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Structural Dynamics and Vibration in Practice This page intentionally left blank Structural Dynamics and Vibration in Practