequency ω = 2πf, in radians per second (rad/s) in order to simplify the writing. Thus, excitations and responses x(t) will be written in the form (see Figure 1.1 for illustration): x(t ) = X (ω ) sin(ω t + ϕ (ω ))

ω

[1.1]

Circular frequency (rad/s)

X (ω ) Amplitude (in the same units as x); this can be function of the circular frequency ω

ϕ(ω) Phase (rad) relative to a reference; this can also be a function of ω. x X

−

ϕ ω

2π − ϕ

ω

t

Figure 1.1. Pure sine of frequency f = ω /2π

NOTE.– the sine argument is often written (ω t − ϕ ) so that the phase of a response in relation to an excitation is positive (the response follows the excitation). We will keep (ω t + ϕ ) here in order to preserve the true sense of variation.

General Introduction to Linear Analysis

3

For the analysis, it is advantageous to consider the complex plane where we have the general relation:

eix = cos x + i sin x

[1.2]

which makes it possible to rewrite relation [1.1] in the form (Figure 1.2): x (t ) = ℜ(x (t ) ) + i ℑ(x (t ) ) = ℜ(x (t ) ) + i x(t ) = X (ω ) e iω t x (t )

[1.3]

Complex function from which we will retain only the imaginary part x(t) (or the real part if the form [1.1] is cosine)

X (ω ) = X (ω ) eiϕ (ω ) , complex function with amplitude |X(ω)| and phase ϕ(ω)

Form [1.3] is handled more easily than [1.1] and prefigures the general case of transient motion. ℑ

x(t )

x x (t ) = X e

iω t

X

ωt ϕ x (0 )

ℜ

t

x (0) = X = X e iϕ

Radius x Figure 1.2. Complex plane

Pure sine motion does not occur very often in reality, but it occupies a privileged place in the analysis because, when it arises, the responses of the structure can be amplified to a great extent for certain frequencies known as resonance frequencies. We can see it more or less in rotating machines or when the excitation is in tune with a natural frequency of the structure. For example, this is the case with the phenomenon commonly called the Pogo effect affecting launchers using liquid fuel, which results from a coupling between the structure, piping and propulsion. The following two alternatives, which are fairly common, involve the same type of analysis.

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1.2.1.2. Swept sine If, in the course of time, the frequency of the sine varies, but sufficiently slowly so that the motion is quasi-stationary at every moment, the analysis reverts to the preceding case. This variation, often monotonic, will sweep a generally wide frequency band: we thus speak of swept sine or sine sweep of the form (see Figure 1.3): x(t ) = X (ω ) sin (ω (t ) t + ϕ (ω ) )

[1.4]

ω (t ) Circular frequency varying with time X (ω ) Amplitude of the sine which can vary with the frequency and thus with time

ϕ (ω ) Phase of the sine which can vary with the frequency and thus with time This case can be observed in a real environment, for example, with engine acceleration. It is also currently used in simulation tests in order to envelope the effects of the real environment in a given frequency band. x

X (ω )

t

Figure 1.3. Example of a swept sine

The function ω(t) can take various forms. The derivative dω /dt is the sweep rate. It has no importance here assuming that the quasi-stationarity hypothesis is satisfied. Various information about the sweep types and about their velocity impact on the responses can be found in [LAL 02], volume 1. Some practical considerations are given in section 11.4.2 concerning tests with shakers.

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5

1.2.1.3. Periodic motion A motion that is repeated at regular intervals of T duration is defined as periodic. Under most conditions, it can be decomposed into a sum of sinusoids whose frequencies are multiples of the fundamental frequency equal to the inverse of the period T (Figure 1.4). Assuming linearity, each frequency will be analyzed separately before reconstructing the time signal, hence the same analysis as for a pure sine. These frequency components, known as harmonics, are given by the decomposition in Fourier series:

x(t) =

an =

∞

⎛

t⎞

∞

⎛

t⎞

n=0

⎝

T⎠

n=0

⎝

T⎠

∑ an cos⎜ 2π n ⎟ + ∑ bn sin⎜ 2π n ⎟ ⎛

2

t⎞

T x(t) cos⎜ 2π n ⎟ dt ∫ 0 ⎝ T T⎠

bn =

[1.5]

t ⎞ 2 T ⎛ x (t ) sin ⎜ 2π n ⎟dt ∫ 0 T T⎠ ⎝

x

t T Figure 1.4. Example of periodic motion

1.2.2. Transient motion

The decomposition of a periodic function into a sum of sinusoids can be extended to transient functions. The discrete sum given by the Fourier series is in this case replaced by a continuous sum given by the Fourier integral, also called Fourier transform, which is a reversible transformation defined by: +∞

X (ω ) = ∫− ∞ x(t ) e − iω t dt

⇔

x(t ) =

1 +∞ iω t ∫ X (ω ) e dω 2π − ∞

[1.6]

The first relation expresses the decomposition of the time function in sinusoids whereas the second relation reconstitutes this same function starting from its sinusoids. X (ω ) is a complex function, with real and imaginary parts, or amplitude

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Structural Dynamics in Industry

and phase. It is the equivalent of relation [1.3]. It should be noted that certain authors write the sign of the exponent or place the term 2π differently. This is only a question of convention, with [1.6] being the most common because of the association ω /2π. Figure 1.5 gives some typical examples: – impulse function, whose Fourier transform is a positive real constant; – step function, whose Fourier transform is in (i ω ) −1 ; – half-sine function, which represents a simple form of shock often taken as a reference; – damped sine function, which is the response of the 1-DOF system to an impulse and which will be found again in Chapter 2. The amplitudes of the Fourier transforms, like the other frequency functions in this book, are plotted with logarithmic rather than linear scales, as they provide more information, particularly the asymptotic behavior in (i ω ) n . This reversibility allows good communication between the time domain where we consider the phenomena x(t ) as a function of time, and the frequency domain where we consider the phenomena X (ω ) as a function of frequency.

Impulse

Step

Half-sine

Damped sine

Figure 1.5. Examples of transients and their Fourier transform (amplitude and phase – logarithmic scales)

Since all real motion is transient, this case is general. However, its analysis is reserved for relatively short durations in order to maintain a reasonable time resolution for x(t ) . For greater durations, we will call upon the concept of random motion.

General Introduction to Linear Analysis

7

1.2.3. Random motion 1.2.3.1. Random process A phenomenon x is described as random (or stochastic) as opposed to determinist if it is irregular (no periodicity in time) and non-repetitive (two time records of x(t ) are necessarily different). The set of data x(t ) relating to the phenomenon is then called a random process. Random processes are very frequent in practice. The typical example is that of mechanical vibrations produced on a vehicle by irregularities in the road surface. This is also the case for acoustic fields of various sources denoted as noise in which all frequencies are represented, as opposed to acoustics of a rather periodic nature. x1 (t )

x1 (t j )

x2 ( t )

t x2 (t j )

t

x3 (t ) x3 (t j )

tj

t Figure 1.6. Random process

In order to analyze a process, it should be first characterized in an adequate manner (detailed considerations on this subject can be found in [CRA 63] or [BEN 86]). Let us consider a phenomenon x which varies according to the time t, for which a series of time records xi (t ) were obtained as indicated in Figure 1.6. This phenomenon can be analyzed in two ways: – either by considering a given record i which provides a random function of the time xi (t ) : this is the temporal perspective; – or by considering the values of the various recordings for a given time t j , which provides a random variable x(t j ) : this is the statistical perspective.

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1.2.3.2. Time analysis Time analysis concerns the function x(t ) on a time interval T. On this function, as illustrated by Figure 1.7, we will define: – the mean x : x=

1 T ∫ x(t )dt T 0

[1.7]

2

– the mean square x : x2 =

1 T 2 ∫ x (t )dt T 0

[1.8]

from which we deduce the root mean square or rms value: xrms = x 2

[1.9]

– the variance σx2 or mean square of the centered function x(t ) − x :

σ x2 =

1 T

T 2 2 2 ∫0 [ x(t ) − x ] dt = x − x

[1.10]

from which we deduce the standard deviation σ x by the square root. σ x thus represents the mean deviation, in the quadratic sense, between x(t ) and its mean. x

x rms x

σ σ t

Figure 1.7. Means and standard deviation

General Introduction to Linear Analysis

9

We can define other properties, such as the moments of order n relative to the mean of x n (t ) , or the correlations that will follow. If all these properties are independent of the considered time interval T, the process is considered stationary, which will be assumed in what follows. If we consider the mean of the product x(t ) x(t + τ ) , for a given delay, τ , we obtain a function of τ called the autocorrelation of x : Rxx (τ ) =

1 T ∫ x(t ) x(t + τ ) dt T 0

[1.11]

We will note that the function R xx (τ ) is even and that: R xx (0) = x 2

[1.12]

For two processes x(t ) and y(t) , the average of the product x(t ) y (t + τ ) is the cross-correlation between x and y : Rxy (τ ) =

1 T ∫ x(t ) y (t + τ ) dt T 0

[1.13]

We will note that Rxy (τ ) is generally not an even function. 1.2.3.3. Statistical analysis Statistical analysis is relative to the random variable x in which case we have a collection of values xi (Figure 1.8) from which we can define a probability density function p (x) by: p( x) =

dP ( x) dx

[1.14]

P(x) being the probability of having xi smaller than a given value x , equal to the proportion of values xi > x . The function p(x) often has a bell shape appearance, as indicated in Figure 1.8 and explained later.

We can thus define the same type of properties as for the time analysis: – the mean x : +∞

x = ∫− ∞ x p( x) dx

[1.15]

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– the mean square x 2 : +∞ x 2 = ∫− ∞ x 2 p ( x) dx

[1.16]

– the variance σ x 2 , square of the standard deviation σ x : +∞

σ x 2 = ∫− ∞ ( x − x ) 2 p( x) dt = x 2 − x 2 xi

[1.17]

P

p

1

0

x

x

Figure 1.8. Probability density

Many random processes result from the cumulative effects of a large number of independent causes, and thus have a tendency to follow a “normal” or Gaussian distribution. In this case, the probability density depends only on the mean x and the standard deviation σ x :

p( x) =

1

σ x 2π

−

e

(x− x)2 2σ x

2

[1.18]

and has the appearance of the bell shaped curve mentioned before. In Table 1.1 different numerical values relative to the normal distribution for the dimensionless variable ξ = ( x − x ) / σ x are given. In practice, we often retain the “ nσ ” probabilities: n

P ( x ≤ x + nσ x )

P ( x − x ≤ nσ x )

1 2 3

84.13% 97.72% 99.87%

68.27% 95.45% 99.73%

General Introduction to Linear Analysis

ξ

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 3.5 4.0 4.5 5.0

p(ξ ) =

−

1 2π

ξ2

e 2

0.3989423 0.3969525 0.3910427 0.3813878 0.3682701 0.3520653 0.3332246 0.3122539 0.2896919 0.2660852 0.2419707 0.2178522 0.1941861 0.1713686 0.1497275 0.1295176 0.1109208 0.0940491 0.0789502 0.0656158 0.0539910 0.0439836 0.0354746 0.0283270 0.0223945 0.0175283 0.0135830 0.0104209 0.0079155 0.0059525 0.0044318 0.0008727 0.0001338 0.0000160 0.0000015

ξ

P (ξ ) = ∫−∞ p (ζ ) dζ

0.5000000 0.5398278 0.5792597 0.6179114 0.6554217 0.6914625 0.7257469 0.7580363 0.7881446 0.8159399 0.8413447 0.8643339 0.8849303 0.9031995 0.9192433 0.9331928 0.9452007 0.9554345 0.9640697 0.9712834 0.9772499 0.9821356 0.9860966 0.9892759 0.9918025 0.9937903 0.9953388 0.9965330 0.9974449 0.9981342 0.9986501 0.9997674 0.9999683 0.9999966 0.9999997

1 − P(ξ )

2 [1 − P(ξ )]

0.5000000 0.4601722 0.4207403 0.3820886 0.3445783 0.3085375 0.2742531 0.2419637 0.2118554 0.1840601 0.1586553 0.1356661 0.1150697 0.0968005 0.0807567 0.0668072 0.0547993 0.0445655 0.0359303 0.0287166 0.0227501 0.0178644 0.0139034 0.0107241 0.0081975 0.0062097 0.0046612 0.0034670 0.0025551 0.0018658 0.0013499 0.0002326 0.0000317 0.0000034 0.0000003

1.0000000 0.9203443 0.8414806 0.7641772 0.6891565 0.6170751 0.5485062 0.4839273 0.4237108 0.3681203 0.3173105 0.2713321 0.2301393 0.1936010 0.1615133 0.1336144 0.1095986 0.0891309 0.0718606 0.0574331 0.0455003 0.0357288 0.0278069 0.0214482 0.0163951 0.0124193 0.0093224 0.0069339 0.0051103 0.0037316 0.0026998 0.0004653 0.0000633 0.0000068 0.0000006

Table 1.1. Numerical values for normal distribution

11

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Inversely, it can be useful to know the standard deviation corresponding to a given probability. Some typical values are as follows: P ( x ≤ x + nσ x ) 95.0% 97.5% 99.0% 99.5% 99.9% 99.95%

P ( x − x ≤ nσ x )

n

90.0% 95.0% 98.0% 99.0% 99.8% 99.9%

1.6449 1.9600 2.3263 2.5758 3.0902 3.2905

The probability density concept can be generalized into two (or more) variables, hence the autocorrelation of the same random variable x considered at two different moments, x1 = x(t1 ) and x2 = x(t 2 ) , with t 2 − t1 = τ :

+∞ +∞

Rxx (τ ) = ∫− ∞ ∫− ∞ x1 x2 p( x1 ) p( x2 ) dx1 dx2

[1.19]

and the cross-correlation between x1 = x(t1 ) and y 2 = y (t 2 ) :

+∞ +∞

Rxy (τ ) = ∫− ∞ ∫− ∞ x1 y2 p( x1 ) p( y2 ) dx1 dy2

[1.20]

If all the statistical properties from [1.15] to [1.20] are equal to the time properties from [1.7] to [1.13], the process is called ergodic. From this property it can be shown that a single time record is completely representative of the process. This will be assumed hereafter. As an example of an ergodic process, consider the vibrations produced on a vehicle by a road built according to a given technique. If we extended this process to all types of roadways, from a dirt track to a motorway, it would no longer be ergodic. 1.2.3.4. Power spectral densities The passage from the time domain to the frequency domain, already mentioned for transient motions, can be done here using correlation functions. We thus define

General Introduction to Linear Analysis

13

the power spectral density (PSD) of a process x(t ) as the Fourier transform of its autocorrelation: +∞

S xx (ω ) = ∫− ∞ Rxx (τ ) e −iωτ dτ

[1.21]

The reversibility of the Fourier transform makes it possible to write: Rxx (τ ) =

1 +∞ iωτ ∫ S xx (ω ) e dω 2π − ∞

[1.22]

which implies, according to [1.12]: x2 =

1 +∞ ∫ S xx (ω ) dω 2π −∞

[1.23]

This mathematical relation may not be completely satisfactory for an engineer who prefers to consider only the positive frequencies, in which case we can write: +∞

x 2 = ∫0

W xx ( f ) df

[1.24]

This PSD Wxx ( f ) , considered more “physical”, is commonly used in practice. It is linked to the mathematical PSD S xx (ω ) by the relation: Wxx ( f ) = 2 S xx (ω )

[1.25]

Factor 2 comes from the contribution of the negative frequencies added to the contribution of the positive frequencies. From relation [1.24] it can be deduced that Wxx ( f ) represents the energy distribution according to the frequency. Since Rxx (τ ) is a real and even function, Wxx ( f ) is a real positive function. Inversely, the mean square x 2 is given by the integration of Wxx ( f ) on the considered frequency band.

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A particular case of PSD is white noise, corresponding to a constant value. Thus, for a white noise of an intensity Wxx ( f ) = W0 limited to the frequency band [ f1 , f 2 ] , we have: x 2 = W0 ( f 2 − f1)

[1.26]

For two processes x(t ) and y(t) , we define the cross-spectral density function between x(t ) and y(t) , as the Fourier transform of their cross-correlation: +∞

S xy (ω ) = ∫− ∞ Rxy (τ ) e −iωτ dτ

[1.27]

The preceding comments on the physical aspect remain valid. However, since R xy (τ ) is not an even function, W xy ( f ) is complex-valued and comprises an inphase spectrum (co-spectrum) and an in-quadrature spectrum (quad-spectrum). In addition, it can be shown that: Wxy (ω ) = W yx (ω )*

(* = complex conjugate)

[1.28]

Between x(t ) and y(t) , the coherence is defined by:

γ

2 xy ( f )

=

W xy ( f )

2

W xx ( f ) W yy ( f )

[1.29]

The coherence is necessarily between 0 and 1. The value 1 means that x(t ) and y(t) are perfectly coherent, or dependent: they are linked by a determinist law. The value 0 means that x(t ) and y(t) are perfectly incoherent or independent. This concept of coherence is important in practice to evaluate the degree of dependence between the processes, particularly in an experimental context.

1.3. Time domain and frequency domain 1.3.1. Introduction

We saw in the preceding section that we could communicate between the time domain and the frequency domain. Thus, we can choose an analysis in one or other of these domains. This choice implies calculations of a different nature, which will be clarified here.

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15

Generally, calculating a given response y of a structure requires two types of information: – the excitation x, which we will assume here to be single without being detrimental to the generality because, for several simultaneous excitations, the linearity hypothesis makes it possible to superpose the effects of each excitation taken separately. This excitation is x(t ) or X (ω ) according to the domain; – the way in which the structure transforms the excitation in order to provide the response. The linearity hypothesis still makes it possible to say that this manner is independent from the excitation. This structural characteristic for a given excitation and response will depend on the domain considered.

1.3.2. The time domain

In the time domain, any excitation x(t ) can be decomposed in sub-domains as depicted in Figure 1.9. x

x x(t i )

= ti

∆t i

t

n

∑

i =1

ti

∆t i

t

Figure 1.9. Decomposition of a time function in impulses

When ∆ti → 0 , each excitation tends towards an impulse, or a Dirac function, i.e. a zero level everywhere except x(ti ) at ti . The linearity hypothesis makes it possible to express the response y(t) at the excitation x(t) as the sum of the responses to impulses x(ti ) . Therefore it is sufficient to know the response of the structure to an impulse, called the unit impulse response. By writing h yx (t ) the response y(t) of the structure to a unit impulse I (satisfying ∫ I (t ) dt = 1 ) at the moment t = 0 , the total response is then equal to: y (t ) =

t

∫−∞ x(τ )hyx (t − τ )dτ

[1.30]

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Structural Dynamics in Industry

This integral, known as a convolution integral, is also called Duhamel’s integral. Thus, this approach requires the calculation of an integral.

1.3.3. The frequency domain

In the frequency domain, any excitation x(t ) can be represented by its Fourier transform X (ω ) . The linearity hypothesis allows us to express the response Y (ω ) to the excitation X (ω ) by considering the response at each ω . Therefore, we only need to know the response of the structure to a sine excitation for any ω present in the excitation. By denoting H yx (ω ) the response Y (ω ) of the structure to a sine of circular frequency ω , of unit amplitude and zero phase (reference relative to the excitation), the desired response is given by: Y (ω ) = H yx (ω ) X (ω )

[1.31]

H yx (ω ) is a complex function called a transfer function, or frequency response

function (FRF), whose amplitude is the ratio between the excitation and response amplitudes, and phase describes the lag between excitation and response. It is obtained by the Fourier transform of the unit impulse response: +∞

H (ω ) = ∫− ∞ h(t ) e − iω t dt

⇔ h(t ) =

1 +∞ iω t ∫ H (ω ) e dω 2π − ∞

[1.32]

This approach introduces a simple product instead of an integral in the time domain making the expressions easier to manage. The following development will therefore be carried out mainly in the frequency domain with emphasis on the FRF, allowing us to identify behavior specific to resonances. It should be noted with expression [1.3] that the derivation in the time domain is accomplished in the frequency domain through multiplication by iω . Thus, between displacements u , velocities u and accelerations u , we have: u(ω ) = iω u (ω ) = − ω 2 u (ω )

[1.33]

In the case of a random motion, relation [1.33] implies for the PSDs: S uu (ω ) = ω 2 S uu (ω ) = ω 4 S uu (ω )

[1.34]

General Introduction to Linear Analysis

17

1.4. Frequency Response Functions 1.4.1. Introduction

Introduced in section 1.3.3, the FRF is by definition the ratio between an excitation and a response in the frequency domain, as indicated by relation [1.31]. It should be noted that it is not a ratio between two responses, as could be envisaged under certain circumstances; this would generate errors when using the developments that follow. In general, several excitations x and responses y may exist simultaneously. In the case of a discretized structure, i.e. whose state is defined by a finite number of parameters referred to as degrees of freedom (DOF), each DOF may be an excitation and/or response point. The excitations x can then be grouped into the vector X x and the responses y into the vector Y y , each component corresponding to a DOF. Between these vectors, we have the relation: Yy (ω ) = H yx (ω ) X x (ω )

[1.35]

H yx (ω ) being a matrix whose lines relate to the response DOF y and the columns

relate to the excitation DOF x (see the conventions on the matrices, clarified in the glossary). It is the generalization of relation [1.31] valid for an excitation and a response, assuming linearity, which makes it possible to superpose the contributions of each excitation. This concept of frequency response function is natural in the context of the degrees of freedom of a discretized structure: for N DOF, each of the N × N pairs (x, y) can be separately considered in order to define a FRF whose nature will be specified in section 1.4.4. It is also possible to generalize the excitations and responses implying several DOF simultaneously. For example, in the context of a discretized structure: – the case of a pressure excitation distributed over a surface, and represented by a set of point forces on the concerned DOF after discretization. The FRF between this pressure and any other structural response will be obtained by adequate linear combination of the FRF between the equivalent forces and the considered response; – the case of a global reaction force at a continuous interface results in a set of point reaction forces on the concerned DOF after discretization. The FRF between any excitation and this reaction will be obtained again by the adequate linear combination of the implied FRF. Finally, we can extend this reasoning to a continuous structure by considering, for example, the FRF between a pressure and an interface reaction. We just need to

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suitably solve the equations of motion in the frequency domain, just like the simple cases discussed in Chapter 6. Hereafter, we will suppose discrete responses and excitations reduced to the DOF concerned, the discretization methods being discussed in Chapter 3.

1.4.2. Frequency Response Functions and responses

By definition of the FRFs, the responses of the structure are deduced from excitations by multiplying them by the frequency response functions. Thus, for a given excitation and response: – for a sine motion, relation [1.31] directly applies in order to give: - the response amplitude as product of the excitation amplitude by the FRF amplitude, - the phase of the response as sum of the excitation phase (possibly zero) and the FRF phase: Y (ω ) = H (ω ) X (ω )

x (t ) = X (ω ) e iω t ⎯⎯ ⎯ ⎯yx⎯ ⎯ ⎯⎯→ y (t ) = Y (ω ) eiω t

[1.36]

– for transient motion, relation [1.31] is put into the time domain by the Fourier transform (FT) and then back again, as shown schematically: Y (ω ) = H ( ω ) X (ω )

−1

FT FT yx x(t ) ⎯⎯ → X (ω ) ⎯⎯⎯⎯⎯⎯⎯ → Y (ω ) ⎯⎯⎯ → y (t )

[1.37]

– for random motion, relation [1.31] must be adapted in order to apply it to the PSD defined in section 1.2.3.4. We can show the relations: S yx (ω ) = S xy * (ω ) = H yx* (ω ) S xx (ω ) 2

S yy (ω ) = H yx (ω ) S xx (ω )

[1.38]

[1.39]

Relation [1.38] is rather used to determine the FRF H yx (ω ) starting from S xx (ω ) and S yx (ω ) or S xy (ω ) . Relation [1.39] is relation [1.31] adapted to the

random environment. It should be noted that the two relations [1.38] and [1.39] give a coherence (relation [1.29]) equal to 1, and this translates the perfect dependence between x and y, the system which links them being deterministic.

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19

In the case of several excitations and responses, relations [1.36] and [1.37] become matrix relations, like [1.35]. For the random motion, the excitations are described by the matrix of their power and cross spectra S xx (ω ) : power spectra on the diagonal, cross spectra outside the diagonal. According to [1.38], this is a Hermitian matrix:

[S xx (ω )]T = [S xx (ω )]*

[1.40]

It should be noted that this matrix does not have the property of symmetry implied by the conventions in this book. This characteristic is not inconvenient since it remains limited to the random domain. Relations [1.38] and [1.39] are then generalized as follows:

[

S yx (ω ) = S xy * (ω )

]

T

= H yx * (ω ) S xx (ω )

S yy (ω ) = H yx* (ω ) S xx (ω ) H xy (ω )

[1.41] [1.42]

H xy (ω ) indicating the transpose of H yx (ω ) , according to established conventions. The sequence of the subscripts should be noted, as for the preceding relations, which avoids written errors.

We will recall from this section that, in the frequency domain, the responses of all natures result from excitations and from FRF by simple matrix products.

1.4.3. The nature of excitations and responses

Before determining the nature of the frequency response functions, it is necessary to specify the nature of the excitations and responses that define them. This nature is a function of the implied DOF.

Figure 1.10. The nature of structural DOF

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Structural Dynamics in Industry

All DOF of the structure belong to one of the two exclusive categories that follow, as shown in Figure 1.10: – support DOF (possibly absent) which are implied in what we commonly call the boundary conditions, or interfaces with the ground or with adjacent structures. They will be called junction DOF here in order to assign them the mnemonic subscript j; – other DOF, i.e. all those which are not implied in the supports. They will be called internal DOF here in order to assign them the mnemonic subscript i. The junction DOF are often reserved to the parts of the structure where motion is blocked. It can be generalized as any imposed motion, defined by displacements u , velocities u or accelerations u , which will be assumed here systematically. Thus, for excitations that we can impose and for responses that we can obtain: – for excitations: we can impose forces Fi , possibly zero, on internal DOF i, j possibly zero, on junction DOF j; and/or motions u j , u j , u i on internal DOF i and reaction – for the responses: we obtain motions u i , u i , u forces F j on junction DOF j.

We will here give the duality between force and motion, expressed by i and j and revised in Table 1.2. DOF i j

Excitation

Response

Fi j u j , u j , u

i u i , u i , u Fj

Table 1.2. The nature of excitations and responses

Let us take the simple example of the beam in Figure 1.11, which schematically represents any elongated structure, such as a pylon or launcher. Let us suppose that its state is defined only by the three translation degrees of freedom and the three rotation degrees of freedom at each end, as indicated.

(a)

(b)

(c)

Figure 1.11. Examples of support configurations

General Introduction to Linear Analysis

21

We can imagine various situations, for example: (a) the beam is free, i.e. without support: all the DOF are of internal type i, on which we will be able to impose forces Fi . For example: a launcher in space with a component in the axis on the lower node for the thrust, on which 12 components of aerodynamic forces can superpose, etc. All the responses are of the motion type i ; u i , u i , u (b) the beam is clamped on the lower node. The corresponding six DOF are of j . The six DOF junction type j, on which we will be able to impose motions u j , u j , u of the upper node are of internal type i, on which we will be able to impose forces j caused by an earthquake, and Fi . For example: a pylon with motions u j , u j , u forces Fi caused by the wind, etc. Under the possibly simultaneous action of these i and the lower types of stresses, the upper node will respond by motions u i , u i , u node by reaction forces F j ; (c) the lower node is guided axially and excited along the axis. The five DOF concerned by the bearings are of junction type j, with the imposed zero motions j . The sixth can be excited either by imposed motion, equivalent to the u j , u j , u preceding case with five zero components, or by force, hence a DOF of internal type i. For example: a specimen tested on a shaker with imposed motion (e.g. base acceleration) or a force (e.g. of an electrodynamic origin). Depending on the case, the response on this DOF is the reaction force or the motion. On the other DOF of the lower node, the responses are the reactions whereas on the DOF of the upper node, the responses are the motions.

1.4.4. The nature of Frequency Response Functions

Depending on the nature of the excitations and responses, forces, displacements, velocities or accelerations, as discussed in section 1.4.3, we obtain the FRFs in Table 1.3 which indicates the denominations used in what follows (possibly followed by the “dynamic” qualifier: dynamic flexibilities, etc.) as well as the most common synonyms. Each case gives rise to a matrix of the frequency response functions whose lines relate to the responses and columns to the excitations. The following comments can be made on these matrices: – all matrices with the same subscripts for the lines and columns are symmetric according to the reciprocity principle which stipulates that the ratio between the excitation on a DOF x and the response on a DOF y is the same as the ratio between the excitation in y and the response in x. This is true under certain conditions, in particular the absence of gyroscopic effects caused by rotating parts;

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Structural Dynamics in Industry

– again due to reciprocity, the transmissibilities in forces are equal to the transmissibilities in displacements, and this implies T ji = (Tij )T . As indicated in the table, the sign (-) which appears between the excitation forces and the reaction forces is due to the fact that the latter are equal and opposed to the forces transmitted by the structure to the supports. We can see below the relevance of the convention on the subscripts. Fi

uj

u j = iω u j

j = −ω 2 u j u

Transmissibilities

Stiffnesses (2)

Impedances

Masses (5)

(-) T ji

K jj

Z jj = K jj / iω

M jj = K jj /( −ω 2 )

Tij / iω

Tij /( −ω 2 )

Excitations Responses

Fj

ui

*

Flexibilities (1)

Transmissibilities

G ii

Tij

u i

Admittances (3)

= iω u i

Yii = iω G ii

i u

Accelerances (4)

= −ω 2 ui

A ii = −ω 2 G ii

Transmissibilities

iω Tij

Tij

Tij / iω Transmissibilities

− ω 2 Tij

iω Tij

Tij

* Sign (-), the considered responses being the reaction forces, opposed to the transmitted forces Other names:

(1) compliances, receptances (2) rigidities (3) mobilities (4) inertances (ambiguous term, to be avoided) (5) apparent masses

Table 1.3. Frequency response functions

General Introduction to Linear Analysis

23

If we choose to represent the motions by the displacements, we generally write: Tij (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎡ u i (ω ) ⎤ ⎡ G ii (ω ) ⎢F (ω )⎥ = ⎢− T (ω ) K (ω )⎥ ⎢u (ω )⎥ ji jj ⎣ j ⎦ ⎣ ⎦⎣ j ⎦

[1.43]

This matrix relation (see conventions) contains the following matrices: G ii (ω )

dynamic flexibility matrix, symmetric by reciprocity

Tij (ω )

dynamic transmissibility matrix in displacements

T ji (ω )

dynamic transmissibility matrix in forces, with T ji = (Tij )T by reciprocity, and sign (-) due to the reaction forces opposed to the transmitted forces

K jj (ω )

dynamic stiffness matrix, symmetric by reciprocity

It should be noted that by considering the transmitted forces instead of the reaction forces, we would have obtained a completely symmetric FRF matrix. This property is actually not very useful in the developments and we will prefer to keep reactions F j which, just like Fi , are forces applied to the considered structure, and this makes the stiffnesses appear directly in relation [1.43] and not their opposites. Apart from these matrices, we will frequently use the matrix of the dynamic masses M jj = K jj /(−ω 2 ) , or its inverse, the matrix of the accelerances A ii = −ω 2 G ii , in certain cases the accelerations being more directly implied than the displacements, particularly in an experimental context.

Relation [1.43] is a detailed description of relation [1.35], which will be used according to the motion types, as indicated in section 1.4.2. We will thus have the following situations for the examples in Figure 1.11: (a) all the FRF are dynamic flexibilities G ii (ω ) (matrix 12 × 12); (b) - on the six top DOF: dynamic flexibilities G ii (ω ) (matrix 6 × 6), - on the six bottom DOF: dynamic stiffnesses K jj (ω ) (matrix 6 × 6), - between the six top DOF and the six bottom DOF: dynamic transmissibilities in displacements Tij (ω ) (matrix 6 × 6) and in forces T ji (ω ) (matrix 6 × 6);

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Structural Dynamics in Industry

(c) excitation in motion: identical to case (b). Excitation in force: - on the six top DOF and the bottom axial DOF: dynamic flexibilities G ii (ω ) (matrix 7 × 7), - on the five constrained bottom DOF: dynamic stiffnesses K jj (ω ) (matrix 5 × 5), - between the two preceding groups: dynamic transmissibilities displacements Tij (ω ) (matrix 7 × 5) and in forces T ji (ω ) (matrix 5 × 7),

in

(all these FRF are not necessarily interesting: for example in the case of a perfectly constrained DOF, the stiffnesses and the transmissibilities in displacements will be useless). The problem is now to determine these frequency response functions according to the physical properties of the structure. A common case is that of the rigid mass m. Its dynamic mass is obviously m(ω ) = m in any direction. We deduce the other FRF from this, according to Table 1.3, for example its flexibility G (ω ) = 1 /( −ω 2 m) , and this is evident only after having thoroughly assimilated the preceding concepts. Generally, the frequency response functions will be determined by solving the motion equations in the frequency domain, as indicated below.

1.5. Equations of motion and solution 1.5.1. Equations of motion

We can trace the equations of motion to Hamilton’s Principle, which says that the motion of a conservative system between two moments t1 and t2 is performed so that the Hamiltonian action t

∫t12 L(t )dt

[1.44]

is stationary, L being the Lagrangian defined by the difference between the kinetic energy T and the total potential energy V, itself the difference between the elastic energy U and the work of the external forces W: L = T −V

V = U −W

[1.45]

General Introduction to Linear Analysis

25

This is particularly true for a continuous system. In the case of a N-DOF discrete system g (general) and formed by the set of i and j DOF describing the status of the system (g = i+j), the Lagrangian is a function of the displacements u g and the velocities u g , and the calculation of the variations makes it possible to write for each DOF g (underlined subscript = fixed subscript: see conventions):

t

δ ∫t 2 L(t ) dt = 0 1

⇔

d ⎛⎜ ∂L dt ⎜ ∂ u g ⎝

⎞ ∂L ⎟− =0 ⎟ ∂u g ⎠

[1.46]

With the following hypotheses: – T is a quadratic form of the velocities, thus of the form: T=

1 u g T M gg u g 2

[1.47]

M gg matrix describing the distribution of the masses on the DOF g, symmetric by

reciprocity; – U is a quadratic form of the displacements, thus of the form: U=

1 u g T K gg u g 2

[1.48]

K gg matrix describing the distribution of the stiffnesses on the DOF g, symmetric

by reciprocity; – W is given by: W = u g T Fg

[1.49]

F g vector of the external forces on the DOF g;

Thus, relation [1.46] implies: g + K gg u g = F g M gg u

[1.50]

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Structural Dynamics in Industry

If we add a dissipation power D for a dissipative system with internal forces proportional to the velocities (viscous damping), thus of the form:

D=

1 u g T C gg u g 2

[1.51]

C gg matrix describing the distribution of the viscous damping on the DOF g,

symmetric by reciprocity; With damping, equation [1.50] becomes: g + C gg u g + K gg u g = F g M gg u

[1.52]

This matrix equation expresses the equilibrium of the forces on each DOF: , dissipation forces C u , elastic forces K u and external forces inertia forces M u F . Obtaining the matrices M, C and K describing the physical properties of the structures implied in its dynamic behavior should be determined starting from various structural parameters of the elements that compose it. It should be noted that, in the event of DOF transformation represented by a given relation between initial DOF u x and new DOF u y , the independence of the energies in relation to the transformation makes it possible to obtain the new matrices M, C, K and the new forces F by: M yy = T yx M xx Txy u x = Txy u y

⇒

C yy = T yx C xx Txy K yy = T yx K xx Txy

F y = T yx F x

[1.53]

1.5.2. Solution using the direct frequency approach

Any FRF X (ω ) may be determined frequency by frequency. The direct approach consists of directly deducing X (ω ) from the physical properties of the structure, i.e. from the matrices M, C and K introduced in section 1.5.1.

General Introduction to Linear Analysis

27

By detailing the partition between the i and j DOF in the frequency domain, equation [1.52] becomes: ⎛ ⎡M ⎜ − ω 2 ⎢ ii ⎜ ⎣M ji ⎝

M ij ⎤ + iω M jj ⎥⎦

⎡ Cii ⎢C ⎣ ji

Cij ⎤ ⎡ K ii + C jj ⎥⎦ ⎢⎣K ji

K ij ⎤ ⎞ ⎡ u i (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎟ [1.54] = K jj ⎥⎦ ⎟ ⎢⎣u j (ω )⎥⎦ ⎢⎣F j (ω )⎥⎦ ⎠

The solution of equation [1.54] provides the responses u i (ω ) and F j (ω ) to the excitations Fi (ω ) and u j (ω ) , hence the frequency response functions of equation [1.43]:

(

G ii (ω ) = − ω 2 M ii + iω Cii + K ii

)

(

−1

Tij (ω ) = − − ω 2 M ii + iω C ii + K ii

[1.55]

) (− ω −1

(

2

M ij + iω C ij + K ij

K jj (ω ) = −ω 2 M jj + iω C jj + K jj − − ω 2 M ji + iω C ji + K ji

(− ω

2

M ii + iω C ii + K ii

) (− ω −1

2

)

)

M ij + iω C ij + K ij

[1.56] [1.57]

)

This involves the decomposition of a matrix of a size equal to the number of internal DOF i of the structure at each frequency, which can be very time consuming for industrial models, hence the interest in the alternative modal approach.

1.5.3. Solution using the modal approach

The modal approach consists of making a spectral decomposition of the matrix to be inversed, and this amounts to solving the equations of motion first in the absence of excitation, whose solutions then make it possible to put all FRF X (ω ) in the form: X (ω ) = ∑ X k (ω )

[1.58]

k

Each term k of the sum is the contribution of what we will call a normal mode of the structure, hence the technique called mode superposition. The normal modes are the solutions of the eigenvalue problem relating to the matrix to decompose. Each

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mode is a shape on the internal DOF i, given by the mode shape (or eigenvector) and associated with a frequency given by the corresponding eigenvalue. Mathematically, there are as many normal modes k as DOF i, but this approach is interesting only if a small number of them contribute significantly, otherwise the calculation time will not be competitive in relation to the direct approach. This is the case of the “low frequency” domain where only the first modes are considered, mainly those whose associated frequency is situated in the excitation band. We can thus speak about modal truncation. The sum of relation [1.58] is then limited to these modes, but preferably completed by a residual term in order to globally represent the contribution of the higher modes, which can be significant:

X (ω ) =

N

n << N

k =1

k =1

∑ X k (ω ) ≈ ∑ X k (ω ) + X res

[1.59]

This modal approach, which can also be applied to the time domain, is the subject of Chapter 4.

1.5.4. Modes and 1-DOF system

The modal approach consists of projecting the motion of the structure on the basis of modes, and thus uncoupling the equations of motion. Each mode verifies a scalar equation whose form is the same as that of the motion of the 1-DOF system. A mode can thus be represented by a 1-DOF system, i.e. a system whose state is defined by a unique parameter. Its traditional form is shown in Figure 1.12, with the unique parameter being the mass position in relation to the rest position in the direction of the motion. It is composed of: – a directional mass (acting only in the direction of motion) m, providing kinetic energy to the system; – a stiffness spring k providing elastic energy to the system; – a damper of constant c, allowing the system to dissipate energy; comprising all the energy terms of the equations of motion in section 1.5.1.

General Introduction to Linear Analysis

m

29

DOF i

c

k

DOF j Figure 1.12. The 1-DOF system

This DOF is of the type i (internal). Following the considerations of section 1.4.3, it is in fact accompanied by a DOF of the type j (junction) representing the motion imposed at the base. We then have, as indicated in Figure 1.12, all the possible excitations and responses of equation [1.43] where each term is here a scalar (i and j of size 1), hence the presence of the three types of frequency response functions: flexibility, stiffness and transmissibility.

i

⇔

…

j Figure 1.13. Structure – 1-DOF-system equivalence

The 1-DOF system is thus the keystone of the analysis because any structure can be reduced to a collection of 1-DOF systems through its modes, as illustrated in Figure 1.13. Its analysis is detailed in Chapter 2. This equivalence will be used on various occasions for practical purposes: understanding different phenomena, elaborating equivalent models, substructuring by modal synthesis, etc.

1.6. Analysis and tests

According to the preceding considerations, the analysis by mode superposition is performed schematically in three steps, as indicated in Figure 1.14: 1) assembling the matrices representing the mass, stiffness and damping properties of the discretized structure;

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Structural Dynamics in Industry

2) obtaining the modes by solving an eigenvalue problem; 3) determining the frequency response functions by mode superposition. Each step corresponds to a type of characteristic of the structure which is independent of the excitation: matrices, modes and frequency response functions. Each of these types are related to a substructuring technique, i.e. the analysis of a structure starting from its substructures: 1) starting from the matrices of the substructures, we obtain the matrices of the structure by matrix assembly. It is the same technique used to assemble the element matrices in a model by finite elements, as described in section 3.2.4. Its advantage is its robustness, but its drawback is the size of the obtained model; 2) starting from the modes of the substructures, we obtain the modes of the structure by adequate processing: this is what is called “modal synthesis”, which is the subject of Chapter 8. Its advantage is the substantial reduction of the size of the model, its weakness being the processing of the model and the mode truncation errors; 3) starting from the frequency response functions of the substructures, we obtain the frequency response functions of the structure by adequate processing: this is the “frequency synthesis”, the subject of Chapter 9, sometimes called impedance coupling or FRF coupling. This technique can prove to be interesting, according to the case, but it requires a rather involved frequency solution. In the experimental domain, access to the characteristics of the structure is performed in the opposite direction: 1) starting from the measurements of the structure excitations and responses, we obtain the frequency response functions between excitations and responses by determining their ratio in the frequency domain; 2) starting from the frequency response functions between excitations and responses, we obtain the modes of the structure by analytically identifying the modal parameters which give the best approximation of the experimental FRF. It is the inverse operation of the mode superposition, and is much more delicate (other identification techniques are possible, as mentioned in section 11.3); 3) starting from the modes, we can possibly recover “experimental” matrices relating to the measured DOF. The comparison between analysis and tests in order to update the mathematical models can be made on all levels, but the modal level is particularly attractive due to the value of the information they represent. These considerations will be discussed again in Chapter 12.

General Introduction to Linear Analysis

ANALYSIS

Matrices M, C, K

Eigenvalue problem

Recovery

Modes

Mode superposition

Modal identification

Frequency response functions

TESTS

Figure 1.14. Analysis and tests

31

Chapter 2

The Single-Degree-of-Freedom System

2.1. Introduction The 1-DOF (degree of freedom) system schematized in Figure 2.1 has already been introduced in section 1.5.4. It is the simplest dynamic system because its state depends only on one parameter, the mass position on the axis where the motion is made. m k

DOF i c DOF j

Figure 2.1. The 1-DOF system

The notations introduced in section 1.4.3 will be kept in order to consider the general case beforehand. Thus, let the mass displacement be ui , since this DOF is of an internal nature, and the displacement of the base that forms the junction be u j . Both the j and i DOF contain only one element. The two vectors u i and u j have only one component here and they are thus to be handled like the scalars ui and u j . The same goes for all the matrices related to subscripts i or j which do not need to be underlined in this particular case since they can take only one value (see conventions).

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The three physical characteristics of the system are as follows: – the mass m. This gives kinetic energy to the system. This undeformable mass is to be considered only in the direction of motion (the vertical in Figure 2.1): it will be described as directional mass in order to distinguish it from the usual concept of mass having the same property in all the directions. Its kinetic energy T is written according to the speed u!i of the mass in the considered direction:

T=

[

1 1 m u! i2 = u! i 2 2

⎡m 0⎤ ⎡ u! i ⎤ u! j ⎢ ⎥⎢ ⎥ ⎣ 0 0⎦ ⎣u! j ⎦

]

[2.1]

– the stiffness k of the spring which also acts only in the direction of motion. Its elastic energy U is written as follows:

U=

[

1 1 k (u i − u j ) 2 = ui 2 2

]

⎡k uj ⎢ ⎣− k

− k⎤ k ⎥⎦

⎡ ui ⎤ ⎢u ⎥ ⎣ j⎦

[2.2]

– the constant c of the damper which also acts only in the direction of motion. In the event of a viscous damping where the force is proportional to the speed with the proportionality coefficient c, its dissipation power D is written as follows:

D=

[

1 1 c (u! i − u! j ) 2 = u! i 2 2

⎡ c − c ⎤ ⎡ u! i ⎤ u! j ⎢ ⎥⎢ ⎥ ⎣− c c ⎦ ⎣u! j ⎦

]

[2.3]

The motion can be defined by an energy-based equation, according to the considerations of section 1.5.1. This equation will then be solved progressively, starting with motion in the absence of excitation in order to introduce the modal approach on this basic case, as indicated in section 1.5.3. Motion expressed by the equation and its solution in the frequency domain, according to the considerations of sections 1.3 and 1.4, is the subject of section 2.2. The characteristics of the time domain, including the concept of response spectrum to characterize an environment, will then be covered in section 2.3. Finally, the problem of the representation of dissipation phenomena, made up until now with a viscous damping, will be tackled in section 2.4.

The Single-Degree-of-Freedom System

35

2.2. The equation of motion and the solution in the frequency domain 2.2.1. Equations of motion The considerations of section 1.5.1 lead to the equation of motion: ⎡m 0⎤ ⎡ u!!i ⎤ ⎡ c − c ⎤ ⎡ u!i ⎤ ⎡ k ⎢ 0 0⎥ ⎢u!! ⎥ + ⎢− c c ⎥ ⎢u! ⎥ + ⎢− k ⎣ ⎦ ⎣ j⎦ ⎣ ⎦ ⎣ j⎦ ⎣

− k⎤ k ⎥⎦

⎡ ui ⎤ ⎡ Fi ⎤ ⎢u ⎥ = ⎢ F ⎥ ⎣ j⎦ ⎣ j⎦

[2.4]

or by developing: m u!!i + c u! i + k u i = Fi + c u! j + k u j

[2.5]

− c (u! i − u! j ) − k (u i − u j ) = F j = − Fi + m u!!i

[2.6]

Equation [2.5] makes it possible to solve ui knowing Fi and uj, whereas equation [2.6] restores Fj knowing ui more. When the excitations in force Fi and in displacement uj are separated, equation [2.5] is reduced to: Fi alone: m u!!i + c u! i + k u i = Fi

[2.7]

uj alone: m (u!!i − u!! j ) + c (u! i − u! j ) + k (ui − u j ) = −m u!! j

[2.8]

Equation [2.7] is the traditional equation of motion of the system limited to DOF i (DOF j blocked). Equation [2.8], corresponding to the system excited only by its base, shows that this case is equivalent to the preceding case, ui being replaced by the relative displacement (ui − u j ) and Fi by the inertia force (−m u!! j ) .

2.2.2. Motion without excitation 2.2.2.1. The conservative system The analysis of the conservative system (i.e. preserving its energy, thus undamped) in the absence of excitation will introduce a first dynamic characteristic of the system: its natural frequency.

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Structural Dynamics in Industry

The equation of motion [2.5] in the absence of excitation, thus with Fi = 0 and uj = 0, and without damping, thus with c = 0, is reduced to: m u!!i + k u i = 0

[2.9]

If we assume, as seen in section 1.2.1.1: ui (t ) = ℜ(u i (t ) ) + i u i (t ) = U i e iω t

[2.10]

the following equation is obtained:

(− ω

2

)

m + k Ui = 0

[2.11]

with the unique solution:

ω = ω k = 2π f k =

k m

Ui being arbitrary

[2.12]

Subscript k introduced in section 1.5.3 prefigures the general case, as for subscripts i and j. It is unnecessary to underline it since it can take only one value. We can then write: ui (t ) = U i sin (ω k t + ϕ i )

[2.13]

the amplitude U i and the phase ϕi being arbitrary. This result means that, in the absence of excitation, the undamped system vibrates all alone at frequency f k with an arbitrary amplitude and phase. f k is called a natural frequency ( ω k circular natural frequency): it is a characteristic of the system since it depends only on itself and not on the excitation. In order to come back to the concept of normal modes introduced in section 1.5.4, it can be said that the system with 1-DOF has only one normal mode k, of frequency f k , whose component on the single DOF is arbitrary.

The Single-Degree-of-Freedom System

37

2.2.2.2. Dissipative system The analysis of the dissipative system, i.e. damped and in the absence of excitation, will introduce a second dynamic characteristic of the system: its damping. The equation of motion [2.5] in the absence of excitation, therefore with Fi = 0 and uj = 0, but with a viscous damping, therefore c ≠ 0, is written as follows: m u!!i + c u! i + k u i = 0

[2.14]

The same approach as for the conservative system leads to the equation:

(− ω

2

)

m + iω c + k U i = 0

[2.15]

The solution is now complex. If we assume:

ζk =

c

[2.16]

2 km

which makes it possible to rewrite equation [2.15] with the circular natural frequency ω k of the conservative system:

(− ω

2

)

+ iω 2 ζ k ω k + ω k 2 U i = 0

[2.17]

assuming ζ k < 1 , the following solution is obtained:

ω = ± ω k 1 − ζ k2 + iζ k ω k

Ui being arbitrary

[2.18]

which, reintroduced in equation [2.10], makes it possible to write: u i (t ) = U i e −ζ k ω k t sin ⎛⎜ ω k 1 − ζ k2 t + ϕ i ⎞⎟ ⎝ ⎠

[2.19]

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Structural Dynamics in Industry

This result means that in the absence of excitation, the damped system vibrates according to a damped sinusoid, which will be specified. The term ζ k is called the viscous damping factor: it is dimensionless and much smaller than 1 for most structures, i.e. for structures without damping devices. In certain very particular cases, which will not be considered hereafter, this term can exceed the value 1 (overdamped case; see the majority of references in dynamics [CLO 75, FER 95, GER 92, etc.]) and solution [2.18] no longer applies. ζ k is a second characteristic of the system, along with frequency f k , relating to the dissipation of energy. The term ω k 1 − ζ k2 is called the damped circular natural frequency of the

system and remains close to ω k when ζ k is much smaller than 1. The motion is thus a damped sinusoid of damped circular natural frequency

ω k 1 − ζ k2 (addressed again in section 2.3.1). We can check that the logarithmic decrement Δ, logarithm of the ratio between two successive maxima is directly related to the reduced damping ζ k by: Δ=

2π ζ k 1 − ζ k2

≈ 2π ζ k

[2.20]

which allows a simple evaluation of ζ k using the motion in the absence of excitation. We will retain from this development the two characteristics of the system which can be described as modal parameters: – the natural frequency f k of the associated conservative system; – the viscous damping factor ζ k . Note that the solution of equation [2.18] of the dissipative system contains at the same time the information of the frequency (mainly in its real part) and that of the damping (mainly in its imaginary part). This point will be discussed again in Chapter 7 for the study of the so-called “complex” modes of the structures with N DOF.

The Single-Degree-of-Freedom System

39

Also note that it is possible to solve equation [2.14] in Laplace-transform variable, like most of the basic works in dynamics (iω replaced by p complex in equation [2.10]). The advantage of the present approach is to directly interpret the dissipative system as a perturbed conservative system. 2.2.3. Solution in the frequency domain

In the frequency domain, where the forces and displacements are of the form X (ω ) e iω t , equations [2.5] and [2.6] lead to: (−ω 2 m + iω c + k ) u i (ω ) = Fi (ω ) + (iω c + k ) u j (ω )

[2.21]

F j (ω ) = −(iω c + k ) (u i (ω ) − u j (ω )) = − Fi (ω ) − ω 2 m u i (ω )

[2.22]

By introducing parameters ω k and ζ k in equation [2.21] (these parameters were highlighted in section 2.2.2), we obtain: m (−ω 2 + i 2 ζ k ω k ω + ω k 2 ) ui (ω ) = Fi (ω ) + m (i 2 ζ k ω k ω + ω k 2 ) u j (ω ) [2.23]

If we take the general form [1.43] between excitations and responses again, which presents only scalar functions here: ⎡ u i (ω ) ⎤ ⎡ Gii (ω ) Tij (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎢ F (ω )⎥ = ⎢− T (ω ) K (ω )⎥ ⎢u (ω )⎥ jj ⎦⎣ j ⎣ j ⎦ ⎣ ji ⎦

[2.24]

the comparison of equations [2.23] and [2.24] leads to the following expressions of dynamic flexibility Gii (ω ) and of the dynamic transmissibility in displacements Tij (ω ) : Gii (ω ) =

1 m (−ω + i 2 ζ k ω k ω + ω k 2 ) 2

[2.25]

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Structural Dynamics in Industry

Tij (ω ) =

i 2ζ kω k ω + ω k 2 −ω 2 + i 2ζ kω k ω +ω k 2

[2.26]

At this level, it is interesting to rewrite these expressions by making dimensionless quantities appear: Gii (ω ) =

Tij (ω ) =

1

⎛ ω 1 − ⎜⎜ ⎝ωk

⎞ ⎟ ⎟ ⎠

2

1 ⎛ ω ⎞ k ⎟ + i 2 ζ k ⎜⎜ ⎟ ⎝ωk ⎠

⎛ ω 1 + i 2 ζ k ⎜⎜ ⎝ωk ⎛ ω 1 − ⎜⎜ ⎝ωk

2

⎞ ⎟ ⎟ ⎠

⎞ ⎛ ω ⎟ + i 2ζ k ⎜ ⎟ ⎜ω ⎠ ⎝ k

⎞ ⎟ ⎟ ⎠

[2.27]

[2.28]

Equation [2.27] shows that the dynamic flexibility is put under the form of a product of two terms: – term 1/k which can be interpreted as the static flexibility of the system (ω = 0), i.e. the ratio between a static force Fi imposed on the DOF i and the generated static displacement ui; – a frequency-dependent dimensionless term, which one can interpret as a factor representing the dynamic effect, and which will be analyzed later on. Equation [2.28] shows that the dynamic transmissibility in displacement is put in a similar form by considering it as a product of two terms: – term 1 which can be interpreted as the static transmissibility in displacement of the system (ω = 0), i.e. the ratio between the static displacement uj imposed on the DOF j and the static displacement ui transmitted to the DOF i; – a frequency-dependent dimensionless term, different from the preceding one, which can also be interpreted as a factor representing the dynamic effect, and which will also be analyzed later on.

The Single-Degree-of-Freedom System

41

Exploiting equation [2.22] by taking equation [2.23] and its comparison to equation [2.24] into account leads to the following expressions of the dynamic transmissibility in forces T ji (ω ) and of the dynamic stiffness K jj (ω ) :

T ji (ω ) =

⎛ ω 1 + i 2 ζ k ⎜⎜ ⎝ωk ⎛ ω 1 − ⎜⎜ ⎝ωk

K jj (ω ) =

⎞ ⎟ ⎟ ⎠

2

⎞ ⎛ ω ⎟ + i 2ζ k ⎜ ⎟ ⎜ω ⎠ ⎝ k

⎛ ω 1 + i 2 ζ k ⎜⎜ ⎝ωk ⎛ ω 1 − ⎜⎜ ⎝ωk

[2.29]

⎞ ⎟ ⎟ ⎠

⎞ ⎟ ⎟ ⎠

2

⎞ ⎛ ω ⎟ + i 2ζ k ⎜ ⎟ ⎜ω ⎠ ⎝ k

⎞ ⎟ ⎟ ⎠

(−ω 2 m)

[2.30]

Equation [2.29] shows that the dynamic transmissibility in force is the same as that in displacement [2.28], by reciprocity, the implied term 1 being interpreted here as the static transmissibility in force of the system, i.e. the ratio between a static force Fi imposed on the DOF i and the static force − F j (ω ) transmitted to the DOF j. As for equation [2.30], the presence of the term − ω 2 encourages the introduction of the dynamic mass M jj (ω ) = K jj (ω ) /(−ω 2 ) (see Table 1.3):

M jj (ω ) =

⎛ ω 1 + i 2 ζ k ⎜⎜ ⎝ωk ⎛ ω 1 − ⎜⎜ ⎝ωk

2

⎞ ⎟ ⎟ ⎠

⎛ ω ⎞ ⎟ + i 2ζ k ⎜ ⎜ω ⎟ ⎝ k ⎠

⎞ ⎟ ⎟ ⎠

m

[2.31]

Equation [2.31] shows that the dynamic mass is still put under the form of a product of two terms: – term m which can be interpreted as the static mass of the system (ω = 0), i.e. the ratio between a static acceleration u!! j = −ω 2 u j imposed on the DOF j and the static reaction force Fj to DOF j; – a frequency-dependent dimensionless term, the same one as for the dynamic transmissibilities.

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Structural Dynamics in Industry

Finally, we will retain the results of equations [2.27], [2.28], [2.29] and [2.31] under the form:

Gii (ω ) = H k (ω )

1 k

[2.32]

Tij (ω ) = T ji (ω ) = Tk (ω )

[2.33]

M jj (ω ) = Tk (ω ) m

[2.34]

with: H k (ω ) =

1 ⎛ ω 1 − ⎜⎜ ⎝ωk

2

⎛ ω ⎞ ⎟ + i 2ζ k ⎜ ⎜ω ⎟ ⎝ k ⎠

⎞ ⎟ ⎟ ⎠

[2.35]

called the dynamic amplification factor, and:

Tk (ω ) =

⎛ ω 1 + i 2 ζ k ⎜⎜ ⎝ωk ⎛ ω 1 − ⎜⎜ ⎝ωk

2

⎞ ⎟ ⎟ ⎠

⎛ ω ⎞ ⎟ + i 2ζ k ⎜ ⎜ω ⎟ ⎝ k ⎠

⎞ ⎟ ⎟ ⎠

[2.36]

called the dynamic transmissibility factor. Thus, among all the possible frequency response functions listed in section 1.4.4, Gii (ω ) , Tij (ω ) = T ji (ω ) and M jj (ω ) have the same form of a product of two terms: – a frequency-independent term of the same dimension as the considered FRF, 1/k for G, 1 for T, m for M, which can be described as a static parameter (ω = 0); – a frequency-dependent dimensionless term, H k (ω ) or Tk (ω ) according to the FRF, subscript k indicating the single normal mode of the system as introduced in section 1.5.4 and connected to the system with 1 DOF as indicated in section 1.5.5. These two functions, H k (ω ) and Tk (ω ) , connected to the concept of dynamic amplification are worth being analyzed in detail, and this is done in section 2.2.4.

The Single-Degree-of-Freedom System

43

We will notice that the two functions H k (ω ) and Tk (ω ) are related by the relation: ⎛ ω Tk (ω ) − 1 = ⎜⎜ ⎝ωk

2

⎞ ⎟ H k (ω ) ⎟ ⎠

[2.37]

and equation [2.23] makes it possible to write the relative displacement ui – uj for an excitation uj under the following form, which can be compared to equation [2.8]: u i (ω ) − u j (ω ) = (Tk (ω ) − 1) u j (ω ) = H k (ω )

u!! j (ω ) −ωk 2

[2.38]

2.2.4. Dynamic amplifications

The two functions H k (ω ) and Tk (ω ) defined by equations [2.35] and [2.36] can be interpreted as an amplification (generally accepted term but with some conditions which will be clarified further) of the static properties of the system, due to its dynamic behavior. They depend only on ω k and ζ k , the two parameters of the system, which were defined in section 2.2.2 and which are very similar outside the high frequency range. They are represented in amplitude/phase in Figure 2.2, with logarithmic scales for the amplitude (see the note in section 1.2.2) and are detailed as indicated below. The amplitude represents the so-called amplification of the dynamic phenomenon compared to the static phenomenon: – it is close to 1 at frequencies much smaller than the natural frequency fk = ωk/2π : the behavior is quasi-static, determined mainly by the spring; – when the frequency increases, the amplification increases gradually until it passes by a maximum in the vicinity of ω = ωk where it is commonly said that the system is in resonance. This maximum has approximately the value Qk = 1/(2ζk). Qk is the amplification at resonance, sometimes called the quality factor (electrical analogy), hence its notation. This resonance is thus directly determined by the damper. Lightly damped structures typically have Qk from 5 to 50 depending on the nature of materials and especially the joints. It is in this important amplification, which can generate large displacements or forces (and therefore large stresses), that the danger corresponding to the dynamic behavior lies;

44

Structural Dynamics in Industry

– beyond ωk the amplification attenuates quickly. It becomes less than 1 starting from ωk/ 2 (exactly for H k (ω ) , approximately for Tk (ω ) ), which, strictly speaking, is not an amplification but an attenuation (the amplification term is, however, generally preserved because it corresponds to most of the applications). This attenuation varies in 1/ω 2 for H k (ω ) , 1/ω 2, then 1/ω for Tk (ω ) which differs from H k (ω ) only at these high frequencies. The behavior here is mainly determined by the mass. The role of the suspension is present here: the excitation is filtered by the system when its frequency is sufficiently large when compared to its natural frequency. The phase represents the phase difference between excitation and response, which is always negative here (the response follows the excitation): – it is close to 0 at frequencies which are much lower than the natural frequency fk = ωk/2π : excitation and response are nearly in phase; – it rapidly decreases towards ωk while passing by -π/2 at the resonance: excitation and response are in quadrature; – beyond ωk it rapidly reaches -π : excitation and response are out of phase. At high frequencies, Tk (ω ) is different from H k (ω ) with a phase gradually returning towards -π/2. With regard to factor H k (ω ) , it intervenes in the dynamic flexibility Gii (ω ) but also in the derived FRF which are the admittance Yii (ω ) = iω Gii (ω ) and the accelerance Aii (ω ) = −ω 2 Gii (ω ) (see Table 1.3), which can be written as: Yii (ω ) =

Aii (ω ) =

iω 1 H k (ω ) ωk km

−ω 2

ωk

2

H k (ω )

1 m

[2.39]

[2.40]

The Single-Degree-of-Freedom System

ϕ

45

ϕ

Hk

Tk

ζ k = 1%

ζ k = 1%

ω /ωk

ω /ωk

H k (ω ) Tk (ω ) Figure 2.2. Dynamic amplification and transmissibility factors (amplitudes and phases)

H k (ω ) and the two dimensionless factors appearing in equations [2.39] and

[2.40], (iω / ω k ) H k (ω ) and − (ω / ω k ) 2 H k (ω ) , are plotted in Figure 2.3. We will notice the perfect symmetry of the second and the reflective symmetry between the other two. For these three factors, the maximum of the amplitude is reached in the vicinity of ω = ωk, more precisely: H k (ω ) max =

iω

ωk

2ζ k 1 − ζ k

H k (ω ) max =

−ω 2

ωk

1

2

2

1 2ζ k

H k (ω ) max =

1 2ζ k 1 − ζ k

2

for

ω = 1 − 2ζ k 2 ωk

[2.41]

for

ω =1 ωk

[2.42]

for

ω 1 = ωk 1 − 2ζ k 2

[2.43]

46

Structural Dynamics in Industry

Hk (

ω ) ωk

i

ω ω Hk ( ) ωk ωk

⎛ ω − ⎜⎜ ⎝ωk

2

⎞ ω ⎟ Hk ( ) ⎟ ω k ⎠

Figure 2.3. Factor H k (ω ) and its variants (amplitudes and phases ζk = 1%, 10%, 20%)

The same analysis can be made with factor Tk (ω ) , which intervenes in the dynamic transmissibilities Tij (ω ) = T ji (ω ) , the dynamic mass M jj (ω ) and the derived FRF which are the impedance Z jj (ω ) = iω M jj (ω ) and the stiffness K jj (ω ) = −ω 2 M jj (ω ) (see Table 1.3), which can be written as:

Z jj (ω ) =

K jj (ω ) =

iω

ωk

Tk (ω ) k m

−ω 2

ωk 2

H k (ω ) k

[2.44]

[2.45]

Tk (ω ) and the two dimensionless factors appearing in equations [2.44] and

[2.45], given (iω / ω k ) Tk (ω ) and − (ω / ω k ) 2 Tk (ω ) , are plotted in Figure 2.4. The symmetries observed for H k (ω ) are not present here.

The Single-Degree-of-Freedom System

Tk (

ω ) ωk

i

ω ω Tk ( ) ωk ωk

⎛ ω − ⎜⎜ ⎝ωk

47

2

⎞ ω ⎟ Tk ( ) ⎟ ωk ⎠

Figure 2.4. Factor Tk (ω ) and its derivatives (amplitudes and phases ζk = 1%, 10%, 20%)

For these three factors, the maximum of the amplitude is also reached in the vicinity of ω = ωk but its analysis is more complex than for H k (ω ) . Let us simply specify that of Tk (ω ) : Tk (ω ) max =

for

ω = ωk

1 1 + 5ζ k 2 + … 2ζ k

1 + 8ζ k 2 − 1 4ζ k

2

= 1 + 2ζ k 2 + …

[2.46]

When ζk is small, all the maxima are close to Qk = 1/(2ζk). For values closer to 1, this approximation degrades. Table 2.1 gives the values of the maximum of H k (ω ) and Tk (ω ) (amplifications at resonance) when ζk varies, in order to evaluate the error made when using the above approximation.

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Structural Dynamics in Industry

H k (ω ) max for ω / ωk

Tk (ω ) max for ω / ωk

ζk

1/(2ζk)

0.00

∞

0.05

10.00

10.0125 0.9975

10.0622 0.9975

0.10

5.00

5.0252

0.9899

5.1228

0.9903

0.20

2.50

2.5516

0.9592

2.7339

0.9647

0.30

1.67

1.7471

0.9055

1.9946

0.9302

0.40

1.25

1.3639

0.8246

1.6550

0.8926

0.50

1.00

1.1547

0.7071

1.4679

0.8556

∞

1.0

∞

1.0

Table 2.1. Amplifications at resonance for H k (ω ) and Tk (ω )

For all the factors derived from H k (ω ) or Tk (ω ) , which can be seen in Figures 2.3 and 2.4, the sharpness of the peak only depends on parameter ζk which gives a means to evaluate it whatever the considered FRF may be. The simplest method consists of using the “half-power” frequencies, i.e. those corresponding to a level equal to the maximum divided by 2 , as shown in Figure 2.5. When ζk is small, using equations [2.35] or [2.36] makes it possible to write: Qk ≈

1 2ζ k

≈

ωk Δω k

[2.47]

Δω k being the half-power interval.

In practice, this estimation is not very accurate because the sampling of the curve makes one underestimate the value of the maximum. It can be improved by using all the nearby frequencies and a more accurate expression that will now depend on the considered type of FRF. This point will be discussed again in Chapter 11 in the context of modal identification. The preceding considerations apply directly to a sine excitation of a given frequency. The case of a random excitation is to be examined in the light of section 1.4.2.

The Single-Degree-of-Freedom System

H k (ω )

10

49

2

or

Tk (ω ) Δω k

ωk

1 0.5

1

2

Figure 2.5. Evaluation of the damping with half-power frequencies (plotted with Hk(ω) and ζk = 5%)

2.2.5. Response to a random excitation

In the case of a random excitation x defined by its PSD S xx (ω ) (see section 1.2.3.4), the response y defined by its PSD S yy (ω ) is deduced by equation [1.39] knowing the transfer function H yx (ω ) . The extension to several excitations x and responses y is treated by equation [1.42]. Let us postulate here the uncoupled excitations in order to examine the excitation in force and in motion separately. According to equations [2.32] and [2.34] we obtain the following: S uiui (ω ) =

1 k

S F j F j (ω ) =

S uiui (ω ) =

2

2

[2.48]

Tk (ω ) S Fi Fi (ω )

[2.49]

H k (ω ) S Fi Fi (ω ) 2

2

Tk (ω ) S u j u j (ω ) 2

[2.50] 2

S F j F j (ω ) = m 2ω 4 Tk (ω ) S u j u j (ω ) = m 2 Tk (ω ) S u!!j u!!j (ω )

[2.51]

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Structural Dynamics in Industry

The RMS values of the responses are obtained by the integration of the PSDs, according to relations [1.23] or [1.24]. For PSDs of constant excitations (white noise) H k (ω )

2

or Tk (ω )

2

must be integrated which can be done using the

general expressions, indicated in [CRA 63]:

+∞ ∫−∞

2

iω B1 + B0 − ω 2 A2 + iω A1 + A0

dω = π

A2 B0 2 / A0 + B1 2 A1 A2

[2.52]

which, for light damping (2ζk ≈ 1/Qk << 1), leads to the following approximations to be used for the “physical” PSDs W introduced in section 1.2.3.4: 2

2

π

+∞ +∞ ∫0 H k (ω ) dω ≈ ∫0 Tk (ω ) dω ≈ 2 f k Qk

[2.53]

Relation [2.53] is to be used in order to calculate the RMS values relative to responses [2.48] to [2.51] with constant excitation PSDs. The following notes may be added: – since most of the integral comes from the resonance, we can assume a slight variation in the PSD W(f) in the vicinity of the resonance and therefore use the value W(fk); – the integral of the factors derived from H k (ω ) or from Tk (ω ) , which appear in relations [2.39], [2.40], [2.44] and [2.45] and which can be seen in Figures 2.3 and 2.4, leads to the following results: - for the product of H k (ω ) or Tk (ω ) by iω: the integral is convergent with H k (ω ) , but it remains limited with Tk (ω ) only by integrating up to a frequency fmax. Supposing that this frequency is small compared to fk:

+∞ ∫0

iω

ωk

2

f

H k (ω ) dω ≈ ∫0 max

iω

ωk

2

Tk (ω ) dω ≈

π 2

f k Qk

[2.54]

The Single-Degree-of-Freedom System

51

- for the product of H k (ω ) or Tk (ω ) by -ω2: the integral remains limited only by integrating up to a frequency fmax. Supposing that this frequency is not large compared to fk for Tk (ω ) :

f ∫0 max

2

−ω 2

f

H k (ω ) dω ≈ ∫0 max

ωk

−ω 2

ωk

2

Tk (ω ) dω ≈

π 2

f k Q k + f max

[2.55]

Relations [2.48] to [2.55] thus give the following results (Wxx written simply as Wx, and assumed to be nearly constant in the vicinity of fk and possibly limited to fmax): ui 2 ≈

1 π f k Qk W Fi ( f k ) k2 2

π

Fj2 ≈ ui 2 ≈

2

π

F j 2 ≈ m2

1 ⎛π ⎞ ⎜ f Q + f max ⎟ W Fi ( f k ) [2.56] 2 ⎝2 k k ⎠ m

[2.57]

f k Qk W Fi ( f k ) f k Q k Wu j ( f k )

2

u!!i 2 ≈

u!!i 2 ≈

π 2

f k Qk Wu!! j ( f k )

⎞ ⎛π f k Q k Wu!!j ( f k ) ≈ k 2 ⎜ f k Qk + f max ⎟ Wu j ( f k ) 2 ⎠ ⎝2

π

[2.58]

[2.59]

Relation [2.58] with the accelerations is known best by its simplicity and its practical applications. The others are logically deduced from it. It should be noted that all the RMS values are approximately proportional to Q k , instead of Qk for a sine excitation at the resonance. 2.3. Time responses 2.3.1. Response to unit impulse

The response to unit impulse is the base of any transient response calculation (see section 1.3.2). It is the inverse Fourier transform of the frequency response function (see section 1.3.3). Thus, we can consider the FRF in section 2.2.3 again in order to find the corresponding responses, an alternative to direct integration.

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Structural Dynamics in Industry

First, the dynamic amplifications H k (ω ) and Tk (ω ) of relations [2.35] and [2.36] give the following expressions (FT-1 inverse Fourier transform):

1

FT−1 ( H k (ω)) = hk (t ) = ωk e−ζ k ωk t

1− ζ k

2

(

sin ωk 1− ζ k 2 t

)

FT−1 (Tk (ω)) = tk (t ) = −hk (t ) / ωk 2 = ⎡

1− 2ζ k ⎛ ω k e −ζ k ω k t ⎢ sin ⎜ ω k 1 − ζ k 2 ⎢ 2 ⎝ ⎣⎢ 1 − ζ k 2

[2.60]

[2.61] ⎞ ⎛ t ⎟ + 2ζ k cos⎜ ω k 1 − ζ k 2 ⎠ ⎝

⎤ ⎞ t ⎟⎥ ⎠⎥ ⎦⎥

These two functions of time are plotted in Figure 2.6. We obtain in equation [2.60] the damped sinusoid of section 2.2.2.2 within a coefficient. hk (t )

1

ζ k = 1%

ωk

10% 20%

ωk t 2π

0

-1 t k (t )

0

1

1

2

ζ k = 1%

ωk

10% 20%

ωk t 2π

0

-1

0

1

Figure 2.6. Functions hk(t) and tk(t)

2

The Single-Degree-of-Freedom System

53

The maxima are reached approximately at 1/4 of a period t = π /( 2 ω k ) . More exactly: hk max = ω k e −ζ k ω k t

for

ωk t =

1 1−ζ k

2

Arc tan

1−ζ k 2

[2.62]

ζk

t k max ≈ ω k e −ζ k π / 2 ≈ ω k (1 − ζ k π / 2)

for

[2.63]

⎛ 1 − ζ 2 1 − 4ζ 2 k k Arc tan ⎜ ⎜ ζ k 3 − 4ζ 2 2 k 1−ζ k ⎝

1

ωk t =

⎞ ⎟ ⎟ ⎠

For values of ζk closer to 1, the approximations lose accuracy. Table 2.2 gives the values of the maximum of hk (t ) and t k (t ) as a function of ζk.

ζk

e

−ζ k

π

hk max

2

ωk

for

2

π

ωk t

t k max

ωk

for

2

π

ωk t

0.00

1.0000

1.0000

1.0000

1.0000

1.0000

0.05

0.9245

0.9267

0.9693

0.9313

0.9055

0.10

0.8546

0.8626

0.9406

0.8801

0.8119

0.20

0.7304

0.7561

0.8872

0.8209

0.6219

0.30

0.6242

0.6715

0.8358

0.8132

0.4231

0.40

0.5335

0.6025

0.7824

0.8632

0.2153

0.50

0.4559

0.5447

0.7225

1.0000

0.0000

Table 2.2. Maxima for hk(t) and tk(t)

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Structural Dynamics in Industry

Starting from relations [2.60] and [2.61], the following unit impulse responses are deduced according to relations [2.32] to [2.34]: – response to a unit impulse force Fi at t = 0: - displacement ui(t): u i (t ) = hk (t )

1 1 e −ζ k ω k t = k mωk

1 1−ζ k

2

⎛ ⎞ sin ⎜ ω k 1 − ζ k 2 t ⎟ ⎝ ⎠

[2.64]

- reaction Fj(t): F j (t ) = − t k (t ) =

[2.65]

⎡ 1− 2ζ k 2 ⎛ sin ⎜ ω k 1 − ζ k 2 − ω k e −ζ k ω k t ⎢ ⎢ 2 ⎝ ⎢⎣ 1 − ζ k

⎞ ⎛ t ⎟ + 2ζ k cos⎜ ω k 1 − ζ k 2 ⎠ ⎝

⎤ ⎞ t ⎟⎥ ⎠⎥ ⎥⎦

– response to a unit impulse acceleration üj, i.e. an initial velocity of u! j = 1 at

t = 0: - acceleration üi(t): u!! j (t ) = t k (t ) =

ωk e

−ζ k ω k t

[2.66] ⎡ ⎤ 2 ⎢ 1 − 2 ζ k sin ⎛⎜ ω 1 − ζ 2 t ⎞⎟ + 2ζ cos⎛⎜ ω 1 − ζ 2 t ⎞⎟⎥ k k k k k ⎢ 2 ⎝ ⎠ ⎝ ⎠⎥ ⎢⎣ 1 − ζ k ⎥⎦

It should be noted that the relative displacement according to relation [2.38] is given by: u i (t ) − u j (t ) = hk (t )

1 −ωk 2

=

[2.67] −

1

ωk

e −ζ k ω k t

1 1−ζ k

2

⎛ ⎞ sin ⎜ ω k 1 − ζ k 2 t ⎟ ⎝ ⎠

The Single-Degree-of-Freedom System

55

- reaction Fj(t): F j (t ) = t k (t ) m =

[2.68]

⎡ ⎤ 1− 2ζ k 2 ⎛ ⎞ ⎛ ⎞ m ω k e −ζ k ω k t ⎢ sin ⎜ ω k 1 − ζ k 2 t ⎟ + 2ζ k cos⎜ ω k 1 − ζ k 2 t ⎟⎥ ⎢ 2 ⎝ ⎠ ⎝ ⎠⎥ ⎥⎦ ⎣⎢ 1 − ζ k

2.3.2. Response to a general excitation

According to section 1.3, any time response can be obtained: – either by passing into the frequency domain starting from the frequency response function of the system, according to strategy [1.36]. The excitation will then be represented by its Fourier transform. This one can be analytically obtained in some simple cases (for example, see [LAL 02] volume 2) or numerically. Thus, this strategy requires two integrations in general, and this diminishes the interest for it; – or directly in the time domain starting from the system response to unit impulse, using the Duhamel integral [1.30]. In this latter case, we will write: – response to a force Fi: - displacement ui(t): u i (t ) =

1 t ∫ Fi (τ ) hk (t − τ ) dτ k −∞

[2.69]

- reaction Fj(t): t

F j (t ) = − ∫−∞ Fi (τ ) t k (t − τ ) dτ

[2.70]

– response to an acceleration üj: - acceleration üi(t): t u!!i (t ) = ∫−∞ u!! j (τ ) t k (t − τ ) dτ

[2.71]

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Structural Dynamics in Industry

It should be noted that the relative displacement equals, according to relation [2.38]:

u i (t ) − u j (t ) = −

m t ∫ u!! j (τ ) hk (t − τ ) dτ k −∞

[2.72]

- reaction Fj(t): t

F j (t ) = m ∫−∞ u!! j (τ ) t k (t − τ ) dτ

[2.73]

In the cases where the excitations in force Fi and in acceleration üj superpose, we obtain: – displacement ui(t): u i (t ) =

1 t ∫ (Fi (τ ) − m u!!(τ ) ) hk (t − τ ) dτ + u j (t ) k −∞

[2.74]

– reaction Fj(t):

(

)

t F j (t ) = − ∫−∞ Fi (τ ) − m u!! j t k (t − τ ) dτ

[2.75]

2.3.3. Response spectra

As mentioned in Chapter 1, the 1-DOF system is used as a reference structure for the characterization of environments. This characterization responds to the need to quantify the severity of an environment according to different criteria, independently of the considered structure so that comparisons between different environments are allowed or equivalent environments can be specified. In practice, the criterion used for the severity is the maximum response which occurs on the structure, related to the notion of damage by plastic deformation or by failure. This maximum is deduced directly from the calculations of the time response which can be made on the structure and provides what is called the response spectrum (with the condition that will be mentioned later).

