NONLINEARITY IN         STRUCTURAL DYNAMICS         Detection, Identification and Modelling         K Worden and G R Tomlin...
­ IOP Publishing Ltd 2001         c         All rights reserved. No part of this publication may be reproduced, stored    ...
For Heather and MargaretCopyright © 2001 IOP Publishing Ltd
‘As you set out for Ithaka              hope your road is a long one,              full of adventure, full of discovery.  ...
Contents             Preface                                                           xv         1   Linear systems      ...
viii           Contents                      2.4.2 Vibration exciter problems                                 59          ...
Contents          ix         4   The Hilbert transform—a practical approach                          127             4.1 I...
x            Contents                    6.3.2 Parameter uncertainty                       235                    6.3.3 St...
Contents          xi             7.6   System identification using optimization                           355              ...
xii           Contents                     9.3.2 Frequency-domain characteristics of the bilinear beam   518              ...
Contents   xiii                   F.5.2 Convergence to solutions                           586                   F.5.3 Uni...
Preface         Nonlinearity is a frequent visitor to engineering structures which can modify—         sometimes catastrop...
apparently random response. In contrast, a linear system always responds to a         periodic excitation with a periodic ...
the fundamental measured quantities like the impulse response function              (IRF) and the frequency response funct...
complete description in this book of the effects of measurement noise and              the need for rigorous model validit...
A book like this does not spring into being without a lot of help from a         lot of people. It is a pleasure to thank ...
Chapter 1         Linear systems         This chapter is provided more or less as a reminder of linear system theory. It i...
2             Linear systems                                          k                             Free body diagram of t...
Continuous-time models: time domain                 3              The solution of (1.7) is elementary and is given in any...
4           Linear systems         of motion. Such a system is said to have three degrees-of-freedom and would         be ...
Continuous-time models: time domain                            5                    y(t)                                  ...
6           Linear systems                   y(t)                                                                         ...
Continuous-time models: time domain                       7         response of the solution.               For linear sys...
8           Linear systems                       Figure 1.5. SDOF system gain as a function of frequency .               T...
Continuous-time models: time domain               9                         φ(ω)                             π            ...
10              Linear systems         Bode plot of an actual SDOF system,                                               Ý...
Continuous-time models: frequency domain                          11         the equation, i.e.                     ·½    ...
12          Linear systems                Figure 1.8. Nyquist plot for system Ý · ¾¼Ý · ½¼   Ý   ܴص—receptance.         ...
Impulse response             13                 Figure 1.9. Nyquist plot for system Ý · ¾¼Ý · ½¼   Ý   ܴص—mobility.     ...
14           Linear systems               Figure 1.10. Nyquist plot for system Ý · ¾¼Ý · ½¼       Ý     ܴص—accelerance. ...
Impulse response              15                                                  x(t)                                  ε ...
16             Linear systems                 Now, according to equation (1.45), the system response to a Æ -function inpu...
Discrete-time models: time domain                              17         more rigorous approach to evaluating ´Øµ is simp...
18             Linear systems         representation defines the motion at all times. In reality, most observations of     ...
Discrete-time models: time domain        19         Ü  together with values for Ý ½ and ݾ . The specification of the first ...
20          Linear systems               By way of illustration, consider the SDOF system (1.30) again.         Equation (...
Classification of difference equations         21         One point about these equations is worth noting. The expressions ...
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72268096 non-linearity-in-structural-dynamics-detection-identification-and-modelling-copy

  1. 1. NONLINEARITY IN STRUCTURAL DYNAMICS Detection, Identification and Modelling K Worden and G R Tomlinson University of Sheffield, UK Institute of Physics Publishing Bristol and PhiladelphiaCopyright © 2001 IOP Publishing Ltd
  2. 2. ­ IOP Publishing Ltd 2001 c All rights reserved. No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher. Multiple copying is permitted in accordance with the terms of licences issued by the Copyright Licensing Agency under the terms of its agreement with the Committee of Vice-Chancellors and Principals. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. ISBN 0 7503 0356 5 Library of Congress Cataloging-in-Publication Data are available Commissioning Editor: James Revill Production Editor: Simon Laurenson Production Control: Sarah Plenty Cover Design: Victoria Le Billon Marketing Executive: Colin Fenton Published by Institute of Physics Publishing, wholly owned by The Institute of Physics, London Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK US Office: Institute of Physics Publishing, The Public Ledger Building, Suite 1035, 150 South Independence Mall West, Philadelphia, PA 19106, USA Typeset in TEX using the IOP Bookmaker Macros Printed in the UK by J W Arrowsmith Ltd, BristolCopyright © 2001 IOP Publishing Ltd
  3. 3. For Heather and MargaretCopyright © 2001 IOP Publishing Ltd
  4. 4. ‘As you set out for Ithaka hope your road is a long one, full of adventure, full of discovery. Laistrygonians, Cyclops, angry Poseidon—don’t be afraid of them: You’ll never find things like that in your way as long as you keep your thoughts raised high, as long as a rare sensation touches your body and spirit. Laistrygonians, Cyclops, wild Poseidon—you won’t encounter them unless you bring them along inside your soul, Unless your soul sets them up in front of you.’ C P Cavafy, ‘Ithaka’Copyright © 2001 IOP Publishing Ltd
  5. 5. Contents Preface xv 1 Linear systems 1 1.1 Continuous-time models: time domain 1 1.2 Continuous-time models: frequency domain 10 1.3 Impulse response 13 1.4 Discrete-time models: time domain 17 1.5 Classification of difference equations 21 1.5.1 Auto-regressive (AR) models 21 1.5.2 Moving-average (MA) models 21 1.5.3 Auto-regressive moving-average (ARMA) models 22 1.6 Discrete-time models: frequency domain 22 1.7 Multi-degree-of-freedom (MDOF) systems 23 1.8 Modal analysis 29 1.8.1 Free, undamped motion 29 1.8.2 Free, damped motion 35 1.8.3 Forced, damped motion 37 2 From linear to nonlinear 41 2.1 Introduction 41 2.2 Symptoms of nonlinearity 41 2.2.1 Definition of linearity—the principle of superposition 41 2.2.2 Harmonic distortion 46 2.2.3 Homogeneity and FRF distortion 49 2.2.4 Reciprocity 51 2.3 Common types of nonlinearity 52 2.3.1 Cubic stiffness 52 2.3.2 Bilinear stiffness or damping 55 2.3.3 Piecewise linear stiffness 55 2.3.4 Nonlinear damping 56 2.3.5 Coulomb friction 57 2.4 Nonlinearity in the measurement chain 57 2.4.1 Misalignment 58Copyright © 2001 IOP Publishing Ltd
  6. 6. viii Contents 2.4.2 Vibration exciter problems 59 2.5 Two classical means of indicating nonlinearity 59 2.5.1 Use of FRF inspections—Nyquist plot distortions 60 2.5.2 Coherence function 62 2.6 Use of different types of excitation 65 2.6.1 Steady-state sine excitation 66 2.6.2 Impact excitation 67 2.6.3 Chirp excitation 68 2.6.4 Random excitation 68 2.6.5 Conclusions 69 2.7 FRF estimators 69 2.8 Equivalent linearization 72 2.8.1 Theory 72 2.8.2 Application to Duffing’s equation 76 2.8.3 Experimental approach 78 3 FRFs of nonlinear systems 81 3.1 Introduction 81 3.2 Harmonic balance 81 3.3 Harmonic generation in nonlinear systems 88 3.4 Sum and difference frequencies 90 3.5 Harmonic balance revisited 91 3.6 Nonlinear damping 93 3.7 Two systems of particular interest 95 3.7.1 Quadratic stiffness 95 3.7.2 Bilinear stiffness 98 3.8 Application of harmonic balance to an aircraft component ground vibration test 101 3.9 Alternative FRF representations 105 3.9.1 Nyquist plot: linear system 105 3.9.2 Nyquist plot: velocity-squared damping 107 3.9.3 Nyquist plot: Coulomb friction 108 3.9.4 Carpet plots 109 3.10 Inverse FRFs 111 3.11 MDOF systems 112 3.12 Decay envelopes 122 3.12.1 The metho