The Single-Degree-of-Freedom System

57

We should mention at this point another severity criterion complementary to the preceding one, and related to the concept of fatigue damage. It is based on the number of response cycles, associated with the endurance curve of the implied materials, in order to give the fatigue damage spectrum. This point, which implies various considerations on the behavior of materials, is not part of this work and it will not be developed here. [LAL 02] volume 4 can be consulted on this subject. If we return to the response spectrum, as it depends on the properties of the considered structure and yet the characterization of the environment should be independent of the structure, it is necessary to resort to a reference structure. The 1DOF system is perfectly suitable to fulfill this role: it is the simplest dynamic system, which depends on a minimum of physical parameters. Thus, the considered environment will be characterized by the response maxima of the 1-DOF system according to its natural frequency fk. The developments of section 2.3.2 show that the damping ζk also intervenes. However, this second parameter has a more limited influence other than at resonances under a sine excitation, which leads to using a constant value, for example, ζk = 5%, or Qk = 10. This response spectrum which, for an environment of a short duration, or shock, is equivalent to the shock spectrum, is defined by the maximum response of a 1DOF system according to its natural frequency fk and for a given damping ζk. Now it is necessary to specify the responses and excitations. It will be noted, with equations [2.7] and [2.8], that, if the excitation is a force, its absolute displacement spectrum is the same as the relative displacement spectrum with the force −m u!! j . Having said this, the considered excitation and response are generally motions, as illustrated in Figure 2.7. We can thus consider the various types of responses: – the displacement u i , the velocity u! i or the acceleration u!!i ; – the absolute response xi , or the relative response ( x i − x j ) ; – the positive part (positive spectrum) or the negative part (negative spectrum); – the response during the excitation (primary spectrum) or after the excitation (residual spectrum), in the case of a limited duration transient.

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ui

m k

S

+

t

c

+

ζ =ζk

⇒

ωk

uj

t Figure 2.7. Principle of the response spectrum or the shock spectrum

It will be noted that the pseudo-velocity S (u! i ) and the pseudo-accelerations S (u!!i ) spectra defined starting from the displacement spectrum S (u i ) by:

S (u!!i ) = ω S (u! i ) = ω 2 S (u i )

[2.76]

are good approximations of the velocity spectra S (u! i ) and of the acceleration spectra S (u!!i ) for transients which strongly oscillate, as in the case of a pure sine for which the two spectra are identical. The following spectra are of particular interest: – the relative displacement spectrum, which provides information about the strains, and thus the stresses, related to the properties of resistance of material. Some people use the term response spectrum for the relative displacement spectrum multiplied by ω 2 in order to make it homogenous to an acceleration; – the absolute acceleration spectrum, which provides information about the maximum internal forces, and thus also about the stresses via adequate surfaces. The difference between this and the preceding one is the intervention of the function t k (t ) instead of hk (t ) in the calculation, as shown by equations [2.71] and [2.72]. Thus, the lighter the damping is, the smaller this difference is, and it is often negligible in practice. The importance of absolute acceleration in the use of response spectra for the N-DOF systems will be specified in section 5.4.4. The impulse is the simplest transient to be treated: section 2.3.1 makes it possible to establish the analytical solution. Moreover, a 1-DOF system of a very low frequency feels any shock as an impulse (except if its integral is zero): the time to react, the shock is finished and only the integral of the shock then counts. Relations

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59

[2.62] and [2.63] relating to hk (t ) and t k (t ) indicate maxima, which are approximately proportional to ω k for a unit impulse and light damping. Thus, in the case of an acceleration impulse of amplitude

∫ u!! j dt = u! 0 , the shock spectrum of

response u!!i equals approximately ω k u! 0 . This linear behavior constitutes the initial slope of the spectrum of any shock in the acceleration of the integral u! 0 . For a shock of any acceleration, the profile of its spectrum is generally as follows: – the initial slope is given by the integral u! 0 for the previously cited reasons; – then the curve inflects and passes by a series of maxima corresponding to the frequencies which are most represented in the transient; – finally, the curve tends progressively towards a constant given by the shock maximum. The response follows the excitation profile exactly at high frequencies. 2

S

u!! j

u!!max

u!!max

τ

1 0

−1 −2

0

1

2

3

4

τω 5 2π

Figure 2.8. Response spectrum of a half-sine acceleration:

S / u!!max according to τω/(2π) for ζk = 0, 5, 10, 20 and 50%

As an illustration, Figure 2.8 gives the example of the positive and negative spectra of the absolute acceleration of a shock in the form of half-sine of duration τ and amplitude u!!max . The following evolution can be seen there: – the initial slope is given by u! 0 = (2τ / π ) u!!max , both for the positive and for the negative spectrum;

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– then, the curve inflects and passes by a maximum for the positive spectrum, and a minimum for the positive spectrum, which depends on the chosen damping: see Table 2.3. The relatively small impact of the damping can be seen here due to a non-oscillating time function. The same applies for all transients that oscillate little; – finally, the curve tends progressively towards u!!max after a series of weaker and weaker bumps. Each bump corresponds to a change in obtaining the maximum: at first, obtained initially at the first oscillation, then at the second and so on.

τω S+ for u!!max 2π

ζk 0.00 0.05 0.10 0.20 0.50

1.7685 1.6516 1.5584 1.4215 1.2130

0.8099 0.8233 0.8343 0.8505 0.8719

τω S− for u!!max 2π –1.7155 –1.3657 –1.1028 –0.7463 –0.2910

0.6836 0.6842 0.6859 0.6929 0.7454

Table 2.3. Maxima of the shock spectrum for a half-sine

If the maximal spectrum is often used in practice for a transient excitation, it can also be used in the particular cases of a sine or random excitation: – for a sine excitation of a given frequency, the maximum is given directly by the equations of section 2.2.3. For a sine sweep including the natural frequency of the system, they are given directly by the levels at the resonance; – for a random excitation of duration T and represented by its PSD W(f), the calculation requires developments which are beyond the limits of this work. We can retain the following result which approximately relates the mean value of the maximum of a response x to its RMS value x rms = x 2 (according to [LAL 02] volume 3):

x max ≈ x rms

2 ln( f T )

[2.77]

The RMS value will be deduced from the equations in section 2.2.5. The product (f T) represents the number of oscillations of the response. Thus, the maximum increases slowly with the duration of the excitation.

The Single-Degree-of-Freedom System

61

Thus, the various types of excitation can be compared according to the criterion of the maximum responses. However, it is necessary to be aware of the risks incurred in the comparison of two excitations of different nature especially because of the influence of damping on the results: the maxima are proportional to Qk for sine excitation, to

Q k for random excitation and variable for transient excitation.

An incorrect estimation of damping can result in significant errors. One last important note is that the shock spectrum is a transformation of the time history, which retains only the amplitude and thus implies a loss of information, contrary to Fourier transform which provides both an amplitude and a phase. Thus, this transformation is not reversible: the excitation cannot be restored starting from a shock spectrum. There is an infinite number of possible excitations corresponding to a given spectrum. For example, it is always possible to find a sine sweep, which produces the same spectrum as that of a given shock, yet whose impact on the responses of a given structure could be very different. This well-known disadvantage can be reduced by providing other information about the shock, for example about the durations. Advances in this field can be found in [GIR 99b] and [GIR 01b]. 2.4. Representation of the damping 2.4.1. Viscous damping

The energy dissipation was introduced with the dissipated power D of equation [2.3], which corresponds to the hypothesis of a dissipation force Fdiss proportional to the relative velocity Δu! with the proportionality coefficient c: Fdiss = c Δu!

[2.78]

It is the most convenient hypothesis to put into the equation. It is fairly well verified in certain cases like that of a piston traveling in a cylinder full of oil (hence the diagram of the damper), or a large surface displacing air, where the viscosity of the fluid is important. The damping is then known as viscous and it is represented by the dimensionless parameter ζ k introduced by relation [2.16]. It is this parameter which limits the amplifications at resonance to the approximate value Qk = 1/(2ζk) as explained in section 2.2.4. On the contrary, far from resonance, its effect is weak as long as it is much smaller than 1.

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In the case of a sine excitation with Δu = Δu max sin ωt , the energy dissipated during a cycle is proportional to the excitation frequency: 2π / ω

∫0

Fdiss Δu! dt = π c ω (Δu max ) 2

[2.79]

In reality, the dissipation in the structures results from various phenomena whose analysis is very complex: fluid-structure interaction, but also hysteresis in the materials, dry friction in the joints, etc. Many other references cover this subject, for example [LAZ 68, SNO 68]. Only some concepts, which are useful for what comes next, will be retained here. In particular, the energy dissipated during one cycle, when it is measured, is not, in many cases, proportional to the frequency of excitation, but rather independent, hence the necessity of another model of damping. 2.4.2. Structural damping

Structural damping corresponds to an energy dissipated by cycle and which is independent of the frequency. In this case, the dissipation force must be proportional to Δu! / ω = i Δu , thus introducing stiffness k of the system: Fdiss = i η k k Δu

[2.80]

η k is called the structural damping factor (some people use the notation g k ). Just like ζ k , it is dimensionless and can be associated with mode k but, by its physical significance, it is not expressed as a percentage. By comparing relations [2.78] and [2.80], the following equivalence is obtained, taking into account [2.16]:

ηk ⇔ 2ζ k ω / ω k

[2.81]

As mentioned in section 2.4.1, the damping is important only in the vicinity of resonance, i.e. for ω ≈ ω k . Thus, we will retain the equivalence, which is valid when the damping is important for the responses:

ηk ⇔ 2ζ k

[2.82]

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63

This concept of structural damping deserves several notes: – it is applicable only in harmonic regime, and therefore in the frequency domain, in order to be able to write Δu! = iω Δu . Transformation to the time domain causes problems; – if the dissipation force is associated with the elastic force k Δu the force k (1 + iη k ) Δu is obtained which introduces the concept of the complex stiffness k (1 + iη k ) where the real part corresponds to the so-called stiffness, and the imaginary part to the structural damping; – on a material level, the concept of Young’s complex modulus E (1 + iη k ) is obtained where η k designates the loss factor, which can be put under the form η k = tan δ , δ loss angle expressing the phase difference between stresses and strains; – contrary to viscous damping, whose static effect is zero since it affects the velocities, the structural damping poses problem when ω = 0 where it disturbs the static properties of the system. However, this disturbance is only of second order in η k and it remains sufficiently limited in practice to have negligible effects; – the ratio between the energy dissipated per cycle and the maximum elastic energy is equal to 2πηk which provides a method of determining η k . Relation [2.81] allows us to rewrite the dynamic amplification factors H k (ω ) and the dynamic transmissibility factors Tk (ω ) of relations [2.35] and [2.36] using structural damping: H k (ω ) =

Tk (ω ) =

1 2

⎛ ω ⎞ ⎟ + iηk 1 − ⎜⎜ ⎟ ⎝ωk ⎠ 1+ iηk

⎛ ω 1 − ⎜⎜ ⎝ωk

2

⎞ ⎟ + iηk ⎟ ⎠

[2.83]

[2.84]

With regards to the maxima of H k (ω ) and Tk (ω ) , they are reached for

ω = ω k and equal 1/ηk and 1 + η k 2 / η k respectively. The value Qk = 1/ηk will be retained as a good approximation for all the maxima of H k (ω ) , Tk (ω ) and their variants introduced in section 2.2.4, the structural damping must remain sufficiently small compared to 1 so that these approximations remain valid.

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2.4.3. Other representations

Other representations of damping can be considered in order to simulate measured behavior while trying for example to better reproduce the hysteresis curve of the material. However, this type of representation destroys the linearity of the problem and this makes its solution much more complex (see Chapter 10). Except for the particular case where these non-linearities are preponderant in the behavior of the structure, the simplicity of the treatment, at the price of limited errors, is preferable. We shall retain for what follows an equivalent structural or viscous damping producing the same energy dissipated by cycle. Each of these representations has its advantages and disadvantages, which will be highlighted after studying the N-DOF systems.

Chapter 3

Multiple-Degree-of-Freedom Systems

3.1. Introduction The equations of motion of a discrete N-DOF system were presented in section 1.5.1 leading to equation [1.52] expressed on the DOF g = i+j: g + C gg u g + K gg u g = F g M gg u

[3.1]

In order to obtain this equation, it is thus necessary to start by discretizing the structure independently of the boundary conditions, in order to introduce the DOF, which will be put into category i or j later: this is the step known as modeling, or constitution of a representative mathematical model of the structure. Next, it is necessary to establish matrices M, C and K relative to these DOF (subscripts i and j are omitted in order to simplify matters), symmetric by reciprocity, describing the physical properties of the structure, which are implied in its dynamic behavior: – matrix M describing the distribution of mass; – matrix C describing the distribution of the viscous damping; – matrix K describing the distribution of stiffness. These matrices are often obtained by breaking the structure into simple structural elements e. The structure is generally discretized using the finite element method except when it can be directly represented by simple structural elements: lumped masses, dampers or springs. The same steps are always followed and these are described in section 3.2 and illustrated with basic structural elements. The characteristics of the finite element method are then presented in section 3.3.

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Some analysis on the industrial models can be found in section 3.4. It should be noted that there are many references devoted to this subject. Only the aspects which are useful for the representation of the dynamic behavior of structures will be covered here, in coherence with the other chapters of the book. Equation [3.1] is solved in a similar way to that of the 1-DOF system given in section 2.2.1: as with equation [2.4], it is divided into two equations, one to solve u i , the other recovering F j from u i . The treatment can be carried out in a direct way: – in the time domain by direct integration, by discretizing the equation with a time step ∆t, which requires adapted algorithms: this is the subject of section 3.5; – in the frequency domain by direct approach, as indicated in section 1.5.2. It can also be done in two steps by preliminary solution of the equation of motion in the absence of excitation, whose solutions are the structure normal modes already mentioned in section 1.5.3. The solution of the complete equation is then obtained by projecting it on these normal modes, and this leads to uncoupled scalar equations, which will be independently solved before coming back to the physical DOF by “mode superposition” (equation [1.58]). This strategy, particularly effective at low frequencies and therefore very popular, is treated separately in Chapter 4.

3.2. Determining the structural matrices 3.2.1. Introduction Let us assume that the structure is broken down into simple structural elements e making a certain number of nodes appear which comprise one or several translation and/or rotation DOF. The different steps necessary to establish matrices M, C and K relating to these DOF are as follows: – generation of the element matrices in a convenient reference frame, therefore a local reference frame for each element; – transformation to a common reference frame, the reference frame of the structure, or global reference frame; – assembly of the element matrices in global reference frame, just as the elements which form the complete structure are assembled; – addition of the possible linear constraints established among certain DOF of the structure.

Multiple-Degree-of-Freedom Systems

67

These steps are described briefly below. The implied matrix methods are generally detailed in the references relating to the calculations of structures, particularly those about the finite element method, which borrows the same strategy, for example [IMB 91].

3.2.2. Local element matrices The first step consists of determining the element matrices, which distribute the properties of mass, damping and stiffness on the DOF associated with each element. They will be noted by M e , C e and K e respectively, by omitting the subscripts relating to the DOF concerned for the sake of convenience. In order to make the calculation easier, it is preferable to use the reference frame of the element and to consider the local displacements u e . A common technique for the establishment of these matrices is the identification of the associated quadratic forms introduced in section 1.5.1: – kinetic energy Te (relation [1.47]): 1 T u e M e u e 2

Te =

[3.2]

– dissipated power De (relation [1.51]): De =

1 T u e C e u e 2

[3.3]

– elastic energy U e (relation [1.48]): Ue =

1 T ue K e ue 2

[3.4]

This identification is particularly simple for the basic structural elements mentioned in the introduction: – lumped mass m e on a translational DOF of displacement u (or inertia on a rotational DOF): C e = K e = 0 and: Te =

1 me u 2 2

⇒ M e = [ me ] on u e = u

[3.5]

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– damper of constant c e between 2 DOF of displacements u1 and u 2 : M e = K e = 0 and:

De =

1 ce (u2 − u1 ) 2 2

⎡u ⎤ ⎡ 1 −1⎤ ⇒ Ce = ce ⎢ on u e = ⎢ 1 ⎥ ⎥ ⎣ −1 1 ⎦ ⎣u2 ⎦

[3.6]

– spring of stiffness k e between 2 DOF of displacements u1 and u 2 : M e = C e = 0 and: Ue =

1 ke (u2 − u1 ) 2 2

⎡ u1 ⎤ ⎡ 1 −1⎤ ⇒ K e = ke ⎢ ⎥ on u e = ⎢u ⎥ 1 1 − ⎣ ⎦ ⎣ 2⎦

[3.7]

It should be noted that if a spring or damper has one end blocked and only the other end of displacement u is of interest, straight away we have: De =

1 ce u 2 2

⇒ Ce = [ ce ] on u e = u

[3.8]

Ue =

1 ke u 2 2

⇒ K e = [ ke ] on u e = u

[3.9]

3.2.3. Element matrices in global reference form The element matrices established in local reference frame (l) must be translated in the reference frame of the structure, or global reference frame (g) in order to allow the assembly operation, which will follow. If the change of reference frame is written:

(u e )l

= Tlg (u e ) g

[3.10]

the independence of energies with respect to the reference frame implies (relation [1.53]):

( M e or Ce or K e ) g

= Tgl ( M e or Ce or K e )l Tlg

[3.11]

Multiple-Degree-of-Freedom Systems

69

In a plane problem, for example, with nodes comprising 3 DOF (u, v, θ) indicating the translation along the axis x, the translation along axis y and the rotation around axis z, the change of reference frame corresponding to a rotation in the plan xy of an angle ϕ (Figure 3.1) is written for a node: ⎡u ⎤ ⎡ cos ϕ ⎢ v ⎥ = ⎢− sin ϕ ⎢ ⎥ ⎢ ⎢⎣θ ⎥⎦ ⎢⎣ 0 l

sin ϕ cos ϕ 0

0⎤ ⎡u ⎤ 0⎥⎥ ⎢⎢ v ⎥⎥ 1⎥⎦ ⎢⎣θ ⎥⎦ g

[3.12]

ul vg

u

ϕ

θl θg

ug

vl

Figure 3.1. Change of the plane reference frame

For the basic structural elements situated in a plane, this is expressed by: – lumped mass m e : if it is a “physical” mass, it has the same property in all directions and keeps the same value no matter what the reference frame is. On the contrary, if it is a directional mass in the local direction x l , in a direction x g forming an angle ϕ with x l , it will have the value m e cos 2 ϕ . In the reference frame ( x, y ) g , we will have the matrix: ⎡ cos 2 ϕ sin ϕ cos ϕ ⎤ ⎡u ⎤ (M e ) g = me ⎢ ⎥ for u e = ⎢ ⎥ 2 sin ϕ ⎦ ⎣v ⎦ g ⎣sin ϕ cos ϕ

[3.13]

In all the cases, only the translations are affected. The equivalent for rotation is inertia around axis z: – stiffness k e (similar to damper c e ) between the DOF along x l of two nodes 1 and 2:

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⎡K (K e ) g = ⎢ e ⎣ −K e

−K e ⎤ K e ⎥⎦

⎡ cos 2 ϕ sin ϕ cos ϕ ⎤ with K e = ke ⎢ ⎥ sin 2 ϕ ⎦ ⎣sin ϕ cos ϕ

⎡ u1 ⎤ ⎢v ⎥ for u e = ⎢ 1 ⎥ ⎢u2 ⎥ ⎢ ⎥ ⎣ v2 ⎦ g

[3.14]

With three dimensions, the direction cosines of each local direction will be taken into account. Most often, each node of the structure possesses six DOFs: three translational and three rotational.

3.2.4. Assembly of element matrices When all the element matrices M e , C e and K e are written in the global reference frame, they can be “assembled” in order to form matrices M , C and K of the complete structure, in the same way as the elements which make the complete structure when they are assembled. By writing that the energies of the structure are the sums of the element energies, matrices M , C and K are built relative to the global DOF of the structure by positioning the terms of the element matrices on the corresponding global DOF. When two terms have the same position, they are added, which amounts to writing the equilibrium of the forces for each DOF. In order to illustrate this assembly operation, two simple examples will be given: – the one-dimensional mass-damper-spring structure of Figure 3.2 illustrates the so-called assembly. The local and global reference frames are the same and this makes it possible to assemble the local element matrices directly. The assembled matrices are the following (the similarity between springs and dampers should be noted): ⎡m1 ⎡ u1 ⎤ ⎥ ⎢ u = ⎢u 2 ⎥ ⇒ M = ⎢⎢ 0 ⎢⎣ 0 ⎢⎣u 3 ⎥⎦ ⎡ k1 K = ⎢⎢− k1 ⎢⎣ 0

− k1 k1 + k 2 − k2

0 m2 0

0 ⎤ 0 ⎥⎥ m3 ⎥⎦

0 ⎤ ⎡ c1 ⎥ − k 2 ⎥ C = ⎢⎢− c1 ⎢⎣ 0 k 2 ⎥⎦

[3.15]

− c1 c1 + c 2 − c2

0 ⎤ − c 2 ⎥⎥ c 2 ⎥⎦

Multiple-Degree-of-Freedom Systems

71

– the two-dimensional mass-spring structure of Figure 3.3 illustrates the influence of the reference frames. The assembled matrices are the following (the presence of dampers would be treated like that of springs): ⎡m1 ⎡ u1 ⎤ ⎢0 ⎢v ⎥ ⎢ ⎢ 1⎥ ⎢0 ⎢u 2 ⎥ u = ⎢ ⎥ ⇒ M =⎢ ⎢0 ⎢v 2 ⎥ ⎢0 ⎢u 3 ⎥ ⎢ ⎢ ⎥ ⎣⎢ 0 ⎣⎢ v 3 ⎦⎥ ⎡ k1 ⎢ 0 ⎢ ⎢− k K =⎢ 1 ⎢ 0 ⎢ 0 ⎢ ⎣⎢ 0

0 k2 0 0 0 − k2

0

0

0

0

m1

0

0

0

0

m2

0

0

0

0

m2

0

0

0

0

m3

0

0

0

0

0 ⎤ 0 ⎥⎥ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ ⎥ m3 ⎦⎥

[3.16]

− k1

0

0

0

0

0

0

− k2

⎤ ⎥ ⎥ k1 + k 3 / 2 − k 3 / 2 − k 3 / 2 + k3 / 2 ⎥ ⎥ k3 / 2 + k3 / 2 − k3 / 2 − k3 / 2 ⎥ − k3 / 2 + k3 / 2 k3 / 2 − k3 / 2 ⎥ ⎥ + k3 / 2 − k 3 / 2 − k 3 / 2 k 2 + k 3 / 2⎦⎥

m3

u3 c2

k2 m2

u2 c1

k1 x m1

u1

Figure 3.2. Example in 1-dimension

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v3 u3 m3 k3

v1

k2

v2

u1

u2

45°

m1

m2 y

k1 x

Figure 3.3. Example of 2-dimensions

3.2.5. Linear constraints between DOF 3.2.5.1. Introduction On many occasions, it may be necessary to define linear constraints between the displacements of a structure. This question will be more particularly developed in section 3.4.3 when discussing industrial practice. To illustrate the need for this here are two typical cases, which are frequently used for their effectiveness in a variety of situations: – rigid elements: these meet the need of making certain parts of the structure undeformable and therefore infinitely rigid. One of the advantages is that it avoids problems of conditioning in the stiffness matrices due to values which are greatly different. For example, for the structure from Figure 3.2, we can introduce a rigid link between masses 1 and 3 by writing u1 = u 3 which is a linear constraint between DOF 1 and 3; – “average” nodes: these allow the average motion of a part of the structure to be defined. The average motion is represented by a node defined as the average of the nodes in question. For example, for the structure in Figure 3.2, node 4 representing the average of nodes 1 and 3 can be introduced by writing u 4 = (u1 + u 3 ) / 2 , which is a linear constraint among the DOF 1, 3 and 4.

Multiple-Degree-of-Freedom Systems

73

These linear constraints can be treated in two ways: – by DOF elimination: as many DOF can be eliminated from the solution as there are constraints. This strategy has the advantage of reducing the size of the system to be solved, but also presents various disadvantages, which can generally be avoided, as will be seen after its description; – by DOF introduction: as many DOF can be added to the solution as there are constraints. This strategy seems less interesting at first because it artificially increases the size of the system to solve, but it also removes the disadvantages of the preceding method and this can be useful in certain situations. Moreover, this is not only found in modeling but also in other contexts. In all cases, it is a question of taking into account m linear constraints among certain DOF g, be they type i or j. These relations can be written in the general form: C mg u g = 0

[3.17]

C mg being the matrix whose rows are relative to the linear constraints m and the

columns to DOF g. The fewer the number of implied DOF g, the more sparse (filled with zeros) this matrix will be. 3.2.5.2. DOF elimination This strategy consists of eliminating the DOF from the solution due to linear constraints. This elimination is not a direct elimination of DOF but it is related rather to a condensation operation, which will be seen in more detail in section 4.2.3. It removes the selected DOF from the solution phase, but it makes it possible to restore them later on. Starting from the linear constraints m regrouped in form [3.17], the procedure is carried out as follows: – selection of a suitable subset m of DOF to be eliminated in order to keep only the complementary subset n = g-m, hence the relation: 1 C mm u m + C mn u n = 0 ⇒ u m = −C −mm C mn u n

(m is suitable if matrix Cmm is invertible);

[3.18]

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– transformation making it possible to pass from the global DOF g = m+n to the retained DOF n: ⎡u ⎤ u g = ⎢ m ⎥ = Tgn u n ⎣ un ⎦

⎡ −C−1 C ⎤ with Tgn = ⎢ mm mn ⎥ ⎣ I nn ⎦

[3.19]

– transformed matrices, as for operation [3.11], with relation [1.53]:

( M or C or K )nn = Tng ( M or C or K ) g Tgn

[3.20]

The solution will then be done only on the retained DOF n with the transformed matrices. If we want to also have the results on the eliminated DOF m, it will be necessary to restore them starting from the DOF n through relation [3.18]. In order to illustrate this elimination, the already-mentioned example of Figure 3.2 with u1 = u 3 gives, with the elimination of u 3 , the obvious result:

ug

⎡ u1 ⎤ ⎡u ⎤ = ⎢⎢u 2 ⎥⎥ u m = [u 3 ] u n = ⎢ 1 ⎥ ⎣u 2 ⎦ ⎢⎣u 3 ⎥⎦

u1 = u 3

⇒ C mg = [1 0 − 1] ⇒ Tgn

⎡m + m3 ⇒ M nn = ⎢ 1 ⎣ 0

u1

0 ⎤ m 2 ⎥⎦

Mass m Inertia I

k1

− ( k1 + k 2 ) ⎤ ⎡ k + k2 K nn = ⎢ 1 k1 + k 2 ⎥⎦ ⎣ − ( k1 + k 2 )

u2

u3

DOF i

k2

θ2 L

⎡1 0 ⎤ = ⎢⎢0 1⎥⎥ ⎢⎣1 0⎥⎦

L

Figure 3.4. Example with linear constraints

[3.21]

Multiple-Degree-of-Freedom Systems

75

Another less common example is that of Figure 3.4 where the two springs are blocked at the bottom (hence the application of relation [3.9]) and they are linked at the top by a rigid link, which supports a mass and an inertia at its mid-point: in brief this is the situation of a car on its front and rear wheels. With the DOF elimination relating to u1 and u 3 , for example, to keep u 2 and θ 2 , we get:

ug

⎡ u1 ⎤ ⎢u ⎥ ⎡u ⎤ ⎡u ⎤ = ⎢ 2 ⎥ um = ⎢ 1 ⎥ un = ⎢ 2 ⎥ ⎢θ 2 ⎥ u ⎣θ 2 ⎦ ⎣ 3⎦ ⎢ ⎥ ⎣u 3 ⎦

M gg

⎡0 0 ⎢0 M =⎢ ⎢0 0 ⎢ ⎣0 0

u1 = u2 − Lθ 2 u3 = u2 + Lθ 2 ⎡M ⇒ M nn = ⎢ ⎣0

0 0⎤ 0 0⎥⎥ I 0⎥ ⎥ 0 0⎦

K gg

⎡ k1 ⎢0 =⎢ ⎢0 ⎢ ⎣0

0⎤ 0 0 0 ⎥⎥ 0 0 0⎥ ⎥ 0 0 k2 ⎦ 0 0

⎡1 − 1 + L 0 ⎤ ⇒ Cmg = ⎢ ⎥ ⎣0 − 1 − L 1 ⎦ 0⎤ I ⎥⎦

L ( − k1 + k 2 ) ⎤ ⎡ k + k2 K nn = ⎢ 1 − + L ( k k ) L2 (k1 + k 2 ) ⎥⎦ 1 2 ⎣

[3.22]

A very useful result that we can establish with this technique is that of the mass matrix of the rigid solid body offset with respect to a node P, which we will assume to have six DOFs, three translational and three rotational in a local reference frame Pxyz. This result is illustrated in Figure 3.5.

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z

Mass m Center of mass (x,y,z)G Inertias (Ixx, Iyy, Izz, Ixy, Ixz, Iyz)

G

y

P x

Figure 3.5. Mass, center of mass and inertias of a solid body

If the mass is denoted by M, the coordinates of the center of mass G by ( x, y, z ) G as compared to Pxyz, and the components of the inertia tensor in the Gxyz reference frame by ( I xx , I yy , I zz , I xy , I xz , I yz ) G , the rigid body mass matrix relative to the DOF of P will be deduced from the mass relative to the DOF of G: ⎡M ⎢0 ⎢ ⎢0 Me/G = ⎢ ⎢0 ⎢0 ⎢ ⎣⎢ 0

0

0

0

0

M

0

0

0

0

M

0

0

⎡u⎤ ⎡1 0 ⎢v⎥ ⎢0 1 ⎢ ⎥ ⎢ ⎢w⎥ ⎢0 0 ⎢ ⎥ =⎢ ⎢θ x ⎥ ⎢0 0 ⎢θ y ⎥ ⎢0 0 ⎢ ⎥ ⎢ ⎢⎣θ z ⎥⎦ G ⎢⎣0 0

0

0

I xx / G

− I xy / G

0

0

− I xy / G

I yy / G

0

0

− I xz / G

− I yz / G

0

0

0 − zG 1 yG 0 1 0 0 0

0

zG 0 − xG 0 1 0

⎤ ⎥ ⎥ 0 ⎥ ⎥ − I xz / G ⎥ − I yz / G ⎥ ⎥ I zz / G ⎦⎥ 0

0

− yG ⎤ ⎡ u ⎤ ⎢ ⎥ x G ⎥⎥ ⎢ v ⎥ 0 ⎥⎢w⎥ ⎥⎢ ⎥ 0 ⎥ ⎢θ x ⎥ 0 ⎥ ⎢θ y ⎥ ⎥⎢ ⎥ 1 ⎥⎦ ⎢⎣θ z ⎥⎦ P

[3.23]

Multiple-Degree-of-Freedom Systems

⎡ M ⎢ 0 ⎢ ⎢ 0 ⇒ Me/ P = ⎢ ⎢ 0 ⎢ M zG ⎢ ⎣⎢− M y G

0 M 0 − M zG 0 M xG

0 0 M M yG − M xG 0

0 − M zG M yG I xx / P − I xy / P − I xz / P

M zG 0 − M xG − I xy / P I yy / P − I yz / P

77

− M yG ⎤ M x G ⎥⎥ 0 ⎥ ⎥ − I xz / P ⎥ − I yz / P ⎥ ⎥ I zz / P ⎦⎥

This treatment of the linear constraints by DOF elimination naturally reduces the size of the system to solve. In exchange, it is necessary to choose the DOF to be eliminated and this is sometimes delicate knowing that, once eliminated, these DOF will not be directly available any longer for operations such as taking into account the boundary conditions (and this means that future DOF j must be left in the retained DOF n) or of the excitation forces. In the same way, the calculation of a response will be possible only by recovery (relation [3.18]). Suitable management of DOF and treatments can attenuate these disadvantages. 3.2.5.3. DOF introduction Linear constraints m [3.17] can be taken into account by introducing supplementary m DOF λ m called Lagrange multipliers. This mathematical technique is well-known in optimization for transforming a minimization problem with constraints into a minimization problem without constraints. Therefore, it can be applied here, starting from the particular case of static analysis, which corresponds to a total minimum potential energy V (see section 1.5.1):

K gg u g = F g

⇔ V=

[

]

1 u g T K gg u g − u g T F g minimal 2

[3.24]

The minimization problem [3.24] with the constraints [3.17] is equivalent to the minimization without constraints of:

V =

[

1 u g T K gg u g − u g T F g + (C mg u g ) T λ m 2

]

The extremum conditions of V give: ∇ u g (V ) = K gg u g − Fg + C gm λ m = 0

∇λ m (V ) = C mg u g = 0

[3.25]

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or

⎡K gg ⎢C ⎣ mg

C gm ⎤ ⎡ u g ⎤ ⎡ F g ⎤ = 0 mm ⎥⎦ ⎢⎣λ m ⎥⎦ ⎢⎣0 m ⎥⎦

[3.26]

Whereas matrix K gg is normally singular because it corresponds to the structure before introduction of the supports, the matrix of system [3.26] is regular if there are sufficient linear constraints m to block the structure. The system solution then gives the displacements u g and Lagrange multipliers λ m , which can be interpreted as forces conjugated to the linear constraints. It should be noted that this technique can cause matrix-conditioning problems. This result can be extended to the dynamic analysis by replacing static stiffness K gg in the frequency domain with dynamic stiffness K gg (ω ) which results from equation [1.54]: K gg (ω ) = −ω 2 M gg + iω C gg + K gg

[3.27]

System [3.26] is then to be solved for each ω. This equates to considering the new structural matrices X ( g + m)( g + m) relative to the DOF g+m deduced from the original matrices X gg (with X = M, C or K) by: ⎡M gg X ( g + m )( g + m) = ⎢ ⎣ 0 mg

0 gm ⎤ ⎡C gg , 0 mm ⎥⎦ ⎢⎣ 0 mg

0 gm ⎤ ⎡K gg , 0 mm ⎥⎦ ⎢⎣C mg

C gm ⎤ 0 mm ⎥⎦

[3.28]

(note here the notation: C gg damping matrix, C mg linear constraint matrix). The number of linear constraints in fact increases the size of the system. In exchange, the disadvantages of the elimination have disappeared: the DOF no longer need to be chosen and they can all be used for the excitation, boundary conditions or response calculations. It should be noted for system [3.26] that if the matrix K gg is regular, which is not always the case in static analysis but is generally the case in dynamic analysis for K gg (ω ) , the elimination of Lagrange multipliers leads to the solution:

(

)

−1 ⎡ ⎤ u g = ⎢K gg −1 − K gg −1 C gm C mg K gg −1 C gm C mg K gg −1 ⎥ F g ⎣ ⎦

[3.29]

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79

This result will be discussed again in section 9.5.3 which concerns substructuring by the flexibility approach ( G = K −1 ). In order to illustrate this technique, let us take the very simple example of Figure 3.6, limited to a static analysis with two DOF linked by a simple spring, and the relation u1 + u 2 = 0 which is equivalent to blocking the average displacement of the system. Relation [3.26] then gives: ⎡k ⎢− k ⎢ ⎢⎣ 1

−k k 1

1⎤ ⎡ u1 ⎤ ⎡ F1 ⎤ 1⎥⎥ ⎢⎢u 2 ⎥⎥ = ⎢⎢ F2 ⎥⎥ ⇒ 0⎥⎦ ⎢⎣ λ1 ⎥⎦ ⎢⎣ 0 ⎥⎦

⇒ u1 = −u 2 = ( F1 − F2 ) / 4k

⎡ u1 ⎤ ⎡ 1 / 4k ⎢u ⎥ = ⎢− 1 / 4k ⎢ 2⎥ ⎢ ⎢⎣ λ1 ⎥⎦ ⎢⎣ 1 / 2

− 1 / 4k 1 / 2⎤ ⎡ F1 ⎤ 1 / 4k 1 / 2⎥⎥ ⎢⎢ F2 ⎥⎥ 1/ 2 0 ⎥⎦ ⎢⎣ 0 ⎥⎦

λ1 = ( F1 + F2 ) / 2

[3.30]

Relation u1+u2 = 0 can be found in the result, with the choice of exciting each DOF.

F2

u2 u1 + u2 = 0

k F1

u1

Figure 3.6. Example of DOF introduction

3.2.6. Excitation forces The excitation force vector F that appears in equation [3.1] must be established in the same way as matrices M, C and K. This is not a big problem if the excitation forces are directly applied to the DOF: and this is the reason that it was not mentioned up to now. But in certain cases, the forces may be applied elsewhere than on the nodes, like distributed forces relative to linear, surface or volume type elements. For the basic structural elements, this point arises only in case of elimination of excitation DOF. We need only to consider the work by the forces (relation [1.53]) in order to establish the equivalent of transformation [3.20]: u g = Tgn u n

⇒ Fn = Tng F g

[3.31]

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EXAMPLE. – if u 3 = (u1 + u 2 ) / 2 , exciting u 3 with F means exciting u1 and u 2 with F/2. Exciting on an average node thus means equally distributing the excitation on the nodes concerned.

3.3. The finite element method 3.3.1. Introduction Generally, it is not possible to represent the structure under consideration suitably using only the basic elements presented in section 3.2. It is necessary to add more elaborate elements to them, elements that are closer to reality. The finite element method comes into play here and provides a variety of elements describing the physical characteristics of the structure more accurately. Full developments on the subject will be found in other references, for example [BAT 76, BAT 90, GAL 73, HUG 87, IMB 91, ZIE 77]. The theoretical presentation is limited here to the major principles and to the simplest applications for illustrations and elementary calculations. The basic idea is to transform a given structural element, of a continuous nature at the beginning, into a “finite” element, i.e. discrete, by projecting its properties on the DOF associated with the nodes that define it. This discretization is carried out schematically by expressing the displacements of the current point of the element according to the displacements at the nodes: u e (x, t ) = A e (x) u e (t )

[3.32]

with: – u e (x, t ) : vector of the displacements of the current point of the element e, depending on its position x (one or several components) in the element and on time t; – u e (t ) : vector of the displacements at the element nodes, depending only on time t; – A e (x) : matrix of shape functions to interpolate u e (t ) , depending only on the position x. As can be seen, it is a method of separation of the variables x and t, which restricts its domain of validity, but remains well adapted to the low frequency context. A second approximation lies in the interpolation functions used, which are not exact, with some exceptions.

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81

The estimation of energies Te and U e of the element taking into account this relation [3.32] and compared to energetic expressions [3.2] and [3.4] leads to the identification of element matrices M e and K e . As for matrix C e , generally it is not determined at the element level, but on the global level in an approximate way as we will see later on. This leads to: – matrix M e starting from the kinetic energy Te integrated over the volume Ve of the element: 1 u e (x, t ) T u e (x, t ) ρ (x) dV 2

Te = ∫V

e

⇒ M e = ∫V A e (x) T A e (x) ρ (x) dV

[3.33]

[3.34]

e

ρ being the mass density at the current point. This relation provides a full mass matrix, known as “coherent” (with the interpolation functions taken for the element), in opposition to “lumped” if one directly assigns the mass of the element to its nodes in order to obtain a diagonal matrix. The coherent matrix corresponds to a more precise description of the masses of the element. However, it is more difficult to treat because it is no longer diagonal. The choice between these two approaches is to be made according to the objectives desired: accuracy or computation speed: – matrix K e starting from the elastic energy U e integrated over the volume Ve of the element: U e = ∫V

e

1 σ e (x, t ) T ε e (x, t ) dV 2

[3.35]

with σ e and ε e respectively stresses and strains of the current point, verifying relations of the type: σ e = Ee ε ε

e

generalized Hooke’s law

e

= D e u e differential relations between strains and displacements

⇒ K e = ∫V ( A T D T E D A) e (x) dV e

[3.36]

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Relations [3.34] and [3.36] thus provide the element mass and stiffness matrices starting from interpolation matrices. The formulation depends on the type of element considered. It is relatively simple for straight beams with certain hypotheses. It is much more difficult for plates and is beyond the subject matter here. Many other works have been written on this subject and the above-mentioned references are a good introduction to it. Only the principal results on the beams are presented here, allowing simple and rapid calculations, after the simplest case of the rod has been treated in order to illustrate the complete approach.

3.3.2. The rod element The rod element is the simplest to consider: it deforms only along its axis and implies only the two DOF of its extremities, as illustrated in Figure 3.7. Its length, its mass, its Young’s modulus and its cross-section will be indicated by L, M, E, S.

u1

u(x)

u2

M, E, S 1

2

x

L Figure 3.7. The rod finite element

A linear interpolation between the end displacements makes it possible to write: ⎡ x u e ( x, t ) = ⎢1 − ⎣ L

x ⎤ ⎡ u1 (t ) ⎤ ⎢ ⎥ L ⎥⎦ ⎣u 2 (t )⎦

[3.37]

With Hooke’s law σ = E ε and relation ε = du / dx , we get: ⎡u ⎤ M ⎡2 1⎤ ue = ⎢ 1 ⎥ ⇒ Me = ⎢ ⎥ 6 ⎣1 2 ⎦ ⎣u 2 ⎦

( Me =

Ke =

M ⎡1 0 ⎤ ⎢ ⎥ in lumped masses) 2 ⎣0 1 ⎦

E S ⎡ 1 − 1⎤ ⎢ ⎥ L ⎣− 1 1 ⎦

[3.38]

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83

For the mass, the matrix makes a coupling appear between the two DOF, contrary to the lumped masses. For the stiffness, the element behaves like a spring of stiffness k = E S / L along its axis. The closer the true field of displacement in the element is to a linear variation, the better this representation will be.

3.3.3. Beam finite element in bending The beam finite element under bending in the xy plane is illustrated by Figure 3.8. It deforms only under bending, i.e. with the displacements only along the perpendicular axis y, implying at nodes the two displacements v along y and the two rotations θ around z. Its length, its mass, its Young’s modulus and its section inertia will be indicated by L, M, E, I respectively.

y

v(x)

θ1

v1

v2

θ2

M, E, I 1

2

x

L Figure 3.8. The beam finite element under bending in the xy plane

If the shear effect is neglected, the cross-sections remain straight and the rotation of the current point is the derivative of the translation: θ = dv/dx. The four DOF at the nodes lend themselves a polynomial of degree 3 for v(x), which leads to the following interpolation: ⎡ 1 − 3 ( x / L) 2 + 2 ( x / L) 3 ⎢ L x / L − 2 ( x / L) 2 + ( x / L ) 3 v e ( x, t ) = ⎢ ⎢ 3 ( x / L) 2 − 2 ( x / L ) 3 ⎢ ⎢⎣ L − ( x / L) 2 + ( x / L) 3

(

(

)

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

)

T

⎡ v1 (t ) ⎤ ⎢θ (t ) ⎥ ⎢ 1 ⎥ ⎢ v 2 (t ) ⎥ ⎢ ⎥ ⎣θ 2 (t )⎦

[3.39]

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Formulae [3.34] and [3.36] thus give: 22 L ⎡ v1 (t ) ⎤ ⎡ 156 ⎢θ (t ) ⎥ ⎢ 22 L 4 L2 M ⎢ ue = ⎢ 1 ⎥ ⇒ Me = ⎢ v 2 (t ) ⎥ 13L 420 ⎢ 54 ⎢ ⎥ ⎢ 2 ⎣− 13L − 3L ⎣θ 2 (t )⎦ 6 L − 12 6 L ⎤ ⎡ 12 ⎢ 6 L 4 L2 − 6 L 2 L2 ⎥ EI ⎢ ⎥ Ke = L3 ⎢− 12 − 6 L 12 − 6 L ⎥ ⎥ ⎢ 2 − 6 L 4 L2 ⎦ ⎣ 6L 2L

− 13L ⎤ − 3L2 ⎥⎥ 156 − 22 L ⎥ ⎥ − 22 L 4 L2 ⎦ 54

13L

[3.40]

So as not to neglect the shear effect when the beam is relatively short, it is necessary to introduce the dimensionless shear coefficient:

φ=

12 E I k G S L2

[3.41]

with G shear modulus and k shape factor of the section. φ is zero without a shear effect and increases with the shear effect. The values of k for various section forms can be found, for example, in [BLE 79]. We can retain for the simplest forms and by neglecting the influence of Poisson’s coefficient: – 9/10 for the full circular section; – 1/2 for the thin-wall circular section; – 5/6 for the rectangular section. Formula [3.39], which assumes θ = dv / dx , is no longer valid, but we can easily obtain the stiffness of the beam starting from its flexibilities (see [IMB 91], for example). Indeed, one has the following flexibilities on node 2 for a beam clamped at node 1: ⎡v 2 ⎤ 1 ⎡(1 + φ / 4) L3 / 3 L2 / 2⎤ ⎡ Fv 2 ⎤ ⎢ ⎥⎢ ⎥ ⎢θ ⎥ = L2 / 2 L ⎦⎥ ⎣ Fθ 2 ⎦ ⎣ 2 ⎦ E I ⎢⎣

[3.42]

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85

By inversion, we obtain the stiffness on node 2 of the beam clamped at node 1:

− 6L ⎤ ⎡v2 ⎤ ⎡ Fv 2 ⎤ ⎡ 12 EI ⎢F ⎥ = 3 ⎢− 6 L (4 + φ ) L2 ⎥ ⎢θ ⎥ ⎦⎣ 2⎦ ⎣ θ 2 ⎦ L (1 + φ ) ⎣

[3.43]

By writing that the elastic energy remains the same, the stiffness of the free beam can be deduced: − 12 6L ⎤ − 6 L (2 − φ ) L2 ⎥⎥ − 6L ⎥ 12 ⎥ − 6 L (4 + φ ) L2 ⎦

6L ⎡ 12 ⎢ 6 L (4 + φ ) L2 ⎢ Ke = 3 − 6L L (1 + φ ) ⎢− 12 ⎢ 2 ⎣ 6 L (2 − φ ) L EI

[3.44]

As regards the masses, formula [3.40] is often kept. However, the more coherent formula [GEN 95] can also be used. ⎡ µ1 ⎢ Lµ M 2 ⎢ Me = 2 ⎢ µ 420 (1 + φ ) 3 ⎢ − L µ4 ⎣ ⎡ µ7 ⎢L µ ρI ⎢ 8 + 2 ⎢− µ 30 L (1 + φ ) 7 ⎢ L µ ⎣ 8

L µ2 L µ5 2

L µ4 − L2 µ 6

µ3 L µ4 µ1 − L µ2

L µ8

− µ7

L2 µ 9

− L µ8

− L µ8

µ7 − L µ8

− L2 µ10

− L µ4 ⎤ − L2 µ 6 ⎥⎥ − L µ2 ⎥ ⎥ L2 µ 5 ⎦

L µ8 ⎤ − L2 µ10 ⎥⎥ − L µ8 ⎥ ⎥ L2 µ 9 ⎦

µ1 = 156 + 294φ + 140φ 2

µ 6 = 3 + 7φ + 3.5φ 2

µ 2 = 22 + 38.5φ + 17.5φ 2

µ 7 = 36

µ 3 = 54 + 126φ + 70φ 2

µ 8 = 3 − 15φ

µ 4 = 13 + 31.5φ + 17.5φ µ 5 = 4 + 7φ + 3.5φ 2

2

[3.45]

µ 9 = 4 + 5φ + 10φ 2 µ10 = 1 + 5φ − 5φ 2

The first term of formula [3.45] comes from the mass in translation at the current point; the second comes from its inertia in rotation, ρ being the mass density. Formula [3.40] can be found again by making φ = 0.

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3.3.4. The complete beam finite element The complete beam finite element is illustrated in Figure 3.9. It connects two nodes to each of the six DOF, three translations and three rotations.

y

v1 x

θz1

θx1

1

2

v2

θz2

M, E, S, I, J

u1 z

w1

θy1

u2 L

w2

θy2

Figure 3.9. The complete beam finite element

The problem can be broken up into four parts provided that the corresponding deformations are uncoupled, and this requires a certain number of properties of the cross-section (no offsets, etc.) which we will assume to be verified here. The four deformations are the following: – axial, concerning the displacements u of each node. This is the rod element treated in section 3.3.2, hence the matrices X ax. ; – torsion, concerning the rotations θ x of each node. The problem is similar to that of extension. It is enough to replace the mass by the inertia and the extension stiffness E S / L by the torsion stiffness G J / L , hence the matrices X tors. ; – bending in the xy plane, concerning the displacements v and the rotations θ z of each node. This is the beam element under bending treated in section 3.3.3, hence the matrices Xbend . xy ; – the bending in the xz plane, concerning the displacements w and the rotations θ y of each node. The problem is similar to that of bending in the xy plane, but with a different orientation of the reference frame ( z → x instead of x → y ), which is expressed by the following modifications of the sign in the matrices Xbend . xz :

Xbend . xz

⎡+ ⎢− =⎢ ⎢+ ⎢ ⎣−

− + −⎤ + − + ⎥⎥ + unchanged sign as compared to Xbend . xy − + − ⎥ − modified sign ⎥ + − +⎦

[3.46]

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87

Due to the uncoupling, the complete matrices of 12 × 12 size will be written starting from the partial matrices: ⎡ u ax. ⎤ ⎡ X ax. ⎢ u ⎥ ⎢ 0 u e = ⎢ tors. ⎥ ⇒ Xe = ⎢ ⎢ubend . xy ⎥ ⎢ 0 ⎢ ⎥ ⎢ u ⎣ bend . xz ⎦ ⎣ 0

0

0

Xtors.

0

0 0

Xbend . xy 0

⎡u ⎤ ⎡θ ⎤ with u ax. = ⎢ 1 ⎥ utors. = ⎢ x1 ⎥ ubend . xy ⎣u2 ⎦ ⎣θ x 2 ⎦

⎤ 0 ⎥⎥ 0 ⎥ ⎥ Xbend . xz ⎦ 0

[3.47]

⎡ w1 ⎤ ⎡ v1 ⎤ ⎢θ ⎥ ⎢θ ⎥ y1 = ⎢ z1 ⎥ ubend . xz = ⎢ ⎥ ⎢ w2 ⎥ ⎢ v2 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢⎣θ y 2 ⎥⎦ ⎣θ z 2 ⎦

In order to re-establish the order of the DOF by node, it is sufficient to permute the lines and the columns in an adequate way. It should be noted that a cylindrical shell of length L, radius R and thickness t, with undeformable extremities and deforming only in membrane, has the same behavior as the beam element with shear effect, with the following parameters: S = 2π R t

I = J / 2 = π R3 t

k = 1/ 2

[3.48]

The generalization of a conical shell of end radii R1 at node 1 and R 2 at node 2 (half-angle at the vertex α with tan α = ( R1 − R 2 ) / L ), with the same hypotheses, gives the following results in terms of flexibilities (cylinder 6 cone): – flexibility in extension:

L L = E S 2π R E t

6

R ln 1 R2 2π sin α cos 2 α E t

[3.49]

– flexibility in torsion:

L L = G J 2π R 3 G t

6

⎛R 1 − ⎜⎜ 2 ⎝ R1

⎞ ⎟⎟ ⎠

2

4π sin α R 2 2 G t

[3.50]

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– flexibilities under bending (extension of formula [3.42] according to P. Seide [SEI 72], Poisson’s coefficient ν): ⎡ v 2 ⎤ ⎡G vv ⎢θ ⎥ = ⎢G ⎣ 2 ⎦ ⎣ θv

G vθ ⎤ ⎡ Fv 2 ⎤ Gθθ ⎥⎦ ⎢⎣ Fθ 2 ⎥⎦

[3.51]

3

⎛L⎞ L ⎜ ⎟ ⎝R⎠ R G vv = + 3π E t π G t

⎛ R ⎞ ⎛ ⎛R R ln 1 − 2 ⎜⎜1 − 2 ⎟⎟ + ⎜⎜1 − ⎜⎜ 2 R2 R1 ⎠ ⎜ ⎝ R1 ⎝ ⎝

6

2

⎞ 1 ⎟ ⎛ + (1 + ν ) sin 2 α ⎞ ⎟ ⎟⎟ ⎜⎝ 2 ⎠ ⎠

π sin 3 α E t

6

R 2 ⎛⎜ ⎛ R 2 − 1− ⎜ R1 ⎜⎜ ⎜⎝ R1 ⎝

⎛ ⎛R ⎜1 − ⎜ 2 ⎜⎜ ⎜ R ⎝ ⎝ 1

⎞ ⎟⎟ ⎠

2

⎛L⎞ ⎜ ⎟ ⎝R⎠ G vθ = Gθv = 2π R E t

L R Gθθ = π R2 E t

⎞ ⎟⎟ ⎠

6

1−

⎞ ⎟⎟ ⎠

2⎞

⎟ ⎛ 1 + (1 + ν ) sin 2 α ⎞ ⎟ ⎟⎟ ⎜⎝ 2 ⎠ ⎠

π sin 2 α cos α R 2 E t 2⎞

⎟ ⎛ 1 + (1 + ν ) sin 2 α ⎞ ⎟ ⎟⎟ ⎜⎝ 2 ⎠ ⎠

π sin α cos 2 α R 2 2 E t

The cylinder properties are found again by making R 2 / R1 tend towards 1 and correlatively α towards 0. These results make it possible to treat the cylinder and cone like elements in a model on condition that the hypotheses made are well verified, particularly the undeformability of the extreme sections. We can also have a fast estimate of the dynamic properties of a conical support topped by a solid body by combining these relations with the mass matrix [3.23]. Some developments on this subject can be found in [GIR 99a].

3.3.5. Excitation forces As pointed out in section 3.2.6, the problem of excitation forces is trivial if they are applied directly to the DOF. If the forces are applied within the elements,

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it is necessary to find equivalent forces at the nodes. There again, we will consider the work of the forces in order to establish the equivalence:

(F

eq

T

u

)

e

= ∫e u dF ⇒ Feq = ∫e A T dF

[3.52]

Thus, the forces equivalent to a linear force p constant on the beam element under bending without a shear effect, for which the interpolation matrix is given by relation [3.39], are given by: ⎡ 1/ 2 ⎤ ⎢ L / 12 ⎥ ⎥ Feq = p L ⎢ ⎢ 1/ 2 ⎥ ⎢ ⎥ ⎣− L / 12⎦

[3.53]

3.4. Industrial models 3.4.1. Introduction Industrial models are generally elaborated by using commercial codes based on the finite element method offering a wide range of elements and techniques to the user. Automatic mesh generation is essential in order to efficiently generate the nodes and the elements describing the structure. The model size is measured by the number of DOF introduced. It increases with the capacity of the computers and gains an order of magnitude every decade. This race with refinement does not exempt the user of a certain know-how in order to avoid errors and make the model reliable. Only some general ideas will be given here for the elaboration and the verification of the models, which will not be exhaustive. More detailed considerations could be found in references related to the finite elements.

3.4.2. The element types The first quality of a finite element code is to have a library of various elements enabling the most realistic modeling possible, both from a geometric point of view and from the perspective of the dynamic properties. The main element types are the following:

– “scalar” elements on one DOF or between two DOF. They make it possible to directly supply the mass, damping and stiffness matrices of the structure:

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- on one DOF (diagonal terms): directional mass (or inertia), spring or damper which are grounded,

- between two DOF (non-diagonal terms) mass, damper or spring. Thus we can for example introduce the basic elements described in section 3.2.2 or, more generally, reconstruct any structural matrix term by term. It is the last resort when no other element type corresponds to what is needed: – the rigid element associated with a node. It is the element of mass, center of mass and inertias described in section 3.2.5 with Figure 3.5 and matrix [3.23]; – the beam element including all possible deformations: axial, torsion, bending. It is the element described by section 3.3.4 to which we can add coupling between deformations caused, for example, by an offset of the neutral axis. – plate elements from which one should distinguish: - in-plane deformation, which is a 2-dimensional generalization of the rod element in section 3.3.2 and which is formulated without too much difficulty, - bending deformation, which is also a 2-dimensional generalization of the beam element under bending in section 3.3.3, but which encounters important theoretical difficulties which can be avoided in several ways, giving place to many more or less efficient formulations from the convergence point of view.

Having said this, we can generally choose between a triangular form, always applicable, and the quadrilateral form, recommended whenever possible. Certain codes propose elements with intermediate nodes on the sides, which are more accurate but more difficult to handle. Finally, it is necessary to know that these elements often do not supply the rotational DOF perpendicular to the surface, the formulation of which is delicate: – shell elements whose formulation presents numerous difficulties. The problem is often avoided by representing the shells by sufficiently numerous plate elements in order to suitably come close to the geometry: modeling using “faces”. It should be noted that the quadrilateral faces, for example, may not be plane thus degrading the representativity of the model; – volume elements which are a 3-dimensional generalization of the rod element and of the plate element in membrane. The forms which are used most are the tetrahedron, the pentahedron and the hexahedron; – other elements. These respond to particular needs, such as fluid elements to treat the interaction fluid-structure, or axisymmetric elements to represent the revolution structures more efficiently. Rigid elements of various forms can also be considered, but they will be considered using equivalent linear constraints between DOF.

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3.4.3. Linear constraints As introduced in section 3.2.5, the linear constraints between DOF make it possible to represent certain properties of the structure, which cannot be taken into account by the preceding elements. The first property that can be considered is the undeformability, which amounts to considering rigid elements between the considered DOF or even between the considered nodes, hence the “rigid node” notation representing the set of rigidly linked nodes. There can be any number of nodes: 2 will make a rigid beam, 3 or 4 coplanar nodes will make a rigid plate, n will make any form. A typical example is that of a circular interface with n nodes, which will be rigidly linked at the centre. Of course, this strategy overestimates the real stiffness in a generally realistic way. It is up to the user to judge for example if certain equipment on its support is nearly undeformable: if it is, its center of mass could be rigidly linked to the interface made rigid this way, if it is not, it is necessary to consider another modeling technique. The concept of deformability around an average motion can be opposed to the concept of undeformability. If we want to introduce the same equipment in the model without stiffening its support, it is enough to consider the average motion of the support at the interface by creating an “average node” instead of a “rigid node” that will be connected to the center of mass. Thus, the mass properties of the equipment will be taken into account without adding stiffness. Another possible application of the linear constraints: the mechanisms. Two nodes at the same location, whose translations are linked while their rotations are not, will model a ball and socket joint between the two beams. All types of mechanisms can be represented this way: hinges, sliding interfaces, etc. The linear constraints can also be used in order to access information which is not directly available. A relative displacement between two DOF will be given by a defined DOF like the difference between the latter ones. A local displacement may be determined by adequate combination of the global displacements, etc. In a general way, any response resulting from a linear combination of the DOF present in the model could be obtained this way. Various applications will be seen later.

3.4.4. DOF management The various elements introduced in the model will supply the DOF with mass, damping and stiffness. The problem of the damping will not be covered here as it is generally solved at another level, which does not affect the DOF management.

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Regarding the mass or the stiffness, it is possible that certain DOF are not associated with mass or stiffness, which will result in zeros on the diagonal of the corresponding matrix. Typical examples: – for the mass: rotational DOF without inertias; – for the stiffness: rotational DOF with rod and volume elements, like the rotational DOF about an axis perpendicular to a plate represented by coplanar elements if it is not provided in the element formulation. A first operation can possibly eliminate some of these zeros: the DOF elimination due to the linear constraints introduced in the model, for example, three or more lumped masses, rigidly linked to provide masses and inertias on the remaining DOF. Then, the DOF that have no mass can be eliminated by an operation known as static condensation, which will be detailed in section 4.2.3. Let us say here that, because they are massless, they can be deduced from others by static interpolation generating linear constraints, which make their elimination possible. The DOF that are without stiffness remain. Modeling errors, like forgetting an element that generates a mechanism, can cause them and in this case the errors must be corrected after detection of the corresponding singularities. They can also correspond to a normal situation, like the presence of coplanar plate elements, which do not supply certain rotations. This type of singularity can be automatically detected by the code and eliminated by directly condensing the DOF concerned after a possible change of the reference frame. All the DOF eliminated by the linear constraints (section 3.2.5.2) could be restored, after solution on the remaining DOF, starting from these same linear constraints. We should remember the following elementary rules: – a given DOF cannot be eliminated twice. As each linear constraint eliminates a DOF, the DOF should be carefully chosen in order to avoid this error and this is not always evident in the case of multiple constraints; – an eliminated DOF is removed from the solution. It cannot be directly used either as a DOF junction or as a DOF excitation. However, the equivalence for the excitation is given by section 3.2.6 and can be automatically applied by the code. In cases of serious difficulties with the DOF elimination technique, we can resort to the DOF introduction technique (section 3.2.5.3) if this is allowed by the code.

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3.4.5. Rules for modeling and verification of the model The nature of the elements and the refinement of the mesh should be adapted according to need. There are a number of criteria, which include: – good representation of geometrical forms: shells should be modeled using a sufficient number of plate elements to adequately represent the curvature. Be aware of conditioning problems, according to the codes, due to the near coplanarity of the adjacent elements. Also be careful about warping in elements; – good representation of all the structural elements: for example, for a stiffened plate we will choose between an equivalent orthotropic plate and the use of stiffeners by plates or by beams, according to the need; – good representation of the dynamic deformations: a sufficient number of elements for a given wavelength: 4 is not enough, 8 is more suitable; – do not forget the local flexibilities in joints and interfaces, which can notably influence the results; – avoid 2 or 3 dimensional elements which are poorly shaped or nearly degenerate. Each industrial field has its own guidelines and these general rules should be completed or adapted according to the acquired experience. The verification of the obtained model is to be done by all possible means. The following can be mentioned: – direct verification of the data: nodes, elements, etc.; – visualization of the structure: position of nodes, connectivity between elements, continuity of the lines and surfaces, etc.; – automatic verification by the code with warning or error messages: detection of local singularities, coherence and compatibility of linear constraints, conditioning, etc.; – before introducing the boundary conditions, the model is generally free, and this is expressed by singularities in the stiffness matrix corresponding to possible rigid body modes and mechanisms without strain energy. This subject will be developed in section 4.2.2. Let us say here that these global displacements can be determined by the stiffness matrix and compared to the geometry of the model. Combined with the mass matrix, they make it possible to determine the structure properties of mass, center of mass and inertia. These operations comprise the verifications of the validity of the model before its use.

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3.4.6. Industrial examples As an illustration, Figures 3.10 and 3.11 show two examples taken from the space industry: – Figure 3.10: finite element model of the primary structure of the Eurostar 3000 telecommunication satellite. There are about 12,000 physical nodes for a total of 72,000 DOF. The elements are mainly plates (9,600), beams (1,300), springs (2,200) and lumped masses (120). The meshing of the panels is often irregular in order to account for the adjacent structural elements as well as the various instruments they support. The mesh density is adjusted so as to suitably represent 100 normal modes expected up to 100 Hz, a value that traditionally delimits the low frequency domain in this branch of activity. Along with the secondary structures, solar panels, dishes, and equipment, this model will have a total size of several hundred thousand DOF. This is typical for this type of structure for the low frequency range; – Figure 3.11: finite element model of the HRG camera of the Spot satellite, whose structure supports primarily mirrors. There are about 2,800 physical nodes for 16,800 DOF, 4,300 elements including a few volume elements. The mesh density must be sufficient to represent modes up to 150-200 Hz, which will interact with the satellite. Some of the dynamic properties of these models will be examined in the following chapters.

Figure 3.10. The finite element model of the primary structure Eurostar 3000 (with the authorization of Astrium Toulouse)

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95

Y

Z

Figure 3.11. The finite element model of the HRG instrument of the Spot satellite (with the authorization of Atrium Toulouse)

3.5. Solution by direct integration 3.5.1. Introduction The solution of equation [3.1] by direct integration is the most general solution strategy, but also the most difficult. It constitutes the last resort when the other possibilities cannot be applied because, for example, of a too large frequency content or because of the presence of non-linearities. The idea is to remain in the time domain by discretizing the equation with a time step ∆t, which is generally constant. The main difficulty comes from the size of the system to solve, which generates numerical stability problems. Indeed, the highest eigenvalues do not have a physical significance because they are directly generated by the discretization but at the same time they condition the numerical solution.

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We usually distinguish: – the explicit methods for which the solution at step n+1 is obtained starting from equations at step n. They are the simplest and the fastest, but errors can increase at each step and they can lead to a divergence; – the implicit methods for which the solution at step n+1 is obtained starting from the equations at step n+1. This strategy makes it possible to limit the errors at each step and to ensure a convergence with certain reserves, but the calculations are more difficult. We will limit ourselves here to giving an example for each type of method for illustration. The knowledge on the subject could be improved with references such as [BAT 76].

3.5.2. Example of explicit method The simplest explicit method is that of the central differences. The velocities at step n are evaluated by difference of displacements to steps n-1 and n+1, and the accelerations by difference of velocities deduced of steps n-1, n and n+1, which gives:

u n =

1 (u n+1 − u n−1 ) 2∆t [3.54]

n = u

1 ∆t 2

(u n+1 − 2 u n + u n−1 )

With the equation of motion at step n: n + C u n + K u n = Fn Mu

[3.55]

we obtain: ⎛ M ⎛ ⎛ M 2M ⎞ C ⎞ C ⎞ ⎜⎜ ⎟⎟ u n +1 = Fn − ⎜⎜ K − ⎟u −⎜ ⎟ u n −1 + − 2 2 ⎟ n ⎜ 2 ∆ ∆t ⎟⎠ 2 t 2 ∆t ⎠ ⎝ ∆t ⎠ ⎝ ⎝ ∆t

[3.56]

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3.5.3. Example of implicit method One of the most often used implicit methods is that of β-Newmark [NEW 59] where displacements and velocities at step n+1 are given by: ⎛1 ⎞ n + β ∆t 2 u n +1 u n +1 = u n + ∆t u n + ⎜ − β ⎟ ∆t 2 u ⎝2 ⎠ n + γ ∆t u n +1 u n +1 = u n + (1 − γ ) ∆t u

[3.57]

β and γ are two parameters that can be adjusted according to the situation. Parameter β, which gives its name to the method, makes it possible to adjust the contribution of the accelerations to displacements. β = 1/4 corresponds to an average acceleration between the steps n and n+1, β = 1/6 with a linear acceleration. The parameter γ plays a similar role for the velocities. The relation 2 β ≥ γ ≥ 1 / 2 guarantees an unconditionally stable algorithm. The limit values β = 1/4 and γ = 1/2 can be used for a linear problem and this, together with the equations of motion at steps n and n + 1, gives: ⎛ 4M 2C ⎞ ⎛ 4M 2C ⎞ ⎛ 4M ⎞ n [3.58] ⎜⎜ ⎟⎟ u n + ⎜ + + K ⎟⎟ u n +1 = Fn +1 + ⎜⎜ + + C ⎟ u n + M u 2 2 ∆ ∆ t t t ⎝ ∆ ⎠ ⎝ ∆t ⎠ ⎝ ∆t ⎠

For a non-linear problem, the value of β will be increased in order to give a margin of stability.

Chapter 4

The Modal Approach

4.1. Introduction The modal approach was introduced in section 1.5.3 and was mentioned in the introduction of Chapter 3. It consists of solving the equations of motion in two steps: solution of the equations without excitation, which provides the structure normal modes, then superposition of the modes. If we take up again the complete equation [1.52] on the global DOF g = i+j: ⎡ M ii ⎢M ⎣ ji

M ij ⎤ ⎡ u i ⎤ ⎡ C ii + M jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣C ji

C ij ⎤ ⎡ u i ⎤ ⎡ K ii + C jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣K ji

K ij ⎤ ⎡ u i ⎤ ⎡ Fi ⎤ = K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣F j ⎥⎦

[4.1]

the motion in the absence of excitation corresponds to zero forces imposed on the internal DOF, Fi = 0 and to zero displacements imposed on the junction DOF, u j = 0 , if the junction exists, resulting in the following two equations: M ii u i + C ii u i + K ii u i = 0 i

[4.2]

M ji u i + C ji u i + K ji u i = F j

[4.3]

Just as for the complete equation [4.1] divided in two, the first equation [4.2] makes it possible to solve u i , while the second equation [4.3] restores F j given ui .

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Equation [4.2] in u i is homogenous and thus any solution is known to within an arbitrary coefficient. The presence of the second term, related to the viscous damping, complicates its solution. Two strategies are then possible at this level: – eliminating the damping term from the equation in order to simplify its solution and then reintroducing it in the second step. This is the “normal modes” approach which leads to real modal components, which can be easily interpreted and used, but at a cost of some difficulty in the mode superposition step due to this elimination; – keeping this damping term in the solution. This is the “complex modes” approach, which leads to complex modal components in the algebraic sense of the term. The preceding difficulty disappears but to the detriment of the simplicity and the computation effort. The current practice uses the normal modes by accepting the approximation which results from them. The complex modes are reserved for the strongly damped structures for which the normal modes can lead to large errors as well as for particular cases, which have not been discussed, such as the presence of the rotating components where the gyroscopic effect introduces an anti-symmetric matrix to matrix C. This approach by complex modes will be examined in Chapter 7 as an extension of the normal mode approach. The first step dedicated to the determination of the normal modes is discussed in section 4.2. The second step of the mode superposition is presented in section 4.3, leading to results of a form similar to that established for the 1-DOF system and thus making the modal parameters appear. These modal parameters play a major role in the structural responses: the effective modal parameters, which will be the subject of Chapter 5. Finally, this approach presents several advantages that make it interesting to use even when it is not strictly applicable, which is the subject of section 4.4.

4.2. Normal modes 4.2.1. Introduction The normal modes are the solutions of the equation of motion in the absence of excitation of the associated conservative system, i.e. equation [4.2] without the damping term: M ii u i + K ii u i = 0 i

[4.4]

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101

This system is of size N, number of internal DOF i. By searching solutions of the form: u i (t ) = U i e iωt

[4.5]

we obtain the eigenvalue problem:

(− ω

2

)

M ii + K ii U i = 0 i

[4.6]

which is not of the standard form A X = λ X. However, instead of considering the matrix A ii = M ii−1 K ii , which reduces to this, form [4.6] will be maintained in order to keep the physical matrices and their symmetry property. It provides: – N positive normal eigenvalues ω k 2 (underlined subscript = fixed subscript: see the notations) from which N circular natural frequencies ω k or N natural frequencies f k = ω k / 2π are deduced; – N eigenvectors with real components Φi k representing N mode shapes. The frequency f k and the form Φi k define the normal mode k. A mode is thus a form associated with a characteristic frequency of the structure. It represents a harmonic motion, with blocked junction if it exists, where all the internal DOF i vibrate in phase (components of the same sign) or out of phase (components of opposite sign). An N-DOF system i possesses N modes in the hypothesis where all the DOF i have mass (otherwise, the massless DOF can be eliminated). In general, the lower the natural frequency, the simpler the mode shape is and vice-versa, as will be seen in the examples. The mode shapes Φi k satisfy the orthogonality properties with respect to the matrices M ii and K ii which result from their symmetry: Φli M ii Φi k = 0 if l ≠ k

Φki M ii Φi k = m k generalized masses

Φli K ii Φi k = 0 if l ≠ k

Φki K ii Φi k = k k = ω k 2 m k generalized stiffness

[4.7]

by supposing ω l ≠ ω k . For the opposite case, i.e. for the double (or multiple) eigenvalues where all linear combinations of the corresponding eigenvectors are also

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eigenvectors, it is always possible to find two (or n) combinations of Φi k and Φi l which meet these relations. The generalized mass is the second of the modal parameters, after the natural frequency, associated with the mode shape. It is found in all the mode superposition expressions and plays the role of normalization. The generalized stiffness is deduced from the generalized mass using the relation [4.7]. By considering the entire set Φik of mode shapes Φi k , relation [4.7] can be rewritten: Φki M ii Φik = m kk

diagonal matrix of the generalized masses m k

Φki K ii Φik = k kk

diagonal matrix of the generalized stiffness k k

[4.8]

In the second step, these orthogonality properties (with respect to matrices M ii and K ii ) will make it possible to uncouple the equations projected based on the modes and thus to establish the relations expressing a mode superposition. Let us note that any mode Φi k is defined to within a constant, which is found in the square on the generalized masses and the stiffnesses. In order to specify this constant, a method of normalization is chosen for each mode, for example by setting the largest component or the generalized mass to 1. The choice has no importance for the calculation of the structural responses, but it intervenes in the intermediate calculations. Remember that the modal approach is interesting only if a small number of modes contribute to the structural responses. In the “low frequency” domain, we are only interested in the first modes, i.e. those whose frequencies are the lowest. The other modes will be subject to a “modal truncation”, i.e. they will be eliminated from the calculation, however, with the possibility of globally representing them, as will be seen in Chapter 5. For an illustration, let us consider the mass-spring system with 2 internal DOF in Figure 4.1, with two identical masses m and two stiffnesses in a 2/3 ratio in order to obtain the simple expressions for the modal terms. The results are as follows:

⎡1 0 ⎤ M ii = m ⎢ ⎥ ⎣0 1 ⎦

⎡ 5 / 3 − 2 / 3⎤ K ii = k ⎢ ⎥ ⎣− 2 / 3 2 / 3 ⎦

The Modal Approach

ω1 2 =

⎡ 1 ⎤ 1 k Φi1 = ⎢ ⎥ 3m ⎣ 2 ⎦

ω22 = 2

⎡− 2⎤ k Φi 2 = ⎢ ⎥ m ⎣1⎦

m1 = 5 m k1 = 5 k / 3

103

[4.9]

m 2 = 5 m k 2 = 10 k

The normalization of the modes was chosen for simplicity. The mass matrix being proportional to the identity, the eigenvectors are in this particular case directly orthogonal. For the first mode, when the DOF 1 moves a unit of distance, the DOF 2 moves twice the distance in the same direction. For the second mode, when the DOF 2 moves a unit of distance, the DOF 1 moves twice the distance in the opposite direction; its frequency is

6 times higher and corresponds to a less simple shape.

m k2 =

2k 3

u2 c2 u1

m k1 = k

i

c1 u0

j

Figure 4.1. 2 internal DOF system

With the examples of the industrial models in section 3.4.6, we have the following modes, each structure being clamped at its base: – the Eurostar 3000 model in Figure 3.10: 97 modes below 100 Hz; – the Spot HRG model in Figure 3.11: 19 modes below 150 Hz.

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The shape of each mode can be displayed according to the corresponding eigenvector. However, a simple plot does not always make it possible to understand the essential properties of the mode: main action direction, importance in relation to a particular excitation, contribution in terms of mass or stiffness, etc. These questions will be tackled in Chapter 5 with the modal effective parameters.

4.2.2. Free structures In the case of a free structure, i.e. without junction DOF j, there are modes which can be described as global displacements, which do not involve elastic energy in the structure. In the general case with 3-dimensions, a free structure (but without mechanism) possesses six possible independent global modes, for example, three pure translations and three rotations, which will be written Φi r (r for rigid). These modes verify: Ur =

1 Φri K ii Φi r = 0 ⇒ K ii Φi r = 0 i 2

⇒ K ii Φir = 0 ir

[4.10]

This produces a singularity of order 6 in K ii or six zero eigenvalues. Although this singularity prevents a static analysis, a dynamic analysis is possible. If relation [4.10] is compared to relation [4.6], it can be deduced that the overall displacements are modes of the structure with zero natural frequencies. We speak of rigid-body modes in contrast to the “elastic modes”, which generate an elastic energy. These are nonetheless normal modes, verifying the orthogonality properties with the elastic modes. Moreover, any linear combination of rigid-body modes is a rigid-body mode and a set of six orthogonal forms can always be found, the three translations and the three rotations in the inertia main reference frame being a particular case in the infinite number of possible solutions. These rigid-body modes, which are similar to global displacements, represent the rigid motion of the structure, which can be associated with its center of mass with its three translations and its three rotations. The elastic modes themselves represent the elastic motion around the preceding motion, without translation or rotation of the center of mass, and this can be described as motion around the center of mass. In order to eliminate any ambiguity, let us mention that this center of mass is a fictitious point whose motion depends on the entire system, which should not be confused with a model node possibly positioned at the center of mass, and which has its own motion.

The Modal Approach

105

The rigid-body modes can be deduced from the geometry of the structure: in which case we talk about kinematic rigid-body modes. They can also be deduced from the stiffness matrix starting from relation [4.10]. If six independent DOF in the structure are chosen, which are designated by r, for example, the six DOF of a given node P serving as reference, as indicated in Figure 4.2 (one can choose six DOF on several nodes, but the interpretation is more delicate), each DOF r can be associated with a rigid-body mode by considering the overall displacement which it produces when the other DOF r are constrained. Relation [4.10] partitioned on the DOF r and l = i-r is written:

⎡K rr ⎢K ⎣ lr

K rl ⎤ ⎡Φrr ⎤ ⎡0 rr ⎤ = K ll ⎥⎦ ⎢⎣Φlr ⎥⎦ ⎢⎣ 0 lr ⎥⎦

[4.11]

i

P

r

Figure 4.2. Free structure

If we consider the unit displacements on the DOF r, implying Φrr = I rr , identity matrix of size 6, the second line of relation [4.11] implies: Φlr = − K ll −1 K lr

[4.12]

Relation [4.12] shows that the rigid-body modes are deduced from the stiffness matrix: we then speak about static rigid-body modes, which could be compared to the kinematic rigid-body modes in order to verify the coherence between the structure geometry and its repartition of stiffnesses. If we calculate the generalized masses of the unit static rigid-body modes obtained this way, this means projecting all the masses on the DOF r, and this provides the structure rigid body mass matrix including its mass, center of mass and

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inertia properties related to the reference frame formed by the reference node and its DOF, just as for relation [3.23]: Φri M ii Φir =

M rr

⎡ M ⎢ 0 ⎢ ⎢ 0 =⎢ ⎢ 0 ⎢ M zG ⎢ ⎢⎣− M y G

0 M 0 − M zG 0 M xG

0 0 M M yG − M xG 0

0 − M zG M yG I xx / P − I xy / P − I xz / P

M zG 0 − M xG − I xy / P I yy / P − I yz / P

− M yG ⎤ M x G ⎥⎥ 0 ⎥ ⎥ [4.13] − I xz / P ⎥ − I yz / P ⎥ ⎥ I zz / P ⎥⎦

Relation [4.13] shows that these rigid-body modes are not orthogonal to themselves because matrix M rr is not diagonal, except if node P is at the center of mass of the structure ( xG = y G = z G = 0 ) and if x, y and z are the main inertia axes (zero crossed inertias), and this goes with one of the preceding comments. Relation [4.13] is another verification of the model validity. All these considerations on the free structures actually state the validation operations mentioned at the end of section 3.4.5. As an illustration, let us consider the rod and beam element matrices in sections 3.3.2 and 3.3.3: – free rod element with reference node P = 1: u r = u1 ul = u2 M ii =

M 6

K ii =

⎡1⎤ E S ⎡ 1 − 1⎤ ⎢− 1 1 ⎥ ⇒ Φir = ⎢1⎥ L ⎣ ⎣⎦ ⎦

⎡2 1 ⎤ ⎢1 2⎥ ⇒ M rr = M ⎣ ⎦

[4.14]

We clearly find the kinematic rigid-body mode (when u1 = 1 , u 2 = 1 ) and mass M of the element;

The Modal Approach

107

– free beam element with reference node P = 1: ⎡v ⎤ ur = ⎢ 1⎥ ⎣θ 1 ⎦ ⎡v ⎤ ul = ⎢ 2 ⎥ ⎣θ 2 ⎦

6 L − 12 6 L ⎤ ⎡ 12 ⎡1 ⎢ 6 L 4 L2 − 6 L 2 L2 ⎥ ⎢ EI ⎢ ⎥ ⇒ Φ = ⎢0 K ii = ir ⎢1 L3 ⎢− 12 − 6 L 12 − 6 L ⎥ ⎢ ⎢ 2 2⎥ − 6L 4L ⎦ ⎣ 6L 2L ⎣0

0⎤ 1 ⎥⎥ L⎥ ⎥ 1⎦

[4.15] 22 L ⎡ 156 ⎢ 22 L 4 L2 M ⎢ M ii = 13L 420 ⎢ 54 ⎢ L 13 3L2 − − ⎣

− 13L ⎤ ⎡ ⎢ M − 3L2 ⎥⎥ ⇒ M rr = ⎢ 156 − 22 L ⎥ ⎢M L ⎥ ⎣⎢ 2 − 22 L 4 L2 ⎦ 54 13L

L⎤ 2 ⎥ ⎥ L2 ⎥ M 3 ⎦⎥ M

It is also possible to find the two kinematic rigid-body modes generated by a pure translation and by a rotation around node 1, as well as the properties of mass (M), center of mass (L/2) and inertia ( M L2 / 3 ) with respect to node 1.

m

u3

m

u2

m

u1

k

k

Figure 4.3. Free 3-DOF system

i

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The following normal modes are found with the free mass-spring system in Figure 4.3: ⎡1 0 0⎤ M ii = m ⎢⎢0 1 0⎥⎥ ⎢⎣0 0 1⎥⎦ ⎡ k 2 ω1 = 0 Φi1 = ⎢⎢ m ⎢⎣

K ii

1 ⎤ 1 ⎥⎥ 1 ⎥⎦

⎡ 1 −1 0 ⎤ = k ⎢⎢− 1 2 − 1⎥⎥ ⎢⎣ 0 − 1 1 ⎥⎦

m1 = 3 m

⎡ 1 ⎤ k Φi 2 = ⎢⎢ 0 ⎥⎥ m 2 = 2 m m ⎢⎣ − 1 ⎥⎦ ⎡ −1 ⎤ k =3 Φi3 = ⎢⎢ 2 ⎥⎥ m3 = 6 m m ⎢⎣ − 1 ⎥⎦

k1 = 0 k

ω22 = 1

k2 = 2 k

ω32

k 3 = 18 k

[4.16]

The first mode is rigid and is a global translation. It is orthogonal to the two elastic modes (directly, because the mass matrix is proportional to the identity) and its generalized mass is equal to the total mass for a unit translation. The two elastic modes represent deformations around the center of mass, which remains motionless (the masses on the DOF are equal and the sum of the components is zero). As has already been mentioned, the center of mass must not be confused with node 2, which happens to be located at the center of mass.

4.2.3. System static condensation Eigenvalue problem [4.6] can lead to time consuming calculations according to the model size. It is possible to reduce the size of the system beforehand without losing too much precision on the low frequency properties, by a judicious elimination of DOF which is expressed by a condensation of matrices M ii and K ii . This operation was of great help years ago when the available algorithms did not enable the solution of very large systems in a reasonable period of time. With advances in hardware and software, it is no longer used as such, but its mechanism can be found in various situations described in this work, and this motivates its detailed description here.

The Modal Approach

109

The simplest strategy is Guyan condensation [GUY 65] which consists of selecting a reduced set of DOF s (selection) and eliminating the other DOF o = i-s (omitted) by static interpolation. With this partition, relation [4.6] is rewritten (U replaced by u for the sake of convenience): ⎛ ⎜−ω 2 ⎜ ⎝

⎡ M ss ⎢M ⎣ os

M so ⎤ ⎡ K ss + M oo ⎥⎦ ⎢⎣K os

K so ⎤ ⎞ ⎡u s ⎤ ⎡ 0 s ⎤ ⎟ = K oo ⎥⎦ ⎟⎠ ⎢⎣u o ⎥⎦ ⎢⎣0 o ⎥⎦

[4.17]

Assuming that the inertia forces Fo = −ω 2 M oi u i relative to the DOF o are negligible, the second line [4.17] provides: K os u s + K oo u o = 0 o

−1 ⇒ u o = −K oo K os u s

[4.18]

a linear constraint between the DOF o and s expressing the static interpolation of displacements u o between displacements u s and which allows the transformation: ⎡u ⎤ ui = ⎢ s ⎥ = Tis u s ⎣u o ⎦

⎡ I ⎤ with Tis = ⎢ −1ss ⎥ ⎣ −K oo K os ⎦

[4.19]

hence the condensed matrices (relation [1.53]): M ss = Tsi M ii Tis −1 −1 −1 −1 = M ss − K so K oo M os − M so K oo K os + K so K oo M oo K oo K os

[4.20] −1 K ss = Tsi K ii Tis = K ss − K so K oo

K os

Relations [4.20] thus reduce the size of matrices from the number of DOF i to the number of DOF s. The better the retained DOF are adapted to the first modes, the better these modes will be represented. As linear constraints [4.18] have a tendency to stiffen the model, all the natural frequencies will be overestimated. This condensation technique for large models is of interest only if a large number of DOF are eliminated, since the obtained matrices are full, contrary to the initial very sparse matrices, and this limits its efficiency with regards to the eigenvalue problem.

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Moreover, it introduces errors relative to the capacity of the chosen DOF s to represent the desired mode shapes. The choice of these DOF thus plays a fundamental role in the precision of the operation. Thus, priority will be given to the DOF with large masses and/or to the large displacements for the desired modes, retaining the translations rather than the rotations, etc. Limited efficiency, delicate choice and loss of accuracy now dissuade the use of this strategy in order to reduce the size of a model except for particular cases requiring a small matrix model, which can be easily manipulated. On the contrary, the mass and stiffness static properties being respected, we will find this condensation again in section 4.3.2 concerning the mode superposition in order to maintain the static properties relative to the junction DOF. For an illustration, consider the beam in Figure 4.4 clamped at node 1 and free at node 2, and in particular the two first modes in pure bending without shear effect. With different models, the following results are obtained: – with a continuous approach (section 6.3) as reference, the circular natural frequencies and the components (v, θ) of the mode shapes at end 2 are given by:

ω1 = 3.516 ω 2 = 22.03

EI 3

ML

EI M L3

1 ⎡ ⎤ Φi1 = ⎢ ⎥ 1 . 3765 / L ⎣ ⎦ 1 ⎡ ⎤ Φi 2 = ⎢ ⎥ L 4 . 48 / ⎣ ⎦

[4.21]

– with only one beam finite element (section 3.3), thus with the matrices resulted from relations [3.40], we obtain:

M ii =

M ⎡ 156 − 22 L ⎤ ⎥ ⎢ 420 ⎣− 22 L 4 L2 ⎦

ω1 = 3.533 ω 2 = 34.81

EI 3

ML

EI M L3

K ii =

− 22 L ⎤ E I ⎡ 12 3 ⎢ − 22 L 4 L2 ⎥⎦ L ⎣

1 ⎡ ⎤ Φi1 = ⎢ ⎥ 1 . 3750 / L ⎣ ⎦ 1 ⎡ ⎤ Φi 2 = ⎢ ⎥ L 7 . 62 / ⎣ ⎦

[4.22]

The Modal Approach

111

resulting in an error of 0.5 and 58% on the circular frequencies: a polynomial of degree 3 gives a good approximation of the first form and a mediocre approximation of the second; – with the same beam finite element and a condensation of the rotational DOF, we obtain: ⎡ 1 ⎤ Tid = ⎢ ⎥ ⇒ ⎣3 /(2 L)⎦

ω1 = 3.568

EI 3

ML

M dd =

33 M 140

K dd =

3E I L3

1 ⎡ ⎤ Φi1 = ⎢ ⎥ ⎣1.5000 / L ⎦

[4.23]

resulting in an error of 1.0% for the first circular frequency, the second having been eliminated. The static interpolation giving θ 2 = 3 v 2 /( 2 L) is a good approximation of the exact value, hence the justified condensation for the first mode, the error being of the same order as that of the discretization by finite elements. It should be noted here that the lumped mass M/2 at the tip would have given M dd = M / 2 , a very large error (–31% on the circular frequency). v2

θ2

M, E, I, φ = 0 1

2 L Figure 4.4. Beam in pure bending

4.2.4. Eigenvalue problem solution The eigenvalue problem solution, i.e. equation [4.6], possibly condensed as indicated in section 4.2.3, is a costly step for industrial models. In order to carry it out efficiently, it is necessary to have an algorithm available, adapted to the problem whose main characteristics are the system size, its sparsity (proportion of the nonzero terms) and the number of desired modes.

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The references on this subject are relatively numerous (see, for example, [PAR 98, WIL 88]). We usually distinguish: – the direct methods consisting of transforming the considered matrices into diagonal matrices (Jacobi) or tri-diagonal matrices (Givens, Householder). This strategy is well adapted to the small size systems and we look for all their solutions. – the iterative methods, which determine a mode or a set of modes by solving a linear system at each step. The simplest is the direct iteration where the first mode is obtained by the algorithm:

z n = K −1 M y n −1

yn = zn / zn

z n → 1 / ω1 2

[4.24]

the following modes being determined in the same way in a subspace which is orthogonal to the previous modes. A more efficient method is the inverse iteration based on the algorithm: K z n = M y n −1

yn = zn / zn

z n → ω1 2

[4.25]

For each iteration, we can simultaneously consider a set of vectors and search for the eigensolutions in this reduced-dimension subspace (principle of Ritz method). Let us also mention that, as a prelude to the solution, the Strum sequence check allows a rapid evaluation of the number of eigenvalues in a given frequency band. Thus, this makes it possible to verify that the algorithm used has found all the solutions in the considered frequency band. As for the industrial models, the most commonly used method is the Lanczos method which, following several enhancements is particularly effective [PAR 98]. It is a subspace iterative method which converges to the exact solution after a number of iterations equal to the system size. In practice, fewer iterations are used because the convergence is rapid. The algorithm is schematically as follows: – initialization: vectors q 0 = 0 and r0 arbitrary – iteration n: - scalar: β n2 = rn −1T M rn −1 - Lanczos vector: q n = rn −1 / β n

[4.26]

The Modal Approach

113

- scalar: α n = q n T M K −1 M q n - vector: rn = K −1 M q n − α n q n − β n q n −1 – reduction n: - transformation matrix: Q n = [ q1 q 2 … q n

]

⎡α 1 ⎢β - transformed matrix: Q n T M K −1 M Q n = Tn = ⎢ 2 ⎢0 ⎢ ⎣0

β2 α2

0

0

βn

0 ⎤ 0 ⎥⎥ βn ⎥ ⎥ αn ⎦

- eigenvalue problem with solutions (λ , X) k : 1 / λ k → ω k 2 , [ q1 q 2 … q n ] X k → Φk

i u1 k=1

m = 1

u2 k=1

m = 1

u3 k=1

m = 1

k=1

u0

u4 j Figure 4.5. Simple example for Lanczos method

As an illustration, see the example in Figure 4.5, for which we have: ⎡1 0 0 ⎤ ⎡ 2 −1 0 ⎤ ⎡3 2 1 ⎤ 1⎢ ⎢ ⎥ ⎢ ⎥ −1 M ii = ⎢0 1 0⎥ K ii = ⎢− 1 2 − 1⎥ ⇒ K ii M ii = ⎢2 4 2⎥⎥ 4 ⎢⎣0 0 1⎥⎦ ⎢⎣ 0 − 1 2 ⎥⎦ ⎢⎣1 2 3⎥⎦

The iterations give the results in Table 4.1.

[4.27]

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n

0

1

2

3 = exact

βn

–

1

5/4

1/5

qn

⎡0 ⎤ ⎢0 ⎥ ⎢ ⎥ ⎢⎣0 ⎥⎦

⎡1 ⎤ ⎢0⎥ ⎢ ⎥ ⎣⎢ 0⎦⎥

⎡0⎤ 1 ⎢ ⎥ 2 5⎢ ⎥ ⎣⎢ 1 ⎦⎥

⎡0⎤ 1 ⎢ ⎥ −1 5⎢ ⎥ ⎣⎢ 2 ⎦⎥

αn

–

3/4

27/20

2/5

rn

⎡1 ⎤ ⎢0⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦

⎡0⎤ 1⎢ ⎥ 2 4⎢ ⎥ ⎢⎣1 ⎥⎦

⎡0⎤ 1 ⎢ ⎥ −1 5 5⎢ ⎥ ⎢⎣ 2 ⎥⎦

⎡0⎤ ⎢0⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦

⎡ 3/ 4 5/4 0 ⎤ ⎢ ⎥ ⎢ 5 / 4 27 / 20 1/ 5 ⎥ ⎢ ⎥ 1/ 5 2 / 5⎥ ⎢⎣ 0 ⎦

Tn

–

[3/4]

⎡ 3/ 4 5/4⎤ ⎢ ⎥ ⎣⎢ 5 / 4 27 / 20 ⎦⎥

ωk2

–

[0.75]

[0.5937

–

⎡1 ⎤ ⎢0⎥ ⎢ ⎥ ⎢⎣ 0 ⎥⎦

Φk

2.4063]

1 ⎤ ⎡ 1 ⎢ +1.50 −0.54 ⎥ ⎢ ⎥ ⎢⎣ +1.75 −0.27 ⎥⎦

⎡2 − 2 ⎣

2 2 + 2 ⎤⎦

1 1 ⎤ ⎡1 ⎢ ⎥ ⎢ 2 0 − 2⎥ ⎢ 1 −1 1 ⎥ ⎣ ⎦

Table 4.1. Lanczos method applied to the system in Figure 4.5

The Modal Approach

115

4.3. Mode superposition 4.3.1. Introduction After having solved the partial equation [4.4], it is now necessary to solve the complete equation [4.1], which is given again here: ⎡ M ii ⎢M ⎣ ji

M ij ⎤ ⎡ u i ⎤ ⎡ C ii + M jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣C ji

C ij ⎤ ⎡ u i ⎤ ⎡ K ii + C jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣K ji

K ij ⎤ ⎡ u i ⎤ ⎡ Fi ⎤ = K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣F j ⎥⎦

[4.28]

The mode superposition results from the projection of all the physical properties of the structure onto a base of modes Φik of section 4.2. However, these fixed junction modes ( Φ jk = 0 jk ), if the junction exists, can only represent motion with respect to the junction. In the case of an excitation by imposed junction motion u j , they will represent the relative motion of the internal DOF with respect to the junction, but the motion of the junction itself makes it necessary to complete the base using other modes related to the junction DOF. The easiest is to consider the static modes, Ψ, obtained by successively imposing a unit displacement u j = 1, while blocking all other junction displacements, as illustrated by Figure 4.6. These modes, commonly called junction modes or constraint modes, verify: Ψ jj = I jj

and

K ii Ψij + K ij = 0 ij

⇒ Ψij = −K ii−1 K ij

[4.29]

on condition that matrix K ii is regular. This is the case for a structure with a single junction which prevents any rigid body mode (section 4.2.2). The opposite case is that of a structure where the junction does not prevent rigid body motion. In this case we will then take the junction modes Ψij verifying the first equation [4.29] and render them orthogonal to the rigid-body modes Φir resulting from the eigenvalue problem. This can be achieved by linear constraints introduced via Lagrange multipliers (section 3.2.5.3), such that Ψij is solution of [ROY 02]: ⎡ K ii ⎢Φ M ⎣ ri ii

M ii Φir ⎤ ⎡Ψij ⎤ ⎡ - K ij ⎤ ⎢ ⎥=⎢ ⎥ 0 rr ⎥⎦ ⎣λ rj ⎦ ⎣− Φri M ij ⎦

[4.30]

This case is not very frequent in practice, but it may be handled just like the others: the rigid-body modes Φir are to be considered just as those of the free structures, the junction modes Ψ(i + j ) j like those of the structure without a rigidbody mode.

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i uj = 1

j Figure 4.6. Junction modes

It should be noted that these junction modes can be interpreted as the static transmissibility in displacements between the DOF i and j. This interpretation will actually be used later. In the particular case of a rigid or statically determinate junction, i.e. represented in the general case of 3 dimensions by the 6 DOF j, for example the 6 DOF of a unique junction node, the 6 junction modes obtained will be rigid (without deformation) and we will use j = r (rigid) in order to distinguish this important case. Starting from the preceding considerations, we will use the base formed by the normal modes Φ(i + j ) k and the junction modes Ψ(i + j ) j which is equivalent to considering the projection according to the transformation:

[

u i + j = Φ(i + j ) k

]

⎡q k ⎤ Ψ(i + j ) j ⎢ ⎥ ⎣u j ⎦

⇔

⎡ u i ⎤ ⎡Φik ⎢u ⎥ = ⎢0 ⎣ j ⎦ ⎣ jk

Ψij ⎤ ⎡q k ⎤ I jj ⎥⎦ ⎢⎣u j ⎥⎦

[4.31]

The first row expresses the decomposition of the absolute displacement ui in a relative displacement based on the normal modes and the junction displacement due to u j . The second row is an identity expressing the fact that the junction displacements u j are conserved, whereas the internal relative displacements u i − Ψij u j will be represented by the modal displacements q k .

The N normal modes, N equal to the number of internal DOF i, can be limited to the first n by modal truncation, which reduces the size of the new base and thus that of the system to solve after transformation [4.31].

The Modal Approach

117

4.3.2. Equation of motion transformation Transformation [4.31] applied to the matrices intervening in the equation of motion [4.28] gives (relations [1.53]): ⎡Φki ⎢Ψ ⎣ ji

0 kj ⎤ ⎡ M ii I jj ⎥⎦ ⎢⎣M ji

⎡Φki ⎢Ψ ⎣ ji

0 kj ⎤ ⎡ C ii I jj ⎥⎦ ⎢⎣C ji

C ij ⎤ ⎡Φik C jj ⎥⎦ ⎢⎣0 jk

⎡Φki ⎢Ψ ⎣ ji

0 kj ⎤ ⎡ K ii I jj ⎥⎦ ⎢⎣K ji

K ij ⎤ ⎡Φik K jj ⎥⎦ ⎢⎣0 jk

⎡Φki ⎢Ψ ⎣ ji

0 kj ⎤ ⎡ Fi ⎤ ⎡ Φki Fi ⎤ = I jj ⎥⎦ ⎢⎣F j ⎥⎦ ⎢⎣Ψ ji Fi + F j ⎥⎦

M ij ⎤ ⎡Φik M jj ⎥⎦ ⎢⎣0 jk

Ψij ⎤ ⎡m kk = I jj ⎥⎦ ⎢⎣ L jk Ψij ⎤ ⎡ c kk = I jj ⎥⎦ ⎢⎣0 jk K ij ⎤ ⎡k kk = I jj ⎥⎦ ⎢⎣ 0 jk

L kj ⎤ M jj ⎥⎦ 0 kj ⎤ 0 jj ⎥⎦ 0 kj ⎤ K jj ⎥⎦

[4.32]

[4.33]

[4.34]

[4.35]

Thus, the following matrices appear: – m kk = Φki M ii Φik : diagonal matrix of the generalized masses m k , according to relations [4.8]; – c kk = Φki C ii Φik : matrix of the generalized damping a priori full (coupled); – k kk = Φki K ii Φik : diagonal matrix of the generalized stiffnesses k k , according to relations [4.8];

[

]

⎡Ψij ⎤ M ij ⎢ ⎥ = Φki (M ii Ψij + M ij ) : matrix of participation ⎣I jj ⎦ factors (name justified later) expressing the mass coupling between normal modes and junction modes;

– L kj = Φki M ii

[

]

⎡ M ii M ij ⎤ ⎡Ψij ⎤ I jj ⎢ ⎥ ⎢ ⎥ = Ψ ji M ii Ψij + Ψ ji M ij + M ji Ψij + M jj : ⎣M ji M jj ⎦ ⎣I jj ⎦ condensed mass matrix (operation [4.20] with the DOF j like selected DOF). In the previously mentioned case of a (rigid) statically determined junction j = r, it is the structure rigid body mass matrix including its properties of mass, center of mass and inertias relative to the unique junction node reference frame, as for relation [4.13];

– M jj = Ψ ji

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[

]

⎡ K ii K ij ⎤ ⎡Ψij ⎤ −1 I jj ⎢ ⎥ ⎢ ⎥ = K jj − K ji K ii K ij : condensed stiffness ⎣K ji K jj ⎦ ⎣I jj ⎦ matrix (simplified expression due to relation [4.29]). In the particular case of a rigid junction j = r, this matrix is zero, which makes it possible to attribute it to an indeterminate junction;

– K jj = Ψ ji

– the other stiffness terms are zero because of relation [4.29]; – the other damping terms are zero for a rigid junction, but they can be non-zero for an indeterminate junction. However, they are often weak and/or not well known. Moreover, they complicate the solution in an important way and thus are often neglected. This hypothesis is similar to that made in the following section for the uncoupling of equations. In addition to the modal parameters presented in section 4.2.1, the presence of a junction introduces the modal participation factors L kj which play an important role in mode superposition. As an illustration, the system in Figure 4.1 has a 1-DOF rigid junction, and therefore a single junction mode, and the modal participation factors, i.e. the mass coupling between each of the 2 mode shapes [4.9] and the junction mode, given by: ⎡1⎤ ⎡3⎤ Ψij = ⎢ ⎥ ⇒ L kj = m ⎢ ⎥ ⎣1⎦ ⎣− 1⎦

[4.36]

The system in Figure 4.5 gives another illustration with a 2-DOF indeterminate junction. The modal participation factors between the 3 normal modes of Table 4.1 and the 2 junction modes are given by: ⎡2 + 2 ⎡3 1 ⎤ 1⎢ m⎢ ⎥ Ψij = ⎢2 2⎥ ⇒ L kj = ⎢ 1 4 2 ⎢ ⎢⎣1 3⎥⎦ 2− 2 ⎣

2 + 2⎤ ⎥ −1 ⎥ 2 − 2⎥ ⎦

[4.37]

With relations [4.32] to [4.35], equation [4.28] transforms into: ⎡m kk ⎢L ⎣ jk

L kj ⎤ ⎡q k ⎤ ⎡ c kk + M jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣0 jk

0 kj ⎤ ⎡q k ⎤ ⎡k kk + 0 jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣ 0 jk

0 kj ⎤ ⎡q k ⎤ ⎡ Φki Fi ⎤ = [4.38] K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣Ψ ji Fi + F j ⎥⎦

The Modal Approach

119

Just as with relation [4.28], relation [4.38] divides into two equations: m kk q k + c kk q k + k kk q k = Φki Fi − L kj u j

[4.39]

F j = −Ψ ji Fi + L jk q k + M jj u j + K jj u j

[4.40]

Equation [4.39] makes it possible to solve q k , hence u i by relation [4.31]. It is the extension of the 1-DOF system equation [2.5]. The second member represents the excitation of the modes; it results from the sum of the generalized forces due to Fi , i.e. Φki Fi , and to u j , i.e. −L kj u j . Equation [4.40] makes it possible to restore F j starting from q k . It is the extension of the 1-DOF system equation [2.6]. It translates the fact that the vector of the reaction forces results from the sum of the forces opposing the statically transmitted forces, i.e. −Ψ ji Fi , of the modal forces, i.e. L jk q k , of the inertia forces, i.e. M jj u j , and of the indeterminate forces, i.e. K jj u j . We can see there a justification for the name “participation factors” used for terms L jk , which make it possible for the k normal modes to take part in the reaction forces.

4.3.3. Problem caused by the damping Equation [4.39] makes it possible to solve q k , all the other terms being known. The second member represents the modal excitation coming from Fi and from u j . Matrices m kk and k kk are diagonal, but the matrix c kk generally is not and this renders the equations coupled by the viscous damping. There is the counterpart of the normal mode hypothesis consisting of eliminating the damping from the eigenvalue problem. That is, the eigenvectors do not diagonalize c kk as they do for m kk and k kk , with some exceptions. The most well-known exception is Rayleigh damping, which consists of assuming proportional damping , i.e. damping that results from a linear combination of the mass and stiffness matrices: C ii = α M ii + β K ii

[4.41]

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C ii is diagonalized along with M ii and K ii . However, this hypothesis, like others which are more complicated, is not based on physical considerations.

A more general strategy is to retain only the diagonal terms from matrix c kk . It is the Basile hypothesis, which can be justified with the following reserves: – the structure is lightly damped. That implies not only that the non-diagonal terms are smaller than the diagonal terms, but that their influence on the structural responses is weak if the following point is verified; – the natural frequencies are not too close. In the case when two frequencies f k and f l are close, the possible error on the responses to f k due to the term c kl is proportional to f k 2 /( f l 2 − f k 2 ) and can thus be significant. In practice, the Basile hypothesis is currently used by attributing a reduced viscous damping ζ k to each normal mode k, just as in the 1-DOF system (equation [2.16]):

ζk =

ck

[4.42]

2 k k mk

c k being the diagonal term k of the matrix c kk , i.e. the generalized damping of the

mode k. This makes it possible to avoid the introduction of the viscous damping in the physical model, which is generally difficult, in favor of modal terms, which will be estimated either according to the acquired experience or using the experimental results. However, the structural damping model of section 2.4.2 makes it possible to deduce a modal structural damping η k starting from structural damping η e attributed to the elements. Actually, attributing a structural damping η e to the

element e means considering a complex elementary stiffness [K (1 + i η )]e . The

assembly of these complex stiffnesses will produce a global stiffness matrix K ii which transformed by the normal modes Φi k , results in:

(

Φki K ii Φi k = k k + i ∑ η e Φki ℜ(K ii ) Φi k e

⇒ ηk = ∑ηe e

(Φki ℜ(K ii ) Φi k ) e kk

)e ≡ k k (1 + i η k )

= ∑ η e (τ k ) e e

[4.43]

The Modal Approach

121

which expresses the modal structural damping η k as the sum of the element structural damping η e weighted by the fraction (τ k ) e of strain energy in the elements. There again, the Basile hypothesis is necessary in order to uncouple the system. If we come back to the viscous damping, equation [4.39] with the Basile hypothesis generates as many uncoupled equations as modes k: m k q k + c k q k + k k q k = Φki Fi − L k j u j

[4.44]

Each mode k thus verifies an equation similar to that of a 1-DOF system (equation [2.5]), as announced in section 1.5.4, with the generalized mass, damping, stiffness and force parameters (depending here on the normalization of mode Φi k ). All the modal parameters necessary for the solution are found in this equation: the generalized mass m k , the generalized stiffness k k = ω k 2 m k , the generalized damping c k = 2 ζ k ω k m k , mode Φi k and the participation factors L k j . As an illustration, let us take the example in Figure 4.1 with c1 = c 2 = c . Taking the results of [4.9] into account, transformation [4.33] gives the following matrix for the modal dampings: ⎡ 1 2 ⎤ ⎡ 2c − c ⎤ ⎡ 1 − 2 ⎤ ⎡2 1 ⎤ =c⎢ Φki C ii Φik = ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥ ⎣− 2 1 ⎦ ⎣ − c c ⎦ ⎣2 1 ⎦ ⎣1 13⎦

[4.45]

The diagonal selection leads to the following modal damping:

ζ1 =

c11

2 k1 m1

=

2 3 c 5 2 km

ζ2 =

c 22

2 k 2 m2

=

13

c

[4.46]

5 2 2 km

The two natural frequencies being very distant, the Basile hypothesis will be justified insofar as c is small compared to 2 k m .

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4.3.4. Frequency resolution In the frequency domain, equation [4.44] is written: (−ω 2 m k + iω c k + k k ) q k = Φki Fi + ω 2 L k j u j

[4.47]

Just as for the solution of the 1-DOF system described in section 2.2.3, the solution for equation [4.47] can be put in the following form, if k k ≠ 0 : q k = H k (ω )

1 (Φki Fi + ω 2 L k j u j ) kk

[4.48]

where, just as in equation [2.32], the generalized stiffness k k = ω k 2 m k intervenes associated with: H k (ω ) =

1 ⎛ ω 1− ⎜ ⎜ωk ⎝

2

⎞ ⎟ + i 2ζ ω k ⎟ ωk ⎠

[4.49]

a dynamic amplification factor of the mode k, depending on the two modal parameters ω k and ζ k as for equation [2.35]. If k k = 0 , the form [4.48] can no longer be written. That happens for the rigidbody modes r of a free or non-free structure (see section 4.2.2 or section 4.3.1 respectively), and which will be assumed here to be orthogonal to each other in order to uncouple the equations. As they cannot dissipate energy (excluding the case of a grounded damper without associated stiffness), we get c r = 0 as well as k r = 0 , and equation [4.47] gives the degenerated form replacing [4.48]:

qr =

1 − ω 2 mr

Φri Fi

[4.50]

m r generalized mass of the rigid-body mode r (representing the mass or a main

inertia if the unit rigid-body modes were chosen in the main inertia reference frame).

The Modal Approach

123

Starting from the modal results [4.48] and possibly [4.50], u i is deduced by relation [4.31] and F j by relation [4.40]. Relation [4.31], which has allowed the uncoupling of the equations in order to solve them by projecting them onto the normal modes, now serves to restore the physical responses starting from the modal responses. The product Φik q k expresses the mode superposition which provides the internal relative displacements by combining the contributions of each mode k, given as: ∑ Φi k q k , the sum comprising all the considered modes k. Then, by considering all the modes k without truncation, we obtain the solution in the general form with the dynamic flexibility, dynamic transmissibility and dynamic stiffness FRF: ⎡ u i (ω ) ⎤ ⎡ G ii (ω ) Tij (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎢F (ω )⎥ = ⎢− T (ω ) K (ω )⎥ ⎢u (ω )⎥ ji jj ⎦ ⎦⎣ j ⎦ ⎣ ⎣ j G ii (ω ) =

N

Φi k Φki

k =1

ωk 2 m k

∑ H k (ω ) N

Tij (ω ) = Ψij + ∑ H k (ω ) k =1

(if rigid-body modes r: ∑ r

Φi r Φri

−ω 2 m r

ω 2 Φi k L k j ωk 2

[4.52]

mk N

K jj (ω ) = −ω 2 M jj + K jj − ω 2 ∑ H k (ω ) k =1

) [4.51]

ω 2 L jk L k j ωk 2

mk

[4.53]

Equation [4.53] gives the dynamic mass: M jj (ω ) = M jj +

K jj

−ω 2

N

+ ∑ H k (ω ) k =1

ω 2 L jk L k j ωk 2

mk

[4.54]

The following preliminary comments can be made on these results: – in relation [4.51], the contribution of the possible rigid-body modes simply expresses Newton’s law: for each rigid-body mode, the contribution to − ω 2 G ii (ω ) , which is the ratio acceleration/force, is equal to the inverse of its generalized mass conveniently normalized (main mass or inertia for a unit rigidbody mode in the inertia main reference frame); – in relation [4.52], we find the junction modes Ψij ; they are interpreted as static transmissibilities, i.e. Tij (ω ) for ω = 0, which results from their definition;

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– in relation [4.53], we find the statically indeterminate stiffness matrix K jj ; it is logically the stiffness matrix K jj (ω ) at ω = 0; – in relation [4.54], we find the junction condensed mass matrix M jj ; for a rigid junction r, it becomes the rigid body mass matrix equal to M rr (ω ) for ω = 0. Thus, the FRF in transmissibility [4.52] and in mass [4.54] are presented in the form of a term that can be described as static (FRF at ω = 0, outside the possible indeterminate contribution K jj /(−ω 2 ) for M jj (ω ) ), and the superposition of modes, contrary to the flexibilities [4.51], which do not have a static term. The comparison with the 1-DOF system relations from [2.32] to [2.34] motivates the decomposition of the static terms in order to find the dynamic transmissibility factors Tk (ω ) . Starting from the expressions of m kk , L kj and M jj resulting from relation [4.32], we can establish that: Ψij =

Φi k L k j − M ii−1 M ij k =1 m k N

∑

M jj =

N

L jk L k j

k =1

mk

∑

+ M jj − M ji M ii−1 M ij

[4.55]

[4.56]

provided that matrix M ii is regular, which should be the case after the possible condensation of the massless DOF (which amounts to considering the pseudoinverse of M ii ). Equations [4.55] and [4.56] call for the following comments: – the terms outside the sums represent the junction or static mass properties that the normal modes cannot take into account due to their zero components at the junction. These terms are due to the discretization, which directly assigns to the junction a contribution of the junction mode (−M ii−1 M ij ) and the mass (M jj − M ji M ii−1 M ij ) via the adjacent elements. The finer the discretization is, the

smaller these terms are (becoming zero with a continuous approach); – relation [4.55] expresses the junction modes as a linear combination of the normal modes. The generalized masses play only a normalization role here; the coefficients are given by the participation factors, which thus quantify the participation of each normal modal to the junction modes.

The Modal Approach

125

By taking into account relations [4.55] and [4.56] in equations [4.52] and [4.54], and by taking equation [4.51] as such again, the following results are obtained: G ii (ω ) =

Tij (ω ) =

N

∑ H k (ω )

k =1

N

∑ Tk (ω )

k =1

Φi k Φki

(if rigid-body modes r:

ωk 2 m k

∑ r

Φi r Φri

−ω 2 m r

Φi k L k j − M ii−1 M ij mk

N

M jj (ω ) = ∑ Tk (ω ) k =1

) [4.57]

[4.58]

L jk L k j K jj + M jj − M ji M ii−1 M ij + mk −ω 2

[4.59]

with, as for equations [2.36] and [2.37]: 1+ i 2ζ k Tk (ω ) =

⎛ ω 1− ⎜ ⎜ωk ⎝

2

ω ωk

⎛ ω = 1+ ⎜ ⎜ωk ⎝ ω

⎞ ⎟ + i 2ζ k ⎟ ωk ⎠

2

⎞ ⎟ H (ω ) k ⎟ ⎠

[4.60]

Relations [4.57] to [4.59] are the logic extensions of the 1-DOF system relations [2.32] to [2.34] for the N degrees of freedom systems represented by their N modes, i.e. without truncation. The sums express the mode superposition. Each mode behaves like a 1-DOF system and contributes by the product of two terms: – a dimensionless dynamic amplification H k (ω ) or Tk (ω ) following the considered FRF, depending on the modal parameters ω k and ζ k . These amplifications were analyzed in detail in section 2.2.4; – a frequency-independent term which is also of the same dimension as the considered FRF, which was of a static nature for the 1-DOF system and which will here be called the matrix of the modal effective parameters, depending on the modes Φi k and/or the participation factors L k j . In the case of a modal truncation which is necessary in order to maintain interest in the modal approach as already mentioned, the sums will be limited to the first n modes. In this case, the error committed will depend on the relations that are used. It

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can be large and may require a correction, which the modal effective parameters are able to provide. The modal effective parameters thus play a major role in calculating the structural responses. A detailed analysis of this concept is interesting in many respects: good understanding of phenomena, evaluation of the importance of the modes, control of the modal truncation, calculation of all types of responses, development of equivalent models. This will be the subject of the following chapter where all the aspects of the modal approach will be seen through the effective modal parameters.

4.4. From the frequency approach to the modal approach The modal approach presents real advantages when it is applicable: reduced calculation time due to modal truncation, interpretation of the results through the normal modes. However, it requires various conditions, starting with the structural properties enabling form [4.1] with matrices M, C, K which are given as constants. If these evolve at all, the normal mode concept no longer applies. However, the advantages of the modal approach are such that even these incompatibilities can be eliminated with the help of a certain adaptations and some concessions on the results. This is particularly the case for the structure including the properties that depend on the frequency: equation [4.1] can always be the subject of a frequency solution, but this is more difficult when compared to a mode superposition. We can then consider, with some conditions, an approximate approach similar to one which will be described below, used in a context of a structure comprising viscoelastic materials [GIR 89]. Assuming that the frequency dependence, while being significant, remains limited to the considered frequency band, it is possible to take it into account while maintaining a modal approach. Let us assume here that the structure is made up of traditional material of stiffness K c and of structural damping η c independent of the frequency, and of viscoelastic material of stiffness K v (ω ) and structural damping η v (ω ) , which, after assembling the equation of motion, leads to: (−ω 2 M + K c + K v (ω ) + i (η c K c + η v (ω ) K v (ω ) ) ) u(ω ) = F (ω )

[4.61]

The Modal Approach

127

By putting the viscoelastic stiffness in the form: K v (ω ) = α (ω ) K ref v

[4.62]

where α(ω) is a scalar representing the frequency dependence and K ref v the stiffness of the viscoelastic material at the reference frequency ω ref , we can rely on

(

a modal analysis made with K ref v and giving the modes ω k , m k , Φi k

)ref . Each of

these k modes has a part of its strain energy in the viscoelastic material which can be determined by relation [4.43] giving the contribution fraction τ k v . In the hypothesis of the uncoupled equations based on the modes, as discussed in section 4.3.3, these equations are written: (−ω 2 mkref

[4.63]

+ k kref (1 + τ k (α (ω ) − 1) + i (η (1 − τ k ) + η (ω ) τ k α (ω ) ) ) )qk (ω ) = Φ ki Fi (ω ) v

v

c

v

v

v

v

The parameters of mode k modified by the frequency dependence can be estimated with the help of equation [4.63] by relying on the resonance that it implies. The circular natural frequencies should verify the relation: ω k 2 = ω kref 2 (1 + τ k v (α v (ω k ) − 1))

[4.64]

which is an equation in ω k to be solved by iteration. The damping corresponding to this circular frequency will be given by: ηk =

η c (1 − τ k v ) + η v (ω k ) τ k v α v (ω k ) 1 + τ k v (α v (ω k ) − 1)

[4.65]

Relations [4.64] and [4.65] give an approximation of the modal parameters by taking into account the viscoelastic material stiffness and damping variations, in the form of a correction which makes the contribution rate of the latter intervene in the modal strain energy. it is of interest to minimize these corrections by minimizing the set of values τ k v (α v (ω k ) − 1) , and this makes it possible to determine ω ref starting from α v (ω ) and an estimation of τ k v (in the first analysis, the central frequency of the considered band can be taken into account).

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Starting from these corrected modal characteristics, the mode superposition relations will be applied in order to determine the FRF of the desired functions. It should be noted that these corrections enable a good approximation of the contribution of a mode to its own resonance, which is necessary but not to the other frequencies. An overall error results from this in the entire band; this is the price to be paid for using the modal approach in the presence of the frequency-dependent properties. The efficiency of such a strategy must be estimated on a case by case basis. It should be noted that this frequency dependence, characteristic for the properties of the viscoelastic materials, can also be found in other situations, for example, with elasto-acoustic coupling. We will then try to establish whether a strategy similar to the preceding one can be applied in an appropriate way.

Chapter 5

Modal Effective Parameters

5.1. Introduction The modal approach was detailed in Chapter 4 to arrive at the FRF given by relations [4.57] to [4.59] expressing the mode superposition. Each mode contributes by a product of two terms: a dynamic amplification and a matrix of the modal parameters described as effective and which will be analyzed here in detail for a better understanding of the approach. The concept of modal effective parameters was progressively established in order to respond to various problems regarding the understanding of phenomena and the limitation of the truncation errors. A concise history of the matter is that, after various attempts in this field, the modal effective mass concept was developed in the 1970s particularly with the references [BAM 71] and [WAD 72], then it was intensively used, especially in the aerospace industry [IMB 78a, IMB 78b]. This concept was then generalized to the effective parameters in the 1980s with a unified presentation of these parameters [GIR 85, GIR 86, GIR 87]. Since then, various additions have been brought into general use in structural dynamics [GIR 97a], more particularly for the detection and selection of the important modes regarding a given response, the evaluation of the modal truncation effects, the comparison of the modal bases resulting from analysis and/or tests, the elaboration of equivalent models and finally the explicit calculation of all types of responses. The modal effective parameters are introduced in detail in section 5.2, with the summation rules that they follow and which make it possible to evaluate the truncation effects. The particular case of a statically determined structure, resulting in the equivalent effective mass models, is discussed in section 5.3. The relation

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between these parameters and the dynamic responses of all natures is highlighted in section 5.4. Finally, some examples taken from the industrial domain are presented in section 5.5. 5.2. Effective modal parameters and truncation 5.2.1. Definition of the effective modal parameters Consider again the mode superposition expressions without truncation from [4.57] to [4.59] giving the FRF matrices in flexibilities G, transmissibilities T and masses M with similar forms:

G ii (ω ) =

N

Φi k Φki

k =1

ωk 2 m k

∑ H k (ω )

N

Φi k L k j

k =1

mk

Tij (ω ) = ∑ Tk (ω )

r

Φi r Φri −ω 2 m r

)

− M ii−1 M ij

N

L jk L k j

k =1

mk

M jj (ω ) = ∑ Tk (ω )

(if the rigid modes r: ∑

[5.1]

[5.2]

+ M jj − M ji M ii−1 M ij +

K jj −ω 2

[5.3]

The terms outside the sums represent the properties directly related to the junction and resulting from the discretization (see the notes on relations [4.55] and [4.56]), or from the statically indeterminate term K jj . The sums express the mode superposition where each mode k (excluding possible rigid-body modes) contributes by a product of two terms: – the dimensionless dynamic amplification H k (ω ) or Tk (ω ) according to the case:

H k (ω ) =

1 ⎛ ω 1− ⎜ ⎜ωk ⎝

2

⎞ ⎟ + i 2ζ ω k ⎟ ωk ⎠

1+ i 2ζ k Tk (ω ) =

⎛ ω 1− ⎜ ⎜ωk ⎝

2

ω ωk

⎞ ⎟ + i 2ζ ω k ⎟ ωk ⎠

[5.4]

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131

– a frequency-independent matrix of the same dimension as the considered type of FRF: Φi k Φki ~ G ii ,k = ω k 2 mk

matrix of the effective flexibilities (for ω k ≠ 0 )

[5.5]

Φi k L k j ~ Tij ,k = mk

matrix of the effective transmissibilities

[5.6]

L jk L k j ~ M jj ,k = mk

matrix of the effective masses

[5.7]

For the rigid-body modes, for which ω k = 0 , the effective flexibilities are infinite and cannot be taken into consideration as such. These modes should be treated separately, as shown in relation [5.1]. In relations [5.5] to [5.7], we notice that the first expected property of the effective parameters is that they do not depend on the normalization of the mode shapes. We can say here that the generalized mass plays a normalization role in relation to the mode shapes in order to make the effective parameters physical, in direct relation with the static properties such as the static flexibilities, the junction modes or the masses, just as with the 1-DOF system. The dimension of these matrices is the same as that of the FRF, on which they depend. However, their expression shows that they are of rank 1 and this can be interpreted as a scalar property of flexibility, transmissibility or mass in a particular direction. This point will be discussed in more detail in certain particular cases. By regrouping relations [5.1] to [5.7], we get: G ii (ω ) =

Tij (ω ) =

N

~

∑ H k (ω ) G ii,k

k =1 N

~

(if rigid-body modes r: ∑

∑ Tk (ω ) Tij ,k − M ii−1 M ij

k =1

r

Φi r Φri −ω 2 m r

)

[5.8]

[5.9]

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Structural Dynamics in Industry N K jj ~ M jj (ω ) = ∑ Tk (ω ) M jj ,k + M jj − M ji M ii−1 M ij + −ω 2 k =1

[5.10]

Relations [5.8] to [5.10] highlight the fact that for a given FRF X(ω), the ~ corresponding effective parameter X k characterizes the importance of mode k

independently of the excitation frequency. For a given dynamic amplification, the bigger the implied effective parameter is, the bigger the contribution to the FRF will be. As illustration, the 2 internal DOF system in Figure 4.1, starting from the results of [4.9] and [4.36], provides the following effective parameters: ~ 3 ⎡ 1 + 2⎤ G ii ,1 = ⎥ ⎢ 5 k ⎣+ 2 4 ⎦

1 ⎡ + 3⎤ ~ Tij ,1 = ⎢ ⎥ 5 ⎣ + 6⎦

~ M jj ,1 = [ 9 m / 5] [5.11]

~ 1 ⎡ 4 − 2⎤ G ii,2 = ⎢ ⎥ 10 k ⎣− 2 1 ⎦

~ 1 ⎡ + 2⎤ Tij ,2 = ⎢ ⎥ 5 ⎣ − 1⎦

~ M jj ,2 = [ m / 5]

We can see, for example, that the effective mass of the first mode is 9 times greater than that of the second mode: with the shape of the first mode, the two masses act in phase, so they add their effect to the reaction, which does not happen in the second mode. For an equal amplification, it will thus have a contribution to the dynamic mass which is 9 times stronger. Similarly, the 3-DOF free system in Figure 4.3, starting from the results from [4.16], provides the following effective parameters (the first mode being a rigid~ body mode, G ii ,1 is not to be considered, as indicated after relation [5.5]): ⎡ 1 − 2 + 1⎤ ⎡ 1 0 − 1⎤ ~ ~ 1 ⎢ 1 ⎢ ⎥ G ii ,2 = − 2 4 − 2⎥⎥ 0 0 0 ⎥ G ii ,3 = 18 k ⎢ 2k ⎢ ⎢⎣ + 1 − 2 1 ⎥⎦ ⎢⎣− 1 0 1 ⎥⎦

[5.12]

It can be generally noted that all the diagonal terms which correspond to drivingpoint FRF (the DOF response is the same as the excitation) must be positive or zero, contrary to the others, which can take both signs.

Modal Effective Parameters

133

5.2.2. Summation rules The modal effective parameters follow the summation rules resulting from their definition. These rules will make it possible to establish corrections for the truncation effects later (section 5.2.3). The simplest rule will be analyzed first, which will give a first order correction, before examining two particular points, then generalizing for higher orders. 5.2.2.1. Direct summation The direct summation rules are obtained by imposing ω = 0 in relations [5.8] to [5.10] in order to recover the static structure properties (or parameters). For the transmissibilities and the masses, this summation rule is that of relations [4.55] to [4.56], already established in section 4.3.4 in order to homogenize the form of the FRF. For the flexibilities, it is given by relation [5.8] with H k (0) = 1 , in the absence of rigid modes (the case of the rigid-body modes will be discussed separately). Altogether: N

~

∑ G ii,k = G ii (in the absence of rigid-body modes)

[5.13]

k =1 N

~

∑ Tij ,k = Ψij + M ii−1 M ij

[5.14]

k =1 N

(

∑ M jj ,k = M jj − M jj − M ji M ii−1 M ij

k =1

)

[5.15]

G ii is the static flexibility matrix, representing with Ψij (section 4.3.1) and M jj (section 4.3.2) the static properties of the structure with regard to DOF i and j.

This matrix is the inverse of the stiffness matrix, the latter being regular in the absence of the rigid-body modes: G ii = K ii−1

[5.16]

Each column i of G ii represents a static deformation of the structure subjected to a unit static force on DOF i. In practice, only a selection s of DOF i is concerned by the excitation forces and in this case, the problem is limited to the system:

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K ii G is = I is

[5.17]

where I is designates the identity matrix partitioned on DOF s. Equation [5.17] can be efficiently solved by forward-backward substitution on DOF s. In the end, we can retain the following points from the direct summation rules [5.13] to [5.15]: – the sum of the effective flexibilities gives the static flexibilities (in the absence of the rigid-body modes); – the sum of the effective transmissibilities gives the junction modes (to within the term related to the junction, see the note in section 4.3.4 following relation [4.55]), which are actually static transmissibilities between DOF i and j (section 4.3.1); – the sum of effective masses gives the static properties of the mass of the structure condensed at the junction (again to within the term related to the junction). Conversely, it can be said that the static properties are distributed on the modes through the effective parameters: each mode receives a bigger or smaller part of these properties, according to its importance in relation to the considered FRF. From the results of [5.11], we can see that all these summation rules are correctly verified for the 2 internal DOF system in Figure 4.1: ~ ~ 1 ⎡1 1 ⎤ −1 G ii ,1 + G ii,2 = ⎢ ⎥ = K ii k ⎣1 5 / 2⎦ ⎡1⎤ ~ ~ Tij ,1 + Tij ,2 = ⎢ ⎥ = Ψij + 0 ij ⎣1⎦

[5.18]

~ ~ M jj ,1 + M jj ,2 = [2 m] = M jj − 0 jj

5.2.2.2. Flexibilities in the presence of rigid modes Formula [5.5] shows that the effective flexibilities of rigid modes r are infinite. Correlatively, the stiffness matrix K ii is singular and cannot be inverted. The physical interpretation for a free structure is that the rigid-body modes provide the motion in translation and rotation of the structure’s center of mass (see section 4.2.2), hence a contribution to the dynamic flexibilities [5.8] in 1 / ω 2 (see the note

Modal Effective Parameters

135

following relation [4.51]). The elastic modes provide the motion around the center of mass and their flexibilities are relative to this motion, which, statically, leads to the concept of flexibility around the center of mass that we call “pseudo-static flexibility”. The summation rule [5.13] is then to be limited to the elastic modes in order to obtain this pseudo-static flexibility: N

~

∑ G ii,k = G ii (pseudo-static flexibility)

[5.19]

k = r +1

This matrix G ii is obtained by filtering the rigid-body modes from the stiffness matrix K ii in one way or another. This filtering is in fact expressed by linear constraints, which, as shown in section 3.2.5, can be treated by DOF elimination or introduction. The elimination can be carried out by temporarily constraining the structure so that the corresponding flexibilities are obtained, then by filtering the rigid-body modes in order to position them at the center of mass, hence the following development: – when the structure is subjected to a unit static force Fii = I ii on DOF i a constant acceleration in translation and in rotation around the center of mass occurs. !! i i on all the internal DOF and This acceleration generates inertia forces Fˆ ii = M ii u therefore the structure subjected to I ii + Fˆ ii is in equilibrium around the center of mass. If the structure is now constrained on an arbitrary set of DOF r in order to render it statically determinate, a static deformation G l i can be deduced from it on the remaining DOF l = i-r given by G l i = K ll−1 (I l i + Fˆ l i ) . This deformation related to DOF r is to be positioned around the center of mass. First it is necessary to !! i i = Ψir u !! r i with estimate Fˆ l i according to the rigid forms Ψir by writing that u !! r i (force Fii = I i i transmitted to the support r). By repeating the − Ψri = M rr u

operation for all DOF i, we come to: G li = K ll−1 Tli

with

−1 Tli = I li − M li Ψir M rr Ψri

[5.20]

– deformations G li obtained this way, generated by forces Fi and the accelerations resulting from this can be described as inertial and dependent not only

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on the stiffness but also on the mass, as opposed to those of the non-free structures. Tli plays the role of a transformation matrix filtering the rigid modes in K ll . It remains to reposition these deformations, relative to the arbitrary support r, around the center of mass by imposing mass-orthogonality relations with the rigid-body modes, and this is expressed by the premultiplication of G li by Til , hence the pseudo-static flexibility matrix (inertia relief modes): G ii = Til K ll−1 Tli

[5.21]

There again, the solution of relation [5.20] can be limited to a selection s of DOF l: K ll G ls = Tls

then

G is = Til G ls

[5.22]

Regarding the introduction of DOF, this can be done by directly exploiting the orthogonality properties of G ii with the rigid-body modes Φir related to the center of mass, using the Lagrange multipliers (section 3.2.5.3), and this leads to [ROY 02a]: ⎡ K ii ⎢Φ M ⎣ ri ii

M ii Φir ⎤ ⎡ G ii ⎤ ⎡ I ii ⎤ = 0 rr ⎥⎦ ⎢⎣ λ ri ⎥⎦ ⎢⎣0 ri ⎥⎦

[5.23]

The matrix of relation [5.23] is now regular and the system can be solved. Again we find the equivalent of relation [5.20]: K ii G ii = I ii − M ii Φir λ ri

with

λ ri = M rr −1 Φri

[5.24]

The difference between the two approaches is in the rigid modes used: in relation [5.20] they come from Ψir related to the temporary constraint DOF r, while in [5.23] they are rigid-body modes Φir related to the center of mass. The latter do not require any choice of DOF r because they directly result from the eigenvalue problem, and this makes the procedure considerably easier.

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137

With the example of the 3-DOF free system in Figure 4.3 which gives the effective flexibilities [5.12], the two approaches give the following results: ⎡ 5 − 1 − 4⎤ ~ ~ 1 ⎢ − 1 2 − 1⎥⎥ = G ii G ii ,2 + G ii,3 = 9k ⎢ ⎢⎣− 4 − 1 5 ⎥⎦

[5.25]

– temporary constraint approach: G ii = Til K ll−1 Tli with, for example: u r = [ u1 ] ⎡u ⎤ ul = ⎢ 2 ⎥ ⎣u 3 ⎦

⎡2 1⎤ ⇒ K ll = k ⎢ ⎥ ⎣1 1⎦

⎡− 1 − 1⎤ 1⎢ Til = ⎢ 2 − 1⎥⎥ 3 ⎢⎣− 1 2 ⎥⎦

[5.26]

– Lagrange multipliers approach: G ii given by: ⎡k ⎢− k ⎢ ⎢0 ⎢ ⎣m

−k

0

2k

−k

−k

k m

m

m⎤ ⎡# $ ⎢ m⎥⎥ ⎢" G ii m⎥ ⎢% $ ⎥⎢ 0 ⎦ ⎣" λ ri

%⎤ ⎡# $ "⎥⎥ ⎢⎢" I ii = #⎥ ⎢% $ ⎥ ⎢ "⎦ ⎣" 0 ri

%⎤ "⎥⎥ #⎥ ⎥ "⎦

[5.27]

We can check that each column of G ii is a deformation for which the system center of mass has a zero displacement (equal masses on the DOF and sum of the components equal to zero). Other examples of pseudo-static flexibilities will be found in Chapter 6 relative to continuous structures. 5.2.2.3. Transmissibilities and effective masses by zones In the presence of a junction j, relation [4.32] introduces the modal participation !! k factors L kj that, with reference to relation [4.40], produce modal forces L jk q representing the reactions of the entire structure at its junction j when it takes the shape of mode k. If the structure is broken down into zones, we can consider the modal reactions due to each zone and thus introduce the concept of participation factors by zone

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[BER 91]. By defining M (i + j )(i + j ) the mass matrix relative to zone z (with zeros on the other DOF) the participation factors by zone are given by: z

z

z

L kj = Φki (M ii Ψij + M ij )

with

z ∑ L kj = L kj

[5.28]

z

These participation factors by zones naturally lead to the transmissibilities and effective masses by zones, representing for each mode the contributions of each zone to the transmissibilities and effective masses of the entire structure: z

z

Φi k L k j ~z Tij ,k = mk

L jk L k j ~z M jj ,k = mk

[5.29]

According to relation [5.28], these parameters verify the summation rules on the zones, making it possible to recover the effective modal parameters of the structure: ~

~

z ∑ Tij ,k = Tij ,k z

~

~

z ∑ M ij ,k = M ij ,k

[5.30]

z

Similarly, the summation rules can be written on the modes in order to recover the static parameters related to each zone: ~

z ∑ Tij ,k = (Ψij + M ii−1 M ij ) z

[5.31]

k

~

z ∑ M jj ,k = (M jj − M jj + M ji M ii−1 M ij ) z k

If the zone is not directly connected to the junction, the second members reduce z

z

to Ψij and M jj respectively representing the contribution of zone z to the static junction mode and junction condensed mass matrix. In practice, this concept of effective parameters by zones is essentially used in the case of statically determinate structures (section 5.3) with regard to the masses [BER 91]. It makes it possible to judge the importance of each zone for each mode in terms of effective masses.

Modal Effective Parameters

139

As an example, the 2 internal DOF system of Figure 4.1 decomposed in two zones (zone 1 with the bottom mass and spring, and zone 2 with the top mass and spring) provides the following results: ~z M ij,k

z =1

z=2

∑

k =1

3m/5

3m/5

9m/5

k =2

2m/5

–m/5

m/5

m

m

2m

∑

Note that it is possible for a negative sign to express a mass in the opposite motion (negative component of the mode). 5.2.2.4. Other summation rules The direct summation rule was obtained by identifying the terms in ω 0 in relations [5.8] to [5.10]. Other summation rules can be established for other powers of ω. If relations [1.55] to [1.57] are compared to relations [5.1] to [5.4] in the absence of damping:

(

)

≡ ∑

Tij (ω ) = − − ω 2 M ii + K ii

) (− ω

G ii (ω ) = − ω 2 M ii + K ii

(

≡

N

∑

1 ⎛ ω 1− ⎜ ⎜ωk ⎝

k =1

[

⎞ ⎟ ⎟ ⎠

2

N

1 ⎛ ω 1− ⎜ ⎜ωk ⎝

k =1

−1

2

⎞ ⎟ ⎟ ⎠

2

~ G ii ,k

M ij + K ij

[5.32]

)

~ Tij ,k − M ii−1 M ij

M jj (ω ) = − ω 2 M jj + K jj

(

−1

)(

− − ω 2 M ji + K ji − ω 2 M ii + K ii

[5.33]

) (− ω −1

2

)

⎤ M ij + K ij ⎥ (−ω 2 ) ⎦

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Structural Dynamics in Industry

≡

K jj ~ M jj ,k + M jj − M ji M ii−1 M ij + −ω 2 k =1 ⎛ ω ⎞ ⎜ ⎟ 1− ⎜ωk ⎟ ⎝ ⎠ N

∑

1

2

[5.34]

a Taylor series development in ω 2 of each member makes it possible, by identification, to find the summation rules for the effective parameters multiplied by an even power of ω k 2 . The first terms are presented in Table 5.1. We encounter the direct summation rules from [5.13] to [5.15] with the second ˆ = Ψ + M −1 M (static transmissibility excluding the junction row. The term Ψ ij ij ii ij contribution) is present in the other terms along with M ii and K ii−1 = G ii . The transition from one row to the next involves the term K ii −1 M ii resulting from the

(

)

−1

development of I ii − ω 2 K ii −1 M ii . Each row makes it possible to correct the truncation effects on each type of FRF for the corresponding order, as shown in the following section. & X k

& G ii ,k

T& ij ,k

& M jj , k

∑ ωk 2 X& k

M ii−1

ˆ M ii−1 K ii Ψ ij

ˆ K Ψ ˆ Ψ ji ii ij

∑ X& k

K ii−1

ˆ = Ψ + M −1 M Ψ ij ij ii ij

ˆ =Ψ ˆ M Ψ ˆ M jj ji ii ij

∑ X& k / ωk 2

K ii−1 M ii K ii−1

ˆ K ii−1 M ii Ψ ij

ˆ M K −1 M Ψ ˆ Ψ ji ii ii ii ij

N

K =1 N

K =1 N

K =1

Table 5.1. Summation rules for the effective parameters

5.2.3. Correction of the truncation effects

Relations [5.8] to [5.10] are exact only when using all the modes whose number is equal to the number N of internal DOF. As the modal approach is interesting only if a limited number n is calculated from them, the sum is limited to n modes. This is the modal truncation operation.

Modal Effective Parameters

141

On which criteria can one select the modes to be retained? In the low-frequency domain where the excitation frequency is limited to a value f max , relations [5.8] to [5.10] show that two factors intervene: the effective parameters and the dynamic amplifications. For modes k whose frequencies are less than f max , the amplification can reach Q k = 1 /( 2 ζ k ) and it will be dangerous not to retain them, even if the effective parameters are relatively small. In practice, excluding particular cases, we will retain all the modes whose frequencies are less than the frequency f max increased by a certain safety margin in order to avoid a significant amplification due to higher modes beyond yet close to f max . By retaining only the first n modes from N, we can thus eliminate the contribution of all the upper modes from relations [5.8] to [5.10]. This contribution can be important because there are many upper modes. A first order correction consists of assimilating their dynamic amplification H k (ω ) or Tk (ω ) to 1 (term in

ω 0 ) in order to retain only their static contribution. This makes it possible to globally represent them using residual terms deduced from static terms due to the summation rules [5.13] to [5.15], and this can be written: n

~

∑ H k (ω ) G ii,k + G ii,res

G ii (ω ) ≈

[5.35]

k =1 n

~

∑ Tk (ω ) Tij ,k + Tij ,res

Tij (ω ) ≈

[5.36]

k =1

M jj (ω ) ≈

n

~

∑ Tk (ω ) M jj , k + M jj , res +

k =1

K jj

[5.37]

−ω2

with: G ii,res =

N

~

∑ G ii,k

k = n +1

= G ii −

n

~

∑ G ii,k

[5.38]

k =1

residual flexibilities Tij ,res =

N

~

∑ Tij ,k − M ii−1 M ij

k = n +1

residual transmissibilities

= Ψij −

n

~

∑ Tij ,k

k =1

[5.39]

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M jj ,res =

N

n

~

~

∑ M jj ,k + M jj − M ji M ii−1 M ij = M jj − ∑ M jj ,k

k = n +1

[5.40]

k =1

residual masses The residual terms of relations [5.35] to [5.37], or residual parameters, therefore represent the static contribution of the upper modes and that of the junction itself (see the note following relations [4.55] to [4.56]). They are deduced from the static terms by difference with the sum of the effective parameters of the retained modes. It should be noted with regard to the residual term in flexibilities G ii,res that it can be obtained in the same way as the pseudo-static flexibility G ii , of a structure with rigid-body modes (section 5.2.2.2), by writing the orthogonality properties in relation to the retained modes, i.e. relation [5.23] with retained modes Φik . Moreover, we can consider the pseudo-static flexibility of a structure with rigidbody modes as the residual flexibility with respect to the rigid modes only [ROY 02a]. Relations [5.35] to [5.37] are approximate because only their static contribution was retained from the upper modes. However they are statically exact by definition. It should be noted that relations [5.36] and [5.37] are strictly equivalent to relations [4.52] and [4.54] with truncation, which directly include static terms. By replacing the residual terms with static terms, relations [5.35] to [5.37] are written again:

∑ (H k (ω ) − 1)G ii,k n

G ii (ω ) ≈ G ii +

Tij (ω ) ≈ Ψij +

M jj (ω ) ≈

[5.41]

k =1

∑( n

k =1

K jj

−ω

~

2

)

~ Tk (ω ) − 1 Tij ,k

(

⎛ ω with Tk (ω ) − 1 = ⎜ ⎜ωk ⎝

)

n ~ + M jj + ∑ Tk (ω ) − 1 M jj ,k k =1

2

⎞ ⎟ H (ω ) [5.42] k ⎟ ⎠

[5.43]

The residual terms of relations [5.35] to [5.37] constitute a first order correction of the truncation effects by representing the dynamic contribution of the upper modes by their static contribution, which is statically exact. Higher order corrections can also be established using the summation rules resulting from relations [5.32] to [5.34] and presented in Table 5.1 for the first terms. The second order correction for

Modal Effective Parameters

143

example represents the dynamic contribution of the upper modes using A + ω 2 B which results in: G ii (ω ) ≈ n

⎛

~

n

~

∑ H k (ω ) G ii,k + ⎜⎜ G ii − ∑ G ii,k k =1 k =1 ⎝

[5.44] ~ ⎛ ⎞ n G ⎞ ⎟ + ω 2 ⎜ G M G − ∑ ii ,k ⎟ + … ii ii ii ⎜ 2 ⎟ ⎟ k =1 ω k ⎠ ⎠ ⎝

Tij (ω ) ≈

[5.45]

n

⎛

n

k =1

⎝

k =1

~ ~ ∑ Tk (ω ) Tij ,k + ⎜⎜Ψij − ∑ Tij ,k

~ ⎞ ⎛ n T ⎞ ⎟ + ω 2 ⎜G M Ψ ˆ − ∑ ij ,k ⎟ + … ⎜ ii ii ij k =1ω 2 ⎟ ⎟ k ⎠ ⎠ ⎝

[5.46]

M jj (ω ) ≈ K jj −ω

2

+

n

⎛

n

⎞

⎛

n

k =1

⎝

k =1

⎠

⎝

k =1

~

⎞

ωk

⎠

M jj , k ~ ~ ∑ Tk (ω ) M jj , k + ⎜⎜ M jj − ∑ M jj , k ⎟⎟ + ω 2 ⎜⎜Ψˆ ji M ii G ii M ii Ψˆ ij − ∑ 2 ⎟⎟ + …

Relations [5.44] to [5.46] will be used in order to limit the truncation effects beyond the first order. In practice, relations [5.35] to [5.37] with their statically exact first order correction are sufficient in most cases. 5.3. Particular case of a statically determined structure 5.3.1. Introduction The particular case of a rigid or statically determinate junction was introduced in section 4.3.1. The junction is represented in the general 3D case by 6 DOF j, renamed r (rigid). This case does not concern the flexibilities, but it simplifies the expressions of transmissibilities and of masses for which the following properties were already established: – the junction modes Ψij become rigid-body junction modes Ψir (but they are not normal modes as in section 4.2.2); – the condensed mass matrix M jj becomes the rigid-body mass matrix M rr , explicitly comprising the mass, center of mass and inertia properties relating to DOF r; – the condensed stiffness matrix K jj becomes zero.

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The 6 DOF r can be distributed among several nodes, but they can always be represented by a single node by means of a rigid-body transformation; this simplifies the interpretation of the results and will be assumed to be the case from here on. In this case, the effective mass matrix of each mode k is of dimension 6 × 6 and is written: L r k L kr ~ M rr ,k = mk

[5.47]

As mentioned in section 5.2.1, this matrix is of rank 1 and has thus only one nonzero eigenvalue, which can be interpreted as being the mass of the mode acting in the direction of its eigenvector L r k . By assuming the 6 DOF r relative to the same ~ node P (translations 1, 2, 3 and rotations 4, 5, 6), as indicated in Figure 5.1, M rr ,k represents the mass, center of mass and inertia properties with regard to the node P: ~ ~ ~ ~ M k = M 11,k + M 22,k + M 33,k

[5.48]

~ ~ ~ ~ I P k = M 44,k + M 55,k + M 66,k

[5.49]

( ~ ~ where M k is the scalar mass acting in direction t k ( L k1 , Lk 2 , Lk 3 ) and I P k is the ( scalar inertia acting around direction rk ( Lk 4 , Lk 5 , Lk 6 ) [BAM 71, WAD 72]. The ~ interpretation can be pushed further [GIR 91] by attributing to M k a center of mass ~ G k according to its moment as compared to P, and this provides the relation: ( ( t k ∧ rk PG k = ( 2 tk

[5.50]

~ and the inertia around G k is given by:

(

)

( ( 2 t k .rk ~ I Gk = ( 2 t k mk

[5.51]

Modal Effective Parameters

( rk

145

( tk

3

4 5

~ ~ M k , I Gk

2

P

ωk 6 1

ζk

dk

Gk

Figure 5.1. Representation of a mode using its effective masses

A representation of each mode can then be imagined: each mode is equivalent to ~ ~ ~ a 1-DOF system with a mass/inertia ( M k , I G k ) acting in G k with a helicoidal ( motion in the direction t k with the translation and the rotation in a ratio ( ( ( 2 ~ ~ t k .rk / t k = ± I G k / M k on a spring giving the circular frequency ω k and a damper giving the viscous damping ratio ζ k , as illustrated in Figure 5.1.

5.3.2. Effective mass models The structure can thus be represented in relation to node P by a collection of 1DOF systems completed by a residual term according to relation [5.40]. Before treating the general case, which requires some precautions, consider the particular case of 1D and 2D models often used in practice, starting with the axial and lateral models: – the axial model (Figure 5.2a) acts only in one direction, the axial direction given by the unique DOF r (translation 1). The effective mass matrix of each mode is reduced to a scalar, the axial mass. The residual mass, attached to the junction, represents the upper modes and the junction itself so that the total mass of the structure is conserved. Thus, any axial structure, as complex as it may be, is equivalent in relation to its junction to a collection of 1-DOF systems representing ~ its modes, each mass being the effective mass of the mode M 11,k = Lk1 2 / m k , the

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~ spring having the stiffness ω k 2 M 11,k and the damper corresponding to ζ k . In case

of truncation, the springs of the upper modes are considered as rigid and make the corresponding effective masses rigidly connected to the junction; – the lateral model (Figure 5.2b) acts only in one lateral direction, i.e. perpendicularly to the axial direction. There are 2 DOF r: translation 2 and rotation 6. The effective mass matrix is of size 2 × 2. Each mode is equivalent to a 1-DOF ~ system with the mass M 22,k = Lk 2 2 / m k acting laterally at a distance ~ d k = Lk 6 / Lk 2 from the junction, the spring having the stiffness ω k 2 M 22,k and the damper corresponding to ζ k . In case of truncation, the residual terms give a residual mass located at a certain distance, in addition to an inertia in such way that the structure mass, center of mass and lateral inertia of the structure are respected; – the 2D model (Figure 5.2c) combines the two preceding models. There are three DOF r: translations 1, 2 and rotation 6. The effective mass matrix is of size 3 × 3. Each mode is equivalent to a 1-DOF system with the mass ~ M k = ( L k1 2 + Lk 2 2 ) / m k acting in the direction ( L k1 , L k 2 ) at a distance d k = Lk 6 / Lk1 2 + Lk 2 2 from the junction, the spring having the stiffness ~ ω k 2 M k and the damper corresponding to ζ k . In case of truncation, the residual

terms give a residual mass acting in a certain direction at a certain distance, completed by an inertia, in such a way that the structure mass, center of mass and inertia are respected in the plane. The general case adds to the plane case a rotation motion around the translation ~ ~ axis for each mode k, implying the inertia I G k besides the mass M k . The helicoidal motion can be taken into account in the following way, as illustrated by Figure 5.3 resulting from Figure 5.1: ~ – the translation and the rotation of the center of mass G k at the base of the 1~ DOF system are denoted by u and θ respectively, u~ and θ are the translation k

k

k

k

and the rotation of the effective mass/inertia; ( ( ( ~ ~ – u~k and θ k are related by: θ k = u~k /( t k .rk / t k

2

) as mentioned at the end of

section 5.3.1. These two DOF are thus dependant and one can be eliminated for the benefit of the other one. We shall conserve, for example, the translation u~k with ~ ~ ~ mass M k (we could also consider rotation θ k and inertia I G k );

Modal Effective Parameters

147

~ – we then create in G k the intermediate DOF uˆ k , which will form the new base ( ( ( 2 of the 1-DOF system by the linear constraint uˆ k = u k + ( t k .rk / t k ) θ k ; – we introduce between uˆ k and u~k the spring giving the circular frequency ω k and the damper giving the viscous damping ratio ζ k . We can check that the model created this way possesses the correct mass, stiffness and damping properties with regard to the mode k and thus represents it perfectly. With regard to the residual term, rather than representing it by a mass and inertias as in the 2D model, it is simpler here to use a representation similar to that of the modes where the masses/inertias will now be directly connected to the junction. For that, it is enough to diagonalize the residual mass matrix M rr,res by solving ~ (M rr ,res − M ~r ,res I rr ) L r ~r ,res = 0 r in order to find the residual masses/inertias ~ M ~r ,res in the directions Lr ~r ,res , as for the modes. Figure 5.3 shows this. ~ M 11,k

truncation

~

∑ M 11,k

k sup

r

1

M 11 − M 1i M ii −1 M i1 a) Axial model

~ M 22,k

truncation ~

∑ M 22,k

k sup

dk

6 r

2

ΔI 66

M rr − M ri M ii −1 M ir b) Lateral model

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~ Mk

truncation

dk

6 r

1

M rr − M ri M ii −1 M ir

2

c) 2D model Figure 5.2. 1D and 2D effective mass models

~ Mk

ωk

P

Mode k

Gk

u~k ~ (θ k )

u~~r ,res

P

~ M ~r ,res

ζk uk

uˆ k

G ~r ,res Residual ~ r

θk

u ~r ,res

~ (θ ~r ,res )

uˆ r ,res

θ ~r ,res

Figure 5.3. 3D effective mass model

Various examples of equivalent models are given in Chapter 6 relative to continuous structures. Using the 2 internal DOF system in Figure 4.1, the results of [5.11] make it possible to establish the effective mass model in Figure 5.4.

Modal Effective Parameters

149

m 2k 3 9m 5

m

m/5 k

⇔

3k 5

2k 5

Figure 5.4. Simple example of effective mass model

The general case is illustrated by the 3D example in Figure 5.5, which, having no symmetry, cannot be represented by 2D models due to coupling. The example consists of a concentrated mass and 3 inertias located at the end of 3 bending blades without mass: the first along Z and bending only in the X direction, the second along X and bending only in the Y direction, the third along Y and bending only in the Z direction. All the parameters are equal to 1, hence the mass and stiffness matrices verifying:

M ii −1 K ii

0 0 −6 ⎡ 12 0 ⎢ 0 12 0 0 0 ⎢ ⎢0 0 12 − 6 12 =⎢ 0 −6 4 −6 ⎢0 ⎢− 6 0 12 − 6 16 ⎢ 0 −6 ⎣⎢ 12 − 6 0

12 ⎤ − 6⎥⎥ 0⎥ ⎥ 0⎥ − 6⎥ ⎥ 16 ⎦⎥

and the results in Table 5.2 for the 6 modes of the system.

[5.52]

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Z

Y X Figure 5.5. 3D model

The effective mass models can be used as equivalent models with respect to the junction and can be connected to the models of the adjacent structures for a coupled analysis advantageously replacing the Guyan condensation. However, they do not make it possible to directly restore reactions Fr to the junction given the !! r accelerations u Nevertheless, they can also be used in the general case of an excitation Fi and u r , as shown in Figure 5.6. Forces Fi should then be distributed on the effective masses proportionally to the effective transmissibilities:

F& r ,k = T& ri ,k F&i

[5.53]

Modal Effective Parameters Modes k

ωk

2

mk

Φik

( tk

( rk

~ Mk

PG k ~ I Gk ( ( t k rk ( 2 tk

151

1

2

3

4

5

6

0.3461

0.5139

0.9899

12.12

23.35

34.67

1.0

1.0

1.0

1.0

1.0

1.0

+0.6008 -0.1895 -0.5135 -0.1359 +0.4307 -0.3681 +0.6008 -0.1895 -0.5135

-0.3671 +0.2800 -0.5329 -0.2816 +0.3692 +0.5360 -0.3671 +0.2800 -0.5329

-0.0449 -0.0933 -0.0181 -0.8827 -0.4248 -0.1712 -0.0449 -0.0933 -0.0181

+0.4248 +0.8827 +0.1712 -0.0933 -0.0449 -0.0181 +0.4248 +0.8827 +0.1712

-0.3692 +0.2816 -0.5360 +0.2800 -0.3671 -0.5329 -0.3692 +0.2816 -0.5360

+0.4307 -0.1359 -0.3681 +0.1895 -0.6008 +0.5135 +0.4307 -0.1359 -0.3681

-0.4598 +1.5450 -1.1584

-1.0945 +0.5350 +1.1830

-0.8075 -0.4516 -0.2196

-0.8048 +0.2087 +0.4399

-0.5376 -0.2003 +0.1180

-0.0427 +0.1980 -0.0531

0.6605

0.4971

0.0110

0.9890

0.5029

0.3395

+1.5334 +1.4111 +1.2734

+1.2399 +2.0469 +0.2214

+1.1150 +0.4302 -4.9850

+0.3565 -0.3283 +0.8080

-0.1474 +0.6596 +0.4481

+0.2360 +0.1137 +0.2342

0.0010

0.0125

0.6140

0.0069

0.0124

0.0020

+0.0390

-0.1586

+7.4553

-0.0833

+0.1568

-0.0759

Table 5.2. Ingredients of the effective mass model of Figure 5.5

~ ~ This is equivalent to a force and a moment ( Fk , M k ) applied directly to the ~ ~ mass/inertia ( M k , I G k ) according to the associated directions, with: ( tk ~ Fk = Φki Fi mk

( ( t k .rk ~ ~ Mk = ( F 2 k tk

[5.54]

~ The effective masses, excited by Fi and u r , provide displacements u r ,k and

reactions Fr ,k . The physical displacements u i are then restored by combining the

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~ u r ,k weighted by the effective transmissibilities, while the reactions Fi are

obtained by directly accumulating the Fr ,k : ~ ~ u i = ∑ Tir ,k u r ,k k

Fr = ∑ Fr ,k

[5.55]

k

Fi

~ Fr,k

ui

~ u r,k

~ M rr,k

Fr,k

Fr

ur

Figure 5.6. General use of an effective mass model

In the case of Figure 5.4, for example, a force F on the physical DOF 2 will be represented in the effective mass model by a force 6 F / 5 on the first effective mass and a force − F / 5 on the second, according to the results from [5.11]. The physical DOF 1 will result from the combination 3 u~1 / 5 + 2 u~2 / 5 .

In case of modal truncation, the following residual terms will be added: ⎡ u i ,res ⎤ ⎡ G ii ,res ⎢F ⎥=⎢ ⎣ r ,res ⎦ ⎢⎣− Tri ,res

⎤ ⎡ Fi ⎤ ⎥ − ω M rr ,res ⎥⎦ ⎢⎣u r ⎥⎦

Tir ,res 2

[5.56]

Modal Effective Parameters

153

All these results could be extended to statically indeterminate structures but with a more delicate physical interpretation. In practice, effective mass models are only currently used for rigid junctions. 5.4. Modal effective parameters and dynamic responses 5.4.1. Frequency responses

Relations [5.35] to [5.37] show that any FRF X (ω ) coming from G ii (ω ) , Tij (ω ) or M jj (ω ) is of the form: ~ X (ω ) = ∑ Ak (ω ) X k + X res

[5.57]

k

(subscripts of the DOF omitted for the sake of convenience, excluding the statically indeterminate term from [5.37]) with: – Ak (ω ) dynamic amplification H k (ω ) or Tk (ω ) ; ~ ~ ~ ~ – X k modal effective parameter G k , Tk or M k ;

– X res residual parameter G res , Tres or M res . Other types of FRF are deduced by multiplying or dividing by iω. Form [5.57] is therefore general. Graphically, its amplitude using logarithmic scales is illustrated schematically in Figure 5.7. The profile is governed by the following rules: – at a very low frequency, we converge towards the static value equal to the sum of the effective parameters of the retained modes, increased by the residual parameter representing the truncated modes and possibly the junction itself (relations [5.38] to [5.40]). Exceptions are the flexibilities of structures with rigidbody modes and the masses of statically indeterminate structures which have a contribution in 1 / ω 2 ; – when the frequency increases, each mode creates a peak corresponding to its resonance which is predominant if it is relatively isolated and if its effective parameter is not too small. Otherwise, it can combine with the neighboring modes and perhaps disappear visually from the curve;

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– between two peaks, i.e. a minimum occurs between two consecutive modes which are not too close and with similar importance. This minimum will take one of the following two forms according to the corresponding phases: - if the effective parameters of the two modes are of the same sign, the two contributions are antagonist and result in a very small amplitude, hence an “antipeak” corresponding to an anti-resonance, the sharpness of which is similar to that of the neighboring peaks. For a sine motion at this frequency, this is a vibrational node. With reference to the last comment in section 5.2.1, any driving-point FRF should present only anti-resonances, - if the effective parameters of the two modes are of an opposite sign, the two contributions add to each other and give a significant amplitude, hence a local minimum or “trough”. There shouldn't be any in a driving-point FRF. Conversely, a transfer FRF may have both anti-resonances and local minima depending on the signs of the corresponding effective parameters.

~ X k Qk

(log) ⏐X⏐ ~

∑ Xk k

(2) ~ Xk

(1) same sign (2) opposed sign ~ for adjacent X k

~ ⎛ωk Xk ⎜ ⎜ ⎝ ω

(1)

ωk Figure 5.7. FRF profile G(ω), T(ω) or M(ω)

⎞ ⎟ ⎟ ⎠

2

(log) ω

Modal Effective Parameters

155

The FRF profile can vary considerably according to the value of the effective parameters, damping and the proximity of the modes. Generally, the peaks are rather well distinguished at low frequencies where the modal density is low, then become progressively coupled towards the high frequencies. In an experimental context, measurement noise may perturb the profile to a certain degree and mask certain aspects, especially the anti-resonances. Note that for a given structure, the peaks of all FRF will be located at the same frequencies, those of the normal modes, while the frequencies of the anti-resonances will depend on the FRF considered. As an illustration several different possible situations with the simple cases in Figures 4.1 and 4.3, we obtain the following plots: – Figures 5.8 relative to the 2 internal DOF system of Figure 4.1 and to the results of [4.9] and [5.11]: - Figure 5.8a:

G11 (ω ) = (3 / 5k ) H 1 (ω ) + (2 / 5k ) H 2 (ω )

- Figure 5.8b:

G12 (ω ) = (6 / 5k ) H 1 (ω ) + (−1 / 5k ) H 2 (ω )

- Figure 5.8c:

G 22 (ω ) = (12 / 5k ) H 1 (ω ) + (1 / 10k ) H 2 (ω )

- Figure 5.8d:

T10 (ω ) = (3 / 5) T1 (ω ) + (2 / 5) T2 (ω )

- Figure 5.8e:

T20 (ω ) = (6 / 5) T1 (ω ) + (−1 / 5) T2 (ω )

- Figure 5.8f:

M 00 (ω ) = (9m / 5) T1 (ω ) + (m / 5) T2 (ω )

– Figure 5.9 relative to the 3-DOF free system in Figure 4.3 and to the results of [4.16] and [5.12]: - Figure 5.9a:

G11 (ω ) = −1 /(3m ω 2 ) + (1 / 2k ) H 2 (ω ) + (1 / 18k ) H 3 (ω )

- Figure 5.9b:

G13 (ω ) = −1 /(3m ω 2 ) + (−1 / 2k ) H 2 (ω ) + (1 / 18k ) H 3 (ω )

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Structural Dynamics in Industry

π

π

0

0

−π

−π

2

10

2

0

10

10

0

10

−2

10

−2

−1

10

0

10

1

10

10

−1

10

a) G11 (ω )

π

0

0

−π

−π

2

10

2

0

10

−2

10

10

0

10

−2

10

−4

−4

−1

10

0

10

1

10

10

−1

10

b) G12 (ω )

0

10

1

10

e) T20 (ω )

π

π

0

0

−π

−π

2

10

2

0

10

10

0

10

−2

10

1

10

d) T10 (ω )

π

10

0

10

−2

−1

10

0

10

c) G 22 (ω )

1

10

10

−1

10

0

10

d) M 00 (ω )

Figure 5.8. FRF for the 2 internal DOF system in Figure 4.1 (m = k = 1, ζk = 2% ⇔ Qk = 25)

1

10

Modal Effective Parameters

π

π

0

0

−π

−π

2

10

2

0

10

10

0

10

−2

10

157

−2

−1

0

10

10

1

10

10

−1

10

a) G11 (ω )

0

10

1

10

b) G13 (ω )

Figure 5.9. FRF for the 3-DOF free system in Figure 4.3 (m = k = 1, ζk = 2% ⇔ Qk = 25)

5.4.2. Random responses

In the case of a random excitation x defined by its PSD S xx (ω ) (see section 1.2.3.4), the response y defined by its PSD S yy (ω ) is deduced from this using equation [1.39]. Starting from relation [5.57], we can thus write: ~ S yy (ω ) = ∑ Ak (ω ) X yx,k + X yx,res k

2

S xx (ω )

[5.58]

The case of the 1-DOF was already discussed in section 2.2.5 in order to obtain the response PSD and the rms values. Since each mode behaves like a 1-DOF system, it is easy to deduce the following results for the rms values of the responses. With the following hypotheses: – the excitation PSD S xx (ω ) , renamed W x ( f ) as in section 2.2.5 for the practical applications, varies slowly in the vicinity of each natural frequency, so that the value W x ( f k ) is used for the contribution of each mode k; – the natural frequencies f k are well separated, so that it is possible to replace the integral of the sum with the sum of the integrals, as illustrated in Figure 5.10; then, the mean squares are given by the relation: ⎛π ⎞ ~2 2 2 y 2 ≈ ∑ ⎜ f k Q k ⎟ X yx W x ( f k ) + X yx , res x ,k 2 ⎠ k ⎝

[5.59]

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Structural Dynamics in Industry

Each mode contributes to the mean square by the product of the term (π / 2 ) f k Q k , resulting from the integration of its dynamic amplification, with the square of the implied effective parameter and the excitation PSD at its natural frequency. As for the residual term, its contribution is the product of its square by the excitation mean square. Relation [5.59] is easy to interpret more particularly in the light of the scheme in Figure 5.2a. (log) ⏐X⏐

(log) ω Figure 5.10. Integration of the response PSD in order to find the rms values

The preceding results are valid only for one excitation and one response. They may be extrapolated to several excitations and responses based on expression [1.42] replacing [1.39]. With the example of the 2 internal DOF system in Figure 4.1 subjected to a white noise in acceleration Wu!!0 at its base, the mean square of the response on DOF 2 is given by (with Qk = 10 and the results [4.9] and [5.11]) :

u!!2 2 ≈

π 10 2 2π

⎛⎛ 6 ⎞2 1 ⎛ 1 ⎞2 ⎞ k k Wu!!0 ⎜ ⎜ ⎟ + ⎜ − ⎟ 2 ⎟ ≈ (2.08 + 0.14) Wu!!0 [5.60] ⎜ ⎟ m 5 3 ⎝ 5⎠ m ⎝⎝ ⎠ ⎠

Modal Effective Parameters

159

5.4.3. Time responses

For a FRF X(ω) of form [5.57], the response y(t) to an excitation x(t) is given by the convolution integral [1.30] with the unit impulse response given by the inverse Fourier transform FT–1 of X(ω). By writing that the transform of the sum is equal to the sum of the transforms, we obtain:

y (t ) = ∑ k

(∫

t

−∞

)

x(τ ) FT −1 ( Ak (ω ) ) d τ X& yx ,k + yres (t )

[5.61]

we can see that each mode contributes to the response by a product, that of the effective parameter with the convolution of the excitation and the unit impulse response hk (t ) or t k (t ) of relations [2.60] or [2.61]. With regard to the residual term, it provides a residual term for the response in a similar way. Using the example of the 2 internal DOF in Figure 4.1 subjected to an acceleration impulse at its base, the time responses in Figure 5.11 are obtained: – Figure 5.11a: response on DOF 1: u!!1 (t ) = (3 / 5) t1 (t ) + (2 / 5) t 2 (t ) ; – Figure 5.11b: response on DOF 2: u!!2 (t ) = (6 / 5) t1 (t ) + (−1 / 5) t 2 (t ) . 1

1

0.5

0.5

0

0

−0.5

−0.5

−1

0

2

4

6

a) u!!1 (t )

8

10

−1

0

2

4

6

8

10

b) u!!2 (t )

Figure 5.11. Impulse response of the 2 internal DOF system in Figure 4.1 (m = k = 1, ζk = 2% ⇔ Qk = 25)

5.4.4. Time response extrema

The response spectra introduced in section 2.3.3 provide the response extrema of a 1-DOF system subjected to the excitation considered. Since each mode behaves

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Structural Dynamics in Industry

like a 1-DOF system, the information about the response extrema of the structure can be deduced. For example, with an absolute acceleration spectrum S u!! ( f ) of a transient applied to the base of a structure, the use of an effective mass model as illustrated by Figure 5.2 makes it possible to easily establish the following results: – the spectrum gives by definition the maximum acceleration of each effective mass; – by multiplying the maximum acceleration by the effective mass, we obtain the maximum reaction at the base due to each mode; – by multiplying the maximum acceleration by the effective transmissibility between the base and an internal DOF i (relation [5.55]), we obtain the maximum acceleration on this DOF due to each mode. Therefore, this gives us the maximum contributions of each mode. If we now want to obtain the maximum responses of the structure, it is necessary to combine these results. As the maxima of the different modes do not generally have any reason to occur at the same time, the exact recombination is not possible. This is the consequence of the loss of information in the spectrum which retains only the amplitude (last note in section 2.3.3). The modal maxima can only be combined approximately, for example: – by a direct sum, which will necessarily give an overestimation of the levels although this is a conservative approach, it can be very pessimistic; – by a quadratic sum, which will probably be closer to reality, but can also underestimate the levels; – by a mixed sum, i.e. direct for certain terms and quadratic for other terms, for example the highest maximum combined with the square root of the quadratic sum of the others. The quality of the result will depend on the case considered. With the example of the 2 internal DOF in Figure 4.1 subjected to an acceleration impulse at its base, the results on DOF 1 and 2 with a modal viscous damping of 5% (Q = 10) are the following: – exact results given in section 5.4.3: u!!1

max

k/m

≈ 0.7480

u!!2

max

k/m

≈ 0.8281

– direct sum: u!!1

max

k/m

≈ 0.3226 + 0.5268 = 0.8495

u!!2

max

k /m

≈ 0.6453 + 0.2634 = 0.9087

Modal Effective Parameters

161

– quadratic sum: u!!1

max

k/m

≈ 0.6178

u!!2

max

k/m

≈ 0.6970

In this particular case, the quadratic sum clearly underestimates the levels. With a larger number of modes, a judicious combination of sums can provide an acceptable approximation. 5.5. Industrial examples

As an illustration of the modal effective parameters in an industrial context, a first example is given with the model in Figure 5.12, which represents a marine support structure.

Figure 5.12. Marine support structure (with the permission of CTSN Toulon)

We are interested in the transmissibility of the vibrations between the motor interface (average of 4 attachment points using a single node) and the support (rigid junction using a single node). The effective modal transmissibilities between two DOF in the same direction for the first 43 modes (up to 500 Hz) are given by Table 5.3 and plotted in Figure 5.13a, in a presentation that can be generalized to any type of effective parameter. For each mode k, of frequency f k , this table provides the value of the effective

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Structural Dynamics in Industry

parameter and the cumulative sum, in absolute and relative values with respect to the corresponding static term, here 1. The term makes it possible to appreciate the importance of each mode in relation to the considered FRF, the sum makes it possible to verify the convergence toward the static term. The FRF is not a drivingpoint FRF here and can have positive or negative effective parameters, hence a nonmonotone convergence, contrary to what a driving-point FRF would give. The first 5 transmissibilities are positive and quite large: 20% of the static for the first one, 59% for the second, etc. The sixth one is negative, the seventh one is positive again, etc. They can clearly be seen again in Figure 5.13a where each jump of the sum shows the term with its sign.

a) Effective transmissibilities

b) Dynamic transmissibility Figure 5.13. Effective transmissibilities and dynamic transmissibility along Y of the model in Figure 5.12

Modal Effective Parameters k

163

fk Effective parameter effective/static parameter (Hz) Term Sum Term Sum --------------------------------------------------------------------1 73.21 2.02640e-01 2.02640e-01 0.20264 0.20264 2 85.16 5.91011e-01 7.93650e-01 0.59101 0.79365 3 100.57 2.13382e-01 1.00703e+00 0.21338 1.00703 4 115.49 1.18260e-02 1.01886e+00 0.01183 1.01886 5 130.77 1.12945e-01 1.13180e+00 0.11294 1.13180 6 145.24 -1.45066e-01 9.86737e-01 -0.14507 0.98674 7 162.14 2.29125e-01 1.21586e+00 0.22913 1.21586 8 176.12 -9.94615e-03 1.20592e+00 -0.00995 1.20592 9 183.61 -5.35969e-02 1.15232e+00 -0.05360 1.15232 10 191.52 7.24342e-03 1.15956e+00 0.00724 1.15956 11 204.39 -3.83887e-04 1.15918e+00 -0.00038 1.15918 12 220.17 3.05485e-02 1.18973e+00 0.03055 1.18973 13 224.20 -1.45466e-05 1.18971e+00 -0.00001 1.18971 14 228.43 3.41062e-02 1.22382e+00 0.03411 1.22382 15 237.66 -1.45901e-03 1.22236e+00 -0.00146 1.22236 16 256.00 -1.29035e-02 1.20946e+00 -0.01290 1.20946 17 266.28 2.48120e-03 1.21194e+00 0.00248 1.21194 18 281.37 -6.03675e-03 1.20590e+00 -0.00604 1.20590 19 285.96 9.87278e-03 1.21577e+00 0.00987 1.21577 20 293.66 8.08338e-03 1.22386e+00 0.00808 1.22386 21 299.20 4.90771e-03 1.22876e+00 0.00491 1.22876 22 313.17 -7.04854e-04 1.22806e+00 -0.00070 1.22806 23 325.21 -1.14711e-01 1.11335e+00 -0.11471 1.11335 24 330.40 4.22691e-03 1.11758e+00 0.00423 1.11758 25 334.56 -1.68215e-02 1.10075e+00 -0.01682 1.10075 26 343.32 -7.51423e-02 1.02561e+00 -0.07514 1.02561 27 357.37 1.17607e-02 1.03737e+00 0.01176 1.03737 28 361.63 -2.34121e-02 1.01396e+00 -0.02341 1.01396 29 373.38 2.91438e-03 1.01687e+00 0.00291 1.01687 30 375.93 3.79900e-03 1.02067e+00 0.00380 1.02067 31 386.76 2.89954e-03 1.02357e+00 0.00290 1.02357 32 401.78 1.16100e-02 1.03518e+00 0.01161 1.03518 33 412.54 -2.89955e-03 1.03228e+00 -0.00290 1.03228 34 423.13 -3.77455e-03 1.02851e+00 -0.00377 1.02851 35 430.96 -1.48424e-02 1.01367e+00 -0.01484 1.01367 36 435.83 -3.07169e-02 9.82950e-01 -0.03072 0.98295 37 443.61 -8.05702e-03 9.74893e-01 -0.00806 0.97489 38 448.63 -4.36527e-03 9.70527e-01 -0.00437 0.97053 39 456.26 -1.42071e-02 9.56320e-01 -0.01421 0.95632 40 480.29 -2.04802e-03 9.54272e-01 -0.00205 0.95427 41 487.30 -8.61732e-03 9.45655e-01 -0.00862 0.94565 42 489.43 -9.06207e-03 9.36593e-01 -0.00906 0.93659 43 494.36 -5.55478e-02 8.81045e-01 -0.05555 0.88104 ----------------Sum : 8.81045e-01 0.88104 Static term : 1.00000e+00 1.00000 Residual term : 1.18955e-01 0.11896

Table 5.3. Effective transmissibilities along Y of the model in Figure 5.12

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Structural Dynamics in Industry

Figure 5.13b illustrates, with the same frequency scale, the dynamic transmissibility amplitude for a global structural damping of 0.02 (amplification of 50 at resonances for all modes). This parallel clearly illustrates the considerations in section 5.4.1. At a very low frequency, the response converges towards the static value of 1. Then, when the frequency increases, each mode creates a peak, which corresponds to its resonance where it is predominant if it is relatively isolated and if its effective parameter is not too small, which is the case for the first modes. Some anti-resonances or some local maxima are found between the peaks according to the signs of the effective parameters, starting with anti-resonances followed by the appearance of local minima. At higher frequencies, the variations become harder to distinguish. We will take the model in Figure 3.11 with its 19 modes below 150 Hz so as to illustrate the analysis possibilities of a model according to its effective masses with regard to its rigid junction. Table 5.4 presents the diagonal terms of the 6×6 effective mass matrix for each mode. Note that they provide nearly all the necessary information, according to relation [5.47], allowing us to determine the participation factors except for their sign. This table gives the following information, for example: – the first mode is a global lateral mode in direction Y with a directional mass of 92.8 kg, having more than a quarter of the total mass and an inertia of 41.4 m2.kg, hence a center of mass at (41.4/92.8)1/2 = 0.668 m along Z. Therefore, it is a global lateral model along Y, with a small component in torsion around X and a very small lateral component along Z; – the second mode is also lateral in direction Y with a smaller directional mass but a significantly higher center of mass along Z. The third mode is lateral in the direction Z with more than half of the total mass. The fourth mode is an axial mode, the fifth one is relatively secondary, etc.; – the first 19 modes represent about 88% of the mass along Y and Z, but only 48% of the mass along X, hence a more important residual term along X. These properties make it possible to clearly understand the importance of each mode in relation to the rigid junction and to predict the form of the dynamic mass of the model, seen from its base, just like the dynamic transmissibility of the preceding example. We can deduce the effective mass model according to the considerations of section 5.3.2. We arrive at the ingredients of Table 5.5 and at the illustration of Figure 5.14.

Modal Effective Parameters k

fk

Mx

My

Mz

Ix

Iy

165

Iz

--------------------------------------------------------------1

41.36

0.000

92.797

0.004

5.426

0.006

2

45.88

0.001

47.891

0.007

1.267

0.019 192.044

41.380

3

52.16

1.920

4

82.19

5

0.000 181.962

0.001 262.336

0.003

103.594

0.006

0.010

0.000

0.578

0.001

87.25

7.009

0.008

0.222

0.003

54.156

0.022

6

94.49

0.006

28.125

0.000

0.076

0.015

64.449

7

111.48

0.256

46.709

49.894

0.256

0.610

1.841

8

113.11

0.082

49.132

52.544

0.432

0.711

1.498

9

114.24

0.082

8.592

0.000

16.756

0.011

5.703

10

114.65

33.338

0.041

0.770

0.069

6.488

0.042

11

123.26

0.005

4.015

0.000

0.550

0.000

4.159

12

128.01

0.345

0.000

0.003

0.064

0.004

0.001

13

128.66

7.134

0.592

7.741

0.000

0.130

0.022

14

138.10

19.973

0.001

1.335

0.005

0.341

0.001

15

142.86

0.010

0.014

0.013

0.019

0.017

0.002

16

145.25

0.001

0.281

0.003

0.133

0.002

0.004

17

145.31

0.051

15.402

0.005

0.091

0.012

4.063

18

146.67

0.022

0.007

0.029

0.002

0.011

0.003

19

147.42

0.080

0.001

0.042

0.001

0.000

0.007

--------------------------------------------------------------Static

336.063 336.063 336.063

48.414 343.560 331.745

--------------------------------------------------------------residual

162.156

42.449

41.478

23.260

Table 5.4. Effective mass model of Figure 3.11

18.111

16.500

166 k

Structural Dynamics in Industry Mk

tk/|tk|

OG

(t.r)/(t.t)

1

92.80

0.0003 -1.0000 -0.0067

0.668 -0.001

0.242

-0.0035

2

47.90

0.0036

0.9999 -0.0124

2.002 -0.005

0.163

-0.0056

3

183.88

-0.1022

0.0003 -0.9948

1.188 -0.003 -0.122

0.0044

4

103.61

-0.9999

0.0074

0.0097

-0.001

0.004 -0.075

5

7.24

-0.9840

0.0328 -0.1753

-0.478

0.058

6

28.13

0.0144

0.9999 -0.0005

7

96.86

0.0514 -0.6944 -0.7177

8

101.76

9

8.67

10

34.15

11

-0.0785

1.513 -0.022 -0.052

0.0230

0.030 -0.040

-0.0465

0.7186

-0.144 -0.043 -0.048

0.0310

0.9953 -0.0010

0.807 -0.080 -1.380

-0.0284 -0.6949 0.0973

0.0018

2.692

-0.153

0.067

0.429

-0.0543

4.02

0.0335

1.017 -0.035 -0.369

0.0209

12

0.35

0.9960 -0.0278

0.007 -0.089 -0.117

-0.4185

13

15.47

-0.6792 -0.1956 -0.7074

0.061

0.0486

14

21.31

-0.9681 -0.0063 -0.2503

0.032 -0.003 -0.123

0.0161

15

0.04

-0.5213

0.6156 -0.5910

0.518 -0.312 -0.782

-0.0779

16

0.28

-0.0542

0.9934 -0.1007

0.112 -0.062 -0.675

-0.1322

17

15.46

18

0.06

-0.6105

19

0.12

0.8068

0.9994 -0.0024

-0.0573 -0.9982

0.0852

0.041

0.1703

0.9881 -0.0347 -0.1502

-0.057 -0.022

0.0184

0.512 -0.031 -0.075

0.0232

0.3503 -0.7103

0.229 -0.279 -0.335

0.1856

0.0707

0.013

0.5866

0.152 -0.037

-0.1998

--------------------------------------------------------------------r1

165.68

0.9859 -0.0084 -0.1673

0.004

0.001

0.021

-0.0012

r2

40.50

-0.0256 -0.9953 -0.0937

0.376

0.003 -0.130

0.0011

r3

36.46

0.314 -0.004 -0.056

0.0026

r4

0.67

r5

1.48

r6

1.29

0.1725 -0.0973

0.9802

0.0030

0.0002

-0.685

0.001

5.748

-0.1084

-0.1002 -0.0861 -0.9912

-3.089

0.003

0.312

0.0983

-2.852 -0.002 -0.517

-0.0767

0.0052

1.0000

0.9995 -0.0319

Table 5.5. Ingredients of the effective mass model of Figure 3.11

Modal Effective Parameters

Figure 5.14. Effective mass model of Figure 3.11

167

Chapter 6

Continuous Systems

6.1. Introduction The preceding chapters were dedicated to the N-DOF systems that can be obtained by discretization of continuous systems. Conversely, continuous systems can be considered as having an infinite number of DOF, and thus require special considerations and processing. Simple continuous systems such as beams and certain plates or shells are the subject of various developments in the literature. For extensive information on normal modes, see [BLE 79]. From an industrial point of view, these results can serve as a rapid estimation of the real cases, which lend themselves to a basic representation. They can also serve as reference in order to test the accuracy of a discrete approach such as that of finite elements. This is the way that this chapter is written; it does not deal with all cases but rather details the most common examples, more particularly from the angle of FRF and modal effective parameters as were presented for the discrete systems in the preceding chapters. Particular attention will be given to beams that are not only of interest as a reference case being relatively accessible for analysis, but also make it possible to treat truss structures by using continuous beam elements, a subject mentioned in section 9.5.4 which deals with FRF coupling. The equations of motion of a continuous system are partial differential equations that can be deduced from Hamilton’s principle (section 1.5.1). Without elaborating all the theory that could be found in most of the classic references, we will limit ourselves to the general ideas making it possible to transpose the results obtained on the discrete systems to the continuous systems.

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Structural Dynamics in Industry

Thus, the following remarks can be made schematically: – as mentioned at the beginning, a continuous system can be considered as a system having an infinite number of DOF. Any shape is defined by a continuous function Φ(x) of a spatial variable x with one or several components according to the case, instead of a vector with several N components Φi for an N-DOF system; – the solution of the equations without excitation leads to an eigenvalue problem, just as for the discrete systems (section 4.2.1): the eigenvalues ω k 2 provide the natural frequencies f k = ω k / 2π and the associated eigenvectors or mode shapes Φk (x) are such that Φk ( x i ) = Φi k , the discrete values x i providing as many components for mode k. The normal mode concept is thus preserved. They are infinite in number, but the truncation operation which limits them to the first n modes remains valid;

– the matrix products are replaced by integrals. It is this way that the generalized mass m k of mode shape Φk (x) , or its participation factor Lkj in relation to a static junction mode Ψ j (x) related to the discrete junction DOF j will be given by: m k = ∫ Φk ( x) 2 dm

Lkj = ∫ Φk ( x) Ψ j ( x) dm

[6.1]

the integral being extended to the entire system; – the mode shapes Φk (x) verify the same orthogonality properties relating to the mass or the stiffness as the discrete systems, and this allows the same approach by mode superposition for the FRF calculation and the same concept of effective modal parameters, under condition of the following point; – the notion of FRF remains valid insofar as the excitation and the response are of a discrete nature, i.e. the excitation, just like the response, concerns one DOF. It can be an internal or junction DOF just as for the discrete structures, this distinction being preserved; – as for the energy dissipation, it is easy to take this into account in the form of structural damping or loss factor η (see section 2.4.2) by considering a complex Young’s modulus E = E (1 + iη ) .

Under these circumstances, all the results obtained on the discrete systems can be transposed to the continuous systems. This is the subject of this chapter for the simplest systems in the most typical situations.

Continuous Systems

171

The basic axial rod is discussed in section 6.2, with a certain level of detail in order to develop the simplest case. The basic bending beam is presented in section 6.3. The case of the plates is approached in a limited way in section 6.4. Finally, some combined cases are considered in section 6.5. 6.2. The rod element 6.2.1. Introduction

With the same notations as in section 3.3.2 (L, M, E, S length, mass, Young’s modulus and cross-section area of the rod respectively), kinetic energy and strain energy have the expression: 2

T=

1 M L ⎛ ∂u ⎞ ∫ ⎜ ⎟ dx 2 L 0 ⎝ ∂t ⎠

2

U=

1 L ⎛ ∂u ⎞ E S ∫0 ⎜ ⎟ dx 2 ⎝ ∂x ⎠

[6.2]

The application of Hamilton’s principle leads to the equation of motion:

ES

∂ 2u ∂x 2

=

M ∂ 2u L ∂t 2

[6.3]

This equation can be solved by separation of variables, assuming: u ( x, t ) = Φ( x ) f (t )

[6.4]

From this, we obtain:

1 ∂ 2 f (t ) E S L 1 ∂ 2Φ ( x ) = = constant = -ω 2 M Φ ( x) ∂x 2 f (t ) ∂t 2 The solution depends on the boundary conditions: – for a fixed end: u = 0 ; – for a free end: ∂u / ∂x = 0 .

[6.5]

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Non-trivial solutions are given for various cases (in order to simplify the notation, the parameter k representing the mode k will not be underlined in the equations): – free-free rod (mode shapes: Figure 6.1):

ωk = k π

ES ML

Φk ( x) = cos

kπ x L

[6.6]

(2 k − 1) π x 2L

[6.7]

kπ x L

[6.8]

– clamped-free rod (mode shapes: Figure 6.2):

ω k = (2 k − 1)

π 2

ES ML

Φk ( x) = sin

– clamped-clamped rod (mode shapes: Figure 6.3):

ωk = k π

ES ML

Φk ( x) = sin

We will now look at each case separately by limiting it to the two DOF at each end numbered 1 and 2, starting with the mixed case of clamped-free rod entailing a certain degree of generality by introducing all FRF types in an elementary way.

k=0 1

4

3

2

Figure 6.1. Mode shapes of the free-free rod (axial displacements plotted perpendicularly)

Continuous Systems

173

k=1 2 3

4

Figure 6.2. Mode shapes of the clamped-free rod (axial displacements plotted perpendicularly)

k=1 2 4

3

Figure 6.3. Mode shapes of the clamped-clamped rod (axial displacements plotted perpendicularly)

6.2.2. Clamped-free rod

The clamped-free rod introduces the two types of DOF, a junction DOF (statically determinate; see sections 4.3.1 and 5.3), numbered j = r = 1, and an internal DOF, numbered i = 2 (Figure 6.4), hence the three FRF types: Gii (ω ) = G 22 (ω ) , Tir (ω ) = T21 (ω ) = Tri (ω ) = T12 (ω ) and K rr (ω ) = K11 (ω ) . M, E, S 1

2 L Figure 6.4. Clamped-free rod

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Given the mode shapes from relation [6.7] and the junction mode Ψ1 ( x) = 1 , the generalized masses and participation factors according to relations [6.1], are expressed as:

mk =

M 2

L k1 =

(−1) k −1 2 M (2 k − 1) π

[6.9]

The modal effective parameters, based on relations [5.5] to [5.7], are:

~ G 22,k =

1

8 2

(2 k − 1) π

2

L ES

[6.10]

(−1) k −1 4 ~ ~ T21,k = T12,k = (2 k − 1) π

[6.11]

~ M 11,k =

[6.12]

1

8 2

(2 k − 1) π 2

M

The summation rules are verified due to the relations:

∞

1

k =1

(2 k − 1) 2

∑

=

π2 8

(−1) k −1 π = 4 k =1 ( 2 k − 1) ∞

∑

[6.13]

Note the similarity between the masses and the flexibilities (which arises repeatedly) with the same coefficients. The cumulative modal effective parameters are plotted in Figure 6.5.

Continuous Systems

1

1

0

0 .25

.75

.25

1.25

.75

175

1.25

∑ ( T21,k = T12,k )

1 ⎛ES ~ ⎞ G 22,k = M 11,k ⎟ M ⎠ k ⎝ L

~

∑⎜

~

k

ω

(abscissa:

λ = – logarithmic scale) 2π ES

2π

ML

Figure 6.5. Clamped-free rod: cumulative modal effective parameters

L M E S

⇔

Truncation example .8106M 2 ES L

.0901M 2 ES L

.0324M 2 ES L

.0165M 2 ES L

.0504M

Figure 6.6. Clamped-free rod: effective mass model

The effective mass model that results from relation [6.12] is shown in Figure 6.6. Starting from these results that can be called exact and therefore form a reference, we are in the position of estimating the errors introduced by a given discretization. For example, matrices [3.38] of the rod element with the linear displacement field (instead of a sine) lead, for a model with only one element, to a single mode with the following characteristics: – for the natural circular frequency, a factor 3 or 2 depending on whether the mass is coherent or lumped, instead of factor π/2, thus an error of +10.3% and –10.0% respectively. Note that the exact value is very close to the mean of the two

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values, and thus many finite element codes use the mean of the coherent and lumped mass matrices for the element mass matrix; – for the flexibility or the effective mass, a factor 3/4 or 1/2 instead of factor 2 8/π , thus an error of –7.5% and –38% respectively; – for the effective transmissibility, a factor 3/2 or 1 instead of factor 4 / π , thus an error of +18% and –21% respectively.

These errors diminish with the number of elements used to represent the rod, with convergence to the exact solution. The three basic FRF can be deduced from relations [6.10] to [6.12] by using relations [5.8] to [5.10] representing the mode superposition approach. In fact, the sums that intervene are limited series developments of trigonometric functions that can also be obtained directly starting from the complete equation of motion (see for example [KOL 73, NEU 87, SKU 68]). With structural damping or a loss factor η affecting Young’s modulus as mentioned in the introduction, we then get the following results: G 22 (ω ) =

ωL L L 1 tan tan λ = c ωMc ES λ 1

T21 (ω ) = T12 (ω ) = cos M 11 (ω ) =

ωL

=

[6.14]

1 cos λ

[6.15]

c

ωL Mc 1 tan = M tan λ c ωL λ

[6.16]

with:

λ=

ωL c

c=

ESL = M

E

ρ

E = E (1 + i η )

[6.17]

Parameter c represents a propagation velocity affected by energy dissipation: it is the traditional propagation velocity in the material c = E / ρ made complex by Young’s modulus, marked E , in the presence of a loss factor η. This propagation velocity depends only on the material with its Young’s modulus and its mass density

Continuous Systems

177

ρ = M /( S L) . It is characteristic of the propagation phenomenon of a perturbation along the rod. The dimensionless parameter λ is the reduced circular frequency λ = ω L / c made complex by η. The fact that it is complex prevents the introduced trigonometric functions from reaching zero or becoming infinite, and this corresponds to the limitation of the values at resonances or antiresonances by the damping. When ω tends towards zero, we obtain the static values. These trigonometric functions are plotted in Figure 6.7. We see peaks appearing at the natural frequencies (resonances), with amplitudes limited by damping. Between the peaks are “anti-peaks” (anti-resonances) whose sharpness is similar to that of the peaks for the point FRF with the tangent, and flat minima (troughs) for the transfer FRF with the inverse of the cosine. π

π

0

0

−π

−π

2

10

2

1

10

0

10

−1

10

10

1

10

0

10

−1

10

−2

10

−2

−1

10

0

10

1

10

10

ES 1 tan λ G 22 (ω ) = M 11 (ω ) = L M λ

−1

0

10

T21 (ω ) =

ω

(abscissa:

2π

ES ML

1

10

=

λ ) 2π

10

1 cos λ

Figure 6.7. Clamped-free rod FRF (η = 0.05)

When the frequency tends towards infinity, the following behavior is observed: – for the flexibility or the dynamic mass, the amplitude tends towards an asymptote in 1/ω and the phase tends towards −π / 2 . With tan λ → −i , we thus have G (ω ) → 1 /(i ω Z ) and M (ω ) → Z / i ω , by assuming Z = M c / L , given the product of the propagation velocity by the mass per unit length. Z is the characteristic impedance for the rod, independent of its length: it reflects the local property of the element at the excited end. We can say that it characterizes the

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behavior of an infinite length rod where the normal modes, after having progressively mixed, no longer provide any local variation in frequency. The modes reflect the phenomenon of the stationary waves established according to reflections at the extremities and, for an infinite length or frequency, there is no reflection, only a propagation phenomena; – for the dynamic transmissibility, the amplitude tends towards an asymptote ⎛ ωLη⎞ approximately in 2 exp ⎜ − ⎟ and the phase shifts indefinitely. The propagation ⎝ c 2⎠ phenomenon mentioned earlier is carried out at velocity c by dissipating energy proportionally to loss factor η, and this is represented by an exponential reduction of the motion amplitude. For these two types of behavior we can see the progressive mix of modal contributions, which naturally smoothes the curves towards the high frequencies, passing from a stationary behavior with well-established peaks to a purely propagating behavior. This smoothing phenomenon has been the subject of various analyses (see, for example, [GIR 90, SKU 68, SKU 80, SKU 87]). If we come back to the significance of the FRF, relations [6.14] to [6.16] allow us to write the mixed FRF matrix:

⎡u 2 (ω )⎤ ⎢ F (ω ) ⎥ = ⎣ 1 ⎦

ωL ⎡ L sin ⎢ 1 c ⎢ω M c ωL ⎢ cos −1 c ⎢⎣

⎤ ⎥ ⎡ F2 (ω )⎤ ⎥ ωMc ω L ⎥ ⎢⎣ u1 (ω ) ⎥⎦ sin − L c ⎥⎦ 1

or ⎡ L 1 sin λ ⎡u 2 (λ )⎤ 1 ⎢E S λ ⎢ = ⎥ ⎢ ⎣ F1 (λ ) ⎦ cos λ ⎢ −1 ⎢⎣

[6.18] ⎤ 1 ⎥ ⎡ F (λ ) ⎤ ⎥⎢ 2 ⎥ ES u (λ ) ⎦ − λ sin λ ⎥⎥ ⎣ 1 L ⎦

6.2.3. Free-free rod

For the free-free rod, the two DOF at the ends are of internal type i = (1, 2) (Figure 6.8), hence, the dynamic flexibility matrix G ii (ω ) with G11 (ω ) = G 22 (ω ) by symmetry and G12 (ω ) = G 21 (ω ) by reciprocity.

Continuous Systems

179

M, E, S 1

2 L Figure 6.8. Free-free rod

As relation [6.6] gives the normal modes, the generalized masses are the same as in relation [6.9] and the effective flexibilities are based on relation [5.5]: ~ G ii ,k =

2 k2 π 2

L ES

⎡ 1 ⎢ k ⎢⎣(−1)

(−1) k ⎤ ⎥ 1 ⎥⎦

[6.19]

The pseudo-static flexibilities (section 5.2.2) correspond to the loading illustrated by Figure 6.9 with zero average displacements, and this leads to the expressions: x L 1 ⎛⎜ x x2 G1 ( ) = 2−6 +3 L E S 6 ⎜⎝ L L2

⇒ G ii =

⎞ ⎟ ⎟ ⎠

x x L 1 ⎛⎜ x2 G 2 ( ) = G1 (1 − ) = −1+ 3 L L E S 6 ⎜⎝ 2 L2

1 L ⎡ 2 − 1⎤ ⎥ ⎢ 6 E S ⎣− 1 2 ⎦

[6.20]

Figure 6.10 plots the cumulative effective flexibilities.

F

⎞ ⎟ ⎟ ⎠

F/L

F/L

L

L

/3

1/3

0

0

/3

1/3

Figure 6.9. Pseudo-static flexibilities of the free-free rod (axial displacements plotted perpendicularly)

F

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1

1

0

0 .25

.75

1.25

.25

(

3E S ~ ~ G11,k = G 22,k L k

∑

)

2π

1.25

(

−6 E S ~ ~ G12,k = G 21,k L k

∑ ω

(abscissa:

.75

ES ML

=

λ 2π

)

– logarithmic scale)

Figure 6.10. Free-free rod: cumulative effective flexibilities

The summation rules are verified using the following relations: ∞

1

k =1

k2

∑

=

π2 6

∞

(−1) k −1

k =1

k2

∑

=

π2 12

[6.21]

In relation to these reference results, matrices [3.38] of the rod element with linear displacement field (instead of a cosine) lead, for a model with only one element, to a correct rigid mode and to a single elastic mode with: – for the natural circular frequency, a factor 2 3 or 2 depending on whether the mass is coherent or lumped, instead of factor π, thus an error of +10% and –36% respectively; – for the effective flexibilities, a factor ±1 / 4 instead of factor ± 2 / π 2 , thus an error of +23%. Again, the errors decrease with the number of elements used in order to represent the rod, with convergence towards the exact solution.

Continuous Systems

181

By exploiting relation [6.18], the following dynamic flexibilities are found:

⎡ u1 (ω) ⎤ −L 1 ⎢u (ω) ⎥ = ⎣ 2 ⎦ ω M c sin ω L c

ωL ⎡ ⎤ 1 ⎥ ⎢cos c ⎡ F1 (ω) ⎤ ⎢ ⎥⎢ ω L ⎥ ⎣ F2 (ω) ⎥⎦ ⎢ 1 cos c ⎦⎥ ⎣⎢

ωL ⎡ ⎤ cos 1 ⎥ ⎢ 1 −L c ⇒ Gii (ω) = ⎢ ⎥ ω L⎥ ω M c sin ω L ⎢ 1 cos c ⎢⎣ c ⎥⎦

[6.22]

or:

G ii (λ ) =

⎡cos λ 1 −L ⎢ E S λ sin λ ⎣ 1

1 ⎤ ⎥ cos λ ⎦

The trigonometric functions that intervene are plotted in Figure 6.11. The same phenomena appear as for the clamped-free rod. π

π

0

0

−π

−π

2

10

2

1

10

0

10

−1

10

10

1

10

0

10

−1

10

−2

10

−2

−1

10

0

10

1

10

10

−1

ES ES −1 G11 (ω ) = G 22 (ω ) = L L λ tan λ 2π

10

1

10

ES ES −1 G12 (ω ) = G 21 (ω ) = L L λ sin λ

ω

(abscissa:

0

10

ES ML

=

λ ) 2π

Figure 6.11. Free-free rod: dynamic flexibilities (η = 0.05)

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Structural Dynamics in Industry

6.2.4. Clamped-clamped rod

For a clamped-clamped rod, the two DOF at the ends are of junction type j = (1, 2) with a statically indeterminate junction (Figure 6.12), hence the dynamic stiffness matrix K jj (ω ) with K 11 (ω ) = K 22 (ω ) by symmetry and K 12 (ω ) = K 21 (ω ) by reciprocity.

M, E, S

1

2

L Figure 6.12. Clamped-clamped rod

As relation [6.8] gives the normal modes and the two junction modes equal Ψ1 ( x) = 1 − x / L and Ψ2 ( x) = x / L , the generalized masses are the same as in relation [6.9] and the participation factors have for expression:

L k1 =

M kπ

Lk 2 =

(−1) k −1 M kπ

[6.23]

The effective masses are given by, based on relation [5.7]: ~ M jj ,k =

2

k2 π 2

⎡ 1 M⎢ k −1 ⎢⎣(−1)

(−1) k −1 ⎤ ⎥ 1 ⎥⎦

[6.24]

The condensed mass on the two junction DOF (section 4.3.2) are calculated starting from the junction modes and we find mass matrix [3.38] again:

M

jj

⎡Ψ ( x) ⎤ M ⎡2 1 ⎤ = ∫ ⎢ 1 ⎥ [ Ψ1 ( x) Ψ2 ( x)] dm = ⎢ ⎥ ( ) Ψ x 6 ⎣1 2 ⎦ ⎣ 2 ⎦

[6.25]

Continuous Systems

1

1

0

0 .25

∑ k

.75

.25

1.25

(

3 ~ ~ M 11,k = M 22,k M

)

∑ k

ω

(abscissa:

ES ML

2π

=

.75

183

1.25

(

6 ~ ~ M 12,k = M 21,k M

)

λ – logarithmic scale) 2π

Figure 6.13. Clamped-clamped bar: cumulative effective masses

In relation to these reference results, matrices [3.38] of the rod element with linear displacement field (instead of a sine) do not provide any modes using only one element, but they give the same static mass property [6.25]. With two elements, only one mode is obtained with: – for the natural circular frequency, a factor 2 3 or 2 2 depending on whether the mass is coherent or lumped, instead of factor π, thus an error of +10.3% and –10.0% respectively, as for the clamped-fee case; – for the effective masses, a factor 3/16 or 1/8 instead of factor 2 / π 2 , thus an error of –7.5% and –38% respectively, as for the clamped-free case. There again, the errors decrease with the number of elements used in order to represent the rod, with convergence towards the exact solution. By exploiting relation [6.18], the following dynamic stiffnesses are found: ⎡ F1 (ω ) ⎤ ω M c 1 ⎢ F (ω )⎥ = ωL L ⎣ 2 ⎦ sin c

⇒ K jj (ω ) =

ωMc L

ωL ⎡ ⎢cos c ⎢ ⎢ −1 ⎣⎢

⎤ ⎥ ⎡ u1 (ω ) ⎤ ω L ⎥⎥ ⎢⎣u 2 (ω )⎥⎦ cos c ⎦⎥

ωL ⎡ ⎢cos c ⎢ ωL ⎢ −1 sin c ⎢⎣ 1

−1

⎤ ⎥ ω L ⎥⎥ cos c ⎥⎦ −1

[6.26]

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or

K jj (λ ) =

E S λ ⎡cos λ ⎢ L sin λ ⎣ − 1

−1 ⎤ ⎥ cos λ ⎦

We can verify that this matrix is the inverse of dynamic flexibility matrix [6.22]. It tends towards static stiffness matrix [3.38] when ω tends towards 0, which is also the junction condensed stiffness matrix K jj . The trigonometric functions that intervene are plotted in Figure 6.14. The same phenomena appear as for the clamped-free rod. π

π

0

0

−π

−π

2

10

2

1

10

0

10

−1

10

10

1

10

0

10

−1

10

−2

10

−2

−1

0

10

10

1

10

10

−1

L L λ K 11 (ω ) = K 22 (ω ) = ES ES tan λ 2π

10

1

10

L L −λ K 21 (ω ) = K12 (ω ) = ES ES sin λ

ω

(abscissa:

0

10

ES ML

=

λ ) 2π

Figure 6.14. Clamped-clamped rod: dynamic stiffnesses (η = 0.05)

6.3. Bending beam element 6.3.1. Introduction

With the same notations as in section 3.3.3 (L, M, E, I being respectively length, mass, Young’s modulus and beam section inertia) and by neglecting the shear effect, the application of Hamilton’s principle leads to the equation of motion:

Continuous Systems

EI

∂ 4v ∂x 4

+

M ∂ 2v =0 L ∂t 2

185

[6.27]

The solution depends on the boundary conditions: – for a fixed end translation: v = 0 ; – for a fixed end rotation: ∂v / ∂x = 0 ; – for a free end rotation: ∂ 2 v / ∂x 2 = 0 ; – for a free end translation: ∂ 3 v / ∂x 3 = 0 . Four cases can be listed for each end: – free translation and free rotation = free node: ∂ 2 v / ∂x 2 = 0 , ∂ 3 v / ∂x 3 = 0 ; – free translation and fixed rotation = sliding node: ∂v / ∂x = 0 , ∂ 3 v / ∂x 3 = 0 ; – fixed translation and free rotation = pinned node: v = 0 , ∂ 2 v / ∂x 2 = 0 ; – fixed translation and fixed rotation = clamped node: v = 0 , ∂v / ∂x = 0 . In each case the non-trivial solutions are of the form:

ω k = λk 2

EI

[6.28]

M L3

Φk ( x) = Ak cosh λ k

x x ⎛ x x⎞ + B k cos λ k − ⎜ C k sinh λ k + D k sin λ k ⎟ L L ⎝ L L⎠

Coefficients λ k and ( A, B, C , D ) k are given according to the boundary conditions by Tables 6.1 and 6.2.

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Node 1/2

Free

Free

Sliding

Pinned

Ak = Bk = 1

Ak = Bk = 1

Ak = Bk = 1

Ck = Dk =

Ck = Dk =

Ck = Dk =

cosh λk − cos λk

sinh λk − sin λk

cosh λk − cos λk

sinh λk − sin λk

cosh λk + cos λk

sinh λk − sin λ

Bk = 1

Bk = 1

See symmetric Sliding

Pinned

Clamped

x⎞ ⎛x ⎜ → 1− ⎟ L L ⎝ ⎠

( A = C = D) k = 0 ( A = C = D) k = 0

− Dk = 1

Clamped

See symmetric

x⎞ ⎛x ⎜ → 1− ⎟ L⎠ ⎝L

See symmetric

x⎞ ⎛x ⎜ → 1− ⎟ L L ⎝ ⎠

See symmetric

See symmetric

See symmetric

x⎞ ⎛x ⎜ → 1− ⎟ L⎠ ⎝L

x⎞ ⎛x ⎜ → 1− ⎟ L⎠ ⎝L

( A = B = C )k = 0

x⎞ ⎛x ⎜ → 1− ⎟ L⎠ ⎝L

Ak = − Bk = 1

Ak = − Bk = 1

Ak = − Bk = 1

Ak = − Bk = 1

Ck = − Dk =

Ck = − Dk =

Ck = − Dk =

Ck = − Dk =

sinh λk − sin λk

sinh λk − sin λk

cosh λk − cos λk

cosh λk − cos λk

cosh λk + cos λk

cosh λk + cos λk

sinh λk − sin λk

sinh λk − sin λk

Table 6.1. Beam in bending: coefficients ( A, B, C , D ) k of relation [6.28]

Continuous Systems Node 1/2

Free

187

Sliding

Pinned

Clamped

Symmetric

Symmetric

Symmetric

Symmetric

Symmetric

cos λk cosh λk =1 Free

4.730 7.853 10.996 14.137

≈ ( 2 k + 1)

π 2

tan λk + tanh λk =0 Sliding

2.365 5.498 8.639 11.781

≈ ( 4 k − 1)

kπ

π 4

tan λk − tanh λk =0 Pinned

3.927 7.069 10.210 13.352

≈ ( 4 k + 1)

Clamped

sin λk = 0

cos λk = 0

sin λk = 0 Symmetric

π

( 2 k − 1)

π 2

kπ

4

cos λk cosh λk = −1

tan λk + tanh λk =0

tan λk − tanh λk =0

cos λk cosh λk =1

1.875 4.694 7.855 10.996

2.365 5.498 8.639 11.781

3.927 7.069 10.210 13.352

4.730 7.853 10.996 14.137

≈ ( 2 k − 1)

π 2

≈ ( 4 k − 1)

π 4

≈ ( 4 k + 1)

π 4

≈ ( 2 k + 1)

Table 6.2. Beam in bending: coefficients λ k of relation [6.28]

π 2

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We will now look at the most typical cases by limiting ourselves, as was the case with the rod, to the two end nodes numbered 1 and 2, starting with the case of the clamped-free beam. 6.3.2. Clamped-free beam

The clamped-free beam introduces two types of node, a junction node (statically determinate; see sections 4.3.1 and 5.3), numbered j = r = 1, and an internal node, numbered i = 2 (Figure 6.15), each having 2 DOF, one of translation v and one of rotation θ. Hence the three types of FRF matrices: G ii (ω ) = G 22 (ω ) , Tir (ω ) = T21 (ω ) and K rr (ω ) = K 11 (ω ) , each being of dimension 2 × 2.

v

v

θ

θ

M, E, I, φ = 0

1

2

L Figure 6.15. Clamped-free beam

Relation [6.28] and Tables 6.1 and 6.2 provide the normal modes which are plotted in Figure 6.16. k=1 2 4

3

Figure 6.16. Mode shapes of the clamped-free beam

As both junction modes equal Ψv ( x) = 1 and Ψθ ( x) = x , according to relations [6.1], the generalized masses and the participation factors are expressed as: mk = M

L kv =

2 Ck M

λk

Lkθ =

2 Ck M L

λk 2

[6.29]

Continuous Systems

189

The modal effective parameters based on relations [5.5] to [5.7] are given by: ⎡ L3 ⎢ ~ 4 ⎢ EI G 22,k = 4 ⎢ L2 λk λk Ck ⎢ ⎣⎢ E I

⎤ L2 λk Ck ⎥ EI ⎥ ⎥ L λk 2 Ck 2 ⎥ EI ⎦⎥

[6.30]

Ck L / λk ⎤ 4 (−1) k −1 ⎡ ~ T21,k = ⎢ 2 ⎥ C k ⎦⎥ λk ⎣⎢C k λ k / L

[6.31]

2 2 4 ⎡M λ k C k ~ ⎢ M 11,k = λ k 4 ⎢⎣ M L λ k C k

[6.32]

M L λk Ck ⎤ ⎥ M L2 ⎥⎦

The similarity between masses and flexibilities is noted again. The sum of the effective parameters gives the corresponding static parameters (section 5.2.2), which are expressed as:

⎡G G 22 = ⎢ vv ⎣Gθv

⎡T T21 = ⎢ vv ⎣Tθv

M 22

⎡M = ⎢ vv ⎣ M θv

⎡ L3 ⎢ G vθ ⎤ ⎢ 3 E I = Gθθ ⎥⎦ ⎢ L2 ⎢ ⎢⎣ 2 E I

L2 ⎤ ⎥ 2E I⎥ L ⎥ ⎥ E I ⎥⎦

Tvθ ⎤ ⎡1 L ⎤ = Tθθ ⎥⎦ ⎢⎣0 1 ⎥⎦ ⎡ M M vθ ⎤ ⎢ = ⎢ M θθ ⎥⎦ ⎢ M L ⎢⎣ 2

[6.33]

[6.34]

ML⎤ 2 ⎥⎥ M L2 ⎥ 3 ⎥⎦

[6.35]

The numerical values of the effective parameters with respect to the corresponding static values (except for Tθv = 0 ) are given by Table 6.3, which also illustrates the summation rules.

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Structural Dynamics in Industry

Figure 6.17 shows the cumulative effective parameters.

k

1 2 3 4 … >4

~ Gvv

=

L3 3EI

~ Mθθ

ML2 3

0.9707 0.0247 0.0032 0.0008

~ Gvθ

L2 2 EI

~ M vθ ML 2

~ ~ Gθθ M = vv L M EI

~ ~ Tvv = Tθθ

~ Tvθ L

~ Tθv

0.6131 0.1883 0.0647 0.0331

+1.5660 -0.8679 +0.5088 -0.3638

+1.1377 -0.1815 +0.0648 -0.0331

+2.1556 -4.1494 +3.9936 -4.0002

8 (−1) k −1 (2k − 1)π

16 (−1) k −1 (2k − 1) 2 π 2

100%

100%

0.8908 0.0788 0.0165 0.0060

192

64 4 4

16 3 3

(2k − 1) π

Σ

=

2 2

(2k − 1) π

100%

(2k − 1) π

100%

100%

4 (−1) k −1

±2

Table 6.3. Clamped-free beam in bending: modal effective parameters

⎛3E I ~ ⎞ 3 ⎟ Gvv, k = M 3 2 θθ , k ⎟ ML k ⎝ L ⎠

∑ ⎜⎜

⎛ 2E I ~ ⎞ 2 Gvθ ,k = M vθ ,k ⎟⎟ 2 M L ⎠ k ⎝ L

∑ ⎜⎜

∑ ( Tvv,k = Tθθ ,k ) ~

~

1 ⎛ EI ~ ⎞ Gθθ ,k = M vv,k ⎟ L M ⎠ k

∑ ⎜⎝

⎛1 ~ ⎞ Tvθ ,k ⎟ ⎝L ⎠

∑⎜

k

k

ω

(abscissa:

2π

EI

=

∑ ( Tθv,k ) ~

k

2

λ – logarithmic scale) 2π

M L3

Figure 6.17. Clamped-free beam: cumulative modal effective parameters

Continuous Systems

191

7.579EI/L3 .6131M L M E I

.7265L

⇔

Truncation Exemple de example troncature

.1883M .2092L

.0647M .1274L .0909L

.0331M

.0504M .0079ML/2 .0006ML2/3

Figure 6.18. Clamped-free beam: effective mass model

The effective mass model resulting from relation [6.32] is that of Figure 6.18. ~ The masses in translation are directly M vv,k . The distances are given by ~ ~ M vθ ,k / M vv,k . The FRF matrices are found in the same way as for the rod. Relations [6.17] of the rod are transposed in the following way:

λ=

ωL cf

cf =

ω

EIL M

E = E (1 + i η )

[6.36]

The parameter c f is the complex propagation velocity under bending along the beam, depending on the material, but also on the cross-section and, something new in relation to the rod, on the frequency. The loss factor η in the material plays the same role, but with a different intervention on c f . The dimensionless parameter λ , made complex by η, is related to the circular frequency by λ 2 = ω / E I / M L3 and it

thus represents the reduced circular frequency. With this parameter and using ch and sh for cosh and sinh, we obtain the following results: ⎡ L3 ⎢ EI G 22 (λ ) = ⎢ ⎢ ⎢ ⎣⎢

1 ch λ sin λ − sh λ cos λ 1 + ch λ cos λ λ3 2 L 1 sh λ sin λ E I λ 2 1 + ch λ cos λ

L2 1 sh λ sin λ E I λ 2 1 + ch λ cos λ L 1 ch λ sin λ + sh λ cos λ EI λ 1 + ch λ cos λ

⎤ ⎥ ⎥ [6.37] ⎥ ⎥ ⎦⎥

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Structural Dynamics in Industry

L sh λ + sin λ ⎤ ⎥ λ 1 + ch λ cos λ ⎥ ch λ + cos λ ⎥ ⎥ 1 + ch λ cos λ ⎦

⎡ ch λ + cos λ ⎢ 1 + ch λ cos λ T T21 (λ ) = T12 (λ ) = ⎢ ⎢ λ sh λ − sin λ ⎢ ⎣ L 1 + ch λ cos λ ⎡ 1 ch λ sin λ + sh λ cos λ ⎢M λ 1 + ch λ cos λ M 11 (λ ) = ⎢ ⎢ 1 sh λ sin λ ⎢ ML 2 λ 1 + ch λ cos λ ⎣

[6.38]

⎤ ⎥ λ 2 1 + ch λ cos λ ⎥ [6.39] 2 1 ch λ sin λ − sh λ cos λ ⎥ ML ⎥ 1 + ch λ cos λ λ3 ⎦ ML

1

sh λ sin λ

c b a b c

a

b

c

a a)

EI 3

L

b)

EI 2

G vv =

1 M L2

G vθ =

a) Tvv = Tθθ

M θθ

1 M vθ ML

b)

L EI 1 c) Gθθ = M vv L M

1 Tvθ L

c) L Tθv

ω

(abscissa:

2π

= EI

λ2 ) 2π

M L3

Figure 6.19. Clamped-free beam FRF (η = 0.05)

The intervening trigonometric functions are plotted in Figure 6.19 according to parameter λ 2 = ω / E I / M L3 . Their behavior is similar to that of the rods. When the frequency tends towards zero, we obtain the static values [6.33] to [6.35]. When

Continuous Systems

193

it tends towards infinity, tan λ → −i , 1 / cos λ → 2 exp(−λ η / 4) approximately, and the asymptotic behavior is the following [GIR 90]:

⎡E I ⎢ 3 G vv ⎢L ⎢E I ⎢ 2 Gθv ⎣L

⎡ ⎢ Tvv ⎢ ⎢ L Tθv ⎣⎢

⎤ ⎡ 1 M θθ G vθ ⎥ ⎢ 2 L ⎥ = ⎢M L EI ⎥ ⎢ 1 M vθ Gθθ ⎥ ⎢ L ⎦ ⎣ML

EI 2

1 ⎤ − i −1 M θv ⎥ ⎡⎢ 3 ML ⎥→⎢ λ −i 1 ⎥ M vv ⎥ ⎢ 2 ⎢⎣ λ M ⎦

1 ⎡ ⎤ Tθv ⎥ 1 η⎞⎢ ⎛ L ⎥ → 2 exp⎜ − λ ⎟ ⎢ 4⎠⎢ 1 ⎝ Tθθ ⎥ ⎢⎣ λ ⎦⎥ 12

1 ⎡ ⎤ Tvθ ⎥ ⎢ Tvv L ⎥ =⎢ ⎢ L Tvθ Tθθ ⎥ ⎦⎥ 21 ⎢⎣

−i ⎤ ⎥ λ 2 ⎥ [6.40] − i + 1⎥ λ ⎥⎦

⎤

λ⎥ ⎥ 1⎥ ⎥⎦

[6.41]

6.3.3. Free-free beam

For the free-free beam, the four end DOF (v1 , θ 1 , v 2 , θ 2 ) are of internal i type (Figure 6.20), hence the dynamic flexibility matrix G ii (ω ) . v 1

v

θ

M, E, I, φ = 0

θ 2

L Figure 6.20. Free-free beam

Relation [6.28] and Tables 6.1 and 6.2 give the mode shapes. They are plotted in Figure 6.21 (including the two rigid modes marked k = 0).

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Structural Dynamics in Industry

k=0 0 1 3

2

4

Figure 6.21. Mode shapes of the free-free beam

The generalized masses are all equal to M, as in relation [6.29] and the effective flexibilities based on relation [5.5] are given by:

~ 4 G ii ,k =

λk 4

L3 EI

⎡ 1 ⎢ ⎢ ⎢ λk Ck ⎢ − L ⎢ ⎢ (−1) k −1 ⎢ ⎢ ⎢ λk Ck ⎢(−1) k −1 L ⎢⎣

[6.42]

−

λk Ck

λk

2

L Ck 2

L2 (−1) k (−1) k

λk Ck

L λk 2 Ck 2 2

L

(−1) k −1 (−1) k

λk Ck L 1

λk Ck L

(−1) k −1

λk Ck ⎤

⎥ L ⎥ 2 λk Ck ⎥ (−1) k ⎥ ⎥ L2 λk Ck ⎥ ⎥ L ⎥ λk 2 Ck 2 ⎥ ⎥ ⎥⎦ L2 2

The sum of the effective flexibilities gives the pseudo-static flexibilities (section 5.2.2) corresponding to the loading illustrated in Figure 6.22 with zero average displacements and rotations, and leads to the expressions plotted in the same figure:

x L3 1 ⎛⎜ x x2 x4 x5 G v1 ( ) = 4 − 22 + 70 − 70 + 21 L E I 420 ⎜⎝ L L2 L4 L5

⎞ ⎟ ⎟ ⎠

Continuous Systems

195

x L2 1 ⎛⎜ x x2 x4 x 5 ⎞⎟ Gθ1 ( ) = − 22 + 156 − 210 + 105 − 42 L E I 420 ⎜⎝ L L2 L4 L5 ⎟⎠ x x L3 1 ⎛⎜ x x4 x5 G v2 ( ) = G v1 (1 − ) = 3 − 13 + 35 − 21 L L E I 420 ⎜⎝ L L4 L5

⎞ ⎟ ⎟ ⎠

x x L2 1 ⎛⎜ x x4 x 5 ⎞⎟ Gθ 2 ( ) = −Gθ 2 (1 − ) = 13 − 54 + 105 − 42 L L E I 420 ⎜⎝ L L4 L5 ⎟⎠

− 22 / L ⎡ 4 ⎢− 22 / L 156 / L2 L 1 ⎢ ⇒ G ii = − 13 / L E I 420 ⎢ 3 ⎢ 2 ⎣ 13 / L − 54 / L 3

3 13 / L ⎤ − 13 / L − 54 / L2 ⎥⎥ 4 22 / L ⎥ ⎥ 22 / L 156 / L2 ⎦

[6.43]

The analogy of the beam element with mass matrix [3.40] should be noted. Fv

Fv

Fθ

Fθ

Figure 6.22. Pseudo-static flexibilities of the free-free beam (dimensionless and multiplied by 420)

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Structural Dynamics in Industry

The numerical values of the effective flexibilities compared to the corresponding static values are given by Table 6.4, which, in addition, illustrates the summation rules. Figure 6.23 gives the cumulative effective flexibilities. ~ ~ ~ ~ ~ ~ Gv1v1 = Gv2 v2 − Gv1θ1 = Gv2θ 2 Gθ1θ1 = Gθ 2θ 2

k

1 2 3 4 … >4

~ ~ Gv1θ 2 = −Gθ1v2

~ − Gθ 1θ 2

4 L3 420 E I

22 L2 420 E I

156 L 420 E I

3 L3 420 E I

13 L2 420 E I

54 L 420 E I

0.8391 0.1104 0.0287 0.0105

0.7090 0.1578 0.0574 0.0270

0.4646 0.1749 0.0891 0.0539

+1.1187 -0.1472 +0.0383 -0.0140

+1.1998 -0.2670 +0.0972 -0.0457

+1.3423 -0.5052 +0.2573 -0.1557

6720

6720

4 4

(2k + 1) π

Σ

~ Gv1v2

100%

560 3 3

2 2

8960 (−1) k −1

13440 (−1) k −1 1120 (−1) k −1

4 4

13 (2k + 1)3π 3 9 (2k + 1) 2 π 2

11(2k + 1) π

13 (2k + 1) π

(2k + 1) π

100%

100%

100%

100%

100%

Table 6.4. Free-free beam: modal effective flexibilities

∑ k

(

~ 420 E I ~ Gv1v1 , k = Gv2v2 , k 4 L3

(

420 E I ~ G v1v2 ,k 3 k 3L

∑

)

)

∑

420 E I 22 L2

k

∑ k

420 E I 2

13 L

EI ~ ( − G~v θ ,k = G~v θ ,k ) ∑ 420 ( Gθ θ ,k = G~θ θ ,k ) 156 L 1 1

2π

EI

1 2

=

1 2

1 1

k

( G~v θ ,k = − G~θ v ,k )

ω

(abscissa:

2 2

(

2 2

420 E I ~ − Gθ1θ 2 ,k 54 L k

∑

λ2 – logarithmic scale) 2π

M L3

Figure 6.23. Free-free beam: cumulative modal effective flexibilities

)

Continuous Systems

197

Using the same parameters as those in relation [6.36], the dynamic flexibilities are written:

⎡ G (λ ) G 12 (λ ) ⎤ G ii (λ ) = ⎢ 11 ⎥ ⎣G 21 (λ ) G 22 (λ )⎦

⎡ L3 ⎢ EI G 11 (λ ) = − ⎢ ⎢ ⎢ ⎣⎢

1 ch λ sin λ − sh λ cos λ 1 − ch λ cos λ λ3 2 L 1 − sh λ sin λ E I λ 2 1 − ch λ cos λ

⎡ L3 ⎢ EI T (λ ) = − ⎢ 2 G 21 (λ ) = G 12 ⎢L ⎢ ⎢⎣ E I

⎡ L3 ⎢ EI G 22 (λ ) = − ⎢ ⎢ ⎢ ⎣⎢

[6.44]

L2 1 − sh λ sin λ E I λ 2 1 − ch λ cos λ L 1 ch λ sin λ + sh λ cos λ EI λ 1 − ch λ cos λ

1 − (sh λ − sin λ ) λ 3 1 − ch λ cos λ 1 − (ch λ − cos λ )

λ 2 1 − ch λ cos λ

1 ch λ sin λ − sh λ cos λ 1 − ch λ cos λ λ3 2 L 1 sh λ sin λ E I λ 2 1 − ch λ cos λ

L2 1 ch λ − cos λ E I λ 2 1 − ch λ cos λ L 1 (sh λ + sin λ ) E I λ 1 − ch λ cos λ

⎤ ⎥ ⎥ ⎥ ⎥ ⎦⎥

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎦

⎤ L2 1 sh λ sin λ ⎥ 2 E I λ 1 − ch λ cos λ ⎥ L 1 ch λ sin λ + sh λ cos λ ⎥ ⎥ EI λ 1 − ch λ cos λ ⎦⎥

The intervening trigonometric functions are plotted in Figure 6.24 with the parameter λ 2 = ω / E I / M L3 .

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Structural Dynamics in Industry

a b c c

c b

b a

a)

EI

b) −

3

G v1v1 =

L EI 2

EI

G v1θ1 =

3

a) −

G v 2 v2

L EI 2

a

b) −

G v2θ 2

L L EI EI c) Gθ1θ1 = Gθ 2θ 2 L L

EI 2

L

c)

ω

(abscissa:

2π

= EI

EI L3

G v1v2

G v1θ 2 =

EI L2

Gθ1v2

EI Gθ1θ 2 L

λ2 ) 2π

M L3

Figure 6.24. Free-free beam: dynamic flexibilities (η = 0.05)

Their behavior is similar to that of the clamped-free beam. When the frequency tends towards zero, we find the contribution of the rigid modes: ⎡ 4 ⎢ M ⎢ ⎢− 6 1 ⎢ ML G ii (ω → 0) = ⎢ −ω 2 ⎢ − 2 ⎢ M ⎢ 6 ⎢− ⎢⎣ M L

−

6 ML 12

M L2 6 ML 12 M L2

2 M 6 ML 4 M 6 ML

−

6 ⎤ M L⎥ ⎥ 12 ⎥ M L2 ⎥ ⎥ 6 ⎥ ML ⎥ 12 ⎥ ⎥ M L2 ⎥⎦

−

[6.45]

Continuous Systems

199

When the frequency tends towards infinity, tan λ → −i , 1 / cos λ → 2 exp( −λ η / 4) approximately, and the asymptotic behavior is the following: ⎡E I ⎢ 3 G vv ⎢L ⎢E I ⎢ 2 Gθv ⎣L

EI

⎡E I ⎢ 3 G vv ⎢L ⎢E I ⎢ 2 Gθv ⎣L

EI

⎤ ⎡ EI G vθ ⎥ ⎢ 3 G vv L ⎥ =⎢ L EI ⎥ ⎢ EI Gθθ ⎥ Gθv ⎢− L ⎦ 11 ⎣ L2

⎤ ⎡i +1 G vθ ⎥ ⎢ 3 L ⎥ → −⎢ λ EI ⎥ ⎢ −i Gθθ ⎥ ⎢⎣ λ 2 L ⎦ 22

−

2

⎤ ⎡E I G vθ ⎥ ⎢ 3 G vv L ⎥ =⎢ L EI ⎥ ⎢E I Gθθ ⎥ G vθ ⎢ L ⎦ 21 ⎣ L2 2

EI 2

−i ⎤ ⎥ λ2 ⎥ i − 1⎥ λ ⎥⎦

⎤ ⎡ 1 Gθv ⎥ ⎢ 3 η ⎛ ⎞ L ⎥ → −2 exp⎜ − λ ⎟ ⎢ λ EI ⎥ 4⎠⎢ 1 ⎝ Gθθ ⎥ ⎢⎣ λ 2 L ⎦ 12 EI 2

[6.46]

−1 ⎤ ⎥ λ2 ⎥ −1 ⎥ λ ⎥⎦

6.3.4. Clamped-clamped beam

For the clamped-clamped beam, the four end DOF (v1 , θ 1 , v 2 , θ 2 ) are of junction type j (Figure 6.25) which is a statically indeterminate junction. Hence the dynamic stiffness matrix K jj (ω ) . v 1

v

θ

M, E, I, φ = 0

θ 2

L Figure 6.25. Clamped-clamped beam

Relation [6.28] and Tables 6.1 and 6.2 give the mode shapes. They are plotted in Figure 6.26.

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Structural Dynamics in Industry

k=1 2 3 4

Figure 6.26. Mode shapes of the clamped-clamped beam

With the four junction modes given by relation [3.39], according to relations [6.1], the generalized masses and the participation factors are expressed as:

Lkv1 =

mk = M

2 Ck M

Lkθ1 =

λk

Lkv2 = (−1) k −1

2 Ck M

λk

2M L

[6.47]

λk 2

Lkθ 2 = (−1) k

2M L

λk 2

from which we obtain the effective masses: 4 ~ M jj ,k = M

[6.48]

λk 4

⎡ λk 2 Ck 2 ⎢ ⎢ L λk Ck ⎢ k −1 λk 2 Ck 2 ⎢(−1) ⎢ k ⎢⎣ (−1) L λ k C k

L λk Ck

(−1) k −1 λ k 2 C k 2

L2

(−1) k −1 L λ k C k

(−1) k −1 L λ k C k

λk 2 Ck 2

(−1) k L2

L λk Ck

(−1) k L λ k C k ⎤ ⎥ (−1) k L2 ⎥ ⎥ L λk Ck ⎥ ⎥ L2 ⎥⎦

The sum of the effective masses gives the condensed mass matrix (section 5.2.2), i.e. the mass matrix [3.40] of the beam finite element.

Continuous Systems

k

~ ~ M v1 v1 = M v 2 v 2

~

= −M v

θ2

2

22 ML 420

156 M 420

0.4646 1 0.1749 2 0.0891 3 0.0539 4 560 … >4 13 (2k + 1) 2 π 2

Σ

~ M v1θ 1

100%

~ ~ Mθ1θ1 = Mθ 2θ 2 4 M L2 420

~ M v1v2 54 M 420

0.8391 0.1104 0.0287 0.0105 6, 720

~

− Mv θ

1 2

0.7090 0.1578 0.0574 0.0270 6, 720 11(2k + 1)3 π 3

(2k + 1)4 π 4

+1.3423 –0.5052 +0.2573 –0.1557 1,120 (−1)k −1 9 (2k + 1) 2 π 2

100%

100%

100%

=

~ M θ 1v 2

201

~ − M θ 1θ 2

13 ML 420

3 M L2 420

+1.1998 –0.2670 +0.0972 –0.0457 13, 440 (−1) k −1 13(2k + 1)3 π 3

+1.1187 –0.1472 +0.0383 –0.0140 8,960 (−1) k −1

100%

100%

(2k + 1) 4 π 4

Table 6.5. Clamped-clamped beam: modal effective masses

The numerical values of effective mass, related to the corresponding static terms are given by Table 6.5, which is the same as Table 6.4 with the column permutations. The cumulative effective masses, given by Figure 6.27, are thus the same as those of Figure 6.23 with the permutations.

∑ k

(

~ 420 ~ M v1v1 , k = M v2 v2 , k 156 M

(

420 ~ M v1v2 ,k M 54 k

∑

)

)

∑ k

(

~ ~ 420 M v1θ1 ,k = − M v2θ 2 ,k 22 M L

(

( M~θ θ ,k = M~θ θ ,k ) ) ∑ 4420 2 ML

420 ~ ~ − M v1θ 2 , k = Mθ1v2 , k 13 M L k

∑

ω

(abscissa:

2π

EI

=

1 1

k

)

∑ k

420 2

3M L

2 2

( − M~θ θ ,k )

λ2 – logarithmic scale) 2π

M L3

Figure 6.27. Clamped-clamped beam: cumulative modal effective masses

1 2

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Structural Dynamics in Industry

Using the same parameters as those of relation [6.36], the dynamic stiffnesses are written:

⎡ K (λ ) K 12 (λ ) ⎤ K ii (λ ) = ⎢ 11 ⎥ ⎣K 21 (λ ) K 22 (λ )⎦

⎡ E I 3 ch λ sin λ + sh λ cos λ ⎢ 3 λ 1 − ch λ cos λ K 11 (λ ) = ⎢ L ⎢ E I 2 sh λ sin λ λ ⎢ 1 − ch λ cos λ L2 ⎣

[6.49]

⎤ ⎥ 1 − ch λ cos λ L ⎥ E I ch λ sin λ − sh λ cos λ ⎥ λ ⎥ L 1 − ch λ cos λ ⎦ EI

⎡ E I 3 − (sh λ + sin λ ) ⎢ 3 λ 1 − ch λ cos λ T K 21 (λ ) = K 12 (λ ) = ⎢ L ⎢ E I 2 ch λ − cos λ ⎢ 2 λ 1 − ch λ cos λ ⎣ L

⎡ E I 3 ch λ sin λ + sh λ cos λ ⎢ 3 λ 1 − ch λ cos λ K 22 (λ ) = ⎢ L ⎢ E I 2 − sh λ sin λ λ ⎢ 1 − ch λ cos λ L2 ⎣

λ2

2

sh λ sin λ

− (ch λ − cos λ ) ⎤ ⎥ 1 − ch λ cos λ ⎥ L EI sh λ − sin λ ⎥ λ ⎥ L 1 − ch λ cos λ ⎦

EI 2

λ2

⎤ ⎥ 1 − ch λ cos λ L2 ⎥ E I ch λ sin λ − sh λ cos λ ⎥ λ ⎥ L 1 − ch λ cos λ ⎦ EI

λ2

− sh λ sin λ

Continuous Systems

203

a b c a

a

b

b

c

a) b) c)

c

L3 L3 K v1v1 = K v2 v 2 EI EI

a) −

L2 L2 K v1θ1 = − K v2θ 2 EI EI

b)

L2 L2 K v1θ 2 = − K θ1v2 EI EI

L L K θ1θ1 = K θ 2θ 2 EI EI

c)

ω

(abscissa:

= EI

2π

L3 Kv v EI 12

L K θ1θ 2 EI

λ2 ) 2π

M L3

Figure 6.28. Clamped-clamped beam: dynamic stiffnesses (η= 0.05)

The intervening trigonometric functions are plotted in Figure 6.28 using parameter λ 2 = ω / E I / M L3 . Their behavior is similar to that of the free-free

beam. When the frequency tends towards zero, we find stiffness matrix [3.40] of the beam finite element. When it tends towards infinity, tan λ → −i , 1 / cos λ → 2 exp(−λ η / 4) approximately, and the asymptotic behavior is the following: ⎡ L3 K vv ⎢ ⎢E I ⎢ L2 ⎢ K θv ⎢⎣ E I

⎤ ⎡ L3 L2 K vθ ⎥ K vv ⎢ EI ⎥ =⎢ EI ⎥ ⎢ L2 L ⎢− K θθ ⎥ Kθv EI ⎥⎦ 11 ⎢⎣ E I

⎤ L2 ⎡ 3 ⎤ K vθ ⎥ λ (i − 1) λ 2 i ⎥ ⎢ EI ⎥ →⎢ ⎥ ⎥ ⎢ λ2 i L λ (i + 1)⎥⎥ K θθ ⎥ ⎢ ⎣ ⎦ EI ⎥⎦ 22

−

[6.50]

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Structural Dynamics in Industry

6.3.5. Shear and rotary inertia effects

The shear and rotary inertia effects become significant when the dimensions of the cross section are no longer small compared to the beam’s half-wavelength, i.e. the distance between two vibration nodes. The characteristic dimensionless parameters for these two effects are written [KOL 73, NEU 87]: r2 =

I

s2 =

2

SL

EI

[6.51]

k G S L2

r being the radius of gyration of the cross section with respect to the length L, characteristic of the rotary effect, and s related to the parameter φ of relation [3.41] but better suited here, and characteristic of the shear effect. Equation [6.27] in the frequency domain is thus written: ∂ 4v ∂x

4

+

λ4 2

L

(r 2 + s 2 )

∂ 2v ∂x

2

−

λ4 4

L

(1 − λ 4 r 2 s 2 ) v = 0

[6.52]

The higher the frequency, the more parameters r and s modify the beam modes. We will not derive all the preceding results which would result in very complex expressions of little practical use. However, it is interesting to know the modifications regarding the dynamic stiffnesses in section 6.3.4 in order to complete the beam element stiffness formulation, with reference to the considerations given in the introduction. Using the following parameters:

λ '= λ +

λ2 2

(r 2 + s 2 ) +

λ4 4

(r 2 − s 2 ) 2 + 1

μ ' = λ '−

λ4 s2 λ' [6.53]

λ"= λ −

λ2 2

(r 2 + s 2 ) +

λ4 4

(r 2 − s 2 ) 2 + 1

μ " = λ "+

λ4 s2 λ"

Continuous Systems

205

the terms of relations [6.49] become:

λ3

ch λ sin λ + sh λ cos λ 1 − ch λ cos λ

→

λ2

2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ '

1 − ch λ cos λ

μ ' μ " (λ ' μ '−λ " μ " ) (1 − ch λ " cos λ ' ) + μ ' μ " ( μ ' λ "+λ ' μ" ) sh λ " sin λ ' ) 2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ '

ch λ sin λ − sh λ cos λ 1 − ch λ cos λ

→

λ3

λ2

(λ ' 2 +λ " 2 ) ( μ " ch λ " sin λ '− μ ' sh λ " cos λ ' ) 2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ ' sh λ + sin λ

1 − ch λ cos λ

→

μ ' μ " (λ ' 2 +λ " 2 ) ( μ " sh λ "+ μ ' sin λ ' ) 2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ '

ch λ − cos λ 1 − ch λ cos λ

→

λ

μ ' μ " (λ ' 2 +λ " 2 ) ( μ ' ch λ " sin λ '+ μ " sh λ " cos λ ' )

sh λ sin λ

→

λ

[6.54]

μ ' μ " (λ ' 2 +λ " 2 ) (ch λ "− cos λ ' ) 2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ '

sh λ − sin λ 1 − ch λ cos λ

→

(λ ' 2 +λ " 2 ) ( μ ' sh λ "− μ " sin λ ' ) 2 μ ' μ " (1 − ch λ " cos λ ' ) + ( μ " 2 − μ ' 2 ) sh λ " sin λ '

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It should be noted that parameter λ" is only defined for λ 4 ≤ 1 / r 2 s 2 , and this leads to the existence of a cutoff frequency beyond which the beam model is no longer valid. 6.4. Plate element 6.4.1. Introduction

Plates do not offer the same level of interest as the beams within the present context. The multiplicity of possible configurations, the complexity brought by the second dimension, the continuous nature of the boundaries, which prevents an assembly procedure similar to that of the beams, all make it a subject requiring too many developments for practical use. We will limit ourselves to some typical results in bending, which can serve as reference, with the exception of the only case providing simple expressions: the simply supported rectangular plate. In general, as with the beam, the natural circular frequencies are of the form:

ω k = λk 2

D

μ L4

D=

E h3 12 (1 − ν 2 )

[6.55]

D is the plate bending stiffness, similar to the beam stiffness E I: here, Young’s modulus E is divided by (1 − ν 2 ) and h 3 / 12 represents the inertia per unit width, h being the thickness. μ is the area mass density and L the characteristic length. The dimensionless parameter λ k is identified by two parameters representing the mode shapes in the two dimensions. Except for the simply supported rectangular case where it can be expressed analytically, it is generally obtained by a numerical approach, except for some simple cases leading to a transcendent equation. The values of λ k for circular, annular, rectangular and other plates with various boundary conditions can be found in [BLE 79]. Some values of λ1 are listed in the following section.

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207

Regarding the FRF, the analysis is obviously difficult (see for example [CRE 88]). Let us mention a simple result, that of the driving-point impedance Z ∞ of an infinite plate, i.e. the ratio force/velocity for the response in velocity to a point excitation in force at the same point, which is written: Z∞ = 8 D μ

[6.56]

The case of an infinite plate is equivalent to that of the finite plate for an infinite frequency, which, with equation [6.56], makes it possible to verify the asymptotic behavior of a plate at high frequencies, independently of its shape and boundary conditions. 6.4.2. Some plate results in bending

For a circular plate where the characteristic length L is the radius, the parameter 2

λ1 of equation [6.55] takes the following values: – for a constrained edge with v = 0.3 (bell-shaped): simply supported 2

λ1 = 4.977, clamped λ1 2 = 10.22; – for a constrained edge and center (axisymmetric shape): simply supported edge 2

λ1 = 14.8 (ν = 0.3), clamped λ1 2 = 22.7 (independent of ν). For a rectangular plate where the characteristic length L is the length following x, for example, λ1 2 takes the following values for a width L/α: – for the 4 simply supported edges (bell-shaped – see the following section):

λ1 2 = π 2 (1 + α 2 )

[6.57]

– for the 4 clamped edges (with less than 0.5% – bell-shaped):

λ1 2 ≈ π 2

5.14 (1 + α 4 ) + 3.11α 2

[6.58]

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6.4.3. Simply supported rectangular plate

The modes of a simply supported rectangular plate of mass M and of dimensions L x and L y characterized by n x half-wavelengths in direction x and n y halfwavelengths in direction y, as illustrated by Figure 6.29. The natural circular frequencies and the mode shapes are expressed by:

ω k = λk 2

E h3 12 (1 − ν 2 ) M L x L y

⎛n λ k = λ (n x , n y ) = π ⎜⎜ x ⎝ Lx

2 ⎛ ny ⎞ ⎞ ⎟ ⎟ +⎜ ⎟ ⎜ Ly ⎟ ⎠ ⎝ ⎠

2

Lx L y

y⎞ x⎞ ⎛ ⎛ Φk ( x, y ) = Φ(n x , n y ) = sin ⎜ n x π ⎟ sin ⎜ n y π ⎟ L⎠ ⎝ L⎠ ⎝

[6.59]

w(x, y)

y

x

nx = 5 ny = 3

Figure 6.29. Modes of the simply supported rectangular plate

Continuous Systems

209

If we are interested in the displacement at the plate center, only the modes with odd n x and n y contribute. For these modes, by considering the rigid junction mode in translation Ψw ( x, y ) = 1 , the generalized masses and the participation factors have for expression, according to relations [6.1]: m k = m( n x , n y ) =

M 4

L k w = L( n x , n y ) =

4M nx n y π 2

[6.60]

The modal effective parameters relating to the plate center and to the rigid junction based on relations [5.5] to [5.7] are given by: ~ G (n x , n y ) =

2⎞ 2 ⎛⎛ ⎛ Ly ⎞ L x ⎞⎟ ⎟ ⎜⎜ ⎜ ⎜ ⎜ n x L ⎟⎟ + ⎜ n y L ⎟ ⎟ ⎜⎝ x ⎠ y ⎠ ⎟ ⎝ ⎝ ⎠

1 2

64 2

2

π4

E h3

[6.61]

n y −1

nx −1

(−1) 2 (−1) 2 ~ T (n x , n y ) = nx n y

~ M (n x , n y ) =

48 (1 − ν 2 ) L x L y

1

nx n y π 4

M

16

π2

[6.62]

[6.63]

The summation rules regarding the transmissibilities and the masses are verified due to relations [6.13]. Concerning the flexibilities, the static flexibility can be found in [YOU 89], and in particular the coefficient 0.1267 for a square plate with ν = 0.3. The cumulative effective parameters are given by Figure 6.30 for a square plate ( L x = L y = L ) with ν = 0.3. The corresponding FRF, obtained by mode superposition, are plotted in Figure 6.31. Relation [6.56] gives the flexibility asymptotic behavior.

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∑ k

E h3

~ G

2

0.1267 L

~

1 ~ M k M

∑T

∑

k

ω

(abscissa:

2π

=

D

λ2 – logarithmic scale) 2π

M L2

Figure 6.30. Simply supported square plate: cumulative modal effective parameters (ν = 0.3)

E h3 2

L

1 M (ω ) M

T (ω )

G (ω )

ω

(abscissa:

2π

D

=

λ2 ) 2π

M L2

Figure 6.31. Simply supported square plate: FRF with respect to the plate center and to the rigid junction (ν = 0.3) (η = 0.1)

6.5. Combined cases 6.5.1. Introduction

Here we will look at the cases that combine two or more basic configurations, for example a continuous element to which one adds a lumped mass or a spring. Despite an apparent simplicity, most of the cases require approximate methods, described in

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211

various references such as [HAR 61], [TIM 74] or [THO 81], requiring a large numerical effort. The most typical results for the usual modal parameters are listed in [BLE 79]. A presentation of these methods is beyond the scope of the present context in which we are essentially trying to analyze some reference cases. However, one method concerning the natural frequencies should be mentioned. It is distinguished by its simplicity and it can rapidly give a result by combining the cases that were already listed or which are simple to calculate. It is called Dunkerley’s formula and is discussed, for example, in [THO 81]. This formula gives a lower limit of the first natural frequency of a structure including n mass elements i (continuous or discrete), starting from the n frequencies of the same structure where each mass element acts separately: 1

ω1

2

n

1

i =1

(ω1 2 ) i

≈∑

[6.64]

The justification for a lumped mass approach is in the fact that 1 / ω1 2 is less than the trace of matrix K −1 M . Each diagonal element of this matrix represents the 1 / ω i 2 . The larger 1 / ω1 2 is in comparison to the other 1 / ω k 2 of the structure, the

better the approximation is, i.e. if the first natural frequency is much lower than the following ones. We should note that the same reasoning can be applied to the local flexibilities (inverses of the local stiffnesses) instead of the masses: the formula also gives a lower limit of the first natural frequency of a structure including n flexible elements i, starting from the n frequencies of the same structure where each flexible element acts separately. This variant can be just as useful. For example, in the case of the 2-DOF system in Figure 4.1 with results [4.9], we find the following results: – with the masses: - bottom mass m alone (top mass removed): (ω 1 2 )1 m / k = 1 , - top mass m alone (bottom mass removed): (ω1 2 ) 2 m / k = 2 / 5 ,

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- Dunkerley’s formula: (ω1 2 )1+ 2 m / k = 2 / 7 , instead of 1/3, thus an error of –7% on the frequency; – with the flexibilities: - bottom flexibility 1/k alone (infinite top stiffness): (ω1 2 )1 m / k = 1 / 2 , - top mass m alone (infinite bottom stiffness): (ω1 2 )1 m / k = 2 / 3 , - Dunkerley’s formula: (ω1 2 )1+ 2 m / k = 2 / 7 , hence the same error as before. We might wonder whether a similar formula exists for the modal effective parameters. Since they are very sensitive to the mode shapes, the results risk being less accurate. In a first approximation, we could adopt the following simple strategies: – with the masses: ~ G1

~ ⎛ G 1 ⎜ ≈∑ ω 1 2 i =1 ⎜⎝ ω 1 2 n

⎞ ⎟ ⎟ ⎠i

( )

n ~ ~ M1 ≈ ∑ M1 i i =1

~ ~ ~ T1 ≈ ω1 2 G1 M 1

[6.65]

– with the flexibilities:

( )i

n ~ ~ G1 ≈ ∑ G1 i =1

~ M1

ω1

2

≈

~ ⎛M ⎜ ∑ ⎜ 12 i =1 ⎝ ω 1 n

⎞ ⎟ ⎟ ⎠i

~ ~ ~ T1 ≈ ω1 2 G1 M 1

With the preceding example and the results in [5.11], we find: – with the masses: ~ - G1 k = 2.07 instead of 2.4, thus –14%, ~ - M 1 / m = 2 instead of 1.8, thus +11%, ~ - T1 = 1.09 instead of 1.2, thus –9% ;

[6.66]

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213

– with the flexibilities: ~ - G1 k = 2.5 instead of 2.4, thus +4%, ~ - M 1 / m = 1.57 instead of 1.8, thus –13%, ~ - T1 = 1.06 instead of 1.2, thus –13%.

These results display the following tendencies: an underestimation of the flexibilities with the masses and of the masses with the flexibilities, an overestimation of the flexibilities with the flexibilities and of the masses with the masses and underestimation of the transmissibilities. The preceding equations, based on a discrete approach at first, are also applicable to continuous systems, and this gives a rapid estimation of the combined cases. The detailed analysis of the simple case of the clamped-free continuous rod is made in the following section in order to consolidate the preceding tendencies before giving some typical results. 6.5.2. Combination rod + local mass or flexibility

Consider the clamped-free continuous rod combined with: a) a lumped mass m at its free end (Figure 6.32a); b) a flexibility 1/k at its clamped end (Figure 6.32b).

M, E, S L a) lumped mass m

m

k

M, E, S L b) flexibility 1/k

Figure 6.32. The clamped-free rod combined with a mass or flexibility

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With an exact approach, the modes are given by [BLE 79]:

ω k = λk

ES ML

a) Φk ( x) = sin

a) λ k tan λ k =

λk x

M m

b) λ k tan λ k =

b) Φk ( x) = cot λ k cos

L

λk x L

kL ES + sin

λk x

[6.67]

L

The generalized masses and the participation factors are given by:

a) m k =

M 2

⎛ sin 2 λ k ⎜1 − ⎜ 2 λk ⎝

⎞ ⎟ + m sin 2 λ k ⎟ ⎠

L k1 = M

1 − cos λ k

λk

+ m sin λ k

[6.68] b) m k =

⎛ sin 2λ k ⎜1 − 2 2λ k 2 sin λ k ⎜⎝ M

⎞ ⎟ ⎟ ⎠

L k1 =

M

λk

The modal effective parameters are obtained using relations [5.5] to [5.7]. We are now interested in the first mode in the particular cases: a) m = M; b) k = E S/L. hence the natural frequencies: a) and b) ω1 = 0.8603

ES ML

and the modal effective parameters: a)

L G! 22.1 = 0.9861 ES

T!21.1 = 1.1191

M! 11.1 = 1.7160 M

b)

L G! 22.1 = 1.7160 ES

T!21.1 = 1.1191

M! 11.1 = 0.9861 M

Continuous Systems

215

With Dunkerley’s formula [6.64], the following frequencies are found: a) and b)

ω1 = 0.8436

ES (combination of factors π/2 and 1) ML

thus an error of –1.9%, which is relatively small in relation to the results of the preceding section on the 2-DOF system. With equations [6.65] and [6.66], the following effective parameters are found: a)

L G! 22.1 = 0.9454 ES

T!21.1 = 1.0958

M! 11.1 = 1.8106 M

thus an error of –4.1%, –2.1% and +5.5% respectively. b)

L G! 22.1 = 1.8106 ES

T!21.1 = 1.0958

M! 11.1 = 0.9454 M

thus an error of +5.5%, –2.1% and –4.1% respectively, which is of the same order of magnitude as for the frequencies. The tendencies are the same as for the 2-DOF system. Equations [6.64] to [6.66] thus give a rapid estimation of the system to a few %. 6.5.3. Some typical results

This section gathers some results obtained by equations [6.64] to [6.66] for certain basic cases presented in this chapter. Table 6.6 gives a first eigenvalue ω1 2 obtained by Dunkerley’s formula [6.64] for the clamped-free rod in section 6.2.2 (generalization of section 6.5.2), the beam of section 6.3 with various boundary conditions (where we find nearly the same results as those in the references on this subject) and the rectangular simply supported plate in section 6.4.3.

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ω1 2

Case Clamped-free rod (section 6.2.2) with lumped mass m Clamped-free rod (section 6.2.2) with flexibility 1/k=αL/ES

Clamped-free beam (section 6.3.2) with lumped mass m Clamped-free beam (section 6.3.2) with flexibility 1/k = αL/EI

L, M, E, S

k=ES/αL

m

L, M, E, S

L, M, E, I

ES (0.41 + α ) M L

3E I

m

(0.24 M + m) L3 k=EI/αL

L, M, E, I

Pinned-pinned beam (section 6.3.1) with lumped mass m

L, M, E, I

Clamped-clamped beam (section 6.3.4) with lumped mass m

L, M, E, I

Simply supported square plate (section 6.4.3) with lumped mass m

ES (0.41 M + m) L

(0.24 + α ) M L3 48 E I (0.49 M + m) L3

m

192 E I (0.38 M + m) L3

m L, M E, h ν=.3

m

Table 6.6. Estimation of ω 1

3E I

2

E h3 (0.22 M + m) 0.1267 L2

by Dunkerley’s formula

~ ~ Table 6.7 gives the effective flexibility G1 and the effective mass M 1 of the first mode. They were obtained by equation [6.65] or [6.66], for the clamped-free rod, the clamped-free beam and the simply supported plate. The effective ~ ~ ~ transmissibility is deduced from relation T1 ≈ ω1 2 G1 M 1 .

Continuous Systems

Case

~ G1

~ M1

Clamped-free rod (section 6.2.2) with lumped mass m

0.33 M + m L 0.41 M + m E S

(0.81 M + m)

Clamped-free rod (section 6.2.2) with flexibility 1/k=αL/ES Clamped-free beam (section 6.3.2) with lumped mass m Clamped-free beam (section 6.3.2) with flexibility 1/k=αL/EI Simply supported square plate (section 6.4.3) with lumped mass m

(0.81 + α )

L ES

0.23 M + m L3 0.24 M + m 3 E I (0.61 + α )

L3 3E I

0.20 M + m 0.1267 L2 0.22 M + m E h 3 ~

~

0.33 + α M 0.41 + α (0.61 M + m)

0.23 + α M 0.24 + α

(0.66 M + m)

Table 6.7. Estimation of G1 and M 1 by equations [6.65] and [6.66]

217

Chapter 7

Complex Modes

7.1. Introduction The modal approach considered up to now and described in detail in Chapter 4 uses the normal modes resulting from the conservative system. Because of the simplicity brought about by the real value modal parameters, it is current practice that the eigenvalues of the calculation give the natural frequencies directly and the real components of the eigenvectors give motions in phase or out of phase. In the calculation of frequency responses, the phase differences result only from the amplification factors that affect each mode. The main disadvantage is the need for an uncoupling hypothesis of the modal damping in order to be able to fully benefit from the advantages of the approach. This is a suitable hypothesis under the conditions mentioned in section 4.3.3 which are usually respected. However, certain cases produce significant errors and therefore require consideration of the dissipative system when computing the modes, hence the complex modes in the algebraic sense of the term. Moreover, other effects such as the presence of rotating parts or acoustic cavities, introduce additional terms in the equations that need to be handled using complex modes. The mode superposition concept is still valid, without supplementary hypothesis, and leads to efficient response calculations with the same reserves as for the normal modes. This is the subject of this chapter. In section 7.2 we will first look at the case of a viscous damping alone, which makes it possible to introduce the complex mode concept in the continuation of the developments presented so far. Then, with section 7.3, we will examine the gyroscopic effects alone which introduce a new consideration, non-reciprocity, but

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which can be treated in a similar way using certain precautions. Finally, in section 7.4, a more general case that makes it possible to combine various effects without violating the mode superposition principle will be considered [GIR 01a]. 7.2. Dissipative systems 7.2.1. Complex modes The approach for complex modes is the same as that for the normal modes introduced in section 4.2.1. They are the solutions of the equation of motion in the absence of excitation of the system now including the dissipation, i.e.: !! i + C ii u! i + K ii u i = 0 i M ii u

[7.1]

This homogenous system is still of size N equal to the number of internal DOF i, but the information it contains has doubled because, schematically, matrix M −1 C is added to matrix M −1 K . In order to solve it, we can rewrite it as: ⎡ 0 ii ⎢M ⎣ ii

!! i ⎤ ⎡− M ii M ii ⎤ ⎡u + ⎥ ⎢ C ii ⎦ ⎣u! i ⎥⎦ ⎢⎣ 0 ii

0 ii ⎤ ⎡u! i ⎤ ⎡0 i ⎤ = K ii ⎥⎦ ⎢⎣u i ⎥⎦ ⎢⎣0 i ⎥⎦

[7.2]

because the first line is an identity and the second provides the equations of motion. The state vector is now made up not only of displacements but also of velocities and has thus doubled in size (i+i) to fit into a space 2N. It should be noted that system [7.2] is not the only one possible, but it has the advantage of preserving the symmetry of the matrices introduced, a property which will be exploited as for the normal modes, and to make the parallel with the normal modes easier, as will be seen later. In the frequency domain where u! i = i ω u i , equation [7.2] gives: ⎛ ⎡ 0 ii M ii ⎤ ⎡− M ii 0 ii ⎤ ⎞ ⎡i ω u i ⎤ ⎡0 i ⎤ ⎜i ω ⎢ ⎟ = ⎥+⎢ ⎜ K ii ⎥⎦ ⎟⎠ ⎢⎣ u i ⎥⎦ ⎢⎣0 i ⎥⎦ ⎝ ⎣M ii C ii ⎦ ⎣ 0 ii --------------- ---------------A (i +i )(i +i ) B (i +i )(i +i )

[7.3]

Complex Modes

221

This is an eigenvalue problem of size 2N matrices A (i +i )(i +i ) and B (i +i )(i +i ) , which are real and symmetric but indefinite, which implies complex solutions occurring in conjugate pairs: – 2N conjugate complex eigenvalues ±i ω k from which 2N complex circular natural frequencies ω k± are deduced. The latter combine frequency information as well as damping information. If we refer to the associated conservative system (identified by the superscript 0) assuming the positive real circular frequencies ω k0 and the damping factors ζ k0 < 1, as in relation [2.18] established for the 1-DOF damped system, we obtain:

ω k± = ±ω k0 1 − (ζ k0 ) 2 + i ζ k0 ω k0

[7.4]

Inversely, starting from a complex circular frequency, a real circular frequency and a damping factor may be deduced by:

ω k0 = ω k±

( )

ζ k0 = ℑ ω k± ω k0

[7.5]

This correspondence is strict only with normal modes with uncoupled damping; ±

– 2N conjugate complex vectors with 2N components Φ(i +i ) k representing 2N eigenvectors associated with ω k± . The N velocity components are related to the N displacement components by the factor i ω k :

± Φ(i +i ) k

⎡i ω ± Φ ± ⎤ k ik ⎥ =⎢ ⎢ Φ± ⎥ ik ⎢⎣ ⎥⎦

[7.6]

These eigenvectors do not have a direct link with those of the associated conservative system, unless matrix C ii is diagonalized along with M ii and K ii (e.g. proportional damping; see section 4.3.3), hence the normal modes with

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uncoupled damping. In this case, the components become real and the complex modes are identical to the normal modes. Each mode k therefore belongs to a pair k ± having the circular frequencies ω k± ±

and eigenvectors Φ(i +i ) k . Each mode will be individually considered later, be it k + or k − , and this makes it possible not to burden the expressions with the ±. The eigenvectors of mode k made up of displacements Φi k and of the velocities i ω k Φi k represent a sine motion where all the DOF i do not necessarily vibrate in

phase or out of phase, but present among them the phase differences given by the components. The eigenvectors Φ(i +i ) k verify the orthogonality properties in relation to the matrices A (i +i )(i +i ) and B (i +i )(i +i ) which result from the symmetries of the latter, hence, by considering the set of modes Φ(i +i ) k from eigenvectors Φ(i +i ) k : Φk (i +i ) A (i +i )(i +i ) Φ(i +i ) k = a kk diagonal matrix of a k Φk (i +i ) B (i +i )(i +i ) Φ(i +i ) k = b kk diagonal matrix of bk = −i ω k a k

[7.7]

These orthogonality properties will make it possible, as for the normal modes with relations [4.8], to uncouple the projected equations based on the modes and thus to establish the relations of mode superposition. Developing A (i +i )(i +i ) and B (i +i )(i +i ) , the diagonal terms a k and bk verify:

ak iωk

= Φki M ii Φi k +

Φki K ii Φi k

ωk 2

= mk

[7.8]

bk = ω k 2 Φki M ii Φi k + Φki K ii Φi k = ω k 2 m k

These two relations show that a k i ω k

can be interpreted as a complex

generalized mass, which is denoted by m k , and bk as a complex generalized stiffness, resulting from the choice of matrices A (i +i )(i +i ) and B (i +i )(i +i ) .

Complex Modes

223

It should be noted that for a normal mode, m k equals twice its generalized mass, due to the doubling of the number of modes, as will be shown when considering mode superposition. As an illustration, let us again take the example in Figure 4.1 with

c1 = k m = 10 c 2 in order to compare normal and complex modes with a strong contrast in damping, corresponding for example to a lightly dissipative structure except in the vicinity of its junction: – normal modes (see results [4.9] and section 4.3.3):

ω10 = 0.5774 k / m ζ 10

= 19.1 %

⎡ + 1⎤ Φi01 = ⎢ ⎥ ⎣ + 2⎦

m10 = 5 m

[7.9]

ω 20 = 1.4142 ζ 20 = 34.6 %

k/m

⎡ − 2⎤ Φi02 = ⎢ ⎥ ⎣ + 1⎦

m 20 = 5 m

– complex modes:

ω1 = (± 0.5889 + 0.1121 i) k / m ⎡ + 1.0494 ∓ 0.1799 i ⎤ Φi1 = ⎢ ⎥ ⎣+ 2.1048 ± 0.3047 i ⎦

⇒

ω10 = 0.5995 k / m

ζ 10 = 18.70 %

m1 = 10 m

[7.10]

ω 2 = (± 1.2717 + 0.4879 i ) k / m ⎡− 2.0994 ∓ 0.6901 i ⎤ Φi 2 = ⎢ ⎥ ⎣ + 0.9303 ∓ 0.4501 i ⎦

⇒

ω 20 ζ 20

= 1.3620 k / m = 35.82 %

m 2 = 10 m

The effect of highly contrasted damping values on the modal parameters can be seen here. If the frequencies and the damping of the two approaches are not very different, the complex mode shapes present a significant phase difference which will influence the responses.

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The complex mode shapes Φi+1 and Φi+2 are plotted in Figure 7.1 and compared to the real mode shapes.

Normal mode 1

Complex mode 1

Normal mode 2

Complex mode 2

Figure 7.1. Mode shapes [7.9] and [7.10] (motion of the DOF 1 and 2 over one period)

7.2.2. Mode superposition The same approach of mode superposition as that in section 4.3 for the normal modes leads to the following results. By considering complete equation [4.28], with the partition between the DOF i and j: ⎡ M ii ⎢M ⎣ ji

!! i ⎤ ⎡ C ii M ij ⎤ ⎡ u + ⎥ ⎢ !! j ⎥ ⎢C ji M jj ⎦ ⎣u ⎦ ⎣

C ij ⎤ ⎡ u! i ⎤ ⎡ K ii + C jj ⎥⎦ ⎢⎣u! j ⎥⎦ ⎢⎣K ji

K ij ⎤ ⎡ u i ⎤ ⎡ Fi ⎤ = K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣F j ⎥⎦

[7.11]

with a modal base made up of complex modes Φ(i + j ) k completed by the junction modes Ψ(i + j ) j (independent of damping), we obtain the transformation:

[

u i + j = Φ(i + j ) k

]

⎡q k ⎤ Ψ(i + j ) j ⎢ ⎥ ⎣u j ⎦

⇔

⎡ u i ⎤ ⎡Φik ⎢u ⎥ = ⎢0 ⎣ j ⎦ ⎣ jk

Ψij ⎤ ⎡q k ⎤ I jj ⎥⎦ ⎢⎣u j ⎥⎦

[7.12]

Complex Modes

225

This transformation applied to equation [7.11] gives: ⎡M kk ⎢L ⎣ jk

!! k ⎤ ⎡C kk L kj ⎤ ⎡q + ⎥ ⎢ !! j ⎥ ⎢C jk M jj ⎦ ⎣u ⎦ ⎣

C kj ⎤ ⎡q! k ⎤ ⎡K kk + C jj ⎥⎦ ⎢⎣u! j ⎥⎦ ⎢⎣ 0 jk

0 kj ⎤ ⎡q k ⎤ ⎡ Φki Fi ⎤ = [7.13] K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣Ψ ji Fi + F j ⎥⎦

with: – M kk , C kk , K kk : non-diagonal matrices of modal mass, damping and stiffness;

( ) – C kj = Φki (C ii Ψij + C ij ) : matrix of the damping participation factors; – L kj = Φki M ii Ψij + M ij : matrix of the mass participation factors;

– M jj , C jj , K jj : condensed mass, damping and stiffness matrices. It should be noted that the effect of the damping is similar to that of the masses with particularly the same type of participation factor. In the frequency domain, the first row of equation [7.13] is written:

(− ω

2

)

(

)

M kk + iω C kk + K kk q k = Φki Fi + ω 2 L kj − i ω C kj u j

[7.14]

By introducing the orthogonality relations [7.7] and [7.8]:

(i ω a kk + b kk ) q k

(

)

= Φki Fi + ω 2 L kj − i ω C kj u j

[7.15]

Equations [7.15] are uncoupled, hence the solution in q k which, introduced into equations [7.13], gives the following general solution, by using the generalized masses m k given by relation [7.8]: ⎡ u i (ω ) ⎤ ⎡ G ii (ω ) Tij (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎢F (ω )⎥ = ⎢− T (ω ) K (ω )⎥ ⎢u (ω )⎥ ji jj ⎣ j ⎦ ⎣ ⎦⎣ j ⎦

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Structural Dynamics in Industry

Φi k Φki

2N

1

k =1 1 −

ω ω 2m k k ωk

G ii (ω ) = ∑

(if rigid modes r : ∑ r

ω2 Tij (ω ) = Ψij +

2N

∑

ωk

k =1 1 −

2

ω ωk

Φi k L k j mk

+

2N

∑

ω ωk

k =1 1 −

Φi r Φri −ω 2 m r

Φi k C k j

ω iωk m k ωk

)

[7.16]

[7.17]

⎛ ⎞ ω2 ω ⎜ ⎟ 2 ⎜ ⎟ N 2N ω k 2 L L L C + C L ω jk k j k jk k j jk k j ⎟ K jj (ω ) = −ω 2 ⎜ M jj + ∑ +∑ ⎜ ⎟ mk iωk m k k =1 1 − ω k =1 1 − ω ⎜ ⎟ ωk ωk ⎜ ⎟ ⎝ ⎠

ω ⎛ ⎞ ⎜ ⎟ N 2 ω C C ⎜ k jk k j ⎟ + i ω ⎜ C jj + ∑ ⎟ + K jj k =1 1 − ω i ω k m k ⎟ ⎜ ⎜ ⎟ ωk ⎝ ⎠

[7.18]

The sums, extended to 2N terms, represent the mode superposition, but with “doubled” normalization factors m k , as mentioned according to relation [7.8].

7.2.3. Modal effective parameters and dynamic amplifications The comparison of equations [4.51] to the following equations shows that, just as for the normal modes, the various contributions of each complex mode can be put in the form of a product of a dynamic amplification by a matrix of the effective parameters. However, the presence of damping increases the number of these products. Concerning the modal effective parameters, they are now complex valued, in conjugate pairs, and detailed as follows: – for dynamic flexibilities, there is no change: only the effective flexibilities appear, similar to those of normal modes [5.5]:

Complex Modes

Φi k Φki ~ G ii,k = ωk 2 m k

227

[7.19]

– for dynamic transmissibilities, as well as effective transmissibilities, similar to those of normal modes [5.6], transmissibilities related to damping appear along with the mass related transmissibilities: Φi k C k j ~ Tij' ,k = iωk m k

Φi k L k j ~ Tij ,k = mk

[7.20]

– for the dynamic stiffnesses or masses, in addition to the effective masses, similar to those of normal modes [5.7], the masses related to damping which could be called effective damping, appear:

L jk C k j + C jk L k j ~ M 'jj ,k = iωk m k

L jk L k j ~ M jj ,k = mk

C jk Ck j ~ C jj ,k = iωk m k

[7.21]

With regards to the dynamic amplifications, they are now first order terms, with the following description: – a dynamic amplification factor, similar to that of normal modes [4.49]: H k (ω ) =

1 1−

[7.22]

ω ωk

– a dynamic transmissibility factor similar to that of normal modes [4.60]:

Tk (ω ) = 1 +

⎛ ω ⎜ ⎜ωk ⎝

⎞ ⎟ ⎟ ⎠

2

ω 1− ωk

⎛ ω = 1+ ⎜ ⎜ωk ⎝

2

⎞ ⎟ H (ω ) k ⎟ ⎠

[7.23]

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By regrouping the modes into pairs, i.e. by summing up the contribution of the ~ ~ modes k + and k − having the effective parameters X + and X − , we again find an expression similar to that obtained for the normal modes. For example, for the dynamic flexibilities:

G ii (ω ) =

=

N

1

∑

k =1 1 −

N

∑

k =1

ω

~ G ii+,k +

ω k+

~ G ii−,k

1

ω

1−

[7.24]

ω k−

~ ~ ⎛ G ii+,k ω k+ + G ii−,k ω k− ⎜~+ ~− ⎜ G ii ,k + G ii ,k − ω ω k+ ω k− ⎜ 0 ω ⎝ + i 2ζ k

1 ⎛ ⎜ ω 1− ⎜ ⎜ ω k0 ⎝

⎞ ⎟ ⎟ ⎟ ⎠

2

⎞ ⎟ ⎟ ⎟ ⎠

ω k0

The dynamic amplification factor of the normal modes is again present; it multiplies a constant real term completed by an imaginary term in ω, which vanishes in the case of normal modes. In the case of mode truncation, the approach is the same as for the normal modes (section 5.2.3) by retaining the first n pairs of modes. Relations [5.41] to [5.43] including the static terms become:

G ii (ω ) ≈ G ii +

Tij (ω ) ≈ Ψij +

∑ (H k (ω ) − 1)G ii,k 2n

[7.25]

k =1

∑ (Tk (ω ) − 1)Tij ,k + ∑ (H k (ω ) − 1)T' ij ,k 2n

2n

~

k =1

M jj (ω ) ≈ M jj +

+

~

~

[7.26]

k =1

∑ (Tk (ω ) − 1)M jj ,k + ∑ (H k (ω ) − 1)M ' jj ,k 2n

2n

~

k =1

~

k =1

2n ~ 1 ⎛⎜ C jj + ∑ H k (ω ) − 1 C jj ,k ⎜ iω ⎝ k =1

(

)

⎞ K jj ⎟+ ⎟ −ω 2 ⎠

[7.27]

Complex Modes

229

Relations [5.35] to [5.37] including the residual terms become:

G ii (ω ) ≈

Tij (ω ) ≈

2n

~

∑ H k (ω ) G ii,k + G ii,res

[7.28]

k =1

2n

2n

~

~

∑ Tk (ω ) Tij ,k + ∑ H k (ω ) T' ij ,k +Tij ,res

k =1

M jj (ω ) ≈

[7.29]

k =1

n

n

~

~

∑ Tk (ω ) M jj ,k + ∑ H k (ω ) M ' jj ,k +M jj ,res

k =1

+

k =1

⎞ K jj ~ 1 ⎛⎜ n H k (ω ) C jj ,k + C jj ,res ⎟ + ∑ ⎟ −ω 2 i ω ⎜⎝ k =1 ⎠

[7.30]

with the equivalent of relations [5.38] to [5.40]:

G ii ,res = G ii −

Tij ,res = Ψij −

2n

~

∑ G ii,k

[7.31]

k =1

∑ (Tij ,k + T' ij ,k ) 2n

~

~

[7.32]

k =1

M jj ,res = M jj −

∑ (M jj ,k + M ' jj ,k ) 2n

~

~

k =1

C jj ,res = C jj −

2n

~

∑ C jj ,k

[7.33]

k =1

7.2.4. Simple example With the example of the 2-DOF system in section 7.2.1 which provides complex modes [7.10], we find the following participation factors, to be compared to those of normal modes [4.36]: ⎡+ 3.1542 ± 0.1247 i ⎤ L kj = m ⎢ ⎥ ⎣ − 1.1691 ∓ 1.1402 i ⎦

C kj = 0

[7.34]

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Regarding the condensed matrices, M jj = 2 m is the same as in normal modes and both C jj and K jj are zero. The modal effective parameters are the following, to be compared to those of normal modes [5.11]: ~ 1 ⎡+ 0.2380 ∓ 0.2070 i G ii±,1 = ⎢ k ⎣+ 0.5798 ∓ 0.2467 i ~ 1 ⎡+ 0.2620 ∓ 0.0256 i G ii±,2 = ⎢ k ⎣ − 0.0798 ± 0.0937 i

⎡+ 0.3332 ∓ 0.0437 i ⎤ ~ Tij±,1 = ⎢ ⎥ ⎣ + 0.6601 ± 0.1223 i ⎦ ~'± ~'± Tij ,1 = Tij ,2 = 0

+ 0.5798 ∓ 0.2467 i ⎤ + 1.2536 ∓ 0.1116 i ⎥⎦

− 0.0798 ± 0.0937 i ⎤ − 0.0036 ∓ 0.0575 i ⎥⎦

⎡+ 0.1668 ± 0.3200 i ⎤ ~ Tij±,2 = ⎢ ⎥ ⎣ − 0.1601 ∓ 0.0534 i ⎦

~ ~ M ±jj ,1 = m [ + 0.9933 ± 0.0787 i ] M ±jj ,2 = m [ + 0.0067 ± 0.2666 i ] ~ ~ ~ '± ~ '± M jj ,1 = M jj ,2 = 0 C ±jj ,1 = C ±jj ,1 = 0

[7.35]

The corresponding FRF are plotted in Figure 7.2. The comparison with the normal modes shows a different behavior, which is more or less pronounced, depending on the FRF.

Complex Modes π

π

0

0

−π

231

−π

2

2

10

10

0

0

10

10

−2

10

−2

−1

10

0

10

1

10

10

−1

10

G11 (ω )

0

10

1

10

T10 (ω )

π

π

0

0

−π

−π

5

10

2

0

10

10

0

10

−5

10

−2

−1

10

0

10

1

10

10

−1

10

G12 (ω )

1

10

T20 (ω )

π

π

0

0

−π

−π

2

2

10

10

0

0

10

10

−2

10

0

10

−2

−1

10

0

10

G 22 (ω )

1

10

10

−1

10

0

10

M 00 (ω )

Figure 7.2. FRF relating to the 2 internal DOF of section 7.2.1 — complex modes, · · · · normal modes

1

10

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7.3. Gyroscopic effects

7.3.1. Introduction The presence of rotating parts in the structure, e.g. a shaft on its bearings or the wheel of a car, introduces additional terms in the equations of motion. These terms have various effects. We are most particularly interested here in the gyroscopic effects of a rigid body of inertia J about its axis of rotation, which, on the other two rotation DOF θ, is expressed by a matrix proportional to the velocities, just as for the viscous damping, and proportional to the rotational velocity Ω (see references such as [GEN 95]): ⎡0 R θθ (ω ) = Ω ⎢ ⎣J

− J⎤ 0 ⎥⎦

[7.36]

Assuming the presence of various rotating parts at various rotational velocities, but constant in time, the equation of motion of the system without external excitation, without damping, is written: !! i + R ii u! i + K ii u i = 0 i M ii u

[7.37]

Matrix R ii , resulting from the assembly of matrices [7.36] of all rotating parts on all internal DOF i, presents the particularity of being anti-symmetric, which is the first time this has occurred in this reference. That means that the reciprocity principle is no longer applicable. The conventions used for the matrices so far have enabled a coherent use of subscripts with the principle that X ji = ( X ij ) T and X ii is symmetric, which contradicts the anti-symmetry of R ii . This difficulty can be overcome in this particular context by precluding the permutation of the subscripts for the transpose and by assuming that two identical subscripts do not necessarily mean a symmetric matrix. Under this condition, we can preserve these notations in order to retain their advantages in terms of the following development. Equation [7.37] is very similar to [7.1] and it will be solved in a similar way, this time by taking advantage of the anti-symmetry on the velocities. For that, it will have to be rewritten in the frequency domain:

Complex Modes

⎛ ⎡ 0 ii − M ii ⎤ ⎡M ii 0 ii ⎤ ⎞ ⎡i ω u i ⎤ ⎡0 i ⎤ ⎜i ω ⎢ ⎟ + = ⎜ R ii ⎥⎦ ⎢⎣ 0 ii K ii ⎥⎦ ⎟⎠ ⎢⎣ u i ⎥⎦ ⎢⎣0 i ⎥⎦ ⎝ ⎣M ii --------------- ---------------A (i +i )(i +i ) B (i +i )(i +i )

233

[7.38]

with A (i +i )(i +i ) anti-symmetric and B (i +i )(i +i ) symmetric. Again, the 2N solutions come by conjugate pairs: – 2N conjugate complex eigenvalues ±i ω k from which 2N circular natural frequencies ω k± = ±ω k are deduced, which are now real because the system is conservative; ±

– 2N conjugate eigenvectors with 2N components Φ(i +i ) k representing 2N eigenvectors associated with ω k± with the same expression as [7.6] and the same significance. Eigenvectors Φ(i +i ) k also verify the orthogonality properties in relation to the matrices A (i +i )(i +i ) and B (i +i )(i +i ) that result from the symmetry or from the antisymmetry of the latter and which, for set of modes Φ(i +i ) k from eigenvectors Φ(i +i ) k , are written: Φ*k (i +i ) A (i +i )(i +i ) Φ(i +i ) k = a kk diagonal matrix of a k

[7.39]

Φ*k (i +i ) B (i +i )(i +i ) Φ(i +i ) k = b kk diagonal matrix of bk = −i ω k a k

* meaning conjugate. The eigenvectors are thus to be associated not with themselves but with their conjugates. As for the preceding cases, these orthogonality properties will make mode superposition possible. By developing A (i +i )(i +i ) and B (i +i )(i +i ) ,

the diagonal terms a k and bk verify: ak iωk

= Φ*ki M ii Φi k +

Φ*ki K ii Φi k

ωk 2

= mk

bk = ω k 2 Φ*ki M ii Φi k + Φ*ki K ii Φi k = ω k 2 m k

[7.40]

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Structural Dynamics in Industry

Relations [7.8] can be seen again, but with Φki replaced by Φ*ki . The transposed vector should also be conjugate because of the anti-symmetry of A (i +i )(i +i ) . The generalized masses m k , just as with the generalized stiffnesses ω k 2 m k are now real. 7.3.2. Mode superposition The same approach as that in section 7.2.2 can be applied for mode superposition, with, however, the precautions imposed by the non-reciprocity. The equation to be solved is that of [7.11] with the terms C replaced by the terms R. The new base will be made up of complex eigenvectors Φ(i + j ) k completed by junction modes Ψ(i + j ) j (independent of the gyroscopic effects, so the transpose of Ψij is Ψ ji ), hence transformation [7.12]. This transformation applied to matrices X

intervening in the equation now implies the products: ⎡Φ*ki ⎢ ⎢⎣Ψ ji

0 kj ⎤ ⎡ X ii ⎥⎢ I jj ⎥⎦ ⎣ X ji

X ij ⎤ ⎡Φik X jj ⎥⎦ ⎢⎣0 jk

Ψij ⎤ I jj ⎥⎦

[7.41]

which gives the same type of result as equation [7.13]: ⎡M kk ⎢L ⎣ jk

!! k ⎤ ⎡ R kk L kj ⎤ ⎡q + ⎥ ⎢ !! j ⎥ ⎢R jk M jj ⎦ ⎣u ⎦ ⎣

R kj ⎤ ⎡q! k ⎤ ⎡K kk + R jj ⎥⎦ ⎢⎣u! j ⎥⎦ ⎢⎣ 0 jk

0 kj ⎤ ⎡q k ⎤ ⎡ Φ*ki Fi ⎤ =⎢ ⎥ [7.42] K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢Ψ ji Fi + F j ⎥ ⎦ ⎣

with: – M kk , R kk , K kk : non-diagonal modal matrices; – L jk = (Ψ ji M ii + M ji ) Φik : matrix of mass participation factors with Φik ; – L kj = Φ*ki (M ii Ψij + M ij ) : matrix of mass participation factors with Φ*ki , a conjugate transpose of the preceding one; – R jk = (Ψ ji R ii + R ji ) Φik : matrix of gyroscopic effect participation factors with Φik ;

Complex Modes

235

– R kj = Φ*ki (R ii Ψij + R ij ) : matrix of gyroscopic effect participation factors with Φ*ki , an opposite conjugate transpose matrix of the preceding matrix; – M jj , R jj , K jj : condensed junction matrices. The solution is done in the same way as in section 7.2.2, for results similar to relations [7.16] to [7.18]. The Φki are replaced by Φ*ki and it is necessary to distinguish the transmissibilities in forces T ji (ω ) from the transmissibilities in displacements Tij (ω ) , which are now conjugate transposes of each other. Without detailing all these results, which will be discussed again in section 7.4 in a more general way, we obtain for example: Φi k Φ*ki

2N

1

k =1 1 −

ω ω 2m k k ωk

G ii (ω ) = ∑

[7.43]

The contribution of each complex mode is still in the form of a dynamic amplification factor by a matrix of complex effective parameters (it should be noted about relation [7.43] that the diagonal terms are real). By regrouping the modes by pairs, because the 2N circular natural frequencies are real and opposite ( ±ω k , ω k assumed positive) we find:

G ii (ω ) =

N

∑

k =1

ℜ(Φi k Φ*ki ) + i

1

⎛ ω 1− ⎜ ⎜ωk ⎝

⎞ ⎟ ⎟ ⎠

2

ω ℑ(Φi k Φ*ki ) ωk

ω k 2 mk

[7.44]

The dynamic amplification factor of normal modes without damping and a term of the same form as in [7.24] appears. In practice, the damping intervenes as for normal modes and it is better to use the formulation which takes it into account without added complication. This is the subject of the following section.

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7.4. A more general case 7.4.1. Introduction For a dissipative system with gyroscopic effects (Ω constant), the matrix affecting the velocities has a symmetric contribution coming from the viscous damping and an anti-symmetric contribution coming from the gyroscopic effects. It is thus non-symmetric, and this adds an element of complexity to the equations of motion. Other effects can also be expressed by additional terms in the equation of motion, for example: – the presence of a structural damping (see sections 2.4.2 and 4.3.3) which, in the frequency domain, makes the stiffness matrix complex while preserving its symmetry; – the presence of a rotating part which excites the structure with a circular frequency ω directly related to its rotational velocity Ω, thus implying a relation of the type Ω = α ω in the frequency domain. Matrix R ii of equation [7.37] is then of the form α ω J ii , with J ii anti-symmetric matrix representing the inertias in !! i = i ω u! i , the rotation, which no longer affects the velocities but the accelerations u matrix −i α J ii being added to matrix M ii . It should be noted that the resulting complex matrix M ii − i α J ii is Hermitian, i.e. its transpose is equal to its conjugate; – the presence of the anti-symmetric stiffness terms due to phenomena that can be found for example with the bearings for rotating parts [MEI 80], making the stiffness matrix non-symmetric; – the presence of a fluid-structure coupling whose formulation can take various forms depending on the choice of the unknowns [MOR 92, ROY 02b]. All these effects generate real or non-real matrices, symmetric or non-symmetric matrices. In general, we can write the equation of motion in relation to the DOF g = i + j in the frequency domain:

(− ω

2

)

M gg + i ω C gg + K gg u g = F g

[7.45]

M, C and K being the matrices of pseudo-mass, pseudo-damping and pseudostiffness, including all the effects presented, and therefore may be complex and/or non-symmetric.

Complex Modes

237

The solution of equation [7.45] will combine the characteristics discussed in the particular cases in sections 7.2 and 7.3: complex circular frequencies, nonreciprocity as well as new features [GIR 01a]. 7.4.2. Complex modes Complex modes are the solutions of the equations of motion without excitation:

(− ω

2

)

M ii + i ω C ii + K ii u i = 0 i

[7.46]

In order to solve this system, we can rewrite it as: ⎛ ⎡ 0 ii − I ii ⎤ ⎡ I ii 0 ii ⎤ ⎞ ⎡i ω u i ⎤ ⎡0 i ⎤ ⎜i ω ⎢ ⎥+⎢ ⎥ ⎟⎟ ⎢ ⎥=⎢ ⎥ ⎜ ⎝ ⎣M ii C ii ⎦ ⎣0 ii K ii ⎦ ⎠ ⎣ u i ⎦ ⎣0 i ⎦ ---------------- ------------A (i +i )(i +i ) B (i +i )(i +i )

[7.47]

with the non-symmetric complex matrices A (i +i )(i +i ) and B (i +i )(i +i ) in general. However, matrix B (i +i )(i +i ) depends only on the stiffness and it remains symmetric for a dissipative system with gyroscopic effects only. The 2N solutions are now not necessarily conjugate: – 2N complex eigenvalues i ω k which are not necessarily conjugate from which

the 2N complex circular natural frequencies ω k are deduced. The latter combine frequency and damping information as in section 7.2.1; – 2N complex eigenvectors with 2N components Φ(i +i ) k representing 2N

eigenvectors associated with ω k , verifying:

(i ω k A (i+i)(i +i) + B (i +i)(i +i) )Φ(i +i)k = 0 (i +i)k

⎡i ω k Φi k ⎤ Φ(i +i)k = ⎢ ⎥ [7.48] ⎣⎢ Φi k ⎦⎥

However, these eigenvectors alone are not enough to allow mode superposition because they are not orthogonal to each other, just like the gyroscopic case in section 7.3.2 where the conjugate transposed vectors must be used to obtain relations

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Structural Dynamics in Industry

[7.39]. In general, it is necessary to complete these eigenvectors, which will be described as “right eigenvectors”, by the “left eigenvectors” marked Y in order to distinguish them from Φ, verifying:

(

)

Yk (i +i ) i ω k A (i +i )(i +i ) + B (i +i )(i +i ) = 0 k (i +i )

[

Yk (i +i ) = − i ω k Yki M ii

Yki

]

[7.49]

It should be noted that by transposing them, these left eigenvectors are the right eigenvectors of transposed matrices A and B:

(i ω

k

)

A T(i +i )(i +i ) + B T(i +i )(i +i ) Y(i +i ) k = 0 (i +i ) k

[7.50]

In general, they do not have a direct relation with the right eigenvectors. However, simple relations exist in certain particular cases such as that in section 7.2 where they are transposes of each other, or that of section 7.3 where they are conjugate transposes. Table 7.1 presents the various properties of the complex modes in various cases of damping and gyroscopic effects, with: – the minimum size N or 2N of the problem to be solved; – the conjugate or purely imaginary eigenvalues i ω k (hence real circular frequencies ω k ); – real or conjugate right eigenvectors Φi k ; – the left eigenvectors Yki transpose or conjugate transposes of the right eigenvectors. We should note the following points: – the minimum size is N when the equation of motion does not have a term of velocity; – the modes are conjugate in the absence of the complex matrix; – the left eigenvectors are transposes of the right vectors when all the matrices are symmetric. They are conjugate transposes of the right vectors in the absence of damping, as well as for Ω = α ω without structural damping (equation [7.45] can be transformed into an equation with Hermitian A and real symmetric B).

Complex Modes

239

Ω Ω=0

viscous structural damping N

=0

=0

N =0

≠0

≠0

2N

± ω k real

N

ω k real

Φi k real

Φi−k = (Φi+k ) *

–

Yki = (Φi k ) T

Yki = (Φi k ) T *

Yki = (Φi k ) T *

ω k complex

±ωk

2N

ωk

N

–

–

Yki = (Φi k ) T

–

–

2N iω k− = (iω k+ ) *

2N iω k− = (iω k+ ) *

2N iω k− = (iω k+ ) *

Φi−k = (Φi+k ) *

Φi−k = (Φi+k ) *

–

Yki = (Φi k ) T

–

Yki = (Φi k ) T *

=0

≠0

ω k real

Ω=αω

–

2N

≠0

Ω = constant

–

2N

–

2N

–

–

–

–

Yki = (Φi k ) T

–

–

Table 7.1. Properties of complex modes

Eigenvectors Φ(i +i ) k and Yk (i +i ) verify the orthogonality properties with respect to matrices A (i +i )(i +i ) and B (i +i )(i +i ) which result from their definition and which are written for the entire set of modes:

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Yk (i +i ) A (i +i )(i +i ) Φ(i +i ) k = a kk diagonal matrix of a k Yk (i +i ) B (i +i )(i +i ) Φ(i +i ) k = b kk diagonal matrix of bk = −i ω k a k

[7.51]

As for the preceding cases, these orthogonality properties will enable the use of mode superposition. By developing A (i +i )(i +i ) and B (i +i )(i +i ) we obtain: ak iωk

= Yki M ii Φi k +

Yki K ii Φi k

ωk 2

= mk

[7.52]

bk = ω k 2 Yki M ii Φi k + Yki K ii Φi k = ω k 2 m k

Relation [7.52] is the generalization of relations [7.8] and [7.40]. The generalized masses m k and the generalized stiffnesses ω k 2 m k are generally complex.

7.4.3. Mode superposition The same mode superposition approach as that of sections 7.2.2 and 7.3.2 gives the following results. Equation [7.45] to be solved is to be projected on the base made up by the complex eigenvectors Φ(i + j ) k completed by the junction modes Ψ(i + j ) j (depending only on the pseudo-stiffness, according to relation [4.29]), hence the transformation [7.12]:

[

u i + j = Φ(i + j ) k

]

⎡q k ⎤ Ψ(i + j ) j ⎢ ⎥ ⎣u j ⎦

⇔

⎡ u i ⎤ ⎡Φik ⎢u ⎥ = ⎢0 ⎣ j ⎦ ⎣ jk

Ψij ⎤ ⎡q k ⎤ I jj ⎥⎦ ⎢⎣u j ⎥⎦

[7.53]

This transformation applied to matrices X intervening in the equation implies the products: ⎡Yki ⎢Ψ ⎣ ji

0 kj ⎤ ⎡ X ii I jj ⎥⎦ ⎢⎣ X ji

where Ψ ji = −K

ji

X ij ⎤ ⎡Φik X jj ⎥⎦ ⎢⎣0 jk

Ψij ⎤ I jj ⎥⎦

K ii−1 is the transpose of Ψij only if matrix K is symmetric.

[7.54]

Complex Modes

241

This transformation applied to equation [7.45] gives: ⎡M kk ⎢L ⎣ jk

!! k ⎤ ⎡C kk L kj ⎤ ⎡q + ⎥ ⎢ !! j ⎥ ⎢C jk M jj ⎦ ⎣u ⎦ ⎣

C kj ⎤ ⎡q! k ⎤ ⎡K kk + C jj ⎥⎦ ⎢⎣u! j ⎥⎦ ⎢⎣ 0 jk

0 kj ⎤ ⎡q k ⎤ ⎡ Yki Fi ⎤ [7.55] = K jj ⎥⎦ ⎢⎣u j ⎥⎦ ⎢⎣Ψ ji Fi + F j ⎥⎦

with: – M kk , C kk , K kk : non-diagonal modal matrices; – L jk = (Ψ ji M ii + M ji ) Φik : matrix of pseudo-mass participation factors with Φik ;

– L kj = Yki (M ii Ψij + M ij ) : matrix of pseudo-mass participation factors with Yki ;

– C jk = (Ψ ji Cii + C ji ) Φik : matrix of pseudo-damping participation factors with Φik ;

– C kj = Yki (C ii Ψij + C ij ) : matrix of pseudo-damping participation factors with Yki ;

– M jj , C jj , K jj : condensed matrices. The solution is done in the same way as in section 7.2.2, with Φki replaced by Yki and by taking the precautions due to the non-reciprocity. We obtain the

generalization of relations [7.16] to [7.18]: Φi k Yki

2N

1

k =1 1 −

ω ω 2m k k ωk

G ii (ω ) = ∑

(if rigid modes r : ∑ r

ω2 Tij (ω ) = Ψij +

2N

∑

ω k 2 Φi k L k j

k =1 1 −

ω ωk

mk

+

2N

∑

ω ωk

k =1 1 −

Φi r Yri −ω 2 m r

Φi k C k j

ω iωk m k ωk

)

[7.56]

[7.57]

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ω2 T ji (ω ) = Ψ ji +

2N

∑

ω k 2 L j k Yki

k =1 1 −

ω ωk

mk

+

2N

∑

ω ωk

k =1 1 −

C j k Yki

ω iωk m k ωk

[7.58]

⎛ ω2 ω ⎜ 2 ⎜ 2N ω k L jk L k j 2N ω k L jk C k j + C jk L k j K jj (ω ) = −ω 2 ⎜ M jj + ∑ +∑ ⎜ mk iωk m k k =1 1 − ω k =1 1 − ω ⎜ ω ω k k ⎜ ⎝

ω ⎛ ⎜ N 2 ω C jk Ck j ⎜ k + i ω ⎜ C jj + ∑ k =1 1 − ω i ω k m k ⎜ ⎜ ωk ⎝

⎞ ⎟ ⎟ ⎟+K ⎟ ⎟ ⎠

jj

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

[7.59]

7.4.4. Modal effective parameters and dynamic amplifications The interpretation of results [7.56] to [7.59] is the same as that of section 7.2.3 with results [7.16] to [7.18]. Various contributions of each complex mode can be put in the form of a product of a dynamic amplification by a matrix of the effective parameters. With regards to the modal effective parameters: – for dynamic flexibilities: Φi k Yki ~ G ii,k = ωk 2 m k

[7.60]

– for dynamic transmissibilities in displacements:

Φi k L k j ~ Tij ,k = mk

Φi k C k j ~ Tij' ,k = iωk m k

[7.61]

Complex Modes

243

– for dynamic transmissibilities in forces:

L j k Yki ~ T ji ,k = mk

C j k Yki ~ T 'ji ,k = iωk m k

[7.62]

– for dynamic masses or stiffnesses: L jk C k j + C jk L k j ~ M 'jj ,k = iωk m k

L jk L k j ~ M jj ,k = mk

C jk Ck j ~ C jj ,k = iωk m k

[7.63]

Concerning the dynamic amplifications, they are the same as those in section 7.2.3, but with 2N complex circular natural frequencies ω k in pairs without particular properties in general: – a dynamic amplification factor: H k (ω ) =

1

[7.64]

ω 1− ωk

– a dynamic transmissibility factor:

Tk (ω ) = 1 +

⎛ ω ⎜ ⎜ωk ⎝

1−

⎞ ⎟ ⎟ ⎠

2

ω ωk

⎛ ω = 1+ ⎜ ⎜ωk ⎝

2

⎞ ⎟ H (ω ) k ⎟ ⎠

[7.65]

In the presence of modal truncation, we will retain only the first n pairs of modes. The relations including the static terms are written:

G ii (ω ) ≈ G ii +

∑ (H k (ω ) − 1)G ii,k 2n

k =1

~

[7.66]

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Tij (ω ) ≈ Ψij +

∑ (Tk (ω ) − 1)Tij ,k + ∑ (H k (ω ) − 1)T' ij ,k 2n

k =1

T ji (ω ) ≈ Ψ ji +

~

[7.67]

k =1

∑ (Tk (ω ) − 1)T ji,k + ∑ (H k (ω ) − 1)T' ji,k 2n

2n

~

k =1

M jj (ω ) ≈ M jj + +

2n

~

~

[7.68]

k =1

∑ (Tk (ω ) − 1)M jj ,k + ∑ (H k (ω ) − 1)M ' jj ,k 2n

2n

~

k =1

~

k =1

2n ~ 1 ⎛⎜ C jj + ∑ H k (ω ) − 1 C jj ,k i ω ⎜⎝ k =1

(

)

⎞ K jj ⎟+ ⎟ −ω 2 ⎠

[7.69]

The relations including the residual terms are written: G ii (ω ) ≈

Tij (ω ) ≈

T ji (ω ) ≈

2n

~

∑ H k (ω ) G ii,k + G ii,res

[7.70]

k =1

2n

2n

~

~

∑ Tk (ω ) Tij ,k + ∑ H k (ω ) T' ij ,k +Tij ,res

k =1

2n

2n

~

~

∑ Tk (ω ) T ji,k + ∑ H k (ω ) T' ji,k +T ji,res

k =1

M jj (ω ) ≈

[7.71]

k =1

[7.72]

k =1

n

~

n

~

∑ Tk (ω ) M jj ,k + ∑ H k (ω ) M ' jj ,k +M jj ,res

k =1

+

k =1

⎞ K jj ~ 1 ⎛⎜ n H k (ω ) C jj ,k + C jj ,res ⎟ + ∑ ⎟ −ω 2 i ω ⎜⎝ k =1 ⎠

[7.73]

with:

G ii ,res = G ii −

2n

~

∑ G ii,k

k =1

[7.74]

Complex Modes

Tij ,res = Ψij −

∑ (Tij ,k + T' ij ,k ) 2n

~

~

245

[7.75]

k =1

T ji ,res = Ψ ji −

∑ (T ji,k + T' ji,k ) 2n

~

~

[7.76]

k =1

M jj ,res = M jj −

∑ (M jj ,k + M ' jj ,k ) 2n

~

~

k =1

C jj ,res = C jj −

2n

~

∑ C jj ,k

[7.77]

k =1

7.5. Applications 7.5.1. Simple example Figure 7.3 shows a simple example including viscous damping and gyroscopic effects (Ω constant). It includes 4 DOF on each node: u y , θ z , u z , θ y , along with the following properties: – stiffnesses of a massless beam in pure bending of length L =1 and of stiffness (E I) = 1 in the plane xy and (E I) = 2 in the plane xz, in order to distinguish the two planes; – lumped mass m = 1 and inertias I y = I z = 1 , on internal node 2; – viscous damping similar to the stiffness but non-proportional (see equation [7.78]); – possible gyroscopic effect on the internal node 2 due to the inertia J = 2 with a rotational velocity Ω = 2π 0.1.

y

z

Rigid disk

uy

Ω 1

m Ix = J Iy = Iz

θz

L, (E I)y, (E I)z

x

2 uz

θy

Figure 7.3. Simple example with damping and gyroscopic effects

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Structural Dynamics in Industry

Matrices M, K and C of equation [7.45] are then written: ⎡1 ⎢0 M ii = ⎢ ⎢0 ⎢ ⎣0

0 1 0 0

0 0 1 0

0⎤ 0⎥⎥ 0⎥ ⎥ 1⎦

⎡ 12 − 6 0 0 ⎤ ⎢− 6 4 0 0 ⎥⎥ K ii = ⎢ ⎢0 0 24 12⎥ ⎢ ⎥ 0 12 8 ⎦ ⎣0

− 0.25 0 0 ⎡ 0.5 ⎤ ⎢− 0.25 ⎥ π 0 . 5 0 ( − 2 0 . 2 ) ⎥ C ii = ⎢ ⎢ 0 0 0.5 + 0.25 ⎥ ⎢ ⎥ (2π 0.2) + 0.25 0.5 ⎦ ⎣ 0

[7.78]

The first pair of complex modes with and without gyroscopic effects are: – without gyroscopic effects:

ω1± = ± 0.8767 + 0.1462 i

m1± = 1.0

⎡ + 0.3339 ± 0.0330 i ⎤ ⎢+ 0.6267 ± 0.0490 i ⎥ ⎥ Φi±1 = (Y1±i ) T = ⎢ ⎢ + 0.0000 + 0.0000 i ⎥ ⎥ ⎢ ⎣ + 0.0000 + 0.0000 i ⎦

[7.79]

– with gyroscopic effects (please note the same components for Φ and Y excepting the signs, the damping and gyroscopic terms being uncoupled):

ω1± = ± 0.6461 + 0.0895 i ⎡ − 0.0177 ± 0.2208 i ⎤ ⎢ − 0.0303 ± 0.4266 i ⎥ ⎥ Φi±1 = ⎢ ⎢− 0.1212 ∓ 0.0179 i ⎥ ⎢ ⎥ ⎣ + 0.2383 ± 0.0341 i ⎦

m1± = 1.0

(Yi±1 ) T

⎡− 0.0177 ± 0.2208 i ⎤ ⎢ − 0.0303 ± 0.4266 i ⎥ ⎥ =⎢ ⎢+ 0.1212 ± 0.0179 i ⎥ ⎢ ⎥ ⎣ − 0.2383 ∓ 0.0341 i ⎦

[7.80]

Complex Modes

247

The modal effective parameters relative to the DOF θz of nodes 1 and 2 are: – without gyroscopic effects: # ± = +0.4926 ∓ 0.0867 i G 22.1 ± # T = T# ± = +0.5979 ± 0.0985i 21.1

± T# '±21.1 = T# '12.1 =0 ± ± # # M '11.1 = C11.1 = 0

12.1

# ± = +0.9959 ± 0.1575i M 11.1

– with gyroscopic effects: # ± = +0.4262 ∓ 0.0574i G 22.1 T# ± = T# ± = +0.2748 ± 0.0400i

± T# '±21.1 = T# '12.1 = +0.1978 ± 0.0149i

± M11.1 = +0.4169 ± 0.0621i

± M '11.1 = +0.6004 ± 0.0471i

21.1

12.1

± C11.1 = −0.0205 ± 0.1388i

The corresponding FRF are plotted in Figure 7.4. The comparison with and without gyroscopic effects shows their significant influence for the considered case. π

π

0

0

−π

−π

2

10

2

0

10

10

0

10

−2

10

−2

−1

10

0

10

1

10

10

−1

0

10

G 22 (ω )

10

M 11 (ω )

π 0 −π 2

10

0

10

−2

10

−1

10

0

10

1

10

T21 (ω ) = T12 (ω ) Figure 7.4. FRF of the system in Figure 7.3 on the DOF θz — with gyroscopic effects, · · · · without gyroscopic effects

1

10

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Structural Dynamics in Industry

7.5.2. Industrial case A typical industrial application using complex modes is the use of damping mechanisms in order to limit the response levels in lightly damped structures. The presence of strong localized damping favors the use of complex modes because the Basile hypothesis with normal modes (section 4.3.3) is often no longer valid. Figure 7.5 illustrates this situation with an example of results from the model in Figure 5.12. In order to attenuate the structural responses, several damping mechanisms represented by 1-DOF systems were placed in the most favorable places with respect to the undamped main modes responsible for these levels. A first calculation was made using normal modes with the Basile hypothesis, i.e. by neglecting the effect of the non-diagonal modal damping terms. A second calculation used the complex modes by taking into account the coupled damping. The comparison shows that the normal modes highly underestimate the levels of certain peaks and in particular the first one. This first peak is actually generated by two relatively close modes and the Basile hypothesis reduces peak height by a factor greater than 2. In general, the coupling of the modes by the damping will favor the energy coupling and will thus increase the levels.

Figure 7.5. FRF of the model in Figure 5.12 with localized damping. — complex modes · · · · normal modes using Basile hypothesis

Chapter 8

Modal Synthesis

8.1. Introduction Before speaking about modal synthesis, a particular substructuring technique which logically follows from the preceding chapters on modes, let us mention some generalities on substructuring itself. The design and development of complex structures are often carried out by different teams, for various reasons ranging from the geographical distribution necessary in certain areas, to the requirements involving costs and deadlines. This modular approach requires the use of substructuring techniques where the structure under consideration is first broken down into substructures or components. Analysis is then made at the level of each substructure where the calculation effort is less, before being extended to the complete structure by coupling, or synthesis, of the results on the components. It is thus a matter of determining the dynamic behavior of a structure from the characteristics of its substructures. It is also necessary that the structure under consideration lends itself to being decomposed into well defined components. The main criterion is the number of DOF representing the connections in relation to the number of DOF of each substructure. If the proportion is low, the substructuring will be efficient. Otherwise it will be relatively difficult and not very profitable. There is little interest in dividing a square plate with regular meshing into two. On the other hand, a very modular design with statically determinate or slightly indeterminate connections is an ideal situation for substructuring, which can provide substantial gains in computation time and in management of models.

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Structural Dynamics in Industry

These dynamic substructuring techniques were introduced in section 1.6 highlighting the possibility to work at different levels: – at the matrix level, by direct assembly of the mass, stiffness and damping matrices of each substructure, as described in section 3.2.4 with the element matrices we need only to replace the latter with the matrices of the substructures, possibly condensed beforehand. Without condensation, the process is equivalent to the assembly of the elements and, of course, the main disadvantage of this approach is the computation time, which can be prohibitive because of the model size. With a static condensation for example (section 4.2.3), the gain in computation time is dependent on the extent of condensation and the accuracy is dependent on the choice of retained DOF, hence a relative efficiency resulting from offsetting time against accuracy that can be greatly improved by modal synthesis; – at the modal level, by modal synthesis, which is the subject of this chapter, this can actually be considered as a special type of condensation where each component is represented and then reduced by a basis of modes including normal modes as well as static modes. The gain in computation time is then dependent on the number of retained modes and the accuracy is dependent on the quality of the latter. This approach is particularly well adapted to linear analysis at low frequencies where the mode superposition techniques are highly efficient; – at the FRF level, by appropriate manipulation of the frequency response functions, which can be called frequency synthesis. This approach, which is the subject of the following chapter, presents interesting complements in relation to the modal synthesis, but it can be relatively computationally heavy especially in situations where the modal synthesis applies well. The first modal synthesis method was proposed by Hurty in 1965 [HUR 65]. It is based on normal modes with fixed interfaces. In 1968, Craig and Bampton published a similar and numerically equivalent method that remains today one of the most popular and most precise methods used. In the following decade, another class using free interface normal modes was introduced in particular by MacNeal [MAC 71], Rubin [RUB 75] and Craig and Chang [CRA 77]. Even though these two types of methods represent the substantial part of the development efforts, other approaches have been proposed particularly by Benfield and Hruda [BEN 71] with loadedinterface modes, and by Herting [HER 79] with hybrid modes. All these methods make the hypothesis of lightly damped structures by using undamped normal modes, which are therefore real, with coupling only in mass and stiffness. However, damping can be introduced in the coupled system using equivalent modal damping factors. More recent developments regarding the use of complex modes were proposed by Hasselman and Kaplan [HAS 74], Craig and Chung [CRA 82] and Hale [HAL 84]. In spite of their interest for some strongly

Modal Synthesis

251

damped structures, these methods are of limited use in industrial context and they will not be discussed in this chapter. In [CRA 87] there is a general review of the methods published in other references. The comparison of the performances of the modal synthesis with those of other substructuring techniques is discussed in Chapter 9 at the end of the presentation on frequency synthesis. The general step common to all methods will be initially described in section 8.2. In section 8.3 we will then examine the various modes possible for each substructure, the choice of which influences the methods, before presenting the most characteristic methods in section 8.4. Some examples borrowed from industrial practice are the subject of section 8.5. 8.2. General approach 8.2.1. Analysis of substructures Although modal synthesis methods are different in terms of specific formulation, they all follow the same basic procedure that can be divided into three distinct steps: − analysis of each substructure separately; − coupling to obtain the assembled structure; − recovery of the behavior of each substructure. S

i c j

Figure 8.1. DOF of substructure S

Let us consider a substructure identified by exponent S, as illustrated in Figure 8.1. The general set g of its DOF includes the interface DOF meant to be connected to the other substructures and which will be designated in what comes next by connection DOF with the mnemonic subscript c. These connection DOF can be fixed or free according to the choice of the boundary conditions for the substructure. We can thus distinguish the concept of connection (subscript c) used to couple the

252

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substructures, from the junction concept (subscript j) for a substructure before coupling. The equations of motion for substructure S in relation to its general DOF g are written: Sg + C Sgg u Sg + K Sgg u Sg = FgS M Sgg u

[8.1]

The analysis of each substructure S implies the creation of a basis of modes B, so as to represent the physical displacements u by a reduced set of generalized coordinates q, as indicated by relation [8.2]: u Sg = B Sgb q bS

[8.2]

In practice, basis B contains a truncated set of normal modes, completed by static modes related to the connection DOF so as to minimize the truncation effects of the modes. The choice of these modes will be detailed later. This truncation restraining the space where the displacements are expressed will stiffen the complete structure and thus will generate positive errors in frequency. The transformation of relation [8.2] can then be applied to equation [8.1] in order to obtain the equations of motion relating to the generalized coordinates with the corresponding mass and stiffness-condensed matrices: S S S S S M bb q b + C bb q b + K bb q bS = FbS

[8.3]

S FbS = B bg F gS

[8.4]

with

S S M bb = B bg M Sgg B Sgb

S S C bb = B bg C Sgg B Sgb

S S K bb = B bg K Sgg B Sgb

The condensed matrices of equation [8.4] can now be assembled in the following coupling operation.

Modal Synthesis

253

8.2.2. Coupling of substructures The coupling of substructures is the assembly of various components while imposing the compatibility of the displacements and the equilibrium of the forces at the connections. Any number of substructures can be considered. We will limit ourselves here to two in order to simplify the expressions without a loss of generality. Let us consider two substructures S to be coupled, designated by A and B. The equations of motion before coupling in terms of generalized coordinates can be written: b + C bb q b + K bb q b = Fb M bb q

[8.5]

⎡q A ⎤ q b = ⎢ bB ⎥ ⎢⎣q b ⎥⎦

[8.6]

with:

⎡M A M bb = ⎢ bb ⎢⎣ 0

⎡F A ⎤ Fb = ⎢ bB ⎥ ⎢⎣Fb ⎥⎦ 0 ⎤ ⎥ B M bb ⎥⎦

⎡C A C bb = ⎢ bb ⎢⎣ 0

0 ⎤ ⎥ B C bb ⎥⎦

⎡K A K bb = ⎢ bb ⎢⎣ 0

0 ⎤ ⎥ B K bb ⎥⎦

The coupling between A and B imposes two types of relations on the forces and the displacements at the connections: – the compatibility of displacements: when the DOF c of the two substructures are imposed to be identical, we can write: u cA = u cB

[8.7]

which can be generalized in a certain number of linear constraints between displacements, possibly including reference frame changes; – the equilibrium of forces: at these DOF, we can write if FcA and FcB designate the force exerted by B on A and respectively the force exerted by A on B: FcA + FcB = 0 c

[8.8]

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which, again, can be generalized in a certain number of linear constraints between the forces. All these linear constraints can be regrouped into a single matrix relation related to the generalized coordinates qb: C mb q b = 0 m

[8.9]

We saw in section 3.2.5 with physical DOF how linear constraints such as these are taken into account. The extension to generalized coordinates does not change the problem that can always be solved by DOF elimination or introduction. With the elimination strategy (section 3.2.5.2), the introduction of linear constraints [8.9] into the equations before coupling [8.5] is performed by partitioning the generalized DOF b in independent DOF n and dependent DOF m, with b = m + n. Relation [8.9] is then written:

[C mm

⎡q ⎤ C mn ] ⎢ m ⎥ = 0 m ⎣q n ⎦

[8.10]

which gives the transformation that makes it possible to pass from q b to q n by q m elimination: ⎡q ⎤ q b = ⎢ m ⎥ = Tbn q n ⎣q n ⎦

with

1 ⎡− C −mm C mn ⎤ Tbn = ⎢ ⎥ I nn ⎣⎢ ⎦⎥

[8.11]

Transformation [8.11] applied to equation [8.5] provides the equations of motion for the coupled system, with the corresponding physical matrices: n + C nn q n + K nn q n = Fn M nn q

[8.12]

Fn = Tnb Fb

[8.13]

with

(M, C, K ) nn = Tnb (M, C, K ) bb Tbn

Modal Synthesis

255

The alternative approach by DOF introduction (section 3.2.5.3) is carried out using the Lagrange multipliers leading to: ⎡M bb ⎢0 ⎣ mb

b ⎤ ⎡C bb 0 bm ⎤ ⎡ q + ⎥ ⎢ 0 mm ⎦ ⎣λ m ⎥⎦ ⎢⎣0 mb

0 bm ⎤ ⎡ q b ⎤ ⎡ K bb + 0 mm ⎥⎦ ⎢⎣λ m ⎥⎦ ⎢⎣C mb

C bm ⎤ ⎡ q b ⎤ ⎡ Fb ⎤ = [8.14] 0 mm ⎥⎦ ⎢⎣λ m ⎥⎦ ⎢⎣0 m ⎥⎦

The two approaches [8.12] and [8.14] are strictly equivalent. In practice however, formulation [8.14] using Lagrange multipliers, although simpler, requires certain numerical precautions. For this reason, elimination approach [8.12] is more common. The equations of the coupled system [8.12] or [8.14] can be solved directly or by mode superposition in order to obtain the solution in terms of generalized coordinates q n . The physical displacements are now obtained by recovery.

8.2.3. Recovery The recovery is the final step of the modal synthesis and it represents the inverse transformation making it possible to back-transform from the modal coordinates to the physical displacements of each substructure. It is carried out by the same transformations as those used for the preceding steps. The combination of relations [8.2] and [8.11] leads to the transformation:

u g = B gb Tbn q n

⎡u gA ⎤ with u g = ⎢ B ⎥ ⎢⎣u g ⎥⎦

⎡B A ⎤ gb B gb = ⎢ B ⎥ ⎢B ⎥ ⎣ gb ⎦

[8.15]

whereas using Lagrange multipliers, the inverse transformation is performed directly with q b : u g = B gb q b

[8.16]

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In practice, all the physical displacements u g do not have the same interest and therefore transformation [8.15] or [8.16] can be limited to a particular subset or observation DOF o: u o = B ob Tbn q n

or

u o = B ob q b

[8.17]

These recovered displacements can be located in any substructure. They can also be used in order to provide in their turn, by combination, other responses of the substructure, such as relative displacements, internal forces or stresses.

8.3. Choice of mode 8.3.1. Introduction The accuracy of the modal synthesis relies to a great extent on the choice of the mode used in order to represent each substructure according to relation [8.2]. The choice is also governed by the necessity to retain as few modes as possible in order to obtain a significant reduction of the model size. The conflict between accuracy and size may be solved by behavior considerations of each substructure according to the frequency. The idea is to define a set of modes capable of representing this behavior up to a certain frequency f max beyond which a loss in accuracy is acceptable. By limiting the spectral validity of the model, the necessary number of modes can be considerably reduced. The modes are a logical starting point in order to make up the basis B because of their natural association with the frequencies. Moreover, they are simple to compute, they can be experimentally obtained and they satisfy the orthogonality properties, which make numerical implementation easier. Two questions then arise: – how many modes should be retained?; – what are the consequences of eliminating the others? These questions can be answered by analyzing the contribution of each mode to a given FRF X(ω) (of the type G, T or M) of a substructure excited at a connection DOF, as schematized in Figure 8.2.

Modal Synthesis

257

(log) ⏐X⏐

f max

α f max

(log) f

Figure 8.2. Contribution of the modes to a FRF

Modes with natural frequencies less than f max should be retained due to their direct influence on the band under consideration. The same goes for those whose frequencies are only slightly higher than f max because their dynamic amplification at f max is significant. In practice, all the modes up to a cutoff frequency α f max are selected where α is a coefficient between 1.5 and 2 (corresponding dynamic amplification: 9/5 to 4/3, weak enough). All the other modes, whose frequencies are higher, provide a small amplification in the band, and therefore have a quasi-uniform contribution. They can therefore be represented by a unique residual term to be deduced from a static mode related to the excitation DOF of the FRF. By considering all the connection DOF c as excitation DOF for the substructure, the contribution of the upper modes can be taken into account by a set c of static modes. Thus, the basis B will be made up of a truncated set of normal modes completed by a set of static modes globally taking into account the upper modes and thus limiting the truncation errors. The nature of these static modes or normal modes

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depends on the boundary conditions considered for the substructure and it is the subject of the following section.

8.3.2. Boundary conditions The analysis of a substructure begins by the choice of the boundary conditions for the connection DOF c. Although apparently simple, this choice is fundamental because it directly influences the nature of the modes used to represent the substructure and consequently the method itself. In general, each connection DOF c can be for three types, as illustrated in Figure 8.3: – free connection DOF: this DOF is left free for the substructure analysis. Therefore, it is an internal DOF i; – fixed connection DOF: this DOF is constrained for the substructure analysis. Therefore, it is a junction DOF j; – loaded connection DOF: this DOF is left free for the substructure analysis, but loaded in mass and/or in stiffness. It is thus a particular case of internal DOF i.

K a) free

b) fixed

M

c) loaded

Figure 8.3. Boundary conditions for modal synthesis

Each of these types of DOF generates specific normal modes and static modes: – the analysis with fixed connection DOF (c = j) produces normal modes with fixed interface, or clamped modes. These modes are generally without rigid-body modes and have by definition zero components at the connection DOF. The static modes associated with these normal modes are obtained by successively imposing a unit displacement at each connection DOF. These are the junction modes Ψ introduced in section 4.3.1;

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– the analysis with free connection DOF (c = i) produces normal modes with free interface or free modes. These modes include rigid-body modes if the substructure is not constrained elsewhere. The static modes associated with these normal modes are obtained by successively imposing a unit force at each connection DOF. These are the static flexibilities G introduced in section 5.2.2, concerning the summation rules. As already mentioned in section 5.2.2.2, a particular treatment should be done in the presence of the rigid-body modes in order to get to the concept of pseudo-static flexibility; – the analysis with loaded connection DOF is a variant of the preceding analysis, resulting from the idea of representing the adjacent substructure by masses and/or stiffnesses at the connections. It therefore produces free modes which are free but modified by the loading. As a rule, this modification makes the modes more representative of the substructure behavior, and this then improves the truncation errors without resorting to the static modes. However, this advantage is counterbalanced by two major disadvantages. The first is that the substructure analysis now depends on the others to a certain extent and this violates the modularity principle. The second is that the elimination of the static modes no longer guarantees the static behavior and can nevertheless imply a relatively degraded dynamic behavior insofar as the retained modes do not properly represent the high frequencies. Starting from these three basic situations, we can consider the hybrid boundary conditions with certain connection DOF, some of them fixed and others free, or possibly loaded. The normal modes will respect these conditions and each connection DOF will provide a static mode (junction mode or flexibility). In practice, the choice of boundary conditions is based on practical considerations such as a preference for a given method, the project directives or the availability of experimental data. For a given choice, it is important to identify the types of modes to be used and their capacity to properly represent the static and dynamic behavior of the substructure in spite of their limited number. A detailed presentation of these modes can be found below.

8.3.3. Normal modes The normal modes of the substructure S are obtained by solving the eigenvalue problem relating to the mass and stiffness matrices partitioned on the internal DOF i:

(−(ω k2 ) S M iiS + K iiS ) ΦiSk = 0 k

[8.18]

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with – (ω k2 ) S : eigenvalues of substructure S; S

–

ΦSg k

⎡ Φi k ⎤ =⎢ ⎥ : eigenvalues of substructure S. ⎢⎣Φ j k = 0 j k ⎥⎦

In the preceding expressions, the free connection DOF are part of set i, while the constrained connection DOF are part of set j. In the case of the loaded connection DOF, the mass and stiffness matrices are modified by the addition of the mass and/or stiffness of the loading C to the C connection DOF c, represented by matrices M C cc and/or K cc : S +C S M cc = M cc + MC cc

S +C S K cc = K cc + KC cc

[8.19]

C Matrices M C cc and/or K cc are supposed to represent the adjacent structure(s), as mentioned in the preceding section. One possibility is to obtain them by static condensation of the adjacent structure(s) on the DOF c if they are known, but this is not always the best method. More generally we will be able to consider a reference loading by making a certain compromise according to various considerations.

8.3.4. Static flexibilities The static flexibilities, sometimes called attachment modes, are related to the free connection DOF of substructure S and represent its static deformations under the action of the unit forces successively applied on each of these DOF. They are given by:

K iiS

S G ic

= I ic

G Sgc

⎡ G ic ⎤ =⎢ ⎥ G = 0 jc ⎦ ⎣ jc

S

[8.20]

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261

This assumes that matrix K iiS is not singular, i.e. the substructure does not have rigid-body modes or mechanisms. System [8.20] can then be directly solved by S factorization of K iiS in order to obtain the static flexibilities G ic .

If, on the other hand, matrix K iiS is singular, relation [8.20] can no longer be applied. This problem of flexibilities in the presence of rigid-body modes was discussed in section 5.2.2.2 in order to get to the concept of pseudo-static flexibilities (or inertia relief modes), which are interpreted as the flexibilities around the center of mass by eliminating the contribution of the rigid-body modes r. Section 5.2.2.2 explains how to obtain these pseudo-static flexibilities. Let us take here the simplest procedure, i.e. using Lagrange multipliers [ROY 02]. The pseudo-flexibilities defined by relation [5.19] can be written by limiting ourselves to the connection DOF c: S S S −1 S G ic = Φik (k kk ) Φkc

( k ≠ r)

[8.21]

S with k kk diagonal matrix of the generalized stiffnesses: k kS = (ω k 2 ) m kS . S By pre-multiplying relation [8.21] by Φ ri M iiS and by using the orthogonality S S relations between rigid-body and elastic modes Φ ri M iiS Φ ik = 0 rk , we obtain the following relation expressing the orthogonality of the pseudo-flexibilities [8.21] with the rigid-body modes: S S Φ ri M iiS G ic = 0 rc

[8.22]

Orthogonality relations [8.22] provide a set of linear constraints that can be introduced in [8.20] using Lagrange multipliers, just as for [5.23]: ⎡ K ii ⎢Φ M ⎣ ri ii

S

M ii Φ ir ⎤ ⎡G ic ⎤ 0 rr ⎥⎦ ⎢⎣ λ rc ⎥⎦

S

⎡I ⎤ = ⎢ ic ⎥ ⎣0 rc ⎦

[8.23]

S The pseudo-static flexibilities G ic can thus be obtained by solving system [8.23] by factorization of the increased matrix.

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Instead of using the flexibilities or pseudo-static flexibilities defined by the relations [8.20] or [8.23], it is often practical to replace them by the residual flexibilities obtained by eliminating the contribution of the retained elastic modes k: S S S S −1 S G ic ,res = G ic − Φik (k kk ) Φkc

( k ≠ r)

[8.24]

The residual flexibilities have the advantage of being M-orthogonal to the retained modes, including the rigid-body modes r, and they can be deduced from relation [8.20] or [8.23] by replacing r with k: ⎡ K ii ⎢Φ M ⎣ ki ii

S

M ii Φ ik ⎤ ⎡G ic, res ⎤ 0 kk ⎥⎦ ⎢⎣ λ kc ⎥⎦

S

⎡I ⎤ = ⎢ ic ⎥ ⎣0 kc ⎦

[8.25]

Thus, the pseudo-flexibilities of relation [8.23] can be interpreted like the residual flexibilities in relation to the rigid-body modes r. 8.3.5. Junction modes

The junction modes are related to the fixed connection DOF of substructure S and represent its static deformations under the action of the unit displacements successively applied on each of these DOF. They can be interpreted as static transmissibilities with regards to the fixed connection DOF, while the preceding static flexibilities are related to the free connection DOF. They are given by (according to relation [4.29]):

K iiS

S Ψ ic

=

S −K ic

Ψ Sgc

⎡ Ψ ic ⎤ =⎢ ⎥ ⎣Ψ jc = I jc ⎦

S

[8.26]

In most cases, the fixed DOF c eliminate all rigid-body modes, and this makes it possible to directly solve system [8.26] with a non-singular stiffness matrix K iiS . In the opposite case, i.e. with a substructure including rigid-body modes or mechanisms making stiffness matrix K iiS singular, relation [8.26] is no longer applicable. Just as for static flexibilities, the contribution of the rigid-body modes

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263

must be eliminated. The matrix of the junction modes minus the contribution of the rigid-body modes r is written (according to relation [5.14]): S S S Ψic = Φik (m kS ) −1 LSkc − (M ii−1 ) S M ic

(k ≠ r)

[8.27]

S By pre-multiplying relation [8.27] by Φ ri M iiS and by using the orthogonality S S relations between rigid-body and elastic modes Φ ri M iiS Φ ik = 0 rk , the following relation is obtained:

S S S S Φ ri M iiS Ψ ic = −Φ ri M ic

[8.28]

These linear constraints can be introduced in [8.26] using Lagrange multipliers, just as for [4.30]:

⎡ K ii ⎢Φ M ⎣ ri ii

S

M ii Φ ir ⎤ ⎡Ψ ic ⎤ 0 rr ⎥⎦ ⎢⎣ λ rc ⎥⎦

S

⎡ − K ic ⎤ =⎢ ⎥ ⎣− Φ ri M ic ⎦

S

[8.29]

S The junction modes Ψic in the presence of the rigid-body modes can thus be obtained by solving system [8.29] by factorization of the augmented matrix.

8.3.6. Illustration

For an illustration of the normal modes and of static modes previously defined, let us take the example in Figure 8.4 with two beam elements in bending in a plane. In this example, the two beams A and B are to be connected in order to form one clamped-free beam, the clamping being on the side of beam A (Figure 8.4a).

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θ

v

A

c

B

a) example with 2 beams

Φ

Ψ

v=1 b) fixed connection modes

Φ

G

F=1 c) free connection modes

Φ

M K d) loaded connection modes Figure 8.4. Examples of static modes and normal modes

If each of the three cases of the boundary conditions presented in section 8.3.2 and applied to the connection of the two beams are considered: – Figure 8.4b illustrates the normal modes and the static modes corresponding to fixed connection DOF. The normal modes given by equation [8.18] are thus those of the clamped-clamped beam A and those of the clamped-free beam B. The associated static modes are the junction modes given by equation [8.26]: there are two for each beam, one generated by a unit translation at the connection, the other by a unit rotation. We should note that since beam A is statically indeterminate, the deformations are elastic, while for beam B, statically determinate, they are rigid;

Modal Synthesis

265

– Figure 8.4c illustrates the static modes and the normal modes corresponding to the free connection DOF. The normal modes given by equation [8.18] are thus those of the clamped-free beam A and those of the free-free beam B (comprising two rigidbody modes), the connection DOF members of the internal DOF. The associated static modes are the two flexibilities given by equation [8.20] for beam A (without rigid-body modes), and by equation [8.23] for beam B (with two rigid-body modes); one generated by a unit force at the connection, the other by a unit moment. Note that the deformations of beam B are elastic (pseudo-flexibilities around the center of mass), even though it remains free; – Figure 8.4d illustrates the normal modes corresponding to loaded connection DOF with for example a loading in mass for beam A, and in stiffness for beam B. This choice assumes that the influence of B on A comes especially from its mass, while the influence of A on B comes especially from its stiffness. This is the case for an appendix B connected to a main body A of larger size. It should be noted here that the loading in stiffness on beam B eliminates its rigid-body modes. These three cases consider the same boundary conditions for the two substructures. Other choices can be considered leading to a certain number of possibilities listed hereafter. 8.3.7. Possible combinations

Use of the same boundary conditions for the connection DOF of all the considered substructures, as illustrated before, is common in practice because each substructure is handled the same way. The approach with the fixed connection DOF by Craig and Bampton [CRA 68] and that of the free connection DOF by Craig and Chang [CRA 77] and also presented in [CRA 81], are two of the most commonly used methods. In certain cases, hybrid conditions, i.e. combining fixed, free or loaded connection DOF, can be interesting: – with several substructures, the connection DOF between some substructures can be free, and fixed between others, as illustrated in Figure 8.5a with the free DOF between A and B and fixed between A and C; – for a given interface between two substructures, the DOF of one can be fixed and the DOF of the other can be free or loaded. It is the case in the Benfield and Hruda approach [BEN 71] with the loaded connection DOF for one of the substructures considered as main and the fixed connection DOF for the others considered as appendices, as illustrated by Figure 8.5b;

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– finally, for a given substructure, certain connection DOF with the adjacent substructure can be fixed and the others free or loaded, as illustrated in Figure 8.5c with pinned A (fixed translation, free rotation) and sliding B (free translation, fixed rotation). This type of hybrid conditions is not common but it can occur, for example, in testing involving particular boundary conditions. C

B

A

(a)

(b)

(c) Figure 8.5. Examples of hybrid boundary conditions

8.4. Some methods 8.4.1. Craig-Bampton method

The method proposed by Craig and Bampton [CRA 68, CRA 81] is historically among the first methods of modal synthesis and today it remains one of the most popular by combining simplicity and accuracy. It considers the fixed connection DOF for each substructure S. This leads to using a truncated set of fixed connection normal modes as basis B, according to relation [8.18], completed by the junction modes according to relation [8.26]. The transformation [8.2] resulting from this is actually that of [4.31] with j = c:

u Sg = B Sgb q bS

⇒

S ⎡u iS ⎤ ⎡Φ ik ⎢ S⎥ = ⎢ S ⎢⎣u c ⎥⎦ ⎢⎣ 0 ck

S ⎤⎡ S⎤ Ψ ic qk S ⎥⎢ S⎥ I cc ⎥⎦ ⎢⎣u c ⎥⎦

[8.30]

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267

The generalized coordinates q bS therefore include the physical displacements u cS of all the connection DOF. The transformed mass and stiffness matrices are written (see relations [4.32] and [4.34]):

⎡m S S M bb = ⎢ Skk ⎢⎣ L ck

LSkc ⎤ ⎥ S M cc ⎥⎦

⎡k S S = ⎢ kk K bb ⎢⎣ 0

0 ⎤ ⎥ S K cc ⎥⎦

S S m kk = Φkg M Sgg ΦSgk

(diagonal matrix of generalized masses)

S S k kk = Φkg K Sgg ΦSgk

(diagonal matrix of generalized stiffnesses)

S LSkc = Φ kg M Sgg Ψ Sgc

(matrix of participation factors)

S S M cc = Ψ cg M Sgg Ψ Sgc

(condensed mass matrix)

S S K cc = Ψ cg K Sgg Ψ Sgc

(condensed stiffness matrix)

[8.31]

Relations [8.31] show a coupling in mass between the normal modes k and the junction modes c by the matrix of participation factors LSkc , with an uncoupled stiffness. Moreover, in case of a statically determinate junction, the condensed S S stiffness matrix K cc is zero and the stiffness matrix K bb is thus diagonal.

According to the approach in section 8.2.2, the coupling between two substructures A and B is expressed by the compatibility of the displacements u cA = u cB at the connection DOF c, and this is expressed on the vector of the

generalized coordinates q b by the following relation: ⎡q A ⎤ ⎢ kA ⎥ ⎢u ⎥ q b = ⎢ cB ⎥ ⇒ ⎢q k ⎥ ⎢u cB ⎥ ⎣ ⎦

⎡q kA ⎤ ⎢ A⎥ [0 I 0 − I ] ⎢⎢u cB ⎥⎥ = 0 q ⎢ kB ⎥ ⎣⎢u c ⎦⎥

[8.32]

If we choose the elimination strategy with the dependent DOF q m = u cB ( C mm = −I mm in [8.10]), then the transformation [8.11] that makes it possible to consider the independent DOF q n is written:

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⎡q kA ⎤ ⎡ I 0 ⎢ A⎥ ⎢ ⎢u c ⎥ = ⎢0 I ⎢q B ⎥ ⎢0 0 ⎢ kB ⎥ ⎢ ⎣⎢u c ⎦⎥ ⎣0 I

⎡q A ⎤ ⎢ k⎥ q n = ⎢u cA ⎥ ⇒ ⎢ B⎥ q ⎣⎢ k ⎦⎥

0⎤ ⎡ A ⎤ qk 0⎥⎥ ⎢ A ⎥ ⎢u c ⎥ I⎥ ⎢ B ⎥ ⎥ qk 0⎦ ⎢⎣ ⎥⎦

[8.33]

Its application to the mass and stiffness matrices of substructures A and B leads to the following matrices relating to the DOF qn: ⎡ m kA ⎢ M nn = ⎢LAck ⎢ 0 ⎢⎣

+

⎡k kA ⎢ K nn = ⎢ 0 ⎢ 0 ⎢⎣

0 ⎤ ⎥ LBck ⎥ m kB ⎥⎥ ⎦

LAkc

A M cc

B M cc

LBkc

0 ⎤ ⎥ 0 ⎥ [8.34] k kB ⎥⎥ ⎦

0 A K cc

+

B K cc

0

The recovery of the physical displacements of A and B is carried out according to relations [8.30] and [8.33], and this gives: ⎡u iA ⎤ ⎡Φ A ⎢ A ⎥ ⎢ ik ⎢u c ⎥ = ⎢ 0 ⎢u B ⎥ ⎢ 0 ⎢ iB ⎥ ⎢ ⎢⎣u c ⎥⎦ ⎢⎣ 0

Ψ icA I B Ψ ic

I

0 ⎤ ⎡ A⎤ ⎥ qk 0 ⎥ ⎢ A⎥ u B⎥⎢ c ⎥ Φ ik ⎢ ⎥ ⎢q kB ⎥⎥ 0 ⎥⎦ ⎣ ⎦

[8.35]

As an example that will be used for all the methods presented, let us consider the 4-DOF spring-mass system in Figure 8.6. Substructure A is made up of a mass and a spring connecting the DOF 1 and 2, substructure B of 3 masses and 3 springs connecting the DOF 2, 3 and 4 (the mass of the DOF 2 thus belongs to B). All masses and stiffnesses are equal 1: m = k = 1. (m = k = 1)

A

B

c m

u1

k

m

u2

k

m

u3

Figure 8.6. 4-DOF spring-mass system

k

m

u3

Modal Synthesis

269

The mass and stiffness matrices of A, B and A + B are written: ⎡u ⎤ u gA = ⎢ 1 ⎥ ⎣u 2 ⎦

u Bg

⎡ 1 − 1⎤ A K gg =⎢ 1⎥⎦ ⎣− 1

⎡1 0⎤ A M gg =⎢ ⎥ ⎣0 0 ⎦

⎡u 2 ⎤ = ⎢⎢u 3 ⎥⎥ ⎢⎣u 4 ⎥⎦

M Bgg

⎡ u1 ⎤ ⎢u ⎥ u gA+ B = ⎢ 2 ⎥ ⎢u 3 ⎥ ⎢ ⎥ ⎣u 4 ⎦

⎡1 0 0⎤ = ⎢⎢0 1 0⎥⎥ ⎢⎣0 0 1⎥⎦

A+ B M gg

⎡1 ⎢0 =⎢ ⎢0 ⎢ ⎣0

0 0 0⎤ 1 0 0⎥⎥ 0 1 0⎥ ⎥ 0 0 1⎦

K Bgg

[8.36]

⎡ 1 −1 0 ⎤ = ⎢⎢− 1 2 − 1⎥⎥ ⎢⎣ 0 − 1 1 ⎥⎦

A+ B K gg

[8.37]

⎡ 1 −1 0 0 ⎤ ⎢− 1 2 − 1 0 ⎥ ⎥ [8.38] =⎢ ⎢ 0 − 1 2 − 1⎥ ⎢ ⎥ ⎣ 0 0 −1 1 ⎦

The normal modes of A + B normalized to a unit generalized mass are given by Table 8.1 which is the reference for the results of all the modal synthesis methods under consideration. Mode k

1

2

3

4

ω k2

0.0

0.5858

2.0

3.4142

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎢ ⎥ ⎢+ 0.5⎥ ⎢ ⎥ ⎣+ 0.5⎦

⎡ − 0.6533⎤ ⎢ − 0.2706⎥ ⎢ ⎥ ⎢+ 0.2706⎥ ⎢ ⎥ ⎣ + 0.6533⎦

⎡+ 0.5⎤ ⎢ − 0.5⎥ ⎢ ⎥ ⎢ − 0.5⎥ ⎢ ⎥ ⎣+ 0.5⎦

⎡+ 0.2706⎤ ⎢ − 0.6533⎥ ⎢ ⎥ ⎢ + 0.6533⎥ ⎢ ⎥ ⎣ − 0.2706⎦

Φi k

(m k = 1)

Table 8.1. Normal modes of the system in Figure 8.6

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If substructure A is represented by its unique fixed connection normal mode and by its unique junction mode (no modal truncation), transformation [8.30] and the transformed matrices are written: ⎡u1A ⎤ ⎡1 1⎤ ⎡q1A ⎤ ⎢ A⎥ = ⎢ ⎥ ⎢ A⎥ ⎣⎢u 2 ⎦⎥ ⎣0 1⎦ ⎢⎣u 2 ⎦⎥

⎡ 1 0⎤ A K bb =⎢ ⎥ ⎣0 0 ⎦

⎡1 1⎤ A M bb =⎢ ⎥ ⎣1 1⎦

[8.39]

If substructure B is represented by its first fixed connection normal mode (second mode eliminated) and its unique junction mode, transformation [8.30] and the transformed matrices are written: ⎡u 2B ⎤ ⎡ 0 1⎤ B ⎡q ⎤ ⎢ B⎥ ⎢ u 2 1⎥⎥ ⎢ 1B ⎥ = ⎢ 3⎥ ⎢ ⎢u B ⎥ ⎢1 + 5 1⎥ ⎣⎢u 2 ⎦⎥ ⎦ ⎢⎣ 4 ⎥⎦ ⎣ ⎡10 + 2 5 3 + 5 ⎤ B =⎢ M bb ⎥ 3 ⎦⎥ ⎣⎢ 3 + 5

[8.40]

⎡10 − 2 5 B =⎢ K bb 0 ⎣⎢

0⎤ ⎥ 0⎦⎥

The matrices resulting from transformation [8.33] and the recovery relation [8.35] are then written: ⎡q1A ⎤ ⎢ ⎥ q n = ⎢u 2A ⎥ ⇒ ⎢q B ⎥ ⎢⎣ 1 ⎥⎦

M nn

⎡ u1 ⎤ ⎡1 ⎢u ⎥ ⎢ ⎢ 2 ⎥ = ⎢0 ⎢ u 3 ⎥ ⎢0 ⎢ ⎥ ⎢ ⎣u 4 ⎦ ⎣0

1 0 ⎡1 ⎤ ⎢ = ⎢1 4 3 + 5 ⎥⎥ ⎢⎣0 3 + 5 10 + 2 5 ⎥⎦

⎤ ⎡ A⎤ q1 1 0 ⎥⎥ ⎢ A ⎥ ⎢u 2 ⎥ 1 1 ⎥ ⎢ B⎥ ⎥ q1 1 1 + 5 ⎦ ⎢⎣ ⎥⎦

1

0

K nn

[8.41]

0 ⎡1 0 ⎤ ⎢ ⎥ = ⎢0 0 0 ⎥ ⎢⎣0 0 10 − 2 5 ⎥⎦

These matrices lead to the results in Table 8.2. The rigid-body mode is preserved as expected. The first elastic mode is well recovered in spite of the truncation of the second mode of the substructure B. On the other hand, the second elastic mode is relatively degraded by the truncation.

Modal Synthesis

Mode k

1

2

3

ω k2

0.0

0.5871

2.359

(0%)

(0.2%)

(17.7%)

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎢ ⎥ ⎢+ 0.5⎥ ⎢ ⎥ ⎣+ 0.5⎦

⎡ − 0.6582⎤ ⎢ − 0.2718⎥ ⎢ ⎥ ⎢ + 0.2911⎥ ⎢ ⎥ ⎣+ 0.6389⎦

⎡ + 0.5628⎤ ⎢ − 0.7620⎥ ⎢ ⎥ ⎢ − 0.1038⎥ ⎢ ⎥ ⎣+ 0.3030⎦

(error)

Φi k (m k = 1)

271

4

–

–

Table 8.2. Normal modes of the system in Figure 8.6 with the Craig-Bampton method

In conclusion, the Craig-Bampton method possesses the following advantages and disadvantages: – advantages: simple formulation, good accuracy (statically exact at the connection DOF – in case of excitation elsewhere than at the connection, add the concerned DOF to the connection DOF), explicit boundary conditions making a direct assembly of the stiffness type possible, numerically robust and not requiring matrix inversion; – disadvantages: a large number of connection DOF favor the static modes to the detriment of the normal modes for the description of the behavior (limiting case: purely static condensation), experimental incompatibility: a fixed connection is more delicate to manage (see Chapter 11). Altogether, this method, which combines simplicity, robustness and accuracy, is the most frequently used in an analytical context. 8.4.2. Craig-Chang method The method proposed by Craig and Chang [CRA 77, CRA 81], in contrast to the preceding method, uses free connection DOF for each substructure S. It is similar to the MacNeal method [MAC 71] and numerically equivalent to the Rubin method [RUB 75]. This leads to using a truncated set of free connection normal modes as

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basis B, according to relation [8.18], completed by the residual flexibilities to the connection according to relation [8.24] or [8.25]. Transformation [8.2] resulting from this is written:

u Sg = B Sgb q bS

⇒

[

u Sg = ΦSgk

]

⎡q S ⎤ G Sgc,res ⎢ k ⎥ ⎢⎣ f cS ⎥⎦

[8.42]

The generalized coordinates q bS thus include the forces f cS relating to the connection DOF. The transformed mass and stiffness matrices are written: ⎡m S S M bb = ⎢ kk ⎢⎣ 0

⎡k S S K bb = ⎢ kk ⎢⎣ 0

⎤ ⎥ S H cc ,res ⎥⎦ 0

⎤ ⎥ S G cc , res ⎥⎦ 0

S S m kk = Φkg M Sgg ΦSgk

(diagonal matrix of generalized masses)

S S k kk = Φkg K Sgg ΦSgk

(diagonal matrix of generalized stiffnesses)

[8.43]

S S S S H cc , res = G cg , res M gg G gc, res (residual inertia matrix) S S S S G cc ,res = G cg ,res K gg G gc ,res (residual flexibility matrix)

The block-diagonal form of the matrices results directly from the orthogonality S between the normal modes and the residual flexibilities. The term G cc , res can be

obtained by simple partition of the residual flexibility matrix G Sgc, res . The inertial S term H cc , res was identified for the first time by Rubin as an improvement of the

MacNeal method which excludes it. The coupling between two substructures A and B is expressed by the compatibility of displacements

u cA = u cB

and the equilibrium of forces

f cA + f cB = 0 at the connection DOF c, which is expressed on the vector of the

generalized coordinates q b by the relation:

Modal Synthesis

⎡q A ⎤ ⎢ kA ⎥ ⎢f ⎥ q b = ⎢ cB ⎥ ⇒ ⎢q k ⎥ ⎢ f cB ⎥ ⎣ ⎦

A ⎡Φ ck

⎢ ⎢⎣ 0

A G cc , res

B − Φ ck

I cc

0

273

⎡q kA ⎤ ⎢ A⎥ ⎢ f ⎥ ⎡0 ⎤ ⎥ ⎢ cB ⎥ = ⎢ ⎥ [8.44] ⎥⎦ ⎢q k ⎥ ⎣0⎦ ⎢⎣ f cB ⎥⎦

B ⎤ − G cc , res

I cc

If the elimination strategy with the dependent DOF f cA and f cB is chosen, transformation [8.11] is written:

qn =

⎡q A ⎤ ⎢ kB ⎥ ⎢⎣q k ⎥⎦

⇒

⎡q A ⎤ ⎡ I ⎢ kA ⎥ ⎢ A k − ⎢ f c ⎥ ⎢ cc Φck = ⎢ B⎥ ⎢ 0 ⎢q k ⎥ ⎢ A ⎢ f cB ⎥ ⎣⎢ k cc Φck ⎣ ⎦

0 ⎤ B ⎥ ⎡ A⎤ k cc Φck ⎥ ⎢q k ⎥ ⎥ ⎢q B ⎥ I ⎣ k⎦ B ⎥ − k cc Φck ⎦⎥

[8.45]

A B −1 with k cc = (G cc , res + G cc, res ) .

Its application to the mass and stiffness matrices of substructures A and B leads to the following matrices relating to the DOF q n : A B ⎡m A + ΦA m ΦA ⎤ − Φkc m cc Φck ⎥ M nn = ⎢ kk B kc cc A ck B B B + Φkc m kk m cc Φck ⎢⎣ − Φkc m cc Φck ⎥⎦ A B ⎡k A + ΦA k ΦA ⎤ k cc Φck − Φkc ⎥ K nn = ⎢ kk B kc cc A ck B B B k kk + Φkc k cc Φck ⎦⎥ ⎣⎢ − Φkc k cc Φck

[8.46]

A B with m cc = k cc (H cc , res + H cc, res ) k cc .

The recovery of the physical displacements of A and B is done according to relations [8.42] and [8.45], and this gives: A A A ⎡u gA ⎤ ⎡Φ gk − G gc , res k cc Φ ck ⎢ B⎥ = ⎢ A ⎢⎣u g ⎥⎦ ⎢⎣ G Bgc, res k cc Φ ck

⎤ ⎡q A ⎤ ⎥ k B ⎢ B⎥ Φ ck ⎥⎦ ⎣⎢q k ⎦⎥

A B G gc , res k cc Φ ck

Φ Bgk − G Bgc, res k cc

[8.47]

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Let us consider again the example of the 4-DOF system in Figure 8.6. If substructure A is represented by its unique free connection normal mode, which is rigid, and its unique residual flexibility at the connection (no truncation), transformation [8.30] and the transformed matrices are written: ⎡u1A ⎤ ⎡1 0⎤ ⎡ q1A ⎤ ⎢ A⎥ = ⎢ ⎥ ⎢ A⎥ ⎣⎢u 2 ⎦⎥ ⎣1 1⎦ ⎢⎣ f 2 ⎦⎥

⎡1 0⎤ A M bb =⎢ ⎥ ⎣0 0 ⎦

⎡0 0 ⎤ A K bb =⎢ ⎥ ⎣0 1⎦

[8.48]

If substructure B is represented by its first two free connection normal modes (third mode eliminated), including the rigid-body mode, and its unique residual flexibility at the connection, transformation [8.30] and the transformed matrices are written: ⎡u 2B ⎤ ⎡1 1 1 18 ⎤ ⎡ q1B ⎤ ⎢ B⎥ ⎢ ⎥⎢ B⎥ ⎢u3 ⎥ = ⎢1 0 − 1 9⎥ ⎢ q 2 ⎥ ⎢u B ⎥ ⎢1 − 1 1 18 ⎥ ⎢ f B ⎥ ⎦ ⎢⎣ 2 ⎥⎦ ⎢⎣ 4 ⎥⎦ ⎣

B M bb

0 ⎤ ⎡3 0 ⎢ = ⎢0 2 0 ⎥⎥ ⎢⎣0 0 1 54⎥⎦

B K bb

[8.49]

⎡0 0 0 ⎤ = ⎢⎢0 2 0 ⎥⎥ ⎢⎣0 0 1 18⎥⎦

The matrices resulting from transformation [8.33] and the recovery relation [8.35] are then written: ⎡q1A ⎤ ⎢ ⎥ q n = ⎢q1B ⎥ ⇒ ⎢q B ⎥ ⎢⎣ 2 ⎥⎦

M nn

0 ⎤ ⎡ A⎤ ⎡ u1 ⎤ ⎡ 19 0 ⎢u ⎥ ⎢ 1 18 18 ⎥ ⎢q1 ⎥ ⎢ 2⎥ = 1 ⎢ ⎥ ⎢q B ⎥ 1 ⎢u 3 ⎥ 19 ⎢− 2 21 2 ⎥ ⎢ B⎥ q ⎢ ⎥ ⎥⎢ 2 ⎥ ⎢ ⎣ 1 18 − 20⎦ ⎣ ⎦ ⎣u 4 ⎦

⎡367 −6 −6 ⎤ ⎥ 1 ⎢⎢ = −6 1, 089 6 ⎥ ⎢ ⎥ 361 ⎢ ⎥ − 6 6 728 ⎣ ⎦

K nn

⎡ 18 − 18 − 18⎤ 1 ⎢ = − 18 18 18 ⎥⎥ 19 ⎢ ⎢⎣− 18 18 56 ⎥⎦

[8.50]

Modal Synthesis

275

These matrices lead to the results in Table 8.3. The rigid-body mode is again preserved. The first elastic mode is very well recovered in spite of the truncation of the third mode of substructure B. The second is slightly degraded by this truncation. As a whole, the results are better than with the Craig-Bampton method, and this is explained by the fact that the number of modes used at the beginning is greater (the linear constraints making the sizes identical at the end). Mode k

1

2

3

4

ω k2

0.0

0.5862

2.0913

–

(error)

(0%)

(0.07%)

(4.6%)

Φi k

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎥ ⎢ ⎢+ 0.5⎥ ⎥ ⎢ ⎣+ 0.5⎦

⎡ − 0.6533⎤ ⎢ − 0.2664⎥ ⎥ ⎢ ⎢ + 0.2608⎥ ⎥ ⎢ ⎣+ 0.6589⎦

⎡ + 0.5541⎤ ⎢ − 0.6488⎥ ⎥ ⎢ ⎢− 0.3184⎥ ⎥ ⎢ ⎣ + 0.4131⎦

(m k = 1)

–

Table 8.3. Normal modes of the system in Figure 8.6 with the Craig-Chang method

In conclusion, the Craig-Chang method possesses the following advantages and disadvantages: – advantages: good accuracy (statically exact at the connection DOF), number of connection DOF without consequence because they are eliminated from the solution, experimental compatibility: a free connection is easier to manage (see Chapter 11) although there are some difficulties in the case of pseudo-flexibilities; – disadvantages: difficult formulation, implicit boundary conditions, matrix inversion is necessary, in case of excitation elsewhere than at the connection it is necessary to introduce the corresponding static flexibilities. Altogether, this method which is delicate to implement but very efficient, is complementary to the preceding method.

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8.4.3. Benfield-Hruda method The method proposed by Benfield and Hruda [BEN 71] uses a loading in mass and stiffness at the connection DOF in order to improve the normal modes. The coupling of the two substructures can be done in two ways: – with the free or loaded connection normal modes for both substructures; – with the free or loaded connection normal modes for one which is considered as main body, and the fixed connection normal modes for the other which is considered as an appendix, completed by the junction modes. The second case using hybrid conditions at the connection will be examined here. The appendix is denoted by A, the main body by B and its possible loading by C. For appendix A, the approach is the same as that of Craig-Bampton, hence the same transformation as [8.30] and the same transformed matrices as [8.31]:

[

⎡q A ⎤ A Ψ gc ⎢ kA ⎥ ⎣⎢u c ⎦⎥

]

⎡m A A M bb = ⎢ Akk ⎢⎣ L ck

LAkc ⎤ A⎥ M cc ⎥⎦

A u gA = Φgk

[8.51]

⎡k A A K bb = ⎢ kk ⎢⎣ 0

0 ⎤ A⎥ K cc ⎥⎦

[8.52]

For central body B and its possible loading C, the basis B is made up of a ˆ with loaded connection in mass and in stiffness, truncated set of normal modes Φ thus corresponding to the matrices: ˆ B = M B + MC M cc cc cc

ˆ B = K B + KC K cc cc cc

[8.53]

B B M cc , K cc matrices relating to non-loaded connection C MC cc , K cc loading in mass and stiffness at the connection A C A (extension of the [BEN 71] case where M C cc = M cc , K cc = K cc , matrices of A condensed at the connection)

ˆ B, K ˆ B matrices relating to the loaded connection M cc cc

Modal Synthesis

277

The transformation for the main body B is then written: ˆ B qB u Bg = Φ gk k

[8.54]

It should be noted that the absence of the static modes in the basis is made up only by the loaded normal modes. The transformed matrices are written: B ˆ B MB Φ ˆB M bb =Φ kg gg gk

B ˆ B KB Φ ˆB K bb =Φ kg gg gk

[8.55]

B In the absence of the loading in mass or in stiffness, the mass matrix M bb or the B B stiffness matrix K bb remains diagonal with the generalized masses m kk or the B generalized stiffnesses k kk .

The coupling between the two substructures A and B represented by matrices [8.52] and [8.55] is expressed by the compatibility of the displacements u cA = u cB at their connection DOF c, and this is translated by the vector of the generalized coordinates qb by the relation: ⎡q A ⎤ ⎢ k⎥ q b = ⎢u cA ⎥ ⇒ ⎢ B⎥ ⎢⎣q k ⎥⎦

[0

ck

ˆB − I cc Φ ck

]

⎡q A ⎤ ⎢ kA ⎥ ⎢u c ⎥ = 0 ⎢ B⎥ ⎢⎣q k ⎥⎦

If the elimination strategy with the dependent DOF u cA transformation [8.11] is written:

⎡q A ⎤ q n = ⎢ kB ⎥ ⇒ ⎢⎣q k ⎥⎦

⎡q kA ⎤ ⎡ I 0 ⎤ A ⎢ A⎥ ⎢ B ⎥ ⎡q k ⎤ ˆ ⎢u c ⎥ = ⎢0 Φ ck ⎥ ⎢ B ⎥ q ⎢q B ⎥ ⎢0 I ⎥⎦ ⎣⎢ k ⎥⎦ ⎣⎢ k ⎦⎥ ⎣

[8.56]

is chosen,

[8.57]

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Its application to the mass and stiffness matrices of substructures A and B leads to the following mass matrices relating to the DOF qn: ⎡ mA kk M nn = ⎢ ˆ B LA ⎢Φ ⎣ kc ck ⎡k A K nn = ⎢ kk ⎢ 0 ⎣

⎤ ⎥ ˆB ⎥ − MC cc ) Φck ⎦

ˆB L Akc Φ ck A

B ˆ B (M cc ˆ kk +Φ m kc

0 A B ˆ B (K cc +Φ kˆ kk kc

B ˆ B Φ ˆB M ˆB ˆ kk with m =Φ gg gk kg

[8.58]

⎤ ⎥ ˆB ⎥ −KC cc ) Φck ⎦

B ˆB Φ ˆB K ˆB kˆ kk =Φ gg gk kg

If the loading of B is taken as equal to the connection condensed mass and A C A stiffness matrices of A [BEN 71], given: M C cc = M cc and K cc = K cc , then matrices [8.58] reduce to:

⎡ mA M nn = ⎢ B kkA ˆ L ⎢⎣Φ kc ck

ˆB ⎤ L Akc Φ ck ⎥ B ˆ kk m ⎥⎦

⎡k A K nn = ⎢ kk ⎢⎣ 0

0 ⎤ ⎥ B kˆ kk ⎥⎦

[8.59]

The recovery of the physical displacements of A and B is performed according to relations [8.51] and [8.54], which gives: ˆA ⎡u gA ⎤ ⎡Φ ⎢ B ⎥ = ⎢ gk ⎢⎣u g ⎥⎦ ⎢⎣ 0

A ˆ B ⎤ ⎡ A⎤ Ψ gc Φ ck q k ⎥⎢ ⎥ B Φ gk ⎥⎦ ⎣⎢q kB ⎦⎥

[8.60]

Consider again the example of the 4-DOF system in Figure 8.6, with structure A as the appendix, structure B as main body. The transformation and the transformed matrices of substructure A are the same as for Craig-Bampton: ⎡u1A ⎤ ⎡1 1⎤ ⎡q1A ⎤ ⎢ A⎥ = ⎢ ⎥ ⎢ A⎥ ⎣⎢u 2 ⎦⎥ ⎣0 1⎦ ⎢⎣u 2 ⎦⎥

⎡1 1⎤ A M bb =⎢ ⎥ ⎣1 1⎦

⎡ 1 0⎤ A K bb =⎢ ⎥ ⎣0 0 ⎦

[8.61]

Modal Synthesis

279

If substructure B is represented by its first two free connection normal modes (third mode eliminated) and without loading, transformation [8.30] and the transformed matrices are written: ⎡u 2B ⎤ ⎡1 1 ⎤ B ⎢ B⎥ ⎢ ⎥ ⎡q1 ⎤ ⎢u3 ⎥ = ⎢1 0 ⎥ ⎢ B ⎥ ⎢u B ⎥ ⎢1 − 1⎥ ⎢⎣q 2 ⎥⎦ ⎦ ⎢⎣ 4 ⎥⎦ ⎣

⎡3 0 ⎤ B M bb =⎢ ⎥ ⎣0 2 ⎦

⎡0 0 ⎤ B K bb =⎢ ⎥ ⎣0 2 ⎦

[8.62]

The matrices resulting from transformation [8.33] and recovery relation [8.35] are then written:

M nn

⎡1 1 1⎤ = ⎢⎢1 4 1⎥⎥ ⎢⎣1 1 3⎥⎦

K nn

⎡1 0 0⎤ = ⎢⎢0 0 0⎥⎥ ⎢⎣0 0 2⎥⎦

⎡ u1 ⎤ ⎡1 ⎢u ⎥ ⎢ ⎢ 2 ⎥ = ⎢0 ⎢ u 3 ⎥ ⎢0 ⎢ ⎥ ⎢ ⎣u 4 ⎦ ⎣0

1 ⎤ ⎡ A⎤ q1 1 ⎥⎥ ⎢ B ⎥ ⎢q1 ⎥ 1 0 ⎥ ⎢ B⎥ ⎥ q2 1 − 1⎦ ⎣⎢ ⎦⎥ 1 1

[8.63]

These matrices lead to the results in Table 8.4. The rigid-body mode is preserved as expected. The first elastic mode is quite well recovered, but without loading the convergence is slower. The second elastic mode is significantly degraded. On the whole, the results suffer from the lack of static modes for B.

Mode k

1

2

3

4

ω k2

0.0 (0%)

0.5959 (1.7%)

2.2374 (11.9%)

–

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎥ ⎢ ⎢+ 0.5⎥ ⎥ ⎢ ⎣+ 0.5⎦

⎡ − 0.6427 ⎤ ⎢ − 0.2597 ⎥ ⎥ ⎢ ⎢+ 0.2142⎥ ⎥ ⎢ ⎣+ 0.6882⎦

⎡+ 0.5805⎤ ⎢ − 0.7183⎥ ⎥ ⎢ ⎢ − 0.1935⎥ ⎥ ⎢ ⎣ + 0.3313⎦

–

(error) Φi k (m k = 1)

Table 8.4. Normal modes of the system in Figure 8.6 with the Benfield-Hruda method without loading

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If we now load B with the connection condensed mass matrix of A, given MC cc

A = M cc = 1 , results [8.62] and [8.63] become:

⎡u 2B ⎤ ⎡1 − 3 + 17 ⎤ ⎢ B⎥ ⎢ ⎥ ⎡q1B ⎤ ⎢u3 ⎥ = ⎢1 16 − 4 17 ⎥ ⎢ B ⎥ ⎢u B ⎥ ⎢1 − 10 + 2 17 ⎥ ⎢⎣q 2 ⎥⎦ ⎢⎣ 4 ⎥⎦ ⎣ ⎦ 0 ⎡4 ⎤ B M bb =⎢ ⎥ 0 748 180 17 − ⎣ ⎦

M nn

⎡ 1 ⎢ 1 =⎢ ⎢− 3 + 17 ⎣

[8.64]

⎡0 ⎤ 0 B ⎥ = ⎢⎢ K bb ⎥ ⎣⎢ 0 2, 074 − 502 17 ⎥⎦

− 3 + 17 ⎤ ⎥ 4 0 ⎥ 0 748 − 180 17 ⎥ ⎦

1

⎡ u1 ⎤ ⎡1 1 − 3 + 17 ⎤ ⎡ A ⎤ ⎥ q1 ⎢u ⎥ ⎢ ⎢ ⎥ ⎢ 2 ⎥ = ⎢0 1 − 3 + 17 ⎥ ⎢q B ⎥ ⎢u 3 ⎥ ⎢0 1 16 − 4 17 ⎥ ⎢ 1B ⎥ ⎥ ⎢q 2 ⎥ ⎢ ⎥ ⎢ ⎣u 4 ⎦ ⎣⎢0 1 − 10 + 2 17 ⎦⎥ ⎣ ⎦

K nn

⎡1 0 ⎤ 0 ⎢ ⎥ ⎢ ⎥ = ⎢0 0 0 ⎥ ⎢ ⎥ ⎢⎣ 0 0 2, 074 − 502 17 ⎥⎦

[8.65]

These matrices lead to the results in Table 8.5. The first elastic mode is now well recovered because the loading fully plays its role there. However, the second elastic mode remains degraded because the loading is no longer adapted: the dynamic mass of A at this frequency is clearly different from its static mass. The drawback of this approach can be seen here: it gives good results only for the global modes for which the loading of B is representative.

Modal Synthesis

Mode k

1

2

3

4

ω k2

0.0 (0%)

0.5866 (0.14%)

2.2960 (14.8%)

–

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎥ ⎢ ⎢+ 0.5⎥ ⎥ ⎢ ⎣+ 0.5⎦

⎡ − 0.6499 ⎤ ⎢ − 0.2687 ⎥ ⎥ ⎢ ⎢ + 0.2549⎥ ⎥ ⎢ ⎣+ 0.6637 ⎦

⎡+ 0.5724⎤ ⎢ − 0.7418⎥ ⎥ ⎢ ⎢ − 0.1474⎥ ⎥ ⎢ ⎣ + 0.3168⎦

–

(error)

Φi k

(m k = 1)

281

Table 8.5. Normal modes of the system in Figure 8.6 with the Benfield-Hruda method with loading

In conclusion, the Benfield-Hruda method possesses the following advantages and disadvantages: – advantages: simple formulation, well adapted to the coupling of a central body with its appendices, number of connection DOF without consequence because they are eliminated from the solution; – disadvantages: limited accuracy (statically inexact at the connection DOF), implicit boundary conditions. Altogether, this method, which is rather outperformed by the preceding methods, should be used only in very specific situations.

8.4.4. Effective mass models The modal effective parameters developed in Chapter 5 can be used to characterize the dynamic behavior of the structure and, in certain situations, in order to elaborate equivalent modal models that may be connected to each other by an adapted assembly procedure. This strategy is equivalent to modal synthesis. It is particularly well adapted to the effective mass models developed in section 5.3.2 and illustrated by Figures 5.1 to 5.3. In order to illustrate this approach, consider the example of the 4-DOF system in Figure 8.6, for which the effective mass models of A and B are shown by Figure 8.7.

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B

c

A

1

1

1

5+ 5

1+ 2 5

1− 2 5 Figure 8.7. Effective mass models for the system in Figure 8.6

Modal truncation is introduced by retaining only the first mode of B. The effective mass of the second mode, which is now part of the residual term, is associated with connection. The assembly of the corresponding mass and stiffness matrices and the recovery, according to the first relation [5.55], lead to:

M nn

0 0 ⎤ ⎡5 1⎢ = ⎢0 10 − 2 5 0 ⎥⎥ 5 ⎢⎣0 0 5 + 2 5 ⎥⎦

K nn

10 0 ⎤ ⎡ A⎤ ⎡ u1 ⎤ ⎡10 q1 ⎢u ⎥ ⎢0 10 0 ⎥⎥ ⎢ A ⎥ ⎢ 2⎥ = 1 ⎢ u ⎢ ⎥ ⎢u 3 ⎥ 10 ⎢ 0 5 − 5 5 + 5 ⎥ ⎢ 2B ⎥ ⎢ ⎥ ⎥ ⎢⎣q1 ⎥⎦ ⎢ ⎣ 0 5 − 3 5 5 + 3 5⎦ ⎣u 4 ⎦

− 10 ⎡ 10 1 ⎢ = − 10 15 + 5 10 ⎢ ⎢⎣ 0 −5− 5

⎤ − 5 − 5 ⎥⎥ 5 + 5 ⎥⎦ 0

[8.66]

These matrices lead to the results in Table 8.2 because the ingredients are the same as those of the Craig-Bampton method: fixed connection normal modes completed by the junction modes. The only difference lies in the intermediate calculations relating to various DOF: modal displacements generalized for CraigBampton, displacement of the effective masses for the effective mass models, which in the end is only a question of normalization. This method is thus equivalent to the Craig-Bampton method, involving a more elaborate calculation, but offering a better physical interpretation.

Modal Synthesis

283

8.4.5. Reduced models A large number of analysis codes offer modal synthesis. Although it is easy to implement, this black box approach often masks or limits the possibilities of this technique. In an effort to place it in the foreground of the dynamic analysis, while preserving its ease of use, the reduced model concept was introduced as an alternative to the black box approach [ROY 97, ROY 00, ROY 01a]. A reduced model is a modal representation of the substructure in the form of an equivalent finite element model. Moreover, it has physical DOF which make it possible to connect directly to the models of adjacent substructures and to recover the desired internal responses. A simple example of reduced model is the effective mass model defined in section 5.3.2. In the axial case, this model has only scalar elements of mass and stiffness, representing the effective masses, the frequencies and the residual mass of the substructure under consideration. It can be directly connected to its base by the corresponding physical DOF. Although it is of a great practical interest, the effective mass model applies only to the substructures with a fixed and statically determinate interface. By extending this concept to any number of connection DOF, we arrive at the fixed connection reduced model in Figure 8.8. uo Φok

Ψoc

mk I kk

I kk

m c~ m kk −1 L kc

qk

u c~

m c~c~ −1 Φc~c

uc

Figure 8.8. Fixed connection reduced model

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This type of representation, proposed by MacNeal [MAC 71] for the first time, combines physical and generalized coordinates linked to each other by scalar masses and stiffnesses and linear constraints at the same time. The physical coordinates are made up of connection DOF c and observation DOF o, while the generalized coordinates represent the modal behavior of the substructure in terms of normal modes and static modes. The scalar stiffnesses and masses represent the block-diagonal terms of mass and stiffness matrices of relation [8.31]. This requires the diagonalization of the terms M cc and K cc by the eigenvalue problem: (−ω c~2 M cc + K cc ) Φc c~ = 0

m c~c~ = Φ c~c M cc Φ cc~

[8.67]

diagonal matrix of generalized masses

Each rectangular box with rounded corners in Figure 8.8 represents linear constraints corresponding to various transformations between the physical and generalized coordinates, the independent coordinates being below and the dependent coordinates being above. For example, the linear constraints of type [8.35] recovering the observation DOF o starting from the independent DOF k and c are written: u o = Φok q k + Ψoc u c

[8.68]

As this representation by reduced model uses the same modes and independent coordinates as those of the Craig-Bampton method, it is strictly equivalent to this method and can be directly coupled to the adjacent substructures, reduced or not, as an equivalent finite element model. The reduced model can also be represented using the free connection DOF, as shown in Figure 8.9.

Modal Synthesis

285

uo

Φok

G os,res hc~ fc

− G cc,res −1 Φck mk

qk

h c~c~ −1 Φc~c G cc,res −1 uc

Figure 8.9. Free connection reduced model

In this case, the generalized coordinates correspond to the free interface normal modes and to the residual flexibilities of relation [8.42] with the mass and stiffness matrices of relation [8.43], completed by the diagonalization of the terms H cc, res and G cc, res : (G cc, res − ω c~2 H cc, res )Φ c c~ = 0

[8.68]

h c~c~ = Φc~c H cc,res Φcc~ diagonal matrix of generalized masses

Because this representation by reduced model uses the same modes as those of the Craig-Chang method, it is strictly equivalent to this method at the substructure level. However, the coupling by direct assembly requires the presence of connection displacements in the generalized coordinates, which assure the compatibility of the displacements without necessarily satisfying the equilibrium of the forces.

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Because modal synthesis using reduced models directly connects the substructures to their common interfaces, we can easily consider a hybrid coupling between the fixed connection models and the free connection models. We can also combine fixed connection DOF and free connection DOF for a given substructure. As an illustration, consider the case of Figure 8.6 with free connection DOF for each substructure. A comprises its rigid normal mode and its residual flexibility (therefore no truncation) and B, in a similar way, comprises its rigid normal mode and its residual flexibility. Figure 8.10 gives the corresponding reduced models, including the recovery relations for the observation DOF u1, u3 and u4.

A

B u2

1 0 u1

1

9/5

0

0

14/27

0

FcA

FcB

1 1

5/9

1

3 q1A

⎡ −1 / 9 ⎤ ⎢ − 4 / 9⎥ ⎣ ⎦

⎡u 3 ⎤ ⎢u ⎥ ⎣ 4⎦

9/5

q1B

⎡1⎤ ⎢1⎥ ⎣⎦

Figure 8.10. Reduced models relating to Figure 8.6

Table 8.6 gives the normal modes of the coupled system. With the same number of modes of the substructures, the errors are comparable to those of the CraigBampton method (or of the effective mass models) where only the compatibility of displacements is respected.

Modal Synthesis

Mode k

1

2

3

4

ω k2

0.0 (0%)

0.5902 (0.8%)

2.2640 (13.2%)

–

⎡+ 0.5⎤ ⎢+ 0.5⎥ ⎥ ⎢ ⎢+ 0.5⎥ ⎥ ⎢ ⎣+ 0.5⎦

⎡ − 0.6458⎤ ⎢ − 0.2646⎥ ⎥ ⎢ ⎢+ 0.2332⎥ ⎥ ⎢ ⎣+ 0.6772⎦

⎡+ 0.5770⎤ ⎢ − 0.7294⎥ ⎥ ⎢ ⎢ − 0.1723⎥ ⎥ ⎢ ⎣+ 0.3246⎦

–

(error) Φi k (m k = 1)

287

Table 8.6. Normal modes of the system in Figure 8.6 with the reduced models in Figure 8.7

8.5. Case study 8.5.1. Benfield-Hruda truss The truss in Figure 8.11, from Benfield-Hruda [BEN 71], is a relatively simple case but of a sufficient size for testing of performances of a modal synthesis method. It is composed of rod elements (section 3.3.2) in a plane. In order to simplify the definition of its geometry, the sides of each triangle are in a ratio (3, 4, 5). The physical properties and even the element formulation have little importance in a comparison between the methods.

A

B Figure 8.11. Benfield-Hruda truss

The first natural frequencies with various boundary conditions are given by the Table 8.7: A + B free making the reference, A or B free or fixed at the connection, A loaded by the connection condensed mass and stiffness matrices of B. The only DOF being the translations in the plane, the free structures have only three rigidbody modes.

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Table 8.8 provides the results of various methods and the same number of modes representing each substructure: 10 for A, 10 for B: – exact: exact calculation with the complete truss; – C-B: Craig-Bampton method with 6 junction modes and 4 fixed connection normal modes for A and for B; – C-C: Craig-Chang method with 6 residual flexibilities and 4 free connection normal modes for A and for B; – B-H: Benfield-Hruda method with 10 loaded or free connection normal modes for A, 6 junction modes and 4 fixed connection normal modes for B; – MR: method of reduced models with 6 residual flexibilities and 4 free connection normal modes for A and for B (same ingredients as for C-C). The following comments can be made on these results: – for all methods, the rigid-body modes are perfectly respected because they are contained in the chosen modes. Then, the errors, which are always positive as mentioned in section 8.2.1, have a tendency to increase with the mode number because they have increasingly complex shapes and are thus difficult to project on the chosen shapes, particularly on the static modes; – contrary to the other methods, Craig-Chang respects the equilibrium of the forces at the connection, which eliminates 6 generalized DOF. In relation to the Craig-Bampton method, the errors are sometimes lower, sometimes greater. Of course, if we had used 6 more modes to respect same number of modes in the end, the results would have been better; – the Benfield-Hruda method with loading is very good up to mode 9 where B must maintain a behavior close to that of the loading. The frequency then degrades due to decreasing representativity of the loading. Without loading, the errors are rather random even for the first modes since B is poorly represented; – the reduced models give better results than Craig-Chang in spite of the same ingredients because of fewer eliminated DOF which gives a system of larger size (while with a fixed connection they give the same results as Craig-Bampton). The fixed connection/free connection comparison shows no clear advantage of one or the other. These tendencies should apply to most industrial cases. However, certain applications may have characteristics that favor a particular method and we must be careful not to over generalize.

Modal Synthesis A+B free (Hz) 1 2 3 4 5 6 7 8 9 10 11 12 13 14

0. 0. 0. 10.4035 21.5874 27.8136 32.1721 41.4068 50.4581 53.2312 55.5072 62.2279 63.2527 66.7135

289

A free (Hz)

A loaded (Hz)

B free (Hz)

A fixed (Hz)

B fixed (Hz)

0. 0. 0. 24.8068 40.2028 47.8225 55.6214 61.6309 69.0954 79.6212 80.9171 85.5764 95.2630 105.7851

0. 0. 0. 13.9695 28.4230 32.6906 43.7530 53.2989 64.2878 71.2053 71.8728 81.3694 92.7730 102.6505

0. 0. 0. 32.5885 47.2285 57.2081 63.5358 68.5197 81.0047 88.0761 93.8960 102.9245 109.6432 119.1179

6.4514 20.1121 26.7025 39.2350 52.4295 63.6445 70.9946 73.2595 84.5600 96.0799 102.1987 105.5343 110.0704 117.5140

6.8934 26.1417 32.3537 48.8557 62.6108 75.4430 76.9894 94.3740 103.2231 110.1695 116.4430 117.7983 130.5449 139.0167

Table 8.7. Normal modes of the truss in Figure 8.6 with various boundary conditions Exact

(Hz)

C-B 6+4 6+4 (Hz)

C-C 6+4 6+4 (Hz)

B-H 10 loaded 6+4 (Hz)

B-H 10 free 6+4 (Hz)

MR (free) 6+4 6+4 (Hz)

0. 0. 0.

0. 0. 0.

0. 0. 0.

0. 0. 0.

0. 0. 0.

0. 0. 0.

A B 1 2 3 4

10.4035

5

21.5874

6

27.8136

7

32.1721

8

41.4068

9

50.4581

10

53.2312

11

55.5072

12

62.2279

13

63.2527

14

66.7135

10.4064 (0.03%) 21.5961 (0.04%) 27.8224 (0.03%) 32.3143 (0.44%) 41.7780 (0.90%) 53.2010 (5.44%) 58.9658 (10.77%) 76.8855 (38.51%) 96.9119 (55.74%) 139.3241 (120.27%) 208.0630 (211.88%)

10.4050 (0.01%) 21.5935 (0.03%) 28.0474 (0.84%) 32.5945 (1.31%) 49.1579 (18.72%) -

10.4045 (0.01%) 21.5964 (0.04%) 27.8622 (0.17%) 32.2643 (0.29%) 41.5125 (0.26%) 50.5767 (0.24%) 56.5318 (6.20%) 59.1541 (6.57%) 63.7833 (2.50%) 67.4293 (6.60%) 85.6977 (28.46%)

11.0780 (6.48%) 21.9316 (1.59%) 29.3628 (5.57%) 32.4374 (0.82%) 41.8854 (1.16%) 50.7794 (0.64%) 56.0707 (5.33%) 57.3306 (3.29%) 63.8736 (2.64%) 67.4160 (6.58%) 85.2645 (27.81%)

Table 8.8. Modal synthesis results with the case in Figure 8.6

10.4045 (0.01%) 21.5919 (0.02%) 28.0195 (0.74%) 32.4136 (0.75%) 45.3703 (9.57%) 60.3208 (19.55%) 62.6838 (17.76%) 74.7362 (34.64%) 97.6048 (56.85%) 139.7343 (120.91%) 218.2874 (227.20%)

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8.5.2. Industrial cases

Figure 8.12. Industrial case: seat and floorboard (with the authorization of PSA Peugeot Citroen)

In order to illustrate the possibilities of modal synthesis in an industrial context, let us consider the two models in Figure 8.12 relating to a car seat and a floorboard that will be treated by the reduced modeling techniques of section 8.4.5: – the seat comprises 4,293 nodes and 4,098 elements for a total of 25,758 DOF. Its connection with the floorboard consists of 4 nodes with 6 DOF considered to be fixed. The corresponding reduced model thus has 24 junction modes, completed by 15 normal modes below 300 Hz, hence 39 generalized coordinates; – the floorboard (half-model) comprises 4,830 nodes and 4,717 elements for a total of 28,980 DOF. Its 4 connection nodes with 6 DOF with the seat are considered free. Its peripheral nodes are linked to a rigid junction node with 3 DOF in the symmetry plane of the floorboard. The corresponding reduced model thus has 24 static flexibilities, completed by 63 normal modes below 300 Hz, to which we add three junction modes for the excitation, hence 90 generalized coordinates.

Modal Synthesis

291

(c) Seat modes (k)

(c) Floorboard modes

(k)

Seat + floorboard modes Figure 8.13. Seat and floorboard: contribution of the substructure modes to the strain energies of the structure modes

The coupled reduced models thus have 39 + 90 = 129 scalar stiffnesses and masses, completed by the linear constraints in order to recover 205 observation nodes. They provide 77 normal modes below 300 Hz. Table 8.9 compares the results with the exact solution in terms of frequency errors and in terms of MAC on the mode shapes (see section 12.4.2). The differences are small up to 230 Hz. The errors on the mode shapes then progressively degrade due to the truncation effects.

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Direct Mod. reduced Error MAC Mode Freq Mode Freq Freq(%) (%) --------------------------------------1 38.23 1 38.23 ( 0.0) (100.0) 2 45.99 2 45.99 ( 0.0) (100.0) 3 72.55 3 72.57 ( 0.0) (100.0) 4 74.00 4 74.13 ( 0.2) (100.0) 5 88.03 5 88.06 ( 0.0) (100.0) 6 97.59 6 97.61 ( 0.0) (100.0) 7 109.24 7 109.34 ( 0.1) (100.0) 8 116.43 8 116.63 ( 0.2) (100.0) 9 124.19 9 124.61 ( 0.3) ( 99.6) 10 130.71 10 130.79 ( 0.1) ( 99.9) 11 134.64 11 134.70 ( 0.0) (100.0) 12 140.34 12 141.47 ( 0.8) ( 99.4) 13 148.83 13 148.83 ( 0.0) (100.0) 14 158.17 14 158.39 ( 0.1) ( 99.2) 15 160.37 15 160.60 ( 0.1) ( 99.5) 16 164.07 16 164.07 ( 0.0) ( 99.9) 17 168.09 17 168.12 ( 0.0) (100.0) 18 168.93 18 168.95 ( 0.0) (100.0) 19 172.17 19 172.19 ( 0.0) (100.0) 20 175.89 20 175.95 ( 0.0) ( 99.8) 21 176.96 21 177.53 ( 0.3) ( 98.8) 22 182.27 22 182.47 ( 0.1) ( 99.2) 23 187.51 23 187.60 ( 0.0) ( 99.8) 24 189.99 24 190.28 ( 0.2) ( 98.5) 25 195.57 25 195.85 ( 0.1) ( 97.6) 26 197.13 26 197.23 ( 0.1) ( 98.5) 27 198.85 27 198.98 ( 0.1) ( 99.1) 28 201.72 28 201.81 ( 0.0) ( 99.0) 29 204.70 29 205.04 ( 0.2) ( 99.0) 30 206.58 30 206.68 ( 0.1) ( 99.9) 31 209.46 31 209.49 ( 0.0) ( 99.9) 32 211.01 32 211.04 ( 0.0) ( 99.9) 33 214.53 33 214.79 ( 0.1) ( 99.4) 34 216.26 34 216.32 ( 0.0) ( 98.7) 35 218.42 35 218.45 ( 0.0) ( 99.9) 36 223.10 36 223.14 ( 0.0) ( 93.7) 37 223.25 37 223.39 ( 0.1) ( 99.0) 38 223.95 38 223.97 ( 0.0) ( 99.9) 39 226.06 39 226.23 ( 0.1) ( 99.2) 40 227.16 40 227.30 ( 0.1) ( 98.4)

Direct Mod. reduced Error MAC Mode Freq Mode Freq Freq(%) (%) --------------------------------------41 229.36 41 229.49 ( 0.1) ( 98.4) 42 230.67 42 231.28 ( 0.3) ( 90.6) 43 236.61 44 243.91 ( 3.1) ( 23.1) 44 239.37 43 239.13 (-0.1) ( 54.7) 45 244.24 45 244.55 ( 0.1) ( 55.3) 46 245.53 46 247.36 ( 0.7) ( 43.8) 47 248.15 47 248.94 ( 0.3) ( 25.1) 48 249.33 48 249.58 ( 0.1) ( 93.3) 49 250.06 49 250.82 ( 0.3) ( 73.9) 50 251.34 50 253.79 ( 1.0) ( 21.7) 51 256.71 51 256.83 ( 0.0) ( 97.6) 52 258.09 53 259.40 ( 0.5) ( 85.9) 53 258.27 52 258.43 ( 0.1) ( 82.6) 54 262.08 54 262.76 ( 0.3) ( 68.0) 55 262.98 55 264.22 ( 0.5) ( 57.1) 56 265.92 56 266.11 ( 0.1) ( 97.1) 57 266.73 57 266.90 ( 0.1) ( 39.6) 58 267.09 58 267.46 ( 0.1) ( 86.9) 59 270.01 59 270.33 ( 0.1) ( 97.1) 60 272.22 61 273.55 ( 0.5) ( 68.7) 61 272.80 60 272.88 ( 0.0) ( 88.5) 62 274.02 62 274.18 ( 0.1) ( 96.6) 63 274.84 63 275.68 ( 0.3) ( 89.6) 64 277.74 64 277.92 ( 0.1) ( 98.9) 65 279.57 65 279.72 ( 0.1) ( 97.2) 66 281.58 67 283.28 ( 0.6) ( 25.2) 67 282.24 66 282.24 ( 0.0) ( 64.1) 68 284.05 68 284.62 ( 0.2) ( 52.6) 69 285.85 69 286.44 ( 0.2) ( 92.6) 70 287.64 70 287.87 ( 0.1) ( 91.4) 71 289.49 71 291.09 ( 0.6) ( 43.5) 72 291.44 72 292.89 ( 0.5) ( 43.0) 73 292.32 73 293.96 ( 0.6) ( 64.6) 74 294.58 76 297.28 ( 0.9) ( 13.8) 75 296.04 75 296.77 ( 0.2) ( 59.1) 76 296.40 74 296.53 ( 0.0) ( 62.0) 77 297.09 77 298.76 ( 0.6) ( 81.8)

Table 8.9. Industrial case: direct calculation/reduced model comparison

Figure 8.13 illustrates the contribution of each of the 129 normal modes and static modes of the substructures to the 77 normal modes of the assembled structure in terms of strain energies. We can see predominately seat modes and floorboard modes as well as highly coupled seat/floorboard modes. Figure 8.14 shows the cumulative effective transmissibilities between a junction DOF and a connection DOF along the same direction, as well as the corresponding FRF. The comparison with the exact solution shows two responses nearly identical at first (preserved static response), followed by increasing error in accordance with the results of Table 8.9.

Modal Synthesis

Figure 8.14. Seat and floorboard: effective and dynamic transmissibilities between junction and connection

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Chapter 9

Frequency Response Synthesis

9.1. Introduction Frequency response synthesis or impedance coupling, is a substructuring technique like modal synthesis that uses the FRF directly. Before discussing the subject, let us recall the principles of substructuring previously examined. The concept of dynamic substructuring was introduced in section 1.6, and mentioned again in section 8.1 during the discussion on modal synthesis. We saw that substructuring can be performed at various levels and in particular: – at the matrix level, by direct assembly of the mass, stiffness and damping matrices, possibly condensed beforehand: this approach can be prohibitive due to the size of the model or poor efficiency compared to modal synthesis; – at the modal level, by modal synthesis, the subject of Chapter 8: this approach is particularly efficient when it used in good conditions, but it may not be well adapted to all needs; – at the response level, by suitable manipulation of the FRF: this operation is similar to assembly, but must be performed at each frequency due to the frequencydependent properties involved, and therefore may result in lengthy computations. However, it is more direct than modal synthesis because it does not require the use of modes, and interfaces naturally with testing where the FRF are directly obtainable. In practice, it is a complementary approach compared to modal synthesis whose advantages will be described according to the cases. This is the subject of this chapter.

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The procedure is carried out in three steps similar to those in modal synthesis: – characterization of each substructure by its FRF in a given configuration; the DOF relative to the boundary conditions may differ from the connection DOF relative to the adjacent substructures. The choice is to be made depending on the configuration of the coupled substructure in addition to possible experimental constraints; – processing of the implied FRF in order to impose the displacement compatibility and force equilibrium at the connections between substructures and deduce the FRF of the coupled structure at these connections; – recovery at the interior of the substructures from the results at the connections. The formulation depends on the boundary conditions chosen for the characterization of each substructure, on their connectivity and on the typology of the excitations and desired responses, which represents many possible cases. It can be explicitly established case by case in the simplest situations. A typical case is that of two substructures where we are interested in the response of one of them while the other one is excited. It can also be the subject of a generalization with any number of substructures, excitations and responses, using different approaches. First we will discuss the FRF in section 9.2 in terms of their subsequent processing. In section 9.3 we will present the main principles of the coupling formulation, before discussing the basic cases in section 9.4 followed by the possible generalizations in section 9.5. We will finish in section 9.6 by a comparison of these approaches with the other substructuring techniques, matrix assembly and modal synthesis, in order to evaluate their advantages and disadvantages and to formulate some rules of use.

9.2. Frequency Response Functions 9.2.1. FRF and other dynamic characteristics Consider again the general relation [1.43] defining the FRF for a structure partitioned in internal DOF i and in junction DOF j, as illustrated by Figure 9.1: ⎡ u i (ω ) ⎤ ⎡ G ii (ω ) Tij (ω ) ⎤ ⎡ Fi (ω ) ⎤ ⎢F (ω )⎥ = ⎢− T (ω ) K (ω )⎥ ⎢u (ω )⎥ ji jj ⎣ j ⎦ ⎣ ⎦⎣ j ⎦

[9.1]

Frequency Response Synthesis

297

Fi , u i

i Fj ,u j

j Figure 9.1. Structure partition in internal DOF i and in junction DOF j

Relation [9.1] introduces the FRF in flexibilities G ii (ω ) , in transmissibilities Tij (ω ) and T ji (ω ) , and in stiffnesses K jj (ω ) , defined in terms of displacements, those relative to the velocities and to the accelerations being deduced as shown in Table 1.3. These FRF can be obtained analytically or experimentally. Obtaining the FRF experimentally is direct because, by definition, we only need to establish the connection between excitations and responses in the frequency domain. Analytically, they are linked to the other dynamic characteristics of the structure, i.e. the structural matrices and the normal modes, by the relations that follow, making it also possible to determine them by various means. Starting from the structural matrices we use relation [1.54] to deduce (without assuming symmetry):

(

G ii (ω ) = − ω 2 M ii + iω Cii + K ii

)

−1

(

) (− ω

(

ji

Tij (ω ) = − − ω 2 M ii + iω C ii + K ii T ji (ω ) = − − ω 2 M ji + iω C ji + K

−1

)(− ω

2

2

M ii + iω C ii + K ii

(

K jj (ω ) = −ω 2 M jj + iω C jj + K jj − − ω 2 M ji + iω C ji + K ji

(− ω

2

)

M ij + iω C ij + K ij

M ii + iω Cii + K ii

) (− ω −1

2

)

−1

[9.2]

)

M ij + iω Cij + K ij

)

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Starting from the modes that could be deduced from the structural matrices by solution of an eigenvalue problem, it is relations [4.51] to [4.53] that apply to the normal modes and relations [7.16] to [7.18] apply to the complex modes.

9.2.2. Transformation of the FRF The FRF of relation [9.1] are related to the internal DOF i and to the junction DOF j of the structure. This structure thus has by definition free DOF i and constrained DOF j. A modification of these boundary conditions will imply a modification of the FRF that we can easily deduce from their definition, i.e. from relation [9.1]. For example, by rendering the DOF j free, therefore with all the DOF g = i + j free, we obtain dynamic flexibilities G 'gg on all the DOF, whose expression, according to the preceding FRF, can be found by solving system [9.1] which gives:

⎡ u i ⎤ ⎡G ii + Tij K −jj1 T ji ⎢u ⎥ = ⎢ K −jj1 T ji ⎣ j ⎦ ⎢⎣ ⇒

⎡ G ii' ⎢ ' ⎢⎣G ji

Tij K −jj1 ⎤ ⎡ Fi ⎤ ⎥⎢ ⎥ K −jj1 ⎥⎦ ⎣F j ⎦

G ij' ⎤ ⎡G ii + Tij K −jj1 T ji ⎥=⎢ G 'jj ⎥⎦ ⎢⎣ K −jj1 T ji

Tij K −jj1 ⎤ ⎥ K −jj1 ⎥⎦

[9.3]

Similarly, by constraining all the DOF g = i+j, the dynamic stiffnesses K 'gg are obtained:

⎡ K ii' ⎢ ' ⎢⎣K ji

K ij' ⎤ ⎡ G ii−1 ⎥=⎢ K 'jj ⎥⎦ ⎢⎣− T ji G ii−1

− G ii−1 Tij

⎤ ⎥ K jj + T ji G ii−1 Tij ⎥⎦

[9.4]

Frequency Response Synthesis

299

Inversely, if we know the structure flexibilities G 'gg or stiffnesses K 'gg , we can deduce from them the mixed FRF:

⎡ G ii ⎢− T ji ⎣

Tij ⎤ ⎡ K ii'−1 ⎥=⎢ K jj ⎦ ⎢K 'ji K ii'−1 ⎣

− K ii'−1 K ij'

⎤ ⎥ K 'jj − K 'ji K ii'−1 K ij' ⎥⎦

⎡G ii'−1 − G ij' G '−jj1 G 'ji =⎢ − G '−jj1 G 'ji ⎢⎣

G ij' G '−jj1 ⎤ ⎥ G '−jj1 ⎥⎦

[9.5]

Finally, for completeness, the flexibilities G 'gg are obviously the inverse of the stiffnesses K 'gg (globally and not term by term) and vice versa:

⎡ K ii' ⎢ ' ⎢⎣K ji

K ij' ⎤ ⎡ G ii' ⎥=⎢ K 'jj ⎥⎦ ⎢⎣G 'ji

G ij' ⎤ ⎥ G 'jj ⎥⎦

−1

[9.6]

Relations [9.3] to [9.6] thus make the transformation of the FRF possible according to the boundary conditions under consideration; it is possible to deduce each case from one of the others: – the flexibilities from the free structure; – the stiffnesses from the completely constrained structure; – the mixed FRF from the partially constrained structure. We can of course envisage a more general case where the structure remains partially constrained but in a different way. Although more difficult to formulate, it does not pose any problem in particular.

9.2.3. Simple examples The 2-DOF system in Figure 9.2, with DOF denoted i and j in this context in order to directly apply the preceding equations, but without imposing boundary conditions, has the following FRF:

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⎡ K ii' ⎢ ' ⎢⎣K ji

K ij' ⎤ ⎡ G ii' ⎥=⎢ K 'jj ⎥⎦ ⎢⎣G 'ji

G ij' ⎤ ⎥ G 'jj ⎥⎦

−1

⎡− ω 2 m + i ω c + k =⎢ −iω c − k ⎣⎢

⎤ ⎥ − ω m + i ω c + k ⎦⎥ −iω c − k

2

[9.7] ⎡ G ii ⎢− T ji ⎣

iω c + k

⎤ ⎥ 2 −ω m + iω c + k ⎥ ⎥ −ω 2 m ⎥ ⎦

1 ⎡ Tij ⎤ ⎢ − ω 2 m + i ω c + k =⎢ iω c + k K jj ⎥⎦ ⎢ ⎢− 2 ⎣ −ω m + iω c + k

m

i c

k

m

j

Figure 9.2. 2-DOF system i and j for various boundary conditions

Similarly, the continuous rod in section 6.2 with its two endpoint DOF denoted j = 1 and i = 2, leads to the FRF of expressions [6.18], [6.22] and [6.26], which can be summarized: ⎡ K ii' ⎢ ' ⎢⎣K ji

K ij' ⎤ ⎡ G ii' ⎥=⎢ K 'jj ⎥⎦ ⎢⎣G 'ji

G ij' ⎤ ⎥ G 'jj ⎥⎦

−1

=

E S λ ⎡cos λ ⎢ L sin λ ⎣ − 1

−1 ⎤ ⎥ cos λ ⎦

[9.8] ⎡ G ii ⎢− T ji ⎣

⎡ L sin λ Tij ⎤ 1 ⎢E S λ ⎢ = ⎥ K jj ⎦ cos λ ⎢ −1 ⎢ ⎣

⎤ 1 ⎥ ⎥ ⎥ ES λ sin λ ⎥ − L ⎦

With these FRF available, we can use them in order to couple the structure under consideration to other structures themselves represented by their FRF.

Frequency Response Synthesis

301

9.3. Coupling by FRF 9.3.1. FRF necessary for coupling Each substructure taken separately, schematized by Figure 9.3, is subjected to external excitations, either acting directly on the substructure or coming from adjacent substructures. It is thus necessary to identify the FRF relative to all these DOF, i.e.: – the excitation DOF, denoted e (subscript attributed to the elements before, but without ambiguity here because only the substructures are taken into account); – the connection DOF, denoted c.

e

i

o

c

j before coupling

after coupling

Figure 9.3. DOF of a substructure

We should note that these DOF e and c implied in the coupling between the substructures are to be distinguished from the DOF i and j which correspond to the configuration of the substructure before coupling. A typical case is that the DOF e are a subset of the DOF i and that the DOF c are identical to the DOF j, but other situations can occur in practice, as we will see later. The FRF to be considered in order to couple the substructure are thus those relative to the DOF e and c. It should be noted that sets e and c are not necessarily disjoint, as some excitations can be applied on the connection DOF. These FRF will be called X which can be of any nature because the DOF e and c are different from the DOF i and j. The FRF X will be of type G, T, K (or derivative) according to the considered configuration for the substructure before coupling. We will see in the following section that the actual solution step of the coupling procedure requires in addition to the FRF Xce between excitations and connections, the FRF Xcc at the connections themselves that will be combined with those of the

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adjacent structures. This solution will directly give the responses at the connections of each substructure. In order to obtain the other desired responses, i.e. at the interior of each substructure on the DOF that we will describe as observation DOF o, it is necessary to recover them using the FRF between the excitations seen by the substructure at the DOF e and c and the responses at the DOF o (Figure 9.3), i.e. Xoe and Xoc. The DOF o may include the DOF e. In summary, if we denote excitation, connection and observation DOF as e, c and o respectively: – the solution step requires the FRF Xce and Xcc of each substructure and gives the responses at the connection DOF c; – the recovery step with the observation DOF o requires the FRF Xoe and Xoc. For example, in the typical case mentioned in the introduction with two substructures A and B connected by the DOF c where we are interested in the responses of the DOF o of A when B is excited at the DOF e, as indicated in Figure 9.4, the necessary FRF are: B A B – for the resolution and the responses at connections: X ce , X cc and X cc ; A – for the recovery on A : X oc .

A o

c e B Figure 9.4. Simple example of coupling

Frequency Response Synthesis

303

9.3.2. Solution of the coupling We assume here that each substructure S is represented by the FRF mentioned in S S the preceding section for the solution step, i.e. X ce and X cc , to be deduced from

the FRF X Sgg of the configuration before coupling using the processing presented in section 9.2.2. By definition, these FRF relate the forces F and motions u, u or ü for each substructure taken separately, in the same way as in relation [9.1]. By using the same principles as in section 8.2.2 for modal synthesis, the coupling between substructures imposes two types of relations on the forces and the motions at the connections: – displacement compatibility: when two DOF of two adjacent substructures A and B have the same displacements we can schematically write:

uA = uB

[9.9]

This relation can be generalized to several DOF of A rigidly linked to several DOF of B, generating linear constraints between these DOF. We thus arrive at a certain number of linear constraints between DOF, possibly including reference frame changes and thus expressing the compatibility of the motions between two or several substructures; – force equilibrium: at these two same DOF, if F A , F B and F respectively denote the force exerted by B on A, the force exerted by A on B, including a possible external force, we can schematically write:

FA +FB +F =0

[9.10]

This relation can be generalized in the same way as for the displacements, generating as many linear constraints between the forces and thus expressing the equilibrium of forces between two or more substructures. These two types of relations [9.9] and [9.10], along with the relations between forces and displacements for each substructure, will provide as many equations as unknowns to solve.

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Considering again the case in Figure 9.4, FRF with only flexibilities, for example, lead to: – relation with the FRF of A:

A u cA = G cc FcA

– relation with the FRF of B :

B B u cB = G ce FeB + G cc FcB

– displacement compatibility:

u cA = u cB

– force equilibrium:

FcA + FcB = 0 c

from which we obtain the responses at the connections as a function of the excitation FeB , which can be written as: A+ B B FcA = − FcB = Tce Fe

u cA

= u cB

A+ B = G ce

FeB

(

with

A+ B A B Tce = G cc + G cc

with

A+ B G ce

A = G cc

(

)

−1

B G ce

)

[9.11]

B −1 B + G cc G ce

A G cc

A+ B A+ B Tce and G ce represent respectively the transmissibility and flexibility FRF of (A +B) between the excitation DOF e and the connection DOF c.

9.3.3. Recovery

Recovery at the observation DOF of each substructure S is easily carried out with S S the FRF X oe and X oc .

Using the case in Figure 9.4 and the flexibility FRF, we have a relation relative to the recovery in A: A u oA = G oc FcA

from which we obtain the responses at the observation DOF due to excitations FeB , according to result [9.11]: A+ B B u oA = G oe Fe

with

(

A+ B A A B G oe = G oc G cc + G cc

)

−1

B G ce

[9.12]

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A+ B G oe represents the flexibility FRF of (A+B) between the excitation DOF e and the observation DOF c. In relation [9.12] we see how the FRF of each substructure intervene.

9.3.4. Summary

The three steps of the process can be summarized as follows: 1) characterization of each substructure S before coupling by its FRF at the excitation DOF e, connection DOF c and observation DOF o: S S - for the solution: X ce and X cc , S S and X oc ; - for the recovery: X oe

2) solution at the connections using the relations resulting from: - displacement compatibility shown in [9.9], - force equilibrium shown in [9.10]; 3) recovery within each substructure using the results at the connections. A A B Thus, for the simple case of Figure 9.4 (without X ce , X oe and X oe ), the result is given by relation [9.12] using the flexibility FRF for A and B, i.e. assuming the two substructures are free before coupling. With other boundary conditions, we will obtain for example:

– A and B constrained at the connection DOF:

(

A+ B A A B G oe = Toc K cc + K cc

)

−1

B Tce

[9.13]

– A constrained at the connection DOF, B free:

(

A+ B A B A G oe = Toc I cc + G cc K cc

)

−1

B G ce

[9.14]

We can make the following notes about results [9.12] to [9.14] which will appear again in the section on generalization: – the results are logical in that the terms go from excitations to responses passing through connections, as indicated by the successive subscripts;

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– at the connections, the properties of A and B are added to each other, whether they are flexibilities or stiffnesses according to the conditions before coupling; – the solution always inverts a matrix whose dimension is equal to the number of connection DOF. This unavoidable truth illustrates the note made in the introduction of Chapter 8 about the influence of the size of the connections on the efficiency of the process; – if relations [9.12] to [9.14] are strictly equivalent from an analytical point of view, they express very different realities in practice whether related to analysis or test, which will be discussed later. We can verify these relations on simple cases such as those of section 9.2.3. For example, we find the FRF of a continuous rod cut in two using the FRF of its two halves due to the properties of the trigonometric functions.

9.4. The basic cases 9.4.1. Introduction

Among the numerous couplings that we can imagine between substructures, the case in Figure 9.4 given as a simple illustration is actually a very common and instructive basic case. The pair “exciting structure B/excited structure A” or “support/equipment” is found at various levels in many domains. For example in the space domain, it is launcher B, which, excites by its engines, its satellite A. In turn, the satellite excites its subsystem, which then excites its equipment, etc. At each level, we must analyze the behavior of the “equipment” coupled to its “support”, according to the available analytical and/or experimental data. Among the various possible representations of the two substructures A and B by their FRF before coupling, those leading to relations [9.12] to [9.14] represent typical situations that will be discussed now.

9.4.2. Free substructures at the connections

Analyzing the free substructures before coupling at the connections (Figure 9.5) is a typical situation in the experimental world where the excitations are often generated by forces and the responses are often measured by accelerometers, hence FRF in accelerances giving the dynamic flexibilities by dividing by (−ω 2 ) . This leads to freeing the connections rather than constraining them. It should be noted

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that each substructure may also be constrained elsewhere for different reasons without having to consider the constraint reactions. These support conditions are independent of the coupling problem. If there is no support, in order to avoid the local flexibility problems, which may be unavoidable in practice, we can suspend the substructure in a suitable way in order to properly simulate free conditions. These aspects will be discussed in Chapter 11. In this case of boundary conditions before coupling, it is relations [9.11] and [9.12] that apply. The latter can be rewritten as:

(

A A B u oA = G oc G cc + G cc

A

)

−1

u cB

with

B u cB = G ce FeB

[9.15]

A

u oA

u oA u cB

u cB F eB

A B G cc + G cc

B

B before coupling

equivalence by operating deflection shapes

Figure 9.5. Free substructures at the connections

The excitation by force FeB is equivalent to the imposed motion u cB , which is the motion at the connections of substructure B, the only one excited by forces FeB . Thus, when forces FeB are not explicitly known, they can be represented by displacements u cB that they cause on the free substructure. u cB is sometimes called the operating deflection shape. This is the case of an engine for which we can replace its internal excitation forces, which are difficult to determine, by its operating deflection shape. The operating deflection shape is applied to the connections of structure A after its connection flexibilities have been added to those of B, as indicated by relation [9.15] and illustrated by Figure 9.5.

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9.4.3. Substructures constrained at the connections

This is the opposite of the preceding situation where the substructures are constrained at the connections before coupling (Figure 9.6), and generate stiffnesses instead of flexibilities at the connections. It is not as well adapted to testing, but has interesting analytical aspects. For these boundary conditions before coupling, relation [9.13] applies. By using accelerations instead of displacements, relation [9.13] applied to the connection DOF can be rewritten as:

(

A B u cA = M cc + M cc

)

−1

FcB

with

B B FcB = Tce Fe

[9.16]

Newton’s law appears very clearly in relation [9.16] which states that the masses at the connections of the two substructures are added to each other and that (A + B) is excited by equivalent forces FcB , as illustrated by Figure 9.6. These are the forces that substructure B, the only one excited by forces FeB , exerts at the connections when they are constrained. Thus, when forces FeB are not explicitly known, they can be represented by the equivalent forces FcB , which they produce on the constrained substructure. FcB is sometimes called “constraint force”. This is again the case for an engine for which the unknown excitation forces are replaced by its constraint forces at the connections. It should be noted that for a connection by a statically determinate node, the dynamic masses of each substructure may be directly represented by the effective mass models introduced in section 5.3.2. The coupling problem reverts to coupling the two effective mass models at the common node, itself excited by the equivalent forces previously determined by constraining substructure B, as illustrated in Figure 9.6. The responses of substructure A are recovered from the responses of the effective masses as indicated by relation [5.55].

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A u oA

A u oA FcB

FcB F eB

A B M cc + M cc

⇔

FcB …

…

B

B before coupling

equivalence by constraint forces

Figure 9.6. Constrained substructures at the connections

9.4.4. Mixed conditions at the connections

If we consider before coupling substructure A constrained at the connection DOF and free substructure B (Figure 9.7), we find an intermediate situation that conditions the representation of each substructure according to its role in the coupling. Consider for example a small machine with little influence on its support except at the connections. This is the typical of a satellite and its launcher. Before coupling, the satellite is considered to be fixed at the interface while the launcher is considered to be free. For these boundary conditions before coupling, it is relation [9.14] that applies. The concept of operating deflection shapes, mentioned in section 9.4.2 remains valid, the deformed shape is altered at the connections by the influence of substructure A, as relation [9.14] indicates, rewritten at the connection:

(

B A u cA = I cc + G cc K cc

)

−1

u cB

with

B u cB = G ce FeB

[9.17]

A variant of this strategy consists of loading substructure B by a reference substructure with a stiffness K 0cc = −ω 2 M 0cc which represents more or less substructure A. This substructure can be a simple rigid mass or a more complex structure if available. This is the strategy of substituting A with 0 which makes it

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possible for B loaded by 0 to converge to its behavior with A, all the more as 0 is close to A. Relation [9.17] thus becomes:

(

B +0 A u cA = I cc + G cc (K cc − K 0cc )

)

−1

u cB

with

B +0 B u cB = G ce Fe

[9.18]

A A

u oA

u oA u cB u cB

0 F eB

B +0 A I cc + G cc (K cc − K 0cc )

B

B before coupling

equivalence by operating deflection shapes

Figure 9.7. Mixed conditions at the connections

9.5. Generalization 9.5.1. Introduction

With the basic cases, the preceding section illustrated the formulation case by case according to the choice of configurations before coupling. When the number of the substructures increases, this formulation becomes difficult. It is then possible to resort to a more general approach leading to an automated implementation. This is the case of the two approaches described below, one of them based on dynamic stiffnesses, the other one based on dynamic flexibilities, thus opposite and complementary at the same time. They are strictly equivalent from an analytical point of view, but here again, their application domain is very different, as will be discussed later. For each of the two approaches, if necessary, we will begin by transforming the available FRF for each substructure, according to the processing presented in section 9.2.2:

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– for the stiffness approach, all FRF will be converted into stiffnesses using relation [9.4], thus requiring, at this level, an inversion of dimension equal to the number of DOF considered to be free initially; – for the flexibility approach, all FRF will be converted into flexibilities using relation [9.3], thus requiring, at this level, an inversion of dimension equal to the number of DOF considered to be constrained initially. This is the price to pay for the generalization, with the choice of the approach depending on the particular circumstances. We will keep the notation c for the connection DOF in the following and we will use s (selection) for the other substructure DOF implied by the excitations or the desired responses, s = e + o.

9.5.2. Stiffness approach

We assume here that we have at our disposal all the dynamic stiffnesses of each substructure before coupling, at the excitation, connection and observation DOF. The stiffness approach is very easy to understand as is the same as that applied in section 3.2.4 for the assembly of the element matrices. This assembly expressed the equilibrium of forces for each DOF, either for the inertial forces relative to the masses, the dissipation forces relative to the viscous damping or the elastic forces relative to the static stiffnesses. In the present context, this approach applies to: – the substructures instead of the elements, but it is the same assembly strategy; – the dynamic stiffness instead of the static stiffness, but it is the same equilibrium of forces that is expressed. Thus, with the example in Figure 9.4, and the stiffnesses K (c + s )(c + s ) of the two substructures, we obtain the stiffnesses of the set using the traditional assembly: A ⎡K ss ⎢ A ⎢K cs ⎢ 0 ⎣⎢

A K sc A B + K cc K cc B K sc

0 ⎤ ⎡u sA ⎤ ⎡FsA ⎤ ⎥ ⎢ ⎥ B ⎥⎢ K cs ⎥ ⎢ u c ⎥ = ⎢ Fc ⎥ B ⎥⎢ B⎥ ⎢ B⎥ K ss u F ⎦⎥ ⎣⎢ s ⎦⎥ ⎣⎢ s ⎦⎥

[9.19]

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The solution of this system provides all the displacements of A and B for any of the excitation forces on A and/or B. It requires the equivalent of three inversions of size equal to the number of DOF c, sA and sB, in addition to any inversions required for determining the substructure stiffnesses. We see that this approach requires many inversions, due to the transformations between stiffness and flexibility, to be performed frequency by frequency. Although conceivable in an analytic context, it is much less so in an experimental context. Other than this difficulty, the processing of the FRF is performed without any particular problems.

9.5.3. Flexibility approach

We assume here that we have at our disposal all the dynamic flexibilities of each substructure before coupling, at the excitation, connection and observation DOF. These flexibilities can be regrouped in order to form the flexibility matrix relative to the set of implied DOF, denoted g (global). Contrary to the stiffness approach, this set is the sum of the DOF before coupling and not their union since the connections have yet to be defined. We can thus write with the substructures A, B… before coupling, and by noting c + s = i because all the DOF are now of internal type: ⎡u iA ⎤ ⎡G iiA ⎢ B⎥ ⎢ ⎢u i ⎥ = ⎢ 0 ⎢ ⎥ ⎢ 0 ⎢⎣ ⎥⎦ ⎢⎣

0 G iiB 0

0 ⎤ ⎡FiA ⎤ ⎥⎢ ⎥ 0 ⎥ ⎢FiB ⎥ ⎥⎢ ⎥ ⎥⎦ ⎢⎣ ⎥⎦

or

u g = G gg F g

[9.20]

In general, the connections can be represented by linear constraints denoted m (see section 3.2.5), between the DOF c of each substructure, as mentioned after the basic relation [9.9]: C mg u g = 0 m

[9.21]

After coupling, the FRF relative to the DOF g are deduced from relations [9.20] and [9.21]. Using the reasoning of section 3.2.5.3, the problem is equivalent to the minimization of the total potential energy with subsidiary conditions:

Frequency Response Synthesis

V=

1 T u g G gg −1 u g − u g T F g 2

C mg u g = 0 m

and

313

[9.22]

that we can rewrite using the Lagrange multipliers λ: V =

1 T u g G gg −1 u g − u g T F g + (C mg u g ) T λ m 2

[9.23]

The extremum conditions of V give: ∇ u g (V ) = G gg −1 u g − F g + C gm λ m = 0

or

⎡G −1 ⎢ gg ⎢⎣ C mg

∇λ m (V ) = C mg u g = 0

C gm ⎤ ⎡ u g ⎤ ⎡ Fg ⎤ ⎥⎢ ⎥=⎢ ⎥ 0 mm ⎥⎦ ⎣λ m ⎦ ⎣0 m ⎦

[9.24]

The elimination of the Lagrange multipliers that can be interpreted as conjugated forces for the linear constraints, leads to the relation verified by u g after coupling: u g = G gg Fg

with

(

G gg = G gg − G gg C gm C mg G gg C gm

) −1 C mg G gg

[9.25]

The forces exerted after coupling on each substructure considered separately are found due to relation [9.20]: Fg = G gg −1 u g = Tgg F g

with

(

)

Tgg = I gg − C gm C mg G gg C gm −1 C mg G gg

[9.26]

with The following notes can be made on relations [9.25] and [9.26]: – in general they are highly redundant because in the case of equality constraints such as [9.9], the flexibilities on the two DOF will be the same (and with opposite forces in the absence of external force). By selecting the excitation DOF e and the observation DOF o, we obtain the reduced relations:

(

G oe = G oe − G og C gm C mg G gg C gm

) −1 C mg G ge

[9.27]

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(

Toe = I oe − C om C mg G gg C gm

) −1 C mg G ge

[9.28]

– this solution involves a single matrix inversion of size m, m being the number of linear constraints, or the number of connections, which is the minimum, and joins the preceding remarks on the efficiency of the substructuring. With the example in Figure 9.4, and flexibilities G ii of the two substructures, the application of relation [9.27] leads to:

[

u g = u oA

u cA

⎡ u oA ⎤ ⎡ ⎢ A B⎥ ⎢ ⎢u c = u c ⎥ = ⎢ ⎢ uB ⎥ ⎢ e ⎦⎥ ⎣ ⎣⎢ ⎡… ⎢ A+ B G gg = ⎢ … ⎢… ⎣⎢

u cB

u eB

A+ B G gg

]

T

[

C mg = 0 oA

⎤ FoA ⎤⎡ ⎢ ⎥ F A = −F B ⎥ c ⎥ ⎥⎢ c B ⎥ ⎥⎦ ⎢ Fe ⎦⎥ ⎣⎢

A A B −1 B G oc (G cc ) G cc + G cc A A B −1 B G cc (G cc + G cc ) G cc B A B −1 B G ec (G cc ) G cc + G cc

I cA

− I cB

0 eB

] [9.29]

A A B −1 B G oc (G cc ) G ce ⎤ + G cc ⎥ A A B −1 B G cc (G cc + G cc ) G ce ⎥ B B A B −1 B ⎥ G ee (G cc ) G ce ⎥ − G ec + G cc ⎦

In particular we again find the result [9.12] which gives the flexibilities between the excitation DOF of B and the observation DOF of A, i.e., the term in row 1, column 3.

9.5.4. Comparison of the two approaches

The two preceding approaches differ first by the matrix inversions to be performed. The flexibility approach minimizes them by inverting only the matrix of the flexibilities expressed on the connections while the stiffness approach performs several stiffness-flexibility conversions. In an analytical context, the latter would be interesting in the case of a large number of small size substructures, and this is the same as with the finite element approach. Conversely, we will favor the flexibility approach for a small number of large size substructures.

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In an experimental context that will be explained in more detail in Chapter 11, the following considerations must be taken into account concerning the FRF: – regarding the excitation, it is easier to control the forces than it is to control a motion. In order to apply a force on a given translation DOF, we only need to use a small size vibrator equipped with a flexible stinger in order to target the desired DOF and thus minimize the impact on the others. In order to impose a motion at a node along a given DOF, a good guiding system is required to limit the parasitic motion on the other DOF of the excitation node; generally, this can be done only with a large-size vibrator or shaker table on which the specimen is placed. – regarding the responses, measuring a motion is easier than measuring a reaction force. In order to observe a motion following a given translational DOF, we can use a small-size accelerometer, as well as other possible devices. In order to measure a force, a special device placed in series between the imposed motion and the structure can be used provided that it does not perturb the configuration, which may not be easy. We can also resort to the use of calibrated strain gauges or the parameters of the electrodynamic vibrator, but the implementation is often delicate or inaccurate. Thus, in general, it is easier to determine flexibilities experimentally (by accelerations) than transmissibilities or stiffnesses especially for statically indeterminate junctions, which goes against the stiffness approach. Note however, that for a statically determinate junction, it is relatively easy to determine the transmissibilities of a specimen on a vibrator and, to a lesser extent, its dynamic mass for the translational DOF assuming that the parasitic motion is small – in which case we can exploit the mixed approach of relation [9.14] and of section 9.4.4. However, the pure stiffness approach is not very compatible with testing. On the other hand, the flexibility approach is the obvious choice, not only for the availability of FRF but also for the minimum amount of inversions that it requires because the quality of the inversion is affected by the presence of noise in the matrix. Let us mention two examples of applications of these approaches that have already been the subject of various developments in industrial contexts: – the stiffness approach: calculation of a truss structure of continuous beams. Each element is considered continuous, and is thus represented at its extremities by the dynamic stiffnesses presented in sections 6.3.4 and 6.3.5. The assembly is the same as for the finite elements. We can thus determine the frequency responses at the nodes of the truss for any excitation in force or imposed motion. The advantage compared to the finite elements is that there is no discretization by interpolation in the element: the displacement field is exact. We can thus shift the calculation to the high frequencies without any discretization error as long as the beam model remains

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valid. The disadvantage, independent of the frequency resolution aspect discussed later, is that the extension to plate or shell structures is not directly possible because of the continuous character of the interface between elements which is incompatible with the FRF concept. We can thus only analyze beam trusses in a simple manner; – the flexibility approach: coupling of the structures represented by their dynamic flexibilities obtained analytically or by test. The rigid connections are handled by linear constraints between DOF. Flexible connections can be processed like substructures although they must be handled carefully. This is well suited for the introduction of a certain type of localized non-linearity as will be seen in Chapter 10. An example of industrial application of the flexibility approach is given in Figure 9.8 with the model in Figure 5.12. The model has already been the subject of an illustration in section 7.5.2 which discussed the complex modes using localized damping mechanisms represented by 1-DOF systems. This situation is well suited to a calculation by frequency response synthesis: the structure is represented by its FRF at the excitation DOF, response DOF and the damping mechanism locations (limited in number). Therefore, we only need to represent each damping mechanism by its dynamic flexibilities and to couple everything as indicated in section 9.5.3. Figure 9.8 compares one of the responses before and after coupling, which makes it possible to appreciate the attenuation with damping (the response after coupling is nearly the same as that of Figure 7.5 obtained with the complex modes, since the modal truncation effects are negligible).

Figure 9.8. FRF for the model in Figure 5.12 — with damping mechanism, · · · · without damping mechanism

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9.5.5. Particular cases

Certain situations, particularly in an experimental context, lead to a unique structure being considered, but which is perturbed in one way or another by external factors. We can then envisage assimilating the perturbations to a second structure that we will couple to the first using the preceding approach. The following cases illustrate this strategy and lead to the indicated results: – constrained structure. This is the case of a free structure, which is thus represented by its flexibilities. We wish to constrain some DOF of these flexibilities. This is also the case, common in an experimental context, of a poorly clamped or guided structure: if we note a significant motion on the DOF that we wanted to constrain, it is because the constraint is imperfect (see section 11.2.3). If we start from the FRF G oe of the poorly constrained structure and from its flexibilities G cc at the poorly constrained DOF c (grounded connection), we can deduce the FRF G oe of the perfectly constrained structure by applying for example relations [9.27] and [9.28] with u c = 0 (expressed in C mg ): G oe = G oe − G oc G cc −1 G ce

and

Tce = G cc −1 G ce

[9.30]

We thus find results [9.5] with c = j, which, in general, make it possible to transform internal DOF into junction DOF; – freed structure. this is the opposite of the preceding case, which also appears in an experimental context for a structure perturbed by a suspension that we wish to release or eliminate completely. In this case we can use relation [9.3] assuming we can determine the stiffness matrix of the suspended structure at the suspension DOF; – locally perturbed structure. A loading applied to some internal DOF of the structure modifies all its FRF. If the loading ∆ on the DOF c (connection with the loading) is characterized by G ∆cc , the flexibilities G oe of the loaded structure are deduced from those of the non-loaded structure G oe by:

(

G oe = G oe − G oc G cc + G ∆cc

)

−1

G ce

[9.31]

Simple examples: loading by a rigid mass, where G ∆cc = (−ω 2 M∆cc ) −1 with M ∆cc representing the rigid body mass matrix; and loading by a static stiffness

where G ∆cc = (K ∆cc ) −1 with K ∆cc representing the static stiffness matrix.

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Note that, unlike the preceding case, if we wish to remove a part of the structure instead of adding a part, we can simply consider the FRF of the part to be removed with the opposite sign (or with a phase difference of π). Thus, in order to remove a part ∆ characterized by its FRF G ∆cc from the structure under consideration, equation [9.31] becomes:

(

G oe = G oe − G oc G cc − G ∆cc

)

−1

G ce

[9.32]

The preceding case of the suspension can be treated this way if we know how to determine its contribution in mass and stiffness to the concerned DOF. This strategy can be generalized in the case of a structure A including structure B in order to characterize the structure A-B. In a way, this operation can be considered as uncoupling two substructures. Whatever the coupling strategy, we only need to consider the opposite of the FRF for B. In practice that requires some precautions in order not to remove more than what is contained in structure A, which would lead for example to the introduction of negative masses. Analytically, attention should be paid to the round off errors; experimentally, attention should be paid to measurement errors. Theoretically, all of A can be removed, but the more that is removed, the greater the risks will be.

9.6. Comparison with other substructuring techniques

In this section, we will look again at the various types of substructuring in the light of the developments already presented in order to make a final comparison.

9.6.1. The matrix level

The matrix level by direct assembly of the mass, stiffness and damping matrices can be subdivided into two types: – assembly of physical matrices resulting from the modeling by finite elements or by other elements. This is the basic technique that should be reserved for structures that are not very modular or for the small size matrices. The technique is simple, general and no information is lost. On the other hand, it can lead to very large size matrices for complex structures resulting in heavy computations; – assembly of condensed matrices after selection of the physical DOF representative of the global behavior. The information used for each substructure is

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easily obtained in a form easy to manipulate and communicate. On the other hand, there is a loss of information that can generate significant errors for complex shapes, and this limits its efficiency to medium frequencies. The behavior at the nonretained DOF, which may be desirable, requires additional matrices to be associated with the condensed matrices in order to recover the responses at the non-retained DOF from the selected DOF. The efficiency of the computation, i.e. the accuracy with respect to the computational effort, is mediocre compared to modal synthesis. In conclusion, it is a simple technique which has its limitations and requires a certain know-how. In summary, the matrix assembly approach has a simple implementation and may be used in cases that do not require high performances. It should not be used as a substructuring technique for handling complex structures in a modular and efficient manner.

9.6.2. The modal level

Modal synthesis with the careful use of modes of each substructure is a particularly efficient technique if it is used well, as presented in Chapter 8, whatever the implementation may be. The choice of modes is very important because the source of errors lies in the unavoidable truncation of the modes, but it can be more or less automated. If we want to retain the practical aspects particularly, we can distinguish: – the use of effective mass models is a very interesting specific case of modal synthesis when it can be applied, i.e. for statically determinate connections. It makes it easier to understand phenomena using the physical interpretation that it allows; – the Craig-Bampton method which is the mathematical expression of the effective mass models with the use of the fixed connection normal modes. It is the most widely used method due to its simplicity and good convergence. Its drawbacks include increased computational effort when there are a large number of connection DOF and the experimental incompatibility with respect to the constrained DOF; – other methods, using other types of modes, that can be more efficient and/or present a better experimental compatibility according to the cases. It is always difficult to control the truncation errors well. Whichever method is chosen, the calculation of modes is always required and can become time-consuming as their number increases. This is thus reserved for low frequencies. On the other hand, once the modes are available, the calculation is very rapid and enables a good comprehension of phenomena because the modes of the coupled structure may be interpreted as combinations of the modes of the

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substructures hence the possibility of quantifying their contribution. The recovery of the responses on the physical DOF from the modal responses must be performed systematically and is thus an integral part of the method. Compared to the assembly of the condensed matrices, for the same number of selected DOF, modal synthesis is clearly more efficient in the entire frequency band considered because the modes provide more information in low frequencies than the static shapes associated with the physical DOF retained in the condensation. Compared to frequency response synthesis, everything depends on the number of modes necessary in the band. A small number of nodes will give a certain advantage from the analytical point of view. However, in an experimental context, the normal modes are not essential to coupling, yet require an important effort to identify them experimentally, which can be interpreted as a drawback.

9.6.3. The frequency response level

The manipulation of the FRF enabling frequency response synthesis avoids the use of the modes that in the end are only calculation intermediaries. The resulting advantages are the following: – the coupling is performed directly by simple matrix calculation without any truncation problems at this level. The implementation is thus simplified. It should be noted that this advantage is not preserved if the FRF of the substructures are obtained by mode superposition; – this technique is the closest to the test environment because the FRF, by definition, are the first characteristics that can be measured; – its application range is larger because it allows the physical properties to be taken into account depending on the frequency, as for the viscoelastic materials, or some non-linearities as will be seen in Chapter 10. In this case the approach becomes a necessity (except when using approximate methods, as mentioned in section 4.4). On the other hand, contrary to the modes that are of a discrete nature enabling mode superposition, the FRF are of a continuous nature from the start and require a discretization in frequency to be implemented numerically. The matrix computations must be performed frequency by frequency and this involves various difficulties: – first, with the computation effort: at each given frequency, it is necessary to solve at least one linear system of size equal to the number of connection DOF, as indicated for example by relation [9.27]. When we consider a relatively large frequency band requiring several hundred frequency points for a suitable definition, the calculation can become very time consuming;

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321

– next, the discretization in frequency, if not too refined in order to avoid the previous difficulty, which may lead to truncated results, particularly at the resonances where the peaks, according to their sharpness, can be missed, since we do not know their frequencies a priori; – finally, the results cannot be directly interpreted as with the modes: the appearance of a particular peak for example cannot be attributed to a particular substructure property. We can only see the results, hence a limited understanding of the phenomena. Thus, in relation to the modal synthesis, the frequency synthesis is more general and closer to the test environment, and this enables it to analyze more problems. Conversely, we will favor modal synthesis when used under ideal conditions such as for linear structures, with relatively small number of well defined modes, because of its simplicity and efficiency.

9.6.4. Conclusion

When substructuring is necessary in relation to the size of the problem to be solved with a sufficiently modular architecture, we will first try to determine whether all the conditions are present for considering modal synthesis, the modes being of analytical or of experimental origin. In the case of frequency dependent properties disturbing the modal approach, we will verify if, by using the approximations, we can eliminate them in a convenient way (see section 4.4). In the case of major difficulties, too many modes, poorly identified modes or other problems, we will use frequency response synthesis whose range of application is larger.

Chapter 10

Introduction to Non-linear Analysis

10.1. Introduction A system is non-linear if it loses one of the properties that are characteristic of the linear behavior (see section 1.1): – the response to a sum of excitations is no longer the sum of the responses to each excitation and thus we can no longer analyze each excitation separately; – the response is no longer proportional to the excitation: the ratio between excitation and response is no longer constant, and this in particular destroys the Frequency Response Function (FRF) concept in the frequency domain. The analysis of such a system is thus made much more delicate and therefore the means of performing an equivalent linear analysis should be explored whenever possible. In practice, this is the case when the degree of non-linearity is small and/or when the non-linearities are highly localized in a structure which is linear elsewhere. On the other hand, the presence of strong non-linearities can make new phenomena appear such as sudden shifts in equilibrium, making non-linear behavior very different from linear behavior and the analysis more difficult. This chapter is only an introduction to the subject in relation to the developments presented in the previous chapters. The main objective of the chapter is to familiarize the reader with the characteristics of non-linearity on basic cases and present some approximate solutions via linear analysis; these approximate solutions can prove to be sufficient for many industrial applications. Thus, the problems linked to the strong non-linearities, particularly to stability problems, and specific

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solution techniques will not be developed. For more details, see [HAR 61], [TIM 74] and [GEN 95] that dedicate a chapter to the subject. The most traditional non-linear systems are introduced in section 10.2. The 1DOF system, always at the basis of the analysis, is then discussed in section 10.3. Finally, the N-DOF systems are tackled in section 10.4 without being too general.

10.2. Non-linear systems 10.2.1. Introduction Non-linearity and their effects on the structure are very diverse. We can make a first classification by considering the system behavior in relation to a periodic excitation. If the response is not periodic, it is chaotic, in the mathematical sense of the term, and this can involve difficult calculations in the time domain. In the following we will limit ourselves to periodic response systems for periodic excitations. Regarding sources of non-linearity, let us mention more particularly: – large displacements; – links, such as play, stops, bearings, etc.; – dry friction; – material non-linearities. In order to better understand the nature of these phenomena and the equations, which describe them, let us consider a simple connection between two DOF without mass, as shown in Figure 10.1. If this connection is linear with a spring of stiffness k and a constant damper c, it generates a restoring force ϕ between the two DOF. This force is expressed according to the relative displacement x = u 2 − u1 by:

ϕ = k x+c x

[10.1]

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325

A non-linear connection between the two DOF will generate a restoring force

ϕ ( x, x) in a more general manner. Subjected to a relative motion of the form x max sin ω t , the force will evolve as a function of x as indicated by Figure 10.1: – linear system without damping (relation [10.1] with c = 0: ϕ ( x) = k x ): a straight segment that is symmetric in relation to the origin. The stiffness of the connection is constant and given by the gradient of the line k = ϕ ( x) / x ; – non-linear system without damping (ϕ(x) only): curve segment passing through the origin, monotone for a stable system but not necessarily symmetric. The apparent stiffness of the link varies with x and it is given by k ( x) = ϕ ( x) / x . An increasing stiffness corresponds to a hardening spring and a decreasing stiffness corresponds to a softening spring; – linear system with damping (relation [10.1]: ϕ ( x, x) = k x + c x ): ellipse centered at the origin; the smaller the constant c is, the flatter the ellipse is. The ellipse area represents the energy dissipated by cycle; – non-linear system with damping (general case ϕ ( x, x) ): curve enclosing the origin, closed because the motion is periodic, but not necessarily symmetric. This is a combination of the two previous cases.

u1

x = u1- u2

u2 without damping

with damping

+F

-F

F

F x

linear

x

xmax

F

xmax

F x

non-linear

xmax

x xmax

Figure 10.1. Force ϕ between 2-DOF for an imposed sine relative motion

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We recall that in certain cases, the behavior at a given excitation circular frequency ω can be linear but it may vary with ω (see section 4.4) for example with the viscoelastic materials characterized by Young’s moduli E and loss factors η functions of the frequency, as schematized in Figure 10.2.

If we return to the previously mentioned non-linearity sources, they can be illustrated by the following simple examples.

E

η

ω Figure 10.2. Typical evolution of the properties of a viscoelastic material

10.2.2. Simple examples of large displacements A first example, which is somewhat specific in relation to structures, but which can be used to represent the sloshing of liquids for example, is that of a simple pendulum of length l and of mass m represented by Figure 10.3. It is a 1-DOF system, angle θ indicating the position of the pendulum. In the presence of a gravitational field of constant g and in the absence of excitation, θ verifies the equation:

m l 2 θ + ϕ (θ ) = 0

with

ϕ (θ ) = m g l sin θ = m g l θ (1 −

θ2 6

+ …)

[10.2]

ϕ (θ ) is the restoring force with the dimension of a moment corresponding to the rotation DOF θ. The term m l 2 plays the role of inertia and the term m g l that of stiffness in rotation. ϕ (θ ) is non-linear with the intervention of sin θ. It is linear for small angles where sin θ can be replaced by θ, by giving a sine motion of period 2π l / g . For larger angles, the motion remains periodic but non-sinusoidal. The

stiffness m g l sin θ / θ diminishes with θ, hence a softening that makes the period

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327

increase with the motion amplitude. If we only look at the first two terms of the development, its analysis is that shown in section 10.3.3.

ϕ θ

L

θmax

θ m

ϕ(x) for x = x max sin ω t Figure 10.3. Simple pendulum

Another traditional example is that of Figure 10.3 with a mass m on a stretched string. It is a 1-DOF system, the displacement x of the mass. If T is the initial tension, it increases in the presence of the displacement x that causes a string lengthening

L2 + x 2 − L in order to give the equation:

m x + ϕ ( x) = 0

with

ϕ ( x) =

⎞ 2 T ⎛⎜ ES x 2 + …⎟ x 1+ ⎟ L ⎜⎝ 2 T L2 ⎠

[10.3]

The restoring force ϕ(x) is linear for small displacements providing a sine motion of period 2π m L / 2 T . For larger displacements, the motion remains periodic but

non sinusoidal, as for the pendulum. Its stiffness ϕ(x)/x now increases with x, hence a hardening that makes the period decrease with the motion amplitude. If we only look at the first two terms of the development, its analysis is that shown in section 10.3.3.

ϕ L L

x x

m

xmax ϕ(x) for x = x max sin ω t Figure 10.4. The stretched string

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10.2.3. Simple example of variable link

Play is a simple example of variable link as illustrated by Figure 10.5 with a spring.

ϕ x

x k a

xmax

a

ϕ(x) for x = x max sin ω t Figure 10.5. Spring with play

The spring is active only if there is a contact, i.e. if the relative displacement x has a larger absolute value than the play a, in this case the restoring force ϕ(x) equals k ( x − a ) and otherwise it is zero. Thus, the stiffness ϕ(x)/x varies from 0 to k according to the value of x, hence a softening in relation to the absence of play that increases the motion period by a value of 4 a / x 0 , with x 0 the initial velocity.

10.2.4. Simple example of dry friction

Dry friction or Coulomb friction (in contrast to viscous friction) is friction between two rough surfaces which slide against each other. The simplest hypothesis, illustrated by Figure 10.6, is that of a skid when there is no relative motion as long as the force developed between the two surfaces remains lower than a certain threshold ϕ 0 (force inside the friction cone). If this threshold is reached, the force opposes the motion by remaining at this threshold. The restoring force then equals ϕ = ϕ 0 x / x . For a motion x = x max sin ω t , ϕ (x) describes a rectangle bounded by ± x max and ±ϕ 0 , whose surface represents the energy dissipated by cycle, as with a damping. If the skid is connected to a spring in series or in parallel, the rectangle deforms into a parallelogram centered at the origin and similar to the ellipse of the linear connection [10.1].

Introduction to Non-linear Analysis

ϕ

x

329

ϕ0 x xmax

ϕ(x) for x = x max sin ω t Figure 10.6. Basic skid

10.2.5. Material non-linearities

Material non-linearities produce phenomena similar to the previous cases: Young’s modulus is similar to a stiffness that can vary, the plasticity is related to the dry friction at the microscopic level. The hysteresis cycle of a given material, i.e. the stress-strain relation for a cyclic stress is typically as shown in Figure 10.7. Region (a) corresponds to the linear elastic phase, region (b) to a non-linear elastic transition and region (c) to plastic deformation. This viscoelasto-plastic behavior can be represented by a careful combination of simple elements such as springs, dampers and skids in order to form a more or less faithful rheologic model (see, for example, [HAR 61]).

σ

b a

c

ε

Figure 10.7. Hysteresis cycle

10.3. Non-linear 1-DOF system 10.3.1. Introduction

The previous examples illustrating various types of non-linearity produce a link between 2 DOF. We may constrain one and load the other with a mass in order to find the 1-DOF system in Figure 10.8. Denoting the displacement by x, the equation of motion of mass m subjected to a force F is written:

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m x + ϕ ( x, x ) = F

[10.4]

ϕ ( x, x) is the restoring force. It is given by relation [10.1] for the linear system, by relations of the type [10.2] or [10.3] for non-linear systems without dissipation where the velocity x does not appear. The velocity will appear in the presence of a viscous damping. In certain cases, variables x and x are uncoupled, but they are coupled in general. x m k

c

Figure 10.8. Non-linear 1-DOF system

Analysis of the system will be limited to that of equation [10.4] which provides the response x according to F, so it deals only with the extension of dynamic flexibility to the non-linear case. The case of dynamic transmissibility and mass can be deduced without any additional problems. The solution of equation [10.4] will be analyzed progressively by considering simple development cases in order to satisfy the objectives of this chapter: – first the undamped general case in the absence of excitation, which makes it possible to establish the period of motion in relation to its amplitude; – then the application to the case of a stiffness of form k (1 + µ x 2 ) approximately covering the cases of section 10.2.2, in order to mention certain results and to prepare the cases with excitation; – then the previous undamped case with excitation, in order to introduce nonlinear effects on the dynamic amplification; – finally the same case with linear viscous damping in order to discuss the resonance phenomenon.

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331

10.3.2. Undamped motion without excitation

The equation of the undamped motion without excitation is written: m x + ϕ ( x) = 0

[10.5]

The restoring force ϕ (x) is assumed simply to be monotone for the moment. By rewriting equation [10.5]: m d (x 2 ) + ϕ ( x) = 0 2 dx

[10.6]

a first integration gives:

x =

2 xmax ∫ ϕ (ξ ) dξ m x

[10.7]

where x max is the maximum value of x, obtained for x = 0 . With x = dx / dt , and assuming that t = 0 for x = 0, a second integration provides:

t=

m x ∫ 2 0

dζ

[10.8]

x ∫x max ϕ (ξ ) dξ

By assuming now that function ϕ (x) is odd, which is often the case, just as with the examples in section 10.2.2, the period T of the motion is four times the duration between displacement x = 0 and displacement x max , hence:

T=

m 1 =4 f 2

x

∫0 max

dζ

[10.9]

x ∫x max ϕ (ξ ) dξ

For ϕ ( x) = k x , we find T = 2π m / k . For ϕ (x ) of the form a x 3 (case [10.3] with a zero tension), we find a period in inverse proportion to x max . For forms

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[10.2] or [10.3] we obtain elliptic integrals. In general numerical integration must be performed. Result [10.9] shows that the period is not generally constant but depends on the motion amplitude x max , as was mentioned in the examples of section 10.2. This is typical of a non-linear behavior. Note that according to relation [10.7], the velocity at the origin is related to x max by: x0 =

2 xmax ∫ ϕ (ξ ) dξ m 0

[10.10]

hence the value of x max as a function of the velocity at the origin. 10.3.3. Case of a stiffness of form k (1 + µ x 2 )

The case of a stiffness of the form k (1 + µ x 2 ) , i.e. with the restoring force:

ϕ ( x) = k x (1 + µ x 2 )

[10.11]

is the simplest to discuss and it covers various practical cases where the restoring force is odd, as in section 10.2.2. If parameter µ is positive, as in relation [10.3], the spring hardens with the amplitude. If parameter µ is negative, as in relation [10.2], the spring softens. In the latter case, the displacement cannot be higher than 1 /

µ

in order to maintain a restoring force. Relation [10.10], taking into account [10.11], may be easily integrated and provides the explicit result:

x max =

1 + 2 µ x0 2 m / k − 1

µ

[10.12]

The period, on the other hand, requires the numerical integration of an elliptic integral coming from relation [10.9]. The result is illustrated in Figure 10.9 as a

Introduction to Non-linear Analysis

333

function of circular frequency ω k = 2π / T (by using the notation in Chapter 2 for the 1-DOF system) compared to the circular frequency of the linear system (µ = 0), in order to pave the way for the frequency responses. The circular frequency increases with amplitude x max of the motion when the spring hardens (µ > 0) and it diminishes when the spring softens (µ < 0).

µ x max

µ =0 µ >0 µ <0

ωk k/m Figure 10.9. Circular frequency of the system according to xmax and µ

During a period, the motion is not sinusoidal because of the non-linearity. It is located below the sine when the spring hardens (µ > 0), above the sine when the spring softens (µ < 0), as indicated by Figure 10.10 on a half-period. It can be decomposed in odd harmonics ( ωk ,3ωk ,5ωk , etc. ) the amplitudes of which decrease rapidly, which makes the motion very close to a sine except for large negative values of µ.

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x

µ x max 2 = −0.8

x max

4

t Figure 10.10. Half-period of the system

If we limit ourselves to the fundamental frequency, also called the “fundamental” or harmonic 1, of circular frequency ω k = 2π / T , we can obtain an approximation of ω k and of harmonic level 3 in relation to the fundamental. In the equation of motion: m x + k x (1 + µ x 2 ) = 0

[10.13]

we can introduce the displacement limited to the fundamental x = x max sin ω k t , which gives: (−ω k 2 m + k ) x max sin ω k t + k µ ( x max sin ω k t ) 3 = 0

[10.14]

By using the relation: 4 sin 3 ω k t = 3 sin ω k t − sin 3 ω k t

[10.15]

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335

we come to: ⎛ µ ⎛ 3 ⎞⎞ ⎜⎜ − ω k 2 m + k ⎜1 + µ x max 2 ⎟ ⎟⎟ x max sin ω k t − k x max 3 sin 3ω k t = 0 [10.16] 4 ⎝ 4 ⎠⎠ ⎝

This relation being true for any t, the coefficient of the fundamental must be zero, which provides the circular natural frequency:

ωk 2 =

k ⎛ 3 2⎞ ⎜1 + µ x max ⎟ m⎝ 4 ⎠

[10.17]

The first term is that of the linear system, the second is the contribution of the non-linearity that increases or diminishes the circular natural frequency depending on whether µ is positive or negative. This result is only an approximation because the presence of higher harmonics 2 n − 1 adds terms in ( µ x max 2 ) n to the coefficient of the fundamental. The coefficient for n = 2 equals 3/128, and the approximation [10.17] remains accurate as long as µ x max 2 is smaller than 1 (better than 1%). The amplitude of harmonic 3 can be obtained by considering the displacement: x = X 1 sin ω k t + X 3 sin 3 ω k t

with

X 1 − X 3 = x max

[10.18]

Reintroduced in equation [10.13] we find, by canceling the coefficient of harmonic 3:

8 ω k 2 m X 3 = −k

µ 4

x max 3

hence

X3 = −

µ x max 3 ⎛ 3 ⎞ 32 ⎜1 + µ x max 2 ⎟ ⎝ 4 ⎠

[10.19]

Thus, a non-linearity µ x max 2 = 1 , which doubles the stiffness at the amplitude x max , will approximately multiply the natural frequency by

7 / 2 and generate a

harmonic 3 of amplitude X 3 = X 1 / 57 , hence 1.8% of the fundamental. We see

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here the rapid convergence of the harmonics, which makes the system motion (Figure 10.10) close to a sine.

10.3.4. Undamped motion with excitation With the stiffness k (1 + µ x 2 ) , the equation of motion with a sine excitation of amplitude F and of circular frequency ω is called the Duffing equation: m x + k x (1 + µ x 2 ) = F sin ω t

[10.20]

Response x(t) is a periodic motion that can be broken down into odd harmonics (ω, 3 ω, 5 ω, etc.). If we keep to the fundamental, x(t) is of the form: x(t ) = X sin ω t

with

X = ± x max

[10.21]

X being a positive or negative real number because there is no damping to create a phase lag. Equation [10.20] then gives, taking into account the developments of section 10.3.3: 3 ⎛ 2 2 ⎞ ⎜ − ω m + k (1 + µ X ) ⎟ X = F 4 ⎝ ⎠

[10.22]

from which we obtain: X =

1 k

1

ω2 m 3 1+ µ X 2 − k 4

F

[10.23]

in order to note that X is not proportional to F because of the term in µ. Equation [10.23] can be solved iteratively with a convergence that increases with smaller values µ, but we can also directly solve equation [10.22], which is only of the third degree, with the well-known relations that can be written:

Introduction to Non-linear Analysis

y3 + 3a y + 2b = 0

with

⇒

y = A + B, −

3

A = − b + b2 + a3

(if the term under

3

337

A + B ± i 3 ( A − B) 2

3

B = − b − b2 + a3

[10.24]

is real, take the real root)

µ x max against the reduced circular frequency

The dimensionless responses

ω / k / m are plotted in Figure 10.11 for various values of the dimensionless excitation

µ F / k . For smaller values, the behavior is practically linear: a factor

2 on the excitation produces a factor 2 on the response. For higher values, the group of curves restrains or expands according to whether µ is positive or negative. When the circular frequency of the excitation gets closer to the circular natural frequency ω k given by relation [10.17] and which cancels the denominator of relation [10.23], the amplitude of the response increases as for a resonance. This generates an infinite peak tending asymptotically towards the curve ω = ω k , which is actually the response to zero excitation. The fact that the curve folds over to the left or right depending on the sign of µ will create stability problems that will be examined in the following section in the presence of a damping, along with the behavior at the resonance. We can show that the corresponding circular frequency is given by [GEN 95]:

ω2 =

⎛ 2 k ⎜ 3 81 ⎛ F ⎞ 1 µ + ⎜ ⎟ m ⎜⎜ 16 ⎝ k ⎠ ⎝

⎞ ⎟ ⎟ ⎟ ⎠

[10.25]

Figure 10.12 takes the results of Figure 10.11 from the FRF point of view, with the amplitude of the dimensionless ratio k x max / F , which should be compared to the dynamic amplification factor H k (ω ) of the 1-DOF linear system. This dynamic amplification that reproduces H k (ω ) for a zero excitation, progressively folds over when the excitation increases, and more quickly for larger values of µ .

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µ x max

− µ x max 0.4 − µ F / k = 0.2 0.1 0.05 2

µ F /k =1 0.5 0.25

ω

ω

k/m

k/m

µ>0

µ<0

Figure 10.11. Response of the undamped 1-DOF system

x max F /k

x max F /k

µ F /k =0

−µ F /k = 0 0.25

0.1

ω µ>0

k/m

ω µ<0

k/m

Figure 10.12. Dynamic amplification of the undamped 1-DOF system

It should be noted that all these results, limited to the fundamental, are only approximate. Particularly, displacement X due to a static force F obtained by making ω = 0 in equation [10.21] is not that obtained for the initial equation [10.20] with x = 0 , due to the coefficient 3/4 replacing the coefficient 1. Table 10.1 gives the resulting errors.

Introduction to Non-linear Analysis

µ F /k

Exact

Fundamental alone

Error

ε 1/4 1/2 1 2

ε 0.2367 0.4239 0.6823 1.0000

ε 0.2397 0.4373 0.7200 1.0731

+ 0.0% + 1.2% + 3.2% + 5.5% + 7.3%

ε 0.05 0.1 0.2 0.3 0.4

ε 0.05013 0.1010 0.2091 0.3389 –

ε 0.05009 0.1008 0.2066 0.3260 0.4862

– 0.0% – 0.06% – 0.3% – 1.2% – 3.8% –

339

µ>0

µ<0

Table 10.1. Static response

µ x max of the 1-DOF system

By adding harmonic 3 to the fundamental we obtain the following displacement: x = X 1 sin ω t + X 3 sin 3 ω t

[10.26]

and equation [10.20] provides: ⎞ ⎛ 3µ ⎛ ⎞ ⎜⎜ − ω 2 m X 1 + k ⎜ X 1 + ( X 13 − X 1 2 X 3 + 2 X 1 X 3 2 ) ⎟ − F ⎟⎟ sin ω t 4 ⎝ ⎠ ⎠ ⎝

[10.27]

⎛ µ ⎛ ⎞⎞ + ⎜⎜ − (3 ω ) 2 m X 3 + k ⎜ X 3 + (− X 13 + 6 X 1 2 X 3 + 3 X 3 3 ) ⎟ ⎟⎟ sin 3 ω t + … = 0 4 ⎝ ⎠⎠ ⎝

The equilibrium of the two harmonics provides two equations with two unknowns X1 and X3. The same reasoning will provide n equations for n considered harmonics.

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For example, with the excitation force F = 1× k at the circular frequency

ω = 0.5 k / m , we obtain the linear solution X 1 = 2 , whereas a factor µ = 0.1 leads to: – by taking the fundamental alone: X 1 = 1.4969 ; – by taking the fundamental and harmonic 3: X 1 = 1.4924 , X 3 = −0.026245 ; – by taking the first 5 non-zero harmonics: X 1 = 1.4923 , X 3 = −0.026270 , X 5 = 0.00039996 , etc. These results illustrate the rapidity of the convergence.

10.3.5. Damped motion with excitation

If a linear damper of constant c is added to the previous system, equation [10.20] becomes: m x + c x + k x (1 + µ x 2 ) = F sin ω t

[10.28]

The presence of the term c x now introduces a phase lag between excitation and response. By passing to complex notation with the excitation F e iωt and by limiting the response x(t) to the fundamental, the latter is of the form: x(t ) = X e iωt

with

X = x max

[10.29]

X being complex because of the phase lag. Equation [10.22] then becomes: 3 ⎛ 2 ⎞ 2 ⎜ − ω m + iω c + k (1 + µ x max ) ⎟ X = F 4 ⎝ ⎠

[10.30]

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341

from which we obtain:

X =

1 k

1

ω2 3 + i 2ζ k 1 + µ x max 2 − k/m 4

[10.31]

F

ω k/m

with ζ k = c /(2 k m ) the viscous damping factor of the system, that we will assume to be small compared to 1. We see with this equation that the maximum amplitude of the response, i.e. the resonance, corresponds to:

1+

ω2 3 =0 µ x max 2 − k/m 4

hence

X

max

1

=

2ζ k

ω

F k

[10.32]

k/m

We find the circular natural frequency ω k of relation [10.17] and the maximum is given by the intersection of the two curves:

ω2 k/m

= 1+

3 µ x max 2 4

and

ω k/m

=

1 2ζ k

F 1 k x max

[10.33]

as shown in Figure 10.13. Let us now consider, as shown in Figure 10.14, a sine sweep of a given amplitude F. With for example an increasing sweep and µ > 0, the response follows path AB, then arrives at the resonance at point C and then suddenly falls at point E, which represents a sudden change in equilibrium typical of non-linear behavior. Therefore, path CDE is instable. The response then follows path EF without any particularities. Again with µ > 0, but considering a decreasing sweep, the path will follow FEDBA with a jump from D to B at the frequency given by relation [10.25], thus bypassing the true resonance. With µ < 0 the phenomena are similar, with a jump from C to E for an increasing sweep and a jump from D to B for a decreasing sweep. It should be noted that the jumps are attenuated or may even disappear with high damping levels.

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µ x max

− µ x max

ω

ω

k/m

k/m

Figure 10.13. Response of the 1-DOF damped system for a given force F

µ x max

− µ x max C

B

D

C

D

A

E

A E

B

F

F

ω

ω

k/m

k/m

µ>0

µ<0

Figure 10.14. Sine sweep at a given force F

Re-examining the example of an excitation force F = 1× k with a factor

µ = +0.1, we have the following values for a damping ζ k = 5 % : –

resonance

C

at

the

circular

frequency

ω k = 1.8121 k / m

with

x max = 5.5183, instead of ω k = k / m and x max = 10 for the linear system;

– for an increasing sweep, a drop just after the resonance at point E, with x max = 0.4392;

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– for a decreasing sweep, a jump at point D at the circular frequency

ω k = 1.3405 k / m with x max = 2.1970, to point B with x max = 3.7077; – by positioning ourselves at resonance C, if we double force F, x max is multiplied only by 1.0362. If, conversely, we divide force F by 2, x max is divided by 17.824 after suddenly dropping. Figure 10.15 illustrates the results in Figure 10.14 from the FRF point of view in order to complete the information in Figure 10.12 at the resonance with the comments remaining the same. It should be noted that for µ < 0, the two curves [10.33] have an intersection only for a small non-linearity or a high damping. x max F/k

x max F /k

µ F /k =0

0 − µ F / k = 0.03

0.25

µ>0

ω

ω

k/m

k/m

µ<0

Figure 10.15. Dynamic amplification of the 1-DOF damped system

For a more accurate estimation using n harmonics, the processing is similar to that in the preceding section.

10.4. Non-linear N-DOF systems 10.4.1. Introduction

Given the complexity of the analysis of a non-linear 1-DOF system, we see the extreme difficulty of the problem for an N-DOF system. One possible strategy is to linearize the structure as realistically as possible and represent the non-linear effects by additional forces at the point at which they act. This strategy is especially efficient for weak and localized non-linearities.

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The simplest case is that of the non-linear link between 2 DOF as described in section 10.2.1. This will give a set of two additional equal and opposite forces expressing the difference between the linear force of the linearized link and the true non-linear force of the link, as schematized in Figure 10.16. This strategy, applied to the case of the 1-DOF system by grounding a mass m, leads to the results in section 10.3. This non-linear link between 2-DOF can easily be generalized to n non-linear links, each between 2 DOF. The non-linear structure is then equivalent to the linear structure subjected to a set of non-linear forces as well as the excitation forces that are applied to it. Knowing the initial non-linearities, these non-linear forces can be calculated according to the displacements at the links, which leads to an iterative solution in order to reach the equilibrium at all points. We can thus conveniently handle a number of problems related to industrial structures. This is the subject of the following developments.

⇔

-∆F Non-linear link

+∆F

Permanent linear link

Figure 10.16. Dealing with a non-linear link by additional forces

10.4.2. Non-linear link with periodic motion

We are looking at a non-linear link between two linear substructures A and B, limited to a spring, as shown in Figure 10.17, in order to simplify the developments. In the presence of viscous dissipation, we will attach a damper to it to be analyzed in the same way. This can be generalized to several links and several substructures without any problems, as will be seen during the development [GIR 97b]. The excitation consists of a certain number of forces Fe on A and/or B, decomposed into a certain number of harmonics, that we will write for the sake of convenience:

Fe =

+H

∑ Fe (h) e i hω t

h=− H

[10.34]

Introduction to Non-linear Analysis

A

345

B Fe

⇓ k +∆F

-∆F c

Figure 10.17. Non-linear link between 2 linear substructures

According to the strategy previously mentioned, the non-linear spring is replaced by a linear spring of stiffness k (and of a linear damper of constant c in the presence of a viscous dissipation). k can have any value, but it is preferable that it be realistic for a better convergence, taking for example the slope at the origin of the function ϕ (x) introduced previously, which expresses the restoring force according to the relative displacement x = u 2 − u1 . The difference between the chosen linear spring and the original non-linear spring is represented by additional external forces ± ∆F verifying: ∆F = k x − ϕ (x)

[10.35]

Substructures A and B with the linearized link are now excited by the external forces Fe and the forces at the link ± ∆F . Except for the fact that ∆F is not known, it is a traditional linear problem that we can solve for each harmonic by frequency synthesis as presented in Chapter 9. If G oe (ω ) designates the dynamic flexibility matrix of the set A+B after the coupling between the excitation DOF e, including the 2 DOF of the link, and the 2 observation DOF o, we will write: u o ( h) = G oe (hω ) (F + ∆F) e ( h )

[10.36]

This relation easily extends to several links and to several substructures. It is now a case of solving system [10.36] given forces Fe , flexibilities G oe (ω )

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resulting from the linear calculation and relation [10.35] between ∆Fe and the link non-linear forces. Relation [10.35] makes it possible to deduce the harmonics of ∆F for each nonlinear link according to the harmonics of x using the following method: +H

x ( h ) ⇒ x = ∑ x ( h ) e i hω t ⇒ ∆F = k x − ϕ ( x) ⇒ ∆F ( h)

[10.37]

h=− H

which is equivalent to solving the full system, of a size equal to the number of harmonics. Thus, the iterative process is the following: at iteration n, equation [10.36] gives harmonics u o ( h) using harmonics (F + ∆F ) e ( h ) of the excitation ( ∆Fe = 0 initially), hence the harmonics x (h) at each non-linear link. These harmonics x (h) generate harmonics ∆F (h) due to procedure [10.37], hence an actualization of harmonics (F + ∆F ) e ( h ) of the excitation for the iteration n + 1.

The iterations should be performed until convergence corresponding to the equilibrium of the forces present in the links. The stronger the non-linearities, the longer it will take to obtain this convergence, without mentioning sudden changes of equilibrium such as those observed in section 10.3.5 for the 1-DOF system. By taking the example of the 1-DOF system at the end of section 10.3.4, with a stiffness k (1 + 0.1 x 2 ) and an excitation force F = 1× k at the circular frequency

ω = 0.5 k / m , the solution X 1 = 1.4924 , X 3 = −0.026245 with the fundamental and harmonic 3 (keeping the sines instead of the exponentials for the interpretation) is obtained at the end of the iterative process with the additional forces ∆F1 = −0.25379 , ∆F3 = +0.091855 .

10.4.3. Direct integration of equations

The direct integration methods presented in section 3.5 are more or less applicable for solving non-linear problems. Reference books on this subject should be consulted. Let us simply mention a variant of the β-Newmark method presented in section 3.5.3, which can be used with limited non-linearities.

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If we take schema [3.57], but temporarily eliminate the terms in u n +1 : ⎛1 ⎞ u n +1 = u n + ∆t u n + ⎜ − β ⎟ ∆t 2 u n ⎝2 ⎠ u n +1 = u n + (1 − γ )∆t u n

[10.38]

we obtain a first approximation of the displacements and velocities enabling additional non-linear forces ∆F (u, u) to be determined, hence the excitation forces (F + ∆F ) n +1 to be taken into account for the calculation of velocities:

∆t C ⎛ ⎞ + β ∆t 2 K ⎟ u n +1 = Fn +1 − C u n +1 − K u n +1 ⎜M + 2 ⎝ ⎠

[10.39]

We now add the terms of the temporarily eliminated velocities, with estimation [10.39]: u n +1 = u n +1 + β ∆t 2 u n +1 u n +1 = u n +1 + γ ∆t u n +1

[10.40]

for a new determination of the additional forces and we iterate on relations [10.39] and [10.40] up to the convergence for u n +1 , u n +1 , u n +1 . At each iteration in time, we perform an iteration due to the non-linearity. This rapidly converges for a weak non-linearity. According to the case under discussion, we will use a margin on coefficient β in relation to the minimum value 1/4 which assures stability of the process only for a linear system.

Chapter 11

Testing Techniques

11.1. Introduction The previous chapters have shown the possibilities and the limits of analysis. All these techniques make it possible to estimate the behavior of the structure under consideration using various hypotheses and approximations, hence certain reserves on the validity of the results according to the case discussed: appropriateness of the computing resources in relation to the structural complexity, legitimacy of the hypothesis, limitation of the treatments, etc. The limitations concerning the size of the models are pushed back each year with greater computer capacity and improved efficiency of methods. Whereas at the beginning it was necessary to juggle with equivalent models to globally represent complex components in order to reduce the number of physical DOF, today the tendency is to refine the meshing sufficiently for a more realistic structural description. However, despite all the possible refinements, the representativity of the models cannot be guaranteed for various reasons: high complexity of certain structural elements, uncertainties about the actual characteristics, existence of non-linearities that are difficult to describe, lack of understanding of dissipative phenomena, etc. Testing should therefore replace analysis on a certain number of points. Tests are thus a complement that cannot be separated from the analysis. Without testing, an analysis can give completely incorrect results as a consequence of errors on the hypothesis, the data or the processing. Without analysis, the tests can

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represent only a very limited reality or one which is very far from operating conditions. Thus, the analysis and test activities should be perfectly interlinked during the development of a structure. Schematically, analysis will be the starting point with the design and dimensioning phase. The experimentation will then provide information on a certain reality that the analysis will try to take into account by adjusting the models in order to improve their reliability and to extrapolate the calculations in order to demonstrate that the structure will be able to withstand all the conditions likely to be met in service. Finally, verification of the behavior will be performed by testing based on the analysis, which provides justification along with specifications. Depending on the industrial domain, the practices are diverse because of the characteristics of the environment or the mission, because of the experience acquired and sometimes also because of traditions that are not always justified. Despite this diversity, the previous logic makes it possible to obtain some common lines. They are developed in section 11.2 which shows two types of tests according to the objectives to be reached: identification tests for structure characterization, which are the subject of section 11.3 and simulation tests for structure qualification which are the subject of section 11.4. These tests will be presented in relation to the analysis techniques of the previous chapters, without spending too much time on the test facilities that are used.

11.2. Dynamic tests 11.2.1. Development plan of a structure The development plan of a structure involves analysis and test activities that follow the steps described hereafter: – elaboration of specifications using available environment data that the structure is subjected to during all phases of its life cycle. This first step is delicate because it requires sufficient knowledge of the environment in order to allow a good characterization of it (see section 1.2). Moreover, if the structure under consideration is likely to influence its environment, it will be necessary to proceed by iterations. This case is more common than might be imagined: for example, despite its relatively small size, the satellite on its launcher is, to a significant extent, an active part in the behavior of the interface, requiring a coupled analysis with the launcher in order to verify the interactions in flight. The specifications should include not

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only the analysis specifications but also those for the tests, according to the development logic used; – design of the structure according to the specifications. This is the step where it is mainly the experience acquired with similar cases that comes into play in order to minimize the iterations that occur during subsequent analyses. A good extrapolation from a structure having proved its value should be preferred to introducing an innovation that could be risky, influencing costs and deadlines; – from the initial definition of the structure resulting from the previous step, we construct a development model which is designed to verify the most critical analysis points and particularly the associated mathematical models. Identifying the characteristics of the tested structure makes it possible to adjust the physical parameters of the models in the updating phase in order to make them more representative of a certain reality. This is the reality of the tests which have been used and it is not necessarily the same as that of the true environment, firstly because of implementation constraints, and also because the objective is not to verify the behavior of the structure but to obtain its main properties. Updating the model is a particularly important problem that will be discussed in detail in Chapter 12; – the updated mathematical models make it possible to refine the analysis of the structure behavior in the various phases of its life history and to determine its possible influence on the environment. If this is significant, new specifications will probably be necessary at this level. A strategy which enables the avoidance or attenuation of iterations takes this interaction into account in one way or another from the beginning, for example using envelope preset levels avoiding a calling into question, but to the detriment of the performance. Again, the experience is predominant; – when the analysis has convergence, thus finalizing the structure definition, a prototype model is then manufactured in order to verify its behavior using environment simulation tests. The objective is now to qualify the structure in relation to its environment. Some of these tests will be performed again, possibly at subsystem levels, on the final structure and the recurrent structures to increase robustness and for acceptance. Of course the importance of each step varies according to the industrial sector. The quality and number of iterations for the analysis or the number of models manufactured at the subsystem and system level closely depend on the production conditions. Between the mass production of the automobile industry and the almost unit production in the space industry, we go from the multitude of intermediate levels to the notion of a unique proto-flight model assuring all the development, qualification and service functions in order to optimize the costs and deadlines.

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11.2.2. Types of tests The previous development plan leads to two types of tests: – identification tests whose objective is to determine the dynamic characteristics of the tested structure in order to update the mathematical models. The first question that comes up is which are the most appropriate characteristics in this context. The three characterization levels listed in section 1.6 and mentioned again in section 8.1 in the discussion about substructuring, include the following: - the matrix level (mass, stiffness and damping) which is closely related to the analytical world is, on the other hand, very distant from the experimental possibilities and cannot be directly exploited other than for some exceptions (simple or simplified structures, which include rigid structures represented only by their mass, center of mass and inertia properties), - the FRF level is interesting on account of the fact that it is closely related to the experimental world but represents a large volume of information because of the high definition in frequency that it requires, - the intermediate modal level is a satisfactory compromise especially when a limited number of modes are present in the considered range of frequencies.. As we have seen in the previous chapters, modes represent a particularly useful characteristic in the lower frequencies and are defined by a small number of parameters if we consider an appropriate choice of DOF. It is then natural to base the structure identification using its modes. This is the modal survey test which is the subject of section 11.3; – simulation tests whose objective is qualifying and accepting the structure. The test facilities used depend on the nature of the environment. One or several vibration generators or shakers can simulate a mechanical environment of a mechanical or acoustic source in configurations that must be adapted case by case. However, the particular case of a shock is often treated by specific test hardware. Finally, an acoustic environment requires the generation of an appropriate acoustic field using devices among which we can mention the reverberant chamber that makes it possible to obtain accurately controlled high levels. All these tests are the subject of section 11.4. The identification and simulation tests are generally completely dissociated because they do not occur at the same level of development and do not have the same constraints. However, in certain cases, it is technically possible to perform them using the same test facility, as mentioned in section 11.3.4, which allows for substantial savings in costs and time by means of an adapted methodology.

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11.2.3. Test hardware The hardware used for performing tests employ various techniques that will not be mentioned here in detail, the purpose being simply to describe the means of excitation and measurement that are generally available, in order to satisfy the test objectives. For more details, we can refer to books such as [HAR 61] and an introduction to measurement techniques is given by [DUP 00]. Regarding the generation of a mechanical environment, either for identification or for simulation, shakers of various sizes are generally used to provide the excitation: – small shakers, which are generally electrodynamic, apply point forces, typically of 200 N on the structure. The use of a very flexible rod (stinger) between the shaker and the structure makes it possible to favor the component along the rod direction by minimizing the other components, hence an excitation in force mainly along a single internal DOF; – large shakers, generally used to impose motion at the base of specimens attached to them. With suitable bearings, the motion is limited to the DOF that corresponds to the considered direction, the other DOF being set at zero. All the DOF whose motion we control are junction DOF. The technology of the large shakers varies according to the domain in which they are used. Two types are generally used: – the electrodynamic shaker, like the small shaker, where the mobile part is excited by the Laplace force. It allows a good control of the motion up to relatively high frequencies, typically 2,000 Hz. Its limitations is in the stroke, typically 50 mm peak to peak, which precludes very low frequencies, typically below 5 Hz, and the delivered force that rarely exceeds 300 kN in sine mode with current technology. We can couple them to a same table in order to increase their monoaxial excitation capacity, but the added obstruction makes the installation complex. Figure 11.1 shows an installation such as this with four shakers each of 160 kN, two for vertical excitation, two for horizontal excitation, allowing specimens of up to 8,000 kg to be vibrated (MVS facility from Intespace [MAR 01]); – the hydraulic jack shaker, where the force is generated by pressure. Unlike the previous case, it can be used only at relatively low frequencies, typically up to 100 Hz. On the other hand, it allows large displacements, and thus very low frequencies, with a large delivered force. Moreover, correct positioning of the jacks connected to the same table allows multiaxial installations that can control up to 6 DOF. Figure 11.2 shows an installation with 8 jacks, making it possible to vibrate specimens of 25,000 kg (Hydra facility from Estec [BRU 01]).

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Horizontal excitation: – 2 × 160 kN in sine mode – table: 3.0 m × 3.0 m

Vertical excitation: – 2 × 160 kN in sine mode – expansion head: diameter 2.1 m

Figure 11.1. MVS facility from Intespace (Toulouse) (with the authorization of Intespace)

Vertical excitation: 2,520 kN Horizontal excitation: 1,260 kN Table: octagon diameter 5.5 m

Figure 11.2. Hydra facility from Estec with the Envisat satellite (Noordwijk) (with the authorization of the European Space Agency)

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The particular case of a shock environment can be treated using the shakers mentioned previously, controlled appropriately, but most often it requires specific facilities with shock machines adapted to the nature of the shock. Without entering into details, we will mention pyrotechnic generators and impact machines. Figure 11.3 shows an example of a metal-metal pendulum impact machine for specimens up to 40 kg (Intespace facility), making it possible to reach much higher levels than with shakers [DUP 99], but with less control.

Configuration for test specimen’s longitudinal axis

Example of achievable shock spectrum values (Q=10): 100 Hz 200 Hz 500 Hz 1,000 Hz 3,000 Hz 10,000 Hz

85 g 300 g 800 g 1,000 g 5,000 g 6,000 g

Specimen in lateral configuration

Figure 11.3. Shock machine from Intespace (Toulouse) (with the authorization of Intespace)

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Various devices using sirens or other noise generators can be envisaged for the generation of an acoustic environment. The reverberant acoustic chamber enables well-controlled and relatively high levels to be obtained on the condition that an approximately homogenous (equal everywhere) and diffuse (isotropic) field is acceptable. The principle is to have very smooth walls where the sound waves created by sirens, are reflected and lose very little energy; hence, becoming rapidly random, a constant acoustic pressure and a coherence (see section 1.2.3.4) is obtained between two points x and y given by:

γ xy =

sin

ωd c

[11.1]

ωd c

ω being considered as the circular frequency, d the distance between the two points, c the propagation velocity in the fluid (air or other) used. Figure 11.4 shows the example of the reverberant chamber from Intespace [MAR 01]. - Volume : 1,100 m 3 - Acoustic pressure global volume: 156 dB - Excitation frequencies: 22.4-11,200 Hz Local sirens

C 8.2 m 13 m

Specimen access door

B

10. 3 m

Figure 11.4. Reverberant acoustic chamber from Intespace (Toulouse) (with the authorization of Intespace)

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The measurements to be performed in order to determine the excitations and the structure responses depend on the nature of the phenomena to be analyzed. If we put aside the acoustic excitations requiring microphones to determine the acoustic pressure field, the measurements essentially concern forces and motions: – the forces can be measured directly using force sensors, most often of the piezoelectric type. The implementation is simple when the configuration is adapted to it, such as with the excitation forces of small shakers. It is much more delicate when measuring reaction forces for example with the motion imposed by a large shaker because the device, placed in series, must not interfere with the test configuration (see [SAL 01] for an example of implementation in the space domain). We can also measure forces using previously calibrated strain gauges, again if the configuration allows for this, for example with a specimen attached to the shaker by a cylindrical cone which is not too stiff. Finally, in the case of electrodynamic shakers, the delivered force is generally proportional to the intensity of the current in the moving coil, providing information that must be used with caution; – the motions can be measured by displacements, velocities or accelerations, which are strictly equivalent because passing from one to another is carried out by derivation or integration in the time domain or by multiplication or division by iω in the frequency domain. The most commonly used sensors are accelerometers based on the same principles as the forces because they are linked to the accelerations by the mass, combining simple implementation and compactness (with some exceptions) as well as good performances if care is taken. Other solutions exist such as velocity or displacement sensors which can be used for specific needs. From this rapid overview of test hardware, we see: – for excitation, it is easier to control the force than the motion; – for response, it is easier to measure the motion than the reaction forces. We deduce from this that, in an experimental context: – it is relatively easy to determine flexibilities or accelerances. Most often, we can excite the structure by point forces with small shakers (or an impact hammer) and perform accelerometric measurements on all the DOF of interest; – it is more difficult to determine transmissibilities, except in the case of a large shaker imposing a well controlled motion at the specimen base where we can calculate the ratio between the accelerations on the specimen and the injected acceleration. In other cases, particularly with statically indeterminate junctions, the implementation is more delicate; – it is very difficult to directly determine stiffnesses or masses, except in the same case of a large shaker provided that the injected force can be conveniently

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determined. For this, as previously mentioned, we can consider a specific device, strain gauges, the intensity of the current for an electrodynamic shaker or an analytic operator combining the accelerations measured on the specimen with masses associated with the various measurement points. Finally, independent of the excitation and responses, the specimen should be considered with its boundary conditions. Since we know that fixing a structure to the ground, i.e. suppressing all translation and/or rotation motion, is very difficult to carry out, if at all possible, it is preferable to suspend the structure in order to simulate free conditions. These conditions are best if the suspension has frequencies that are much lower than the first elastic mode of interest of the structure (typically a ratio of 10). We will also verify that it introduces very little local stiffness and dissipation at the attachment points. On the other hand, any support introduces a local flexibility that we will try to minimize so as not to significantly influence the behavior of the structure. This aspect can be illustrated by the simple example of Figure 11.5 which symbolizes a lateral mode of the structure, of mass m at a distance d from the interface, affected by a flexibility at the fixed support represented by the rotational stiffness K = α k d 2 , k being the stiffness of the system representing the mode. The natural

frequency of the mode is thus divided by 1+ 1 / α . If the ratio of stiffness α is large compared to 1, the natural frequency will not be affected very much. On the other hand, a ratio of 10 gives a frequency reduction of 5%, which may not be acceptable. k K =α k d2

m d

Figure 11.5. Example illustrating the influence of a support flexibility

11.3. The identification tests 11.3.1. Introduction We saw in section 11.2.2 that the normal modes were the most appropriate dynamic characteristics for the identification of the structure, with the FRF being closer to the test environment but representing a much greater volume of information. The object of this section is thus to determine experimentally the

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parameters defining the normal modes via tests generally called modal analysis tests or simply modal survey tests. We know, from Chapters 4 and 5 which discussed the normal modes, that the FRF are related to the modes by the mode superposition expressions. In a way, the modal survey test is the inverse of this operation which determines the underlying modes starting from measurements in the form of FRF between excitations and responses. We will later see that the excitations can take very particular forms according to the techniques that are used. Before describing the various techniques that can be used, it is necessary to know which modal parameters are to be determined. This question is worth serious consideration in order to completely clarify it. This is the subject of section 11.3.2. These parameters can then be determined in two ways: – by a method with appropriation of modes, sometimes called phase resonance, which consists of successively isolating each mode by an appropriate excitation and measuring its parameters directly. This is the subject of section 11.3.3; – by a method without appropriation of modes, sometimes called phase separation, which consists of exciting a group of modes whose parameters are then determined by processing the measurements. This is the subject of section 11.3.4. These techniques, which are always delicate to implement, have been the subject of various research and publications in order to try to improve their performance. Here, as in the previous chapters, we will discuss the principal issues, without going into the details that can be found in [MAI 97] or [EWI 00]. In [GIR 77] there is an overview of the subject made in the transitional period between the two types of technique.

11.3.2. Modal parameters to be identified In order to know which modal parameters are to be determined, consider the general expression [5.56] which can represent flexibility, transmissibility or mass FRF (the others derived using i ω): ~ X (ω ) = ∑ Ak (ω ) X k + X res k

[11.2]

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with Ak (ω ) the dynamic amplification H k (ω ) or Tk (ω ) of the mode k ~ (expressions [5.4]) as the function of the modal parameters ω k and ζ k , X k as the

modal effective parameter and X res the residual parameter related to the modal truncation. For a given FRF, the parameters to be determined for each mode k are those that are implied in expression [11.2], i.e.: – circular natural frequency ω k ; – damping factor ζ k ; ~ – modal effective parameter X k ,

to be completed by the residual parameter X res for a given modal truncation. This being true for a FRF between a given excitation and response, if several excitations x and several responses y are involved in the structure behavior, hence a FRF matrix X yx , the effective parameters involved in each mode k will be those of ~ matrix X yx,k . Thus, in general, the modal parameters that enable the characterization of a ~ structure relating to its FRF X yx (ω ) are (ω , ζ , X yx ) k , to be completed by the ~ residual parameters X yx,res . However, it may be preferable to replace the matrices of effective parameters ~ X yx,k by the terms that compose them, which is often the case in practice. If we look again at expressions [5.5] to [5.7]: Φi k Φki ~ G ii,k = ωk 2 m k

Φi k L k j ~ Tij ,k = mk

L jk L k j ~ M jj ,k = mk

[11.3]

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We see that the basic ingredients for each mode k are: – generalized mass m k , which plays a normalization role; – modal components Φi k , to be limited to the excitation DOF x and to the response DOF y from all the DOF i; – participation factors L j k , to be limited to the excitation DOF x and to the response DOF y from all the DOF j. In this case, the modal parameters to be determined are given by (ω , ζ , m, Φi , L j ) k (the parameters can be found listed after equation [4.44]), the components of Φ and L being limited to the excitation DOF x and response DOF y, ~ to be completed by the residual parameters X yx,res . According to expressions [11.3], we can deduce the generalized mass of the driving-point FRF, i.e. with the excitation and the response on the same DOF, whether it is a G or M type: the corresponding effective parameter provides m k for a driving-point component equal to 1 for example. The other effective parameters then provide the other components. In the case when the driving-point FRF is not available, only the ratio between the components can be determined. For example, with a specimen excited at its base by a shaker along DOF r, if we measure only transmissibilities between the base and various internal DOF, we obtain only the ratio between components Φi k , the ratio Lr k / m k remaining undetermined. To remove the indetermination, it is necessary for example to measure the dynamic mass M r r (ω ) providing the effective mass ~ M r r ,k , hence m k for a component Lr ,k equal to 1. This situation is perfectly

logical when we notice that the transmissibilities are dimensionless (except for a possible length relating translations and rotations) and that their identification can provide only dimensionless modal parameters, such as ω k , ζ k and Φi k (shape defined to within a coefficient), but not m k which has the dimension of mass and thus requires a dimensional parameter such as the effective mass. It should be noted that the participation factors L j k require the measurement of a reaction force, which, as mentioned in section 11.2.3, is not easy to perform, especially for an indeterminate junction.

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We should also note that coupled modal damping (section 4.3.3) has not yet been mentioned. Its determination requires a great deal of experimental accuracy and the difficulty of taking it into account in the analysis does not motivate us to consider it. This is part of the drawbacks of the normal mode approach. With the complex mode approach, developed in Chapter 7, this difficulty disappears and all the previous results can be extrapolated, with the dynamic amplification being a function of the complex circular natural frequencies including the damping, and of the complex modal effective parameters related to the generalized masses, the modal components and the participation factors.

11.3.3. Phase resonance modal tests The phase resonance modal test consists of exciting each mode of the structure separately and determining its parameters directly. This technique requires a very particular excitation for each mode, hence a long and delicate implementation. On the other hand, the modal characteristics are determined by processing the measurement relatively simply and quickly. It is because of this simplicity of processing that this technique was favored at the beginning of the 1950s, when calculation capabilities were limited, in spite of the difficulties related to the excitation. Ever since, it has been subject to improvements in order to reduce its difficulties by automatic control of the excitation, optimization of the appropriation process, simplified measurement of the parameters, etc. in order to have an efficient operational use by specialized laboratories today. Appropriating a mode requires having a criterion available, which makes it possible to determine if the mode is well isolated. If we consider the approach by normal modes for which all components are real, this criterion is obvious: the mode is isolated if all the points of the structure vibrate in phase (+ sign components) or out of phase (– sign). This criterion is no longer valid with complex modes, which therefore limits the use of this technique for normal modes. Moreover, it is preferable that the excitation be sinusoidal (any properly correlated excitation could actually be used, but without any advantage a priori) and close to the mode's natural frequency in order to obtain higher, thus more accurate, response levels, excluding non-linearities. It is then necessary to distribute the excitation on the structure so as to excite only one mode at a time, and for each desired mode. This must be performed by a specific set of combined forces on the internal DOF. If we look again at equation of motion [4.47] based on the modes, and assuming uncoupled modal damping, and by

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separating the equation verified by the mode k considered from those of the other modes l: (−ω 2 ml + iω c l + k l ) q l = Φli Fi = 0 l

l≠k

(−ω 2 m k + iω c k + k k ) q k = Φki Fi

[11.4]

The first equation, which expresses the non-excitation of the modes other than k, provides as many homogenous relations as modes l. Since theoretically there are as many modes as internal DOF i, all the components of Fi are determined within a coefficient and the second equation relates this coefficient to the modal response level q k . This means that the appropriation of mode k can be done only for a given distribution of the excitation on all the DOF i. Theoretically, we need as many excitation forces as DOF for a given level q k , which is impossible to accomplish. In practice, however, the responses q l only need to be sufficiently small compared to q k , which can be achieved with an excitation frequency close to f k using a small

number of carefully distributed forces, from two to ten depending on the complexity of the considered modes. Having said this, the steps of a phase resonance test are as follows: – preliminary sine sweep (see section 1.2.1.2) with a non-adjusted excitation, in the considered frequency range in order to detect the searched modes using the appearance of the resonances. Each resonance reveals the presence of a mode, but with a contribution of the other modes that it will be necessary to eliminate for a good appropriation; – appropriating each mode separately by adjustment of the excitation forces in order to estimate the theoretical distribution given by equation [11.4]. The phase criterion is applied in order to verify the appropriation and to judge its quality: all points should vibrate in phase or out of phase. Moreover, excitations and responses will be in quadrature at the resonance; – determination of the modal characteristics for each appropriated mode. The natural frequency f k is immediately determined by the resonance, as well as the mode shape Φi k . The damping ζ k can be estimated in various ways, the simplest being that of [2.20]. All these parameters require only motion measures. On the other hand, the determination of the generalized mass m k requires a force measure, as it was explained in section 11.3.2, various methods being possible and possibly

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giving ζ k at the same time (additional masses, forces in quadrature, circle of admittance in the complex plane, complex power, etc.). In the presence of a junction, the measurement of the reactions at the DOF j will provide L j k . The main advantage of this technique is that it preserves the physical aspect of the phenomena. Each mode is appropriated in turn and the phase criterion is there to confirm its existence. The results are thus very reliable, often with very good precision despite certain difficulties such as the presence of measurement noise, non-linearities or of close modes, since all the injected energy is targeted on the mode under consideration. However, mode appropriation has various disadvantages, beginning with the problem of implementing the distributed excitation and its practical limitation to normal modes with a sine excitation. Moreover, the quality of the appropriated modes can suffer if certain parts of the structure are inaccessible, for example in the case of the presence of liquid, as with the fuel tanks of a launcher. This is why various methods which do not use appropriation have been introduced since the 1970s, allowing more flexibility in terms of the excitation at the price of a more complex post processing, but largely compensated by the increased performance of computers.

11.3.4. Phase separation modal tests The phase separation modal tests consist of simultaneously exciting several modes of the structure, then in separating them by post processing. This technique does not require a particular excitation for each mode, hence an implementation that is generally more flexible and of course simpler. The difficulty is now with the processing that must provide the modal characteristics using the measurements. After various attempts based on the experience gained from the phase resonance tests, the investigations have rapidly converged towards the problem of the identification of the FRF themselves [GIR 77]. Using appropriation, we adapted the FRF to the desired modal characteristics. With phase separation, we adapt the search of modes to the available FRF. It is thus the inverse problem to the mode superposition, as is indicated in Figure 1.14: i.e. finding the various modal parameters starting from the FRF with the form of [11.2]. Possibilities for the excitation are now numerous: – from the spatial point of view, we can consider any type of excitation. As previously, it can consist of several forces but also of only one, and this solves the

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accessibility problem and considerably simplifies the implementation because there is no need for adjustment. However, we must excite all the modes which we are searching for, which will possibly require several successive excitations in the case of uncoupled behavior. For example, for a beam with its motion along its axis, its torsion and its two independent bending planes, we could adapt the force to each case in order to study these behaviors separately unless we want to apply only one suitable offset force that excites all directions at the same time; – the excitation can be forces on DOF i, but also motions imposed on DOF j. Despite the inherent difficulties with this type of excitation, as mentioned in section 11.2.3, and which means that it is used less often, it can be envisaged perfectly, particularly with a specimen on a shaker with imposed base motion. The advantages include regrouping of the identification and qualification tests in a single test campaign, minimizing costs and delays, and therefore favor the use of this approach by adaptation of the usual methodology which does not present a major difficulty [GIR 93, GIR 00, ROY 01b]; – there can be any type of excitation not only from the spatial point of view, but also from the temporal point of view. While a sine excitation was necessary to appropriate a mode, the FRF [11.2] can also be obtained using a random or transient excitation. The latter could even be a simple hammer impact. In all cases we simply need to measure the excitation and the responses in order to obtain the corresponding FRF. In practice, the transient excitation with the hammer, the least accurate, is used only to quickly obtain tendencies or to clarify a problem. Random excitation is the simplest to carry out, but a well-controlled sinusoidal excitation is preferred. In addition to the flexibility in the choice of excitation, the processing can be performed later, and this equally diminishes the immobilization period of the specimen. We only need to store the FRF resulting from the measurements after verifying that they are of sufficient quality and that they allow the identification of all the modes for which we are searching. The identification process now consists of extracting the modal parameters from the stored FRF according to the form of the analytical expressions in equation [11.2]. Again, certain choices are possible between the viscous or structural damping, and between normal or complex modes, which represent an important advantage in the characterization procedure. However, the extraction of modal parameters is a delicate process. It requires adapted numerical algorithms and a solid experience in the area because the test conditions with the measurement noise, non-linearities, etc. can seriously perturb the identification process. This operation is the subject of the following sections.

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Because of its numerous advantages, simplicity of implementation, flexibility of utilization and processing, this technique without appropriation is widely used. However, the processing requires a certain know-how and can prove long and delicate, which does not allow it to replace appropriation techniques. Advocates of appropriation point out the reliability of the results due to their physical interpretation, as well as a better overall accuracy and robustness in difficult cases such as for non-linearities or close modes.

11.3.5. Extraction of modal parameters We will limit ourselves here to a synthetic description of the main methods in relation to the analysis techniques set out in the previous chapters. We can schematically divide the methods into two classes: – the methods that consider each mode separately, close to its predominant resonance. The FRF that should be identified is then that of a 1-DOF system, and this makes it possible to extract the corresponding modal parameters in various ways. This class of methods is often denoted by the abbreviation SDOF (Single Degree Of Freedom). The presence of other modes disturbs this analysis but after an initial identification of all modes, we can make a correction and iterate to convergence (iterative SDOF). These methods are discussed in section 11.3.6; – the methods that consider several modes simultaneously, within a given frequency band. The FRF to be identified is then that of a system with several DOF, and the separation of the modes is possible by various methods in the time or frequency domain. This is the MDOF class (Multi-DOF), which is discussed in section 11.3.7. When the modes have conveniently been identified in the considered frequency band, they are completed using residual terms in order to represent the truncation effects. In a general, we can write:

X (ω ) =

∑

k lower

Ak (ω ) X k +

∑ A (ω ) X k

k band

k

+

∑

k upper

Ak (ω ) X k

[11.5]

Normally, their are no lower modes for a non-free structure since we usually search all the modes below a certain frequency. Otherwise, they are the rigid modes for a free structure or the suspension modes for a suspended structure, or particular

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modes, for example sloshing modes of liquid in tanks. If their frequencies are low in relation to the frequency of the beginning of the band, they will have a contribution to the FRF amplitude in 1 / ω 2 approximately (inertial term):

∑

Ak (ω ) X k ≈

k lower

∑

1

k lower

⎛ω −⎜ ⎜ω ⎝ k

⎞ ⎟⎟ ⎠

2

Xk

[11.6]

Regarding the upper modes, they typically correspond to the truncation effect where the amplitudes equal approximately 1, hence the appearance of the residual term related to the static term by the summation rule:

∑

Ak (ω ) X k ≈

k upper

∑

X k ≈ X res = X stat −

k upper

∑X

[11.7]

k

k band

Figure 11.6 schematically illustrates these considerations (similar to Figure 5.7). In practice, the identification of the static term is easier than that of the residual term because it is associated with the response levels at the beginning of the band where the measurements can be better exploited. In the absence of lower modes, it is the low frequency asymptotic value. Otherwise, the constant X stat must be separated from the contribution in 1 / ω 2 of the lower modes. ~

∑ Ak X k

X

∑

~ Xk ⎛ω −⎜ ⎜ ωk ⎝

~

⎞ ⎟ ⎟ ⎠

∑ Xk

2

k lower

k band

k upper

Figure 11.6. Contribution of the modes and of the residual or static terms

ω

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It should be noted according to relation [11.5] and the possible expressions for Ak(ω) that the modal parameters are rather related to the imaginary part of the FRF, while the inertial terms, static or residual are related to the real part (for the G, T or M FRF in the form of [11.2], the others deduced using i ω). Thus, if the method allows it, it would be of interest to identify the modal parameters using the imaginary parts and the other parameters using the real parts. Note that all these considerations apply to all possible types of FRF even if in practice, the FRF are essentially accelerances due to their ease of implementation. The processing usually performed by identification software codes on this type of FRF can be extended to other types of FRF without any particular difficulty.

11.3.6. Single DOF (SDOF) methods

For a SDOF, the FRF is the product of a dynamic amplification including the natural frequency and the damping, by the effective parameter. Its amplitude depending on the frequency has the form of a peak, like those of the dynamic amplifications in Figure 2.2, whose modal parameters we will deduce. Schematically: – the natural frequency is approximately the frequency of the peak. A correction may be added depending on the damping that influences the result, as indicated in section 2.2.4 with Table 2.1; – the damping is directly linked to the peak sharpness as indicated in section 2.2.4 with formula [2.47] and Figure 2.5; – the effective parameter is approximately given by the maximum of the peak divided by the amplification at resonance, inversely proportional to the damping. A rigorous approach will take into account the exact expressions of the dynamic amplifications that we can exploit in various ways, which are more or less sensitive to measurement noise and to the presence of the close modes. Among the most common is the circle fit, which is based on the fact that the plot of the imaginary part of the FRF versus the real part, said to be in the Nyquist plane, is a circle in the following cases: iω u = F

iω ⎛ ω 1− ⎜ ⎜ωk ⎝

2

⎞ ⎟ + i 2ζ ω k ⎟ ωk ⎠

~ Gk

or

u = F

1 ⎛ ω 1− ⎜ ⎜ωk ⎝

2

⎞ ⎟ + iη k ⎟ ⎠

~ Gk

[11.8]

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With measurements at various frequencies around the resonance, we obtain the plot in Figure 11.7 for the case of structural damping. The angle at the center θ is related to the frequency and to the damping by:

tan

θ 2

=

⎛ ω 1− ⎜ ⎜ωk ⎝

ηk

⎞ ⎟ ⎟ ⎠

2

⇒ η k ω k 2 = −2

dω 2 (ω k ) dθ

[11.9]

The modal parameters are then given by: – for the circular natural frequency ω k : real part = 0, or maximum of the imaginary part, or maximum of the amplitude or maximum of dθ / dω (the least perturbed by the close modes); – for the damping η k : formula [11.9]; ~ – for the modal effective parameter G k : product of η k by the diameter. ℑ(u / F ) ℜ(u / F )

ω

θ ωk Figure 11.7. Circle fit in the Nyquist plane

Besides cases [11.8], strategies other than the circle fit can be used [EWI 00, MAI 97] generally leading to a solution of an overdetermined system providing an average solution and the deviation with the measurements that makes it possible to evaluate the quality of the identification. In an iterative approach, in order to take into account the influence of the other modes, step n consists of removing the contribution of modes l ≠ k in the vicinity

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of the circular frequency ω k , followed by a new estimation of the parameters, and then repeating the procedure for each mode. The convergence will be rapid if the modes are relatively well separated. It should be noted that if we fix the circular frequencies ω k at to the frequencies of the peaks, and the damping factors ζ k or ~ η k according to the sharpness of the peaks, the effective parameters X k are solutions of a linear system [GIR 00, ROY 01b]. The quality of the results will be evaluated by comparison between the FRF resulting from the measurements and the synthesized FRF. These SDOF methods are used often for their simplicity and for the direct comprehension of the results. They are similar to an appropriation strategy in that one mode is identified at a time. Their drawbacks appear in the presence of strong damping or close modes where the separation of the modal contribution becomes difficult. They are generally reserved for the normal mode approach which is compatible with their advantages – hence the need to complete these methods by more general methods that can handle several modes simultaneously.

11.3.7. Multi-DOF (MDOF) methods

For a MDOF method, the FRF is that of a system with several DOF, therefore resulting from the contributions of several modes that must be separated by calculation. Various methods have been proposed in order to solve this problem [EWI 00, MAI 97]. We will limit ourselves here to an overview of two examples among the most popular methods. At this level we can work directly on the FRF, thus in the frequency domain, but also in the time domain where we will manipulate damped sinusoids instead of dynamic amplifications. Indeed, in general of complex modes in 2N space (Chapter 7) using the Laplace variable λ = iω , we have the correspondence between the FRF and impulse response (subscript k of the modes is not underlined here for simplicity):

X (λ ) =

2n

Ak k =1 λ − λ k

∑

⇔

x(t ) =

2n

∑ Ak exp(λ k t )

[11.10]

k =1

In the time domain, if the response x(t) is measured at instants i ∆t, we will have:

Testing Techniques

x(t i ) =

2n

∑ Ak Vk i

V k = exp(λ k ∆t )

with

k =1

371

[11.11]

The V k are thus solutions of the polynomial equation:

α 0 + α 1 V + α 2 V 2 + … + α 2n V 2n = 0

and

with

α 2n = 1

[11.12]

x(t 2n −1 ) ⎤ ⎡ α 0 ⎤ ⎡ x(t 2n ) ⎤ x(t 2 n ) ⎥⎥ ⎢⎢ α 1 ⎥⎥ ⎢⎢ x(t 2n +1 )⎥⎥ = ⎥⎢ ⎥ ⎢ ⎥ ⎥⎢ ⎥ ⎢ ⎥ x(t 4 n − 2 )⎦ ⎣α 2 n −1 ⎦ ⎣ x(t 4n −1 ) ⎦

x(t1 ) ⎡ x (t 0 ) ⎢ x(t ) x(t 2 ) 1 ⎢ ⎢ ⎢ ⎣ x(t 2n −1 ) x(t 2 n )

which provides α k then V k , λ k and Ak , from which we deduce the usual modal parameters. This complex exponential method is used when the time response is available. In the case of several excitations and time responses, the simultaneous treatment of the measurements on this basis leads to the polyreference method. The equivalent in the frequency domain is the orthogonal polyreference method. The FRF is written in the form:

∑ bi Ak X (λ ) = ∑ = i k =1 λ − λ k ∑ ak 2n

λi λ

k

=

B (λ ) A(λ )

[11.13]

k

If X(λ) is measured at frequencies corresponding to λ m , m = 1, 2, … M, a k will be solutions of the polynomial equation: a 0 + a1 E (1, p ) + a 2 E (2, p) + … + a 2 n E (2n, p) = 0

[11.14]

with α 2n = 1 , p > 2n and E (k , p ) = [X (λ1 )

X (λ 2 )

⎡λ k ⎢ 1 X (λ M )] ⎢⎢ ⎢ ⎢⎣

λ k2

⎤ ⎡ Q p (λ1 ) ⎤ ⎥⎢ ⎥ ⎥ ⎢ Q p (λ 2 ) ⎥ ⎥⎢ ⎥ ⎥⎢ ⎥ k ⎥ Q (λ ) λ M ⎦ ⎢⎣ p M ⎥⎦

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Q p (λ ) being a basis of orthogonal polynomials such that:

∑ Q (λ * p

m

) Qq (λm ) =

m

0 if p ≠ q 1 if p = q

[11.15]

used to minimize the sensitivity of the roots of equation [11.14] to the perturbations on the a k . We can deduce a k , then λ k and Ak , from which we extract the usual modal parameters. These methods are relatively sensitive to measurement noise and the results that they give must be carefully controlled. Identification software codes that implement these methods generally provide indicators for the detection of the presence of a mode at a given frequency in order to distinguish the physical modes from fictitious modes resulting from numerical or experimental noise. Just like for the SDOF methods, the quality of the results must be evaluated by comparison between the FRF resulting from the measurements and the synthesized FRF.

11.4. Simulation tests 11.4.1. Introduction

Simulation tests should be performed as a function of the characteristics of the real environment and of the available facilities. The test hardware limits the choice to several types of test whose implementation follows specific procedures with specifications resulting from various characterizations of the environment. In the case where the real environment combines excitations of various natures on the given structure, such as an acoustic field and mechanical vibrations, it is rare that we can perform a test that combines these types of excitation. This is typically the case of the satellite on its launcher which, at launch, is exposed to an acoustic field under the fairing as well as to mechanical vibrations via the launcher interface. For this case, installing a large shaker in a reverberant acoustic chamber has not yet been accomplished. Thus from the start, we must separate the tests and possibly choose the most severe: thus, for the simulation of the wideband launch environment, we reserve the acoustic chamber for large satellites with large surfaces exposed to the acoustic field, and the shaker for small compact satellites which are more sensitive to the vibrations at the interface.

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Following the test hardware description in section 11.2.3, we will briefly discuss the implementation of: – shakers in section 11.4.2; – shock machines in section 11.4.3; – reverberant acoustic chambers in section 11.4.4. The elaboration of specifications will then be discussed generally in section 11.4.5 and in section 11.4.6 in terms of the structure's impact on its environment.

11.4.2. Tests with shakers

From the temporal point of view, a shaker can be used according to various modes. The most current are the following. 1) Excitation in sinusoidal mode, and more particularly in swept sine (section 1.2.1.2) with a frequency function of the time and an amplitude function of the frequency. In the case of a standard simulation, the sweep rate is often relatively slow to generate a quasi-stationary motion at each moment but also rapid enough to limit the number of cycles applied if there are not many in service. The sweep law, which can be either, ascending or descending, is generally linear, exponential (often called logarithmic) or hyperbolic, each one having its particularities. Table 11.1 gives some properties for these three laws particularly related to the structure resonances that are generated by its modes, thus related to a 1-DOF system, with two parameters that can be important: the duration ∆t of the passage between the half-power points (see Figure 2.5) and the corresponding number of cycles ∆N, hence the following properties: - linear sweep: no particular property from this point of view, but the advantage of simplicity; - exponential sweep: duration ∆t independent of the natural frequency of the 1DOF system, which explains its widespread use. The sweep rate can then be expressed in octaves/minute, the octave corresponding to a ratio of 2 in frequency. For example a sweep from 5 to 100 Hz with 2 octaves/minute will last for about 130 s; - hyperbolic sweep: number of cycles ∆N independent of the natural frequency of the 1-DOF system, which can be interesting from the fatigue point of view. 2) Excitation in random mode for a given duration, with a power spectral density W (section 1.2.3.4) function of the frequency. In the common case of an imposed acceleration, the function W(f) is generally given in g 2 / Hz , with constant levels by bands, separated by slopes expressed in dB by octave, the octave corresponding to a ratio of 2 in frequency, and 3 dB a factor of 2 in amplitude. Example:

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- from 20 to 150 Hz: 6 dB/octave (evolution in f 2 ), - from 20 to 700 Hz: 0.04 g2/Hz , - from 700 to 2,000 Hz: –3 dB/octave (evolution in l/f). The mean square is obtained by integration of W(f) on all the considered frequency bands (formula [1.24]) which, with the previous example, gives: u2 =

∫

2,000

[11.16]

W ( f ) df

20

⎡ 150 ⎛ f ⎞ 2 ⎤ 2,000 ⎛ 700 ⎞ 2 = 0.04 g 2 / Hz ⎢ ∫ ⎜ ⎟ df ⎥ ≈ (7.3g) ⎟ df + (700 − 150) + ∫700 ⎜ 20 ⎝ 150 ⎠ ⎢⎣ ⎝ f ⎠ ⎥⎦

LAW

Linear

Exponential

f(t)

t f1 + ( f 2 − f1 ) T

⎛t f ⎞ f1 exp⎜⎜ ln 2 ⎟⎟ f1 ⎠ ⎝T

f1 + f 2 T 2

f 2 + f1 T f2 ln f1

Number of cycles

N=

T ∫0

f (t ) dt

Sweep rate

df V= dt

f 2 − f1 T

Duration ∆t between points at half-power

fk T Q ( f 2 − f1 )

Cycles ∆N between points at half-power

fk 2 T Q ( f 2 − f1 )

f f ln 2 f1 T in octaves/minute:

f ln 2 f1 60 ln 2 T

1 ⎛ 1 1 ⎜⎜ − ⎝ f1 f 2 f ln 2 f1 T 1 1 − f1 f 2

1 t − f1 T

⎛ 1 1 ⎞ f2 ⎟⎟ ⎜⎜ − ⎝ f1 f 2 ⎠ T

T

T Q ln

Hyperbolic

f2 f1

fk T f Q ln 2 f1

⎛ 1 1 ⎞ ⎟⎟ Q f k ⎜⎜ − ⎝ f1 f 2 ⎠ T ⎛ 1 1 ⎞ ⎟⎟ Q ⎜⎜ − ⎝ f1 f 2 ⎠

Table 11.1. Properties of the sweep laws in relation to the 1-DOF system (sweep of duration T between the frequencies f1 and f2)

⎞ ⎟⎟ ⎠

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3) Excitation in transient mode. Recent developments in the field of active control allow the reproduction of a given time history, but in practice this is rarely implemented because of control difficulties and especially because of the lack of reproducibility that occurs in the real environment. Instead of specifying a time history, a shock spectrum is preferred whose concept was introduced in section 2.3.3 allowing us to use the envelope of the effects of the environmental on a 1-DOF system, hoping that this remains valid for the structure considered. In this context, the shakers can still be used after the determination of a suitable time function, but their capacity is relatively limited either in available force or in frequency, which most often requires specific devices, which is the subject of section 11.4.3. From the spatial point of view, the simplest case is that of only one excitation in a determined direction. Everything depends on what we wish to impose (see section 11.2.3): – imposing a force can be easily performed using a suspended shaker controlled by force. This shaker is generally of a small size and excites the structure using a slender rod (stinger) in order to direct the force essentially in the desired direction; − imposing a motion is generally performed by placing the specimen on a suitably guided table that is excited using a large shaker controlled by acceleration. This shaker should be of a large size so that it can deliver large forces particularly in the vicinity of the specimen resonances. The table should be sufficiently rigid so as not to significantly deform at the considered frequencies. The guidance should be sufficiently efficient to limit the parasitic motions independently of the specimen behavior. In practice, we are limited to three reference axes, the vertical axis and the two lateral axes. With several shakers, we can excite along several components. The difficulty is to properly control not only each excitation but also the relations among the excitations: the phases in the sinusoidal mode, the coherences in the random mode, the simultaneities in the transient mode. Excitation by forces is performed with several shakers suspended around the specimen, which is often done in the context of an identification test (section 11.3). Excitation by imposed motions must be done using a single table whose DOF are controlled in the same way as shown in Figure 11.2. We can possibly combine the two types of excitation, for example by mounting the specimen on a table and imposing a motion to it and also by exciting it in force to simulate an indeterminate configuration, for example.

11.4.3. Shock device tests

The limitations presented by the shakers in the transient mode lead to the use of specific devices. Other than the pyrotechnic generators that can be implemented in

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order to better reproduce shocks of pyrotechnic origin, impact type shock machines for example can generate relatively high level shocks but with limited adjustment of a few parameters: drop distance, impact material, addition of resonant parts, etc. It should be noted that the shock obtained is very sensitive to the tightening torque of the screws that join the various parts of the hardware, which increases the adjustment possibilities to a certain extent. In this context, the precise reproduction of a given time function is not possible. On the other hand, we can try to envelop a given shock spectrum suitably or to maintain it within certain tolerances, for example −3 dB/+6 dB, which, besides simplifying the specification, makes it possible to compensate for the lack of reproducibility that we often notice for this type of environment. The main difficulty for these tests resides in the measurement of the responses. The acceleration levels can be very high and the frequency content very broad, which requires sensors and an acquisition chain adapted to this type of environment. Values of 20,000 g and 20,000 Hz are commonly measured close to the source and may often be exceeded.

11.4.4. The tests in a reverberant acoustic chamber

The reverberant acoustic chamber is reserved for specimens that offer large surfaces with an intense acoustic field on condition that a diffuse field (coherence given by relation [11.1]) is representative enough. Its volume should be enough not only to receive the specimen (1/10 rule for example), but also to guarantee a low frequency homogenous field due to a sufficient number of acoustic modes. The acoustic field generated by a certain number of sirens is controlled by octave bands (ratio of 2 in frequency) or a 1/3 of octave bands (ratio

3

2 ) at a given level

−5

expressed in pressure dB, i.e. 20 log10 ( p / 2.10 Pa ) . Table 11.2 gives the normalized values of the bands. For example a sound level of 139 dB in the octave centered at 250 Hz corresponds to a RMS pressure of 178 Pa between 250/ 2 and 250 2 Hz. The pressure PSD in each frequency band ∆f i is then given by:

W p ( fi ) =

pi 2 ∆f i

[11.17]

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and the global level N (dB) is obtained from the levels Ni in each band using the expression:

p = ∑ pi 2

2

⇒

N = 10 log10

i

Central frequency (Hz)

Cutoff frequency (Hz)

22.5

----------25

224

28

355 40 ----------50

56

2,800 315 ----------400

450

710 80 ----------100

112

5,600 630 ----------800

900

1,400

5,000 ----------6,300

7,100 8,000 9,000

1,000

125 160 ----------200

4,000 4,500

1,120

140

2,500 ----------3,150

3,550

500

63

Central frequency (Hz)

2,240

560

71

Cutoff frequency (Hz)

250

31.5

180

Central frequency (Hz)

280

35.5

90

[11.18]

i

Cutoff frequency (Hz)

45

∑

Ni 10 10

11,200

10,000 -----------

1,250 ----------1,600

1,800 2,000

Table 11.2. Normalized frequency bands (1/3 octave and octave)

11.4.5. Elaboration of specifications The test specifications should be deduced from what we know about the environment. In a very general manner, we can envisage the following strategies: − tests in real conditions: they will thus be very representative but rarely possible; − tests in laboratories with specifications resulting from pre-established norms: this is the solution when there is insufficient data on the real environment and when the case being considered is close to a situation which has already been encountered.

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These norms exist in various domains and give a first idea of the levels to be applied. However, in order to cover the greatest number of cases, they are generally very conservative (“refuge” severities) and can ignore certain specificities of the environment being considered; − tests in laboratories with specifications adapted to the case considered from a tailoring approach. An accurate reproduction of the real environment is sometimes possible, but it often comes up against various difficulties: prohibitive durations, lack of reproducibility, superposition of several types of excitations, etc. An envelope environment is usually defined based on various criteria and taking into account various safety factors. On this latter point, the practice varies according to the industrial sector and depending on the acquired experience and the degree of analysis authorized. The simplest strategies are often used, for example: − to cover a low frequency environment whose frequency content is poorly known: a sine sweep in the band being considered, with envelope levels and a sweep rate representing a compromise between a desired quasi-stationary motion at each moment and a realistic duration. This approach ignores the influence of the specimen on its environment, which requires additional precautions, as described in the following section; − to cover a wide-band random environment: an envelope of the power spectral densities that are likely to occur under real conditions, applied with a realistic duration. Again, the specimen can influence its environment but generally only to a small extent especially at high frequencies. A more elaborate strategy consists of defining an equivalent environment according to appropriate criteria. The idea is to base the equivalence on the environment effect on a reference system, which joins the spectrum concepts, discussed in section 2.3.3 with the 1-DOF system, more precisely: − the extreme response spectrum, which represents the environment severity in terms of maximum stresses versus the system frequency; − the fatigue damage spectrum, which represents the environment severity in terms of fatigue versus the system frequency. For a given life cycle profile, we may determine an extreme response spectrum and a fatigue damage spectrum enveloping and/or accumulating the spectra corresponding to various events which have been encountered. The equivalent environment is then determined by reproducing these two spectra as well as possible. The difficulty is in the choice of the parameters required for the calculation of the spectra, for example the value of the damping for the extreme response

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spectrum. Values distant from reality risk perturbing the equivalence especially because the simulated environment will be of different nature from the real environment. Thus, replacing a mostly transient environment by a sine sweep or by a random excitation can lead to significant errors because the transient responses are not particularly sensitive to the factor Q, while the sine levels are proportional to Q and the random levels to

Q . Thus, there is always great interest in reproducing the

real environment as well as possible. The levels found starting from the real environment are multiplied by a safety coefficient in order to take into account various factors. In particular: − the safety coefficient that makes it possible to take into account the dispersion of the environmental characteristics and the dispersion of the structural characteristics. Hypotheses on each distribution (most often normal or lognormal) allow the determination of this coefficient, called the guarantee coefficient, according to the desired reliability; − the safety coefficient that makes it possible to guarantee the desired reliability with a reduced number of tests, often only one, instead of an infinity. Again, this coefficient, called the test factor will be determined starting from the assumed distributions. A detailed analysis of these subjects can be found in [LAL 02] volume 5.

11.4.6. Impact of a structure on its environment

The environment imposed on a structure often results from an interaction between two substructures as introduced in section 9.4.1 and schematized by Figure 11.8: the examined structure A, passive, is connected to structure B which is excited by a given source: engine, aerodynamic forces, etc. The excitation applied to structure A is actually the response at the interface of combined A+B subjected to the true excitation source, which thus implies a coupled analysis of combined A+B. Under these conditions, structure A influences the response at the interface in so far as it reacts to the excitation, which, in the frequency domain, occurs more particularly at the natural frequencies of its principal modes.

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A uA u A/ B

B

F

Figure 11.8. Interaction between sub-structures

To convince ourselves of this, let us consider the 1-dimensional problem in Figure 11.8 in order to help understand the phenomena, with for example an excitation by the axial force F B and responses in acceleration in the same direction, particularly u A / B at interface A/B and u A on structure A. In the frequency domain, we will obtain large levels at the resonances of A+B, i.e. in the vicinity of the natural frequencies f kA+ B . A contrario, the levels will be far smaller than these resonances, particularly at natural frequencies f kA of structure

A alone constrained at interface A/B. However, at these frequencies, there is an important dynamic amplification between u A / B and u A governed by the factor

QkA , which means that if u A levels are not high, u A / B levels are small. In other words, if antiresonances are produced at frequencies f kA , the structure A reacts to the excitation by diminishing the level of response at the interface. To what extent? This depends on the “ratio of forces” between structures A and B at these frequencies. This ratio of forces is governed by the dynamic masses, as relation [9.16] shows: a significant dynamic mass for structure A as compared to that of structure B will have a significant influence on the result. Thus, for a given antiresonance, the depth of the antiresonance will be greater if:

Testing Techniques

381

~ − the effective mass M kA of the mode for structure A is large and its damping

ζ kA small; − the dynamic mass of structure B at the considered frequency is small. Under these conditions, structure A can react in a significant manner even though it is much smaller than structure B. For example, in the case of the satellite on its launcher already mentioned, its most important modes should be taken into account in a coupled analysis in order to determine realistic flight levels. Also under these conditions, the strategy consisting of qualifying structure A by a sine sweep on a shaker with envelope levels deduced from a first approximation of the environment should be modified by a level reduction in the vicinity of the main resonances in order to simulate the antiresonances in the coupled configuration. This procedure, called notching can be applied in the following way: − initial specification of acceleration based on the specimen resulting from a standard envelope of the levels in the frequency band being considered; − choice of various sensors at the critical points on the specimen (accelerometers, strain gauges, force sensors, etc.) on which we will limit the levels to prescribed values deduced from a coupled analysis giving the estimations of the real response. This limitation can be performed directly on the initial specification after the preliminary test results at low levels (manual notching) or automatically during the sweep by controlling the levels of the chosen sensors when they have reached their prescribed values, as shown in Figure 11.9. The level reduction thus corresponds to the antiresonances of the coupled response. u Prescribed level

A uA u

A/ B

u

uA

A Specification

u A/ B

u A/ B fk A

Figure 11.9. Automatic level reduction

f

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This strategy of level reduction can be easily applied for mono-axial excitations in the sine mode. We can extrapolate it to other situations, for example in random mode, if the test facilities allow it and may be justified by an appropriate analysis of the behavior under real conditions. In order to illustrate these phenomena and to give some quantitative considerations, let us consider the simple example in Figure 11.10. The discussed structure A is a 1-DOF system representative of an important mode of effective mass m, stiffness k and damping ζ = 1/(2Q), structure B is a simple rigid mass M representative of a certain dynamic mass and directly excited by a force F. The frequency responses u A and u B are plotted in Figure 11.10. We can see: −

ω kA+ B

on

uA

and

uB

a

resonance

with

circular

frequency

= k ( M + m) /( M m) ;

− on u B an antiresonance with circular frequency ω kA = k / m . The latter will require a level reduction in test on a shaker if we adopt the qualification strategy of specimen A by a swept sine based on an envelope of the levels u B . With an initial specification given by the maximum response u B and a prescribed level limitation to the maximum response u A , the extent of the level reduction on the specification with the circular frequency ω kA will be, in the hypothesis m << M: B Q u max A u max

≈

⎛ m⎞ 1+ ⎜Q ⎟ ⎝ M⎠

2

[11.19]

Thus, for a damping ζ of 1% (Q = 50), a ratio 10 between m and M will imply a level reduction of a factor 5, these orders of magnitude being current. A ratio 100 will come to reduce the levels with 12%. On the other hand, a ratio of 1,000 will actually have no influence as the dynamic amplification can no longer generate a significant reaction.

Testing Techniques

uA F

A uA

m << M c=

k

Q

u

F

M +m

km

F

B

Q

1 M +m

uB

M

≈

B

⎛ m⎞ 1+ ⎜Q ⎟ ⎝ M⎠ ≈ M +m

ω

2

1 M

1 M +m k m

k ( M + m) Mm

Figure 11.10. Example illustrating the level reduction strategy

ω

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Model Updating and Optimization

12.1. Introduction When test results that give information about a certain reality of the dynamic behavior of the specimen are available, it is important to compare them to the results of the mathematical models in order to verify their representativity. This is the test/analysis correlation step that makes it possible to judge the quality of the analysis which has already been performed and often highlights the necessity of modifying the models in order to come closer to the measured reality. We then consider model updating that tries to minimize the distance between the results, which relates to an optimization calculation, in the same way as the improvement of the structural performances with respect to dynamic criteria. The problem is then how to modify the model parameters on which we can act in order to improve the representativity of the model or the performances of the parameters? In the traditional context of finite element models, we can think of various strategies: − modification of the physical matrices resulting from models, for example those obtained by mass and stiffness condensation. This approach, considered in the past for its interest from a numerical point of view, should be avoided because it loses the physical meaning and it does not allow extrapolation to configurations other than those of the tests; − modification of the model’s mesh (position of the nodes, topology), which is easy to perform but requires strong interaction with the finite element code at each iteration in order to estimate the new behavior, hence computationally intensive;

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− model modification at the level of the physical parameters involved in the elements, such as the stiffness of the springs, the value of the lumped masses, the properties of the material, the thicknesses of the plates, the sections of the beams, etc. This latter strategy is compatible, to a certain extent, with a reanalysis without returning to the finite elements and it is the essential ingredient of industrial practice. This reanalysis can be considered with various approaches, particularly: − substructuring which considers the modification as a substructure and deduces the behavior of the modified structure by coupling between the initial structure and the modification. This approach can be efficient in certain situations such as adding point masses, but it is generally not well-adapted, for example to modifying the thickness of a plate, which involves a large number of connection DOF; − sensitivity analysis which consists of determining the impact of an infinitesimal parameter modification on the structure’s behavior, starting with the normal modes. Its interest resides first in allowing the detection of influent parameters, then in predicting the modified structure, which will be most accurate for small modifications. It is a good prelude to the updating, but it remains limited in its efficiency for relatively large variations of parameters; − Ritz method, a generic approach that consists of projecting the solution being searched based on available modes, which makes it possible to consider greater modifications than previously, whose accuracy depends on the modal basis used. The literature on model updating is abundant because the subject is important for the reliability of the analyses. A large inventory can be found in [MOT 93]. We will limit ourselves here to several important points in relation to the previous developments, especially the modal approach and its effective parameters in normal modes. We will leave the damping problem aside, which is either common for a well-isolated mode, or very delicate if we want to treat it like the other parameters. Sensitivity analysis is discussed in section 12.2; the Ritz method in covered in section 12.3. Model updating is discussed in section 12.4 with some considerations on the physical parameters that could be considered and on the procedure possibilities starting from the test/analysis correlation. Finally, optimization techniques are introduced in section 12.5 in order to complete the information on the implementation, followed by some applications in section 12.6.

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387

12.2. Sensitivity analysis 12.2.1. Introduction Sensitivity analysis consists of calculating the derivatives of the dynamic properties of the structure under consideration in relation to the physical parameters that can be modified. In the context of modal approach, this naturally starts with the modal parameters involved in the mode superposition. We deduce from them the derivatives of the structural responses that give the direction to which we turn and that can be used in order to estimate the responses of the modified structure using small but finite variations of physical parameters. X,p generally denotes the relative partial derivative of X in relation to the physical parameter xp:

X,p =

∂X ∂X = xp ∂x p / x p ∂x p

[12.1]

The advantage of the relative derivative is that the variation percentage of parameter xp can be calculated without having to know it. With this derivative, we first deduce the value of X after finite modification ∆xp of parameter xp according to its value before the modification:

X ( x p + ∆x p ) ≈ X ( x p ) + X , p

∆x p xp

[12.2]

In order to determine these derivatives X,p, we will first take interest in the basic modal parameters that are natural frequencies and eigenvectors (mode shapes), in order to then pass to the modal effective parameters that constitute the essential ingredients of the dynamic responses.

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12.2.2. Sensitivity of the natural frequencies The sensitivity of the natural frequencies to the parameter xp is obtained by deriving the relations that define the normal modes. By performing the following derivation operation for each mode k on the relation that it verifies:

Φki

∂ ∂x p / x p

(

(−ω k2 M ii + K ii ) Φi k = 0

)

[12.3]

we get to the result:

ω k2, p

=

Φki (−ω k2 M ii , p + K ii, p ) Φi k mk

[12.4]

Mii,p and Kii,p are the matrices derived (term by term) from Mii and Kii in relation to xp. They are thus zero on the DOF i not affected by the considered parameter. One of them can be zero if the parameter affects only the masses or only the stiffnesses, for example Mii,p for an elastic modulus or Kii,p for a density. It is interesting to note that this sensitivity depends only on the mode k, which implies that there is no truncation error: only knowing mode k makes it possible to obtain the sensitivity of its natural frequency in an exact manner. This remarkable property does not apply to the eigenvectors.

12.2.3. Sensitivity of the eigenvectors Contrary to the previous case, the sensitivity of an eigenvector k to parameter xp cannot be deduced uniquely from the mode k, and therefore requires a more complex treatment. The exact derivatives can be determined directly [NEL 76], or in a more efficient way by an iterative process [ALV 97], but the calculation remains difficult. In the context of a large number of iterations on the configurations, for example for updating purposes, an alternative likely to provide a suitable approximation is the projection on a basis of normal modes: Φi k , p = ∑ Φil c l k l

[12.5]

Model Updating and Optimization

389

Coefficients clk for l ≠ k, are determined using the following derivation operation:

Φli

∂ ∂x p

(

(−ω k2 M ii + K ii ) Φi k = 0

)

[12.6]

Coefficients ckk are determined using a normalization choice, by writing for example mk,p = 0, which means maintaining a constant generalized mass mk. We then arrive at relation:

Φi k , p = −Φi k

Φki M ii, p Φi k 2 mk

(rigid modes: Φi r , p = −Φi r

−

∑ Φil

Φli (−ω k2 M ii, p + K ii , p ) Φi k

l ≠k

Φri M ii , p Φi r 2 mr

(ω l2 − ω k2 ) ml

[12.7]

assuming mrr is diagonal)

The sensitivity of eigenvector k thus depends on all the modes, hence the truncation effects that will diminish with the number of retained modes. It is also possible to complete the basis of normal modes using other modes, for example static displacements, as we will see during the description of the Ritz method.

12.2.4. Sensitivity of the modal effective parameters Owing to their formulation (section 5.2.1), the sensitivity of the modal effective parameters passes by that of the previous modal ingredients as well as that of the participation factors, themselves resulting from the junction modes [GIR 92]. As for the normal modes, the sensitivity of the junction modes to the parameter xp can be obtained by deriving the relations that they verify, i.e. [4.29], by projecting them on a basis of normal modes in order to obtain: −1 Ψij , p = −Φik k kk Φki (K ii, p Ψij + K ij , p )

[12.8]

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The expression is sensitive to modal truncation, as for the eigenvector sensitivities. Without truncation, we find the term Φik k −kk1 Φki = G ii = K ii−1 . From this we deduce the sensitivity of the participation factors by deriving their definition relation following equation [4.32], in order to get to:

L kj , p = Φki, p (M ii Ψij + M ij ) + Φki (M ii , p Ψij + M ii Ψij , p + M ij , p )

[12.9]

Thus, the sensitivity of the modal effective parameters:

ω k, p 2 Φi k , p Φki + Φi k Φki, p ~ ~ (G ii,k ) , p = − G ii,k ω k 2 mk ωk 2

[12.10]

Φi k , p L k j + Φi k L k j , p ~ (Tij ,k ) , p = mk

[12.11]

L j k , p L k j + L j k L k j, p ~ (M jj ,k ) , p = mk

[12.12]

Again, these expressions are sensitive to truncation.

12.2.5. Simple example Consider the example of the 2 internal DOF in Figure 4.1 with its results [4.9], [4.36] and [5.11]. By making m = k = 1 as indicated in Figure 12.1, and by taking as physical parameter xp the mass m1 of DOF 1, the previous expressions give: ⎡1 0⎤ M ii , p = ⎢ ⎥ ⎣0 0 ⎦

ω12, p = −

⎡0 0 ⎤ K ii, p = ⎢ ⎥ ⎣0 0 ⎦

1 1 ⎡ 3 ⎤ Φi1, p = ⎥ ⎢ 15 50 ⎣− 14⎦

⎡0 ⎤ Ψij , p = ⎢ ⎥ ⎣0 ⎦

L10, p =

39 50

[12.13]

Model Updating and Optimization

~ 1 ⎡24 18 ⎤ (G ii,1 ) , p = ⎢ ⎥ 125 ⎣18 − 24⎦

~ 1 ⎡24⎤ (Ti 0,1 ) , p = ⎢ ⎥ 125 ⎣18 ⎦

391

~ 117 (M 00,1 ) , p = 125

4 8 1 ⎡32⎤ Φi 2, p = ⎢14 ⎥ L10, p = − 25 5 25 ⎣ ⎦ ~ 1 ⎡− 24 − 18⎤ ~ 1 ⎡− 24⎤ (G ii,2 ) , p = ⎥ ⎢ ⎢ − 18 24 ⎥ (Ti 0,2 ) , p = 125 ⎣ 125 ⎣ − 18 ⎦ ⎦

ω 22, p = −

8 ~ (M 00,1 ) , p = 125

With for example a mass increase of 1%, on the DOF 1, we obtain a reduction of the first eigenvalue of 1/1,500 = 0.00066667 instead of 0.00066640 using an exact calculation, i.e. an error of 0.04%. On the other hand, an increase of 100% leads to 0.0667 instead of 0.0620, i.e. an error of 7%. As we see the error increases with increasing modification. It is even more pronounced with the second mode. From the updating or optimization point of view, this parameter does not influence the first natural frequency much but rather the second. It is thus preferable to use it for the modification of the second frequency in order to maximize it if this is the objective, or to compare it to that of the tests if updating is the objective. On a related note, it could also be considered to diminish the dynamic mass of the system in the vicinity of the first resonance using the effective mass because it has a very high sensitivity.

m2 = 1

k2 =

u2

2 3

i m1 = 1

u1

k1 = 1

u0

j

Figure 12.1. Sensitivity analysis on the 2 internal DOF system

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12.3. Ritz reanalysis 12.3.1. Introduction

The Ritz reanalysis consists of projecting the mass and stiffness matrices for the modified structure based on available modes, and then to solve the eigenvalue problem in this basis, and finally to restore the components of the eigenvectors on the physical DOF. It should be noted that if the basis is complete, i.e. if there are as many independent modes as DOF, the solution is exact but in this case the operation loses its interest. The more truncated the basis is, the more we lose accuracy while gaining computational efficiency. The normal modes of the initial structure naturally provide a good starting point that can be completed with other modes, for example static modes.

12.3.2. Utilization of the normal modes

Let us consider a finite modification ∆xp of parameter xp affecting a subset p of DOF, which implies modifications ∆Mpp and ∆Kpp of the mass and stiffness matrix of the structure under consideration. If we project the internal DOF based on the initial normal modes, i.e. ui = Φik qk, the initial mass and stiffness matrices become: (M + ∆M ) kk = Φki (M + ∆M ) ii Φik = m kk + Φkp ∆M pp Φ pk

[12.14] (K + ∆K ) kk = Φki (K + ∆K ) ii Φik = k kk + Φkp ∆K pp Φ pk

Matrices mkk and kkk are diagonal and modification ∆xp acts as a perturbation term. If it is weak, the resulting matrices will have a dominant diagonal, which makes diagonalization easier. The eigenvalue problem on these new matrices, which are reduced in size to the number of retained modes, thus gives:

[− ω

'2 k

]

(M + ∆M ) kk + (K + ∆K ) kk Φ'k k = 0 k

[12.15]

from which we can obtain the new normal modes on the physical DOF: (ω 2 + ∆ω 2 ) k = ω k'2

(Φ+ ∆Φ) ik = Φik Φ'kk

[12.16]

Model Updating and Optimization

393

Regarding the junction modes Ψij, by projecting their modification ∆Ψij onto the initial normal modes, we obtain: (Ψ + ∆Ψ) ij = Ψij − Φik (k kk + Φkp ∆K pp Φ pk ) −1 Φki (∆K ii Ψij + ∆K ij ) [12.17]

All matrices to be diagonalized or inverted are of a size equal to the number of retained modes, thus considerably reducing the calculations compared to an exact solution using finite elements, which is the objective, but at the cost of introducing truncation errors.

12.3.3. Utilization of additional modes

In order to improve the results, the basis formed by the k normal modes of the initial model can be enriched by additional modes s (selection) of various origins: random modes, static modes, etc. They must be linearly independent, but, their quality will influence accuracy and certain modes are more efficient than others. For reasons related to those mentioned in section 8.3 about modal synthesis, modes such as static flexibilities or junction modes are generally good candidates to add to normal modes. The enriched basis (k + s) approach is the same as previously and equations [12.14] to [12.16] still apply. The only difference is that the matrices (m, k)(k+s)(k+s) are no longer diagonal because the modes do not necessarily have orthogonal properties. However, this does not cause any problem at all.

12.3.4. Simple example

Consider the example of the 3-internal DOF system in Figure 4.5 (the example in Figure 4.1 is too simple to vary the cases). By taking as physical parameter xp the stiffness of the second spring that we double, as indicated in Figure 12.2, the modifications of the matrices are as follows: ⎡0 0 0 ⎤ ∆M ii = ⎢⎢0 0 0⎥⎥ ⎢⎣0 0 0⎥⎦

⎡ 1 − 1 0⎤ ∆K ii = ⎢⎢− 1 1 0⎥⎥ ⎢⎣ 0 0 0⎥⎦

∆K ij

⎡0 0⎤ = ⎢⎢0 0⎥⎥ ⎢⎣0 0⎥⎦

[12.18]

The application of the previous equations gives the results in Table 12.1 with the first two retained normal modes. Table 12.2 indicates the errors in the eigenvalues

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for various choices of the basis among normal modes Φik, static modes Ψij and static flexibilities Gii (Gi1 = Ψi1 and Gi3 = Ψi2). We see that the use of a static mode instead of the second normal mode improves one of the eigenvalues but degrades the other. The case (Gi1 + Gi2) corresponds to the average of the flexibilities at the two extremities of the modified spring and makes a compromise between the two errors. The static modes alone are not very efficient. i u1 m = 1

k=1

u2 k=1 ∆k = 1

u3 k=1

m = 1

m = 1

k=1

u0

u4 j Figure 12.2. Ritz reanalysis on a 3-internal DOF system

Non-modified structure Exact

Modified structure 2 retained modes

ωk2

0.586

2.000

3.414

0.609

2.227

5.164

0.617

2.511

Φik

+0.500 +0.707 +0.500

+0.707 +0.000 -0.707

+0.500 -0.707 +0.500

+0.562 +0.672 +0.483

+0.497 +0.192 -0.846

+0.661 -0.715 +0.226

+0.553 +0.705 +0.444

+0.666 -0.055 -0.744

+0.736 +0.547 +0.331

+0.264 +0.453 +0.669

Ψij

+0.750 +0.500 +0.250

+0.250 +0.500 +0.750

+0.714 +0.571 +0.286

+0.286 +0.429 +0.714

Table 12.1. Ritz reanalysis of the system in Figure 12.2

Model Updating and Optimization Error on ω12

Selected modes 3 independent modes Φi1 Φi2 Φi1 Ψi1 or Gi1 Φi1 Ψi2 or Gi3 Φi1 Gi2 Φi1 Gi1 + Gi2 Ψi1 Ψi2 Gi1 Gi2 Φi1 Ψi1 or Gi1 Ψi2 or Gi3 Gi2

0 1.4% 0.2% 3.1% 0.9% 2.3% 9.5% 5.6% 3.3% 52.6% 52.6% 36.9%

395

Error on ω22 0 12.8% 55.8% 0.8% 119.4% 2.8% 12.3% 95.7% – – – –

Table 12.2. Errors of Ritz reanalysis for various bases

12.4. Model updating 12.4.1. Physical parameters

The physical parameters intended to be modified in the structure can be chosen from various types. They can involve a particular element or a set of elements such as those of a plate whose thickness is the parameter being discussed. We will reason here in relation to element e but this can be extended to any structural part (which takes us back to the zone concept in section 5.2.2.3). The developments of the previous chapters make derivatives X,p of the mass or stiffness matrices intervene for the sensitivity analysis and modified matrices (X + ∆X) for finite variations. The analysis will be simplified if X,p or (X + ∆X) is simply expressed according to X. That eliminates parameters such as the shear or Poisson coefficient. The simplest case is where the matrices X are proportional to a given power of parameter xp: to the power of αp for the masses and to the power of βp for the stiffnesses. We then have the following relations for the element matrices: − for the sensitivity analysis: M ,ep = α p M e

K ,ep = β p K e

[12.19]

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Structural Dynamics in Industry

− for the Ritz analysis: α

β

⎛ x + ∆x ⎞ p e ( M + ∆M ) e = ⎜ ⎟ M ⎝ x ⎠p

⎛ x + ∆x ⎞ p e (K + ∆K ) e = ⎜ ⎟ K ⎝ x ⎠p

[12.20]

A simple example, which is often used in practice, is that of the mass density ρ and Young’s modulus E of a homogenous and isotropic material, for which: xp = ρ

⇒ αp =1 βp = 0

xp = E

⇒ αp =0

β p = 1 [12.21]

Certain physical parameters intervene differently according to how the structural element deforms. Thus, the stiffness of a plate is proportional to its thickness if it deforms in its plane and proportional to the cube of its thickness if it deforms out of its plane. In order to be able to treat this type of case, which is limited in practice to the stiffness but a very common case, it is necessary to partition the element matrices according to the various types of deformation d, in order to attribute the corresponding exponent βp to each type of deformation. We will then have [GIR 92]: − for the sensitivity analysis: K ,ep = ∑ ( β p ) d K ed

[12.22]

d

− for the Ritz analysis: ( β p )d

⎛ x + ∆x ⎞ (K + ∆K ) e = ∑ ⎜ ⎟ x ⎠p d ⎝

K ed

[12.23]

Regarding the plates, which are simpler to analyze than the beams, there are two possible types of deformation: in plane (membrane) and out of plane (bending). With the thickness as the physical parameter and a mid-plane offset proportional to the thickness to take into account a possible membrane-bending coupling, αp and (βp)dd take the following values:

Model Updating and Optimization

α p =1

⎡1 2⎤ (β p ) dd = ⎢ ⎥ ⎣ 2 3⎦

397

[12.24]

We should note that the non-diagonal term, i.e. the coupling exponent, is the average of the diagonal terms, i.e. the exponents of the membrane and of the bending. In the case of a honeycomb plate of height h and with face thickness t which is small compared to h, each of these parameters has the exponents: xp = t (h constant) : αp = 1 βp membrane = 1 βp bending = 1 xp = h (t constant) : αp = 0 βp membrane = 0 βp bending = 2

[12.25]

Other cases may be derived such as the case where the membrane thickness and the bending inertia are considered separately, or the case of parameter xp whose exponents are given for the membrane and bending. Finally, a partitioning according to the orthotropic parameters with simplifying hypotheses (no coupling strain, rectangular shape with orthotropic directions parallel to the sides, etc.) can be envisaged for an orthotropic plate. As for beams, there are four possible types of deformation: along the axis (axial), around the axis (torsion) and perpendicular to the axis (bending in each plane), hence a 4x4 matrix for exponents (βp)d. In order to simplify this situation, we can revert to the example of a plate using the often-verified hypotheses: average exponent of the two bending exponents for the torsion, average exponent for the coupling. Typical cases: − full section of characteristic dimension r: xp = r : αp = 2 βp axial = 2 βp bending = 4

[12.26]

− closed thin section of characteristic dimension r and of thickness t: xp = r (t constant) : αp = 1 βp axial = 1 βp bending = 3 xp = t (r constant) : αp = 1 βp axial = 1 βp bending = 1

[12.27]

As with the plate, other cases can be established, such as the case where the section and the inertias are considered separately, or the case of parameter xp with prescribed axial and bending exponents.

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Table 12.3 summarizes the main cases already mentioned. Plate thickness or beam section

Parameter xp

bending

h t t

h

h = ∫ dz

I = ∫ z 2 dz

h = ∫ dz

I = ∫ z 2 dz

r

t

r

S, I

S, I

α = β membrane β

xp = h

1

3

xp = t (h constant)

1

1

xp = h (t constant)

0

2

xp = h (I constant)

1

0

xp = I (h constant)

0

1

xp such as

α

β

xp = r

2

4

xp = r (t constant)

1

3

xp = t (r constant)

1

1

xp = S (I constant)

1

0

xp = I (I constant)

0

1

xp such as

α

β

h = xpα

S = xpα

t = xpβ

I = xpβ

Table 12.3. Exponents α and β for various plate or beam parameters

12.4.2. Test/analysis correlation The variation of the physical parameters modifies the properties of the model that should now be compared to the test results. This comparison concerns the normal modes, if a modal identification was performed on the structure or the FRF directly.

Model Updating and Optimization

399

Comparison of modal parameters such as the natural frequencies is easily carried out, but comparing mode shapes, for example, poses the problem of the comparison of two vectors. These modes satisfy orthogonality properties [4.7] that we can exploit but the presence of the mass and stiffness matrices means that the operation depends on the model, which is not a desirable situation. It is generally preferable to use the direct orthogonality with the widely accepted MAC (Modal Assurance Criterion) [ALL 82] which is nothing more than the conveniently normalized scalar product of the two vectors:

MAC(Φi k , Φil ) =

(Φki .Φil ) 2 (Φki .Φi k ) (Φli .Φi l )

[12.28]

The MAC has a value between 0 and 1 and thus measures the degree of correlation between the two vectors. Two identical modes (to within a scaling factor) result in a MAC of 1, two purely orthogonal modes give 0. Knowing that this direct orthogonality is not exact, we should moderate our interpretation of the obtained values: an MAC of 0.9 indicates a good correlation, an MAC smaller than 0.7 indicates a rather mediocre correlation. The MAC matrix between analytical modes Φi k and experimental modes Φil (non-symmetric) makes it possible to have an overview of the correlation of the modes. For example:

⎡0.9 0.2 0.3⎤ MAC kl = ⎢⎢ 0.1 0.3 0.8⎥⎥ ⎢⎣0.2 0.7 0.4 ⎥⎦

[12.29]

indicates a good correlation between the first analytic mode and the first experimental mode, but the second analytic mode corresponds more to the third experimental mode and vice versa. Having said this, the test/analysis correlation can be made on: – the natural frequencies, whose relative errors can be easily calculated; – the mode shapes, using their MAC, as indicated previously. The MAC matrix makes it possible to pair the modes, which is sometimes difficult, with inversions as in example [12.29], or when several analytical modes are paired to the same

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experimental mode, which is quite common if there are a small number of measurement points; – the modal effective parameters, whose relative errors can be easily computed. Due to their role in the FRF, they are privileged ingredients for correlation. It should be noted that the effective transmissibilities of a mode for a given junction DOF are proportional to the components of its mode shape; – the FRF which, as with the mode shapes, pose the problem of the comparison of two vectors, here representing discretized functions. Various publications have suggested solutions on this point, but they have been difficult to integrate into industrial practice. We should recall that the correlation is easily performed on frequencies and mode shapes using the MAC and that the effective parameters are a good compromise between the mode shapes and the FRF for subsequent model updating.

12.4.3. Updating procedure After having seen how to modify the model with its physical parameters and how to correlate it with the test results, the updating procedure can be performed in the following way: – choice of the modifiable physical parameters. This can rely on various physical considerations and also put emphasis on a sensitivity analysis in order to detect the parameters, which are likely to be efficient for updating; – evaluation of the dynamic properties of the model by any type of reanalysis for a given configuration of physical parameters; – evaluation of the distance between the results obtained and the test data using an objective function based on the various possible correlations mentioned in the previous section. Each correlation provides a particular error and the global distance will be deduced by combination, for example quadratic combination, of these basic errors; – iterative process for the minimization of the objective function. If a systematic return to the finite elements is too difficult, which is often the case for large models, this process can start with limited modifications with an approximate but rapid reanalysis, such as the Ritz reanalysis, followed by an exact solution using finite elements in order to verify the results. In the case of unsatisfactory results, the procedure can be repeated as necessary.

Model Updating and Optimization

401

It now remains to define this optimization process, which, in addition to updating, can be used to improve the model performances in relation to dynamic criteria. This is the subject of the following section.

12.5. Optimization processes 12.5.1. Introduction Optimization, in its broadest sense, consists of finding the best solution while respecting the imposed constraints. If the objective to be optimized and the constraints can be written in a functional form in relation to a set of variables, the optimization can be defined as process used in order to obtain the extremum of a function F, the objective function, also called the quality function or the cost function. Usually, we often consider the minimum, knowing that the optimal solution of the maximum of F strictly corresponds to the optimal solution of the minimum of –F. The formulation of the problem with constraints can be summarized as follows: Finding xp which minimizes F(xp) under the constraints gm(xp) ≤ 0

[12.30]

vector xp regrouping variables xp and gm regrouping m inequality constraints. “Box” constraints are a specific case which are written xp min ≤ xp ≤ xp max and are relatively easy to handle. We could add the equality constraints but they can often be eliminated and they are considered very little at this level. Since the objective function and constraints of the problem are usually nonlinear, we will focus on non-linear optimization methods. There is abundant literature on this subject; see for example [BAZ 79], [RAO 84] and [VAN 84]. We will limit ourselves here to several general considerations that follow and to a method, that of the non-linear simplex, which deserves particular attention due to its suitability to industrial problems.

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12.5.2. Non-linear optimization methods Non-linear optimization methods can be classified into two categories according to the presence or absence of constraints: – methods without constraints which are divided into two groups according to whether gradients are used: - gradient-based methods (calculation of the derivatives of F) are generally the most efficient. The steepest-descent method, although attractive by its simplicity is actually rarely used because of a poor final convergence. Newton’s method, which guarantees an immediate convergence for a quadratic objective function, is rarely used because of its inherent instability. Other methods try to eliminate these disadvantages but they can remain delicate and require a certain implementation effort, - direct methods (calculation only of F) have the advantage of simplicity for the formulation and implementation, but they are less efficient and have a slow final convergence. Random search methods free themselves from a large number of difficulties such as discontinuities, local minima or complex surfaces, but at the price of a number of iterations that increases very rapidly with the number of parameters. The progression can be improved using genetic algorithms where particular operators such as reproduction, crossover and mutation are used. The sequential search methods, considering one variable at a time, are similar to the gradient methods and can be relatively efficient. Finally, the simplex method, based on a very intuitive algorithm, offers a simple formulation and implementation along with a good initial convergence and robustness that distinguishes it from the others; – constraint methods which, due to the presence of constraints, create additional problems related to feasibility, boundaries, constrained minima, non-linearities of the constraint functions, calculation of the gradients of the constraint functions, etc. They can be classified into two categories according to the use of the objective function: - indirect methods which use methods without constraints after having transformed the objective function to take into account the constraints – assuming that the constraint functions are suitable for this. The transformation can be applied to the variables or by introducing internal or external penalty functions to the objective function, - direct methods which take the constraints into account explicitly. They are less efficient in the presence of non-linear constraints. Depending on the problem at hand, a particular method will be better adapted. In the context of an industrial model updating in relation to a certain number of experimental modes, hence an objective function with a complex surface, the gradient based methods are often not very efficient and the direct methods require

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403

efficient search algorithms in order to minimize the computational effort – for which the non-linear simplex method is well suited under certain conditions.

12.5.3. Non-linear simplex method The non-linear simplex method [NEL64] consists of making a geometrical polygon (simplex) with n + 1 vertices. The simplex thus evolves in a space with n dimensions corresponding to the number of parameters xp. When n = 2, we obtain a triangle whose motion in the plane (x1, x2) can be easily visualized as we show in order to describe the method. The simplex can move in various ways, as indicated by Figure 12.3. At each step, it starts by performing a reflection from the highest vertex with respect to the center or average of the other points, then tries a movement according to the result which is found either an extension in order to increase and accelerate, or a contraction to diminish and slow down. If none of these maneuvers is successful, it shrinks. Thus each step requires two evaluations of the objective function, except for shrinking that requires n. Reflection

Expansion

Contraction

worse

worse

worse

better

better

better

new

new

Shrinking worse new better

new

worse new new better

x 2 Initial simplex 3

2

1

8...

5 4 6

7 x1

Figure 12.3. Movements of the simplex (n = 2)

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Constraints on the parameters may be taken into account by applying a penalty at the boundaries, but it must not be too abrupt in order to maintain progress of the simplex within the limits of the feasible region, at the risk of slightly violating the constraints. We might also increase the number of vertices in order to improve behavior near the constraints. The initial position of the simplex can be chosen randomly. A combination with regularly distributed vertices and an adjustable size c is given using the following process [BAZ 79]: – selection of a starting point (Xi)1; – addition of n points (Xi)j using: (Xi)j+1 = (Xi)1 + (∆i)j with: (∆ j ) j =

c n 2

( n + 1 + n − 1)

(∆ i ≠ j ) j =

c

( n + 1 − 1)

[12.31]

n 2

After the simplex has moved in order to minimize the objective function, it can be restarted at its final point using an appropriate size in order to verify that it is not located in a local minimum. As we see, this method is based on very simple principles that make it robust in most situations. The weak points are a rapid loss of efficiency when the number of parameters increases and a relatively slow final convergence. If we are not looking for too much accuracy in terms of the optimal solution and if we can limit the number of parameters to the most influential, the simplex is a good choice for difficult industrial cases.

12.6. Applications 12.6.1. Optimization of a simple system Figure 12.4 gives a simple case of optimization with n parameters. The distribution of the stiffnesses ki is determined, which maximizes the first natural frequency, the sum of stiffnesses assumed constant. Table 12.4 gives the optimal distribution as a function of n. The case n = 2 is plotted in Figure 12.5. This case can be used to verify the performances of an optimization process. It should be noted that the objective function, which is very regular, is not a severe case.

Model Updating and Optimization

m = 1

k1

m = 1

k2

…

m = 1

kn

m = 1

405

kn+1 n

= 1− ∑ kn i =1

Figure 12.4. Simple example of optimization

n

ω12 max

1 2 3 … n–1

1/5 1/14 1/30

ki 2/5 3/5 3/14 5/14 4/30 7/30

6/14 9/30

10/30

3 i (2 n + 1 − i) n (n + 1) (2n + 1)

6 n (n + 1) (2n + 1)

Table 12.4. Optimal distribution of the stiffnesses of the system in Figure 12.4

k2 1

5/14

3/14

1

k1

Figure 12.5. Optimum for n = 2

12.6.2. Updating a simple system Consider the 2-DOF system in Figure 12.6 providing a reference model with its “experimental” modes, and the model to be updated by the modification of only one parameter, the stiffness k of the upper spring. We will note that the modification is not consistent with the reference model, which is often the case since the number of considered parameters is usually much smaller than the number of dynamic properties that we want to improve.

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Let us take a look, for example, at the natural frequencies and the effective masses of the two modes of the system. The values of the reference model, assumed to represent the test results that we are trying to find, are the following: ref

fk

⎡0.1048⎤ =⎢ ⎥ ⎣0.2647⎦

~ ref ⎡1.8575 ⎤ Mk = ⎢ ⎥ ⎣0.1425⎦

Reference

m2 = 1

[12.32]

m2 = 1

Model to update

k2 = 1

k m1 = 1

k1 = 1.2

m1 = 1

1

Figure 12.6. Simple example of updating

If we only consider the frequencies, using the relative error

ref

∆f k / f k

as

objective function F(k), this produces the results indicated in Figure 12.7. The minimum is reached for k = 1.086 and equals 1.94%: the frequencies are improved but sum error remains. The modal parameters have the following values: ⎡0.0994⎤ fk = ⎢ ⎥ ⎣0.2654⎦

⎡1.9083 ⎤ ~ Mk = ⎢ ⎥ ⎣0.0917 ⎦

[12.33]

If we consider only the effective masses, using the relative error ~ ~ ref as objective function F(k), this produces the results shown in ∆M k / M k Figure 12.7. The minimum is reached for k = 1/1.2 and equals 0: the test values are reached because the stiffness distribution is correct. The modal parameters are in this case: ⎡0.0957 ⎤ fk = ⎢ ⎥ ⎣0.2416⎦

⎡1.8575 ⎤ ~ Mk = ⎢ ⎥ ⎣0.1425⎦

[12.34]

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rigid modes and 16 elastic modes up to 150 Hz. The experimental model corresponds to 91 triaxial accelerometers providing 273 response points. A sine sweep using a small shaker attached to the right lower corner of the door leads to the FRF in Figure 12.9, from which 10 global modes from 50 to 140 Hz were identified.

Analytical model

Experimental model

Figure 12.8. Industrial model for updating

Figure 12.9. Experimental FRF (imaginary part)

Model Updating and Optimization

409

The parameters considered for the updating are the plate thicknesses and the stiffness of the weld spots. Table 12.5 gives the correlation in frequencies and MAC of the initial model and of the final model after updating. The representativity of the model in relation to the test results has been noticeably improved.

ESSAI INITIAL Err Freq MAC FINAL Err Freq MAC Mode Freq Mode Freq (%) (%) Mode Freq (%) (%) --------------------------------------------------------------------1 52.86 1 43.68 (-17.4) (80.7) 1 52.93 ( 0.1) (92.3) 2 57.61 3 54.88 ( -4.7) (86.0) 2 55.95 ( -2.9) (87.1) 3 64.97 4 63.81 ( -1.8) (49.9) 3 64.66 ( -0.5) (80.8) 4 75.41 7 79.67 ( 5.7) (58.8) 6 80.78 ( 7.1) (73.2) 5 87.34 6 78.09 (-10.6) (27.5) 7 87.10 ( -0.3) (70.4) 6 106.35 11 104.02 ( -2.2) (44.7) 9 107.23 ( 0.8) (68.0) 7 112.92 12 105.55 ( -6.5) (57.6) 10 107.75 ( -4.6) (74.1) 8 126.50 9 90.92 (-28.1) (56.0) 12 124.25 ( -1.8) (73.1) 9 134.27 16 127.51 ( -5.0) (47.9) 13 133.36 ( -0.7) (71.4) 10 137.85 15 126.48 ( -8.2) (44.6) 14 136.09 ( -1.3) (44.4)

Table 12.5. Correlation before and after updating

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Index

A

C

acceleration 16, 20 accelerometer 357 admittance 22, 44 amplification at resonance 43, 48 dynamic amplification 43, 125, 227, 243, 337 analysis 29 frequency analysis 16 linear analysis 1 non-linear analysis 323 sensitivity analysis 386 statistical analysis 9 time analysis 8, 14 antiresonance 154, 380 autocorrelation 9, 12

change of reference frame 68 circular frequency 2 circular natural frequency 36, 101 complex 221, 237 damped 38 coherence 14, 356 continuous elements bending beam 184 combined 210 plate 206 rod 171 correlation 385 coupling fluid-structure coupling 236 impedance coupling 30, 295 cross spectrum 14, 19 cross-correlation 9, 12 cylindrical shell 87

B Basile hypothesis 120, 248 boundary conditions 20, 258, 298, 358

D damper 34, 68 damping 29, 119, 325 structural 62, 120, 170 viscous 26, 61, 119 viscous damping ratio 37, 119

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degree of freedom (DOF) 17 connection 251, 258 elimination 73 excitation 301 internal 20 introduction 77 junction 20 management 91 observation 256, 302 density power spectral (PSD) 13 probability 9 displacement 16, 20 dissipated power 26, 67 domain see analysis Duffing equation 336 Duhamel integral 16, 55 dry friction (Coulomb) 328

E effect gyroscopic effect 232 shear effect 84, 204 eigenvalue problem 101, 170 energy kinetic 24, 81 elastic 24, 81 equations of motion 24

F factor dynamic amplification 42, 63, 122, 227, 243 dynamic transmissibility 42, 63, 124, 227, 243 loss 63, 170 participation 117, 124, 170, 225, 234 quality 43 shape 84

finite elements bending beam 83 complete beam 86 plate 90 rigid 72, 91 rod 82 types of 89 flexibility dynamic 23, 39, 123 effective 131 pseudo-static 135, 259 residual 141 static 40, 133, 260 force 20 additional 344 constraint 308 dissipation 26 equivalent 89 external 26 inertia 26 reaction 17 restoring 324 formula Dunkerley 211 Seide 88 Fourier series 35 transform 5 frequency 2 fundamental frequency 5, 334 half-power frequency 49 low frequency 28 natural frequency 36, 101, 170 response function (FRF) 16, 170, 296

H Hamilton principle 24 harmonic 5, 334, 344 hysteresis 329

Index

I impedance 22, 46 characteristic 177 drive point impedance of an infinite plate 207 uncoupling 318 impulse 15

J junction statically determinate 143, 267, 315 statically indeterminate 182, 199, 315, 357

L Lagrange multipliers 77, 115, 136, 313 Lagrangian 24 level reduction (notching) 381 linear constraints 72, 91 logarithmic decrement 38

M MAC (Modal Assurance Criterion) 399 mass 24 coherent 81 directional 28, 34, 69 dynamic 23, 41, 123 effective 131 by zone 137 generalized 101, 170 lumped 67, 81 residual 142 static 41, 134 matrix assembly 70, 250, 318 element 66, 81

419

mass 24 rigid body mass 75, 105, 117, 143 stiffness 24 transformation 26 mean square 8, 10 mechanisms 91 method Benfield-Hruda 276 centered difference 96 complex exponential 371 Craig-Bampton 266 Craig-Chang 271 finite element 80 Lanczos 112 multi-DOF (MDOF) 370 Newmark 97, 346 non-linear simplex 403 orthogonal polyreference 371 single DOF (SDOF) 368 modal truncation 28, 102, 116, 125, 140 mode attachment 260 complex 100, 219, 235 elastic 104 inertia 135, 261 normal 27, 100, 170, 259 orthogonality 101, 170, 222, 233, 240 rigid (body) 104, 115, 122, 131, 134, 142 static junction 115, 170, 262 superposition 27, 115, 170, 224, 234, 240 model effective mass 145, 281, 308 finite element 89 reduced 283 updating 351, 385 verification 93

420

Structural Dynamics in Industry

modulus complex 63, 170, 176 Shear 84 Young’s 82, 83 motion 20 parasitic motion 315, 375 periodic motion 35 random motion 7 sine motion 2, 373 transient motion 5, 375

N nσ probabilities 10 node 66 average 72 clamped 185 free 185 pinned (simply supported) 185 rigid 91 sliding 185 normal law (Gauss) 10 Nyquist plane 368

O operating deflection shape 307 optimization techniques 386

P parameters modal 38, 102, 118, 121, 359 modal effective 125, 129, 169, 226, 242 residual 142 static 42, 138 period 2, 5, 332 process ergodic 12 random 7 stationary 9

R reanalysis 386 reciprocity principle 21, 232 resonance 3, 16, 43, 153, 380 response frequency 39, 153 random 49, 157 time 51 to unit impulse 15, 51 Ritz method 386 reanalysis 392, 400 root mean square (rms) value 8 rules modeling 93 summation 133, 138, 139

S safety coefficients 379 shaker 352, 372 shape see mode shock device 355, 375 (simple) pendulum 326 sine sweep 4, 373 solution by direct integration 95 by modal approach 27, 99, 126 in the frequency domain 26, 122, 126 to eigenvalue problem 111 spectrum extreme response 57, 159, 378 fatigue damage 57, 378 shock 57, 375 spring 28, 34, 68, 324, 332 standard deviation 8, 10 static condensation (Guyan) 108 stiffness dynamic 23, 41, 123 generalized 101

Index

static 78, 311, 317 stretched string 327 structure free 104, 122 statically determinate 143 substructuring 29, 249, 295, 318 suspension 44 synthesis frequency 30, 250, 295, 320 modal 30, 250, 320 system 1-DOF 28, 33 conservative 24 continuous 24 discrete 25 N-DOF 25, 65 non-linear 1-DOF 329 sweep rate 4, 373

T tests 29, 349 identification 350 modal survey 359 phase resonance modal 362 phase separation modal 364 simulation 352, 372

test specifications 377 dissipative system 26, 219 transmissibility dynamic 23, 39, 40, 123 effective 131 by zone 137 static 40, 116, 123, 134 residual 141

V variance 8, 10 velocity 16, 20 propagation 176, 191 vibration see motion

W white noise 14

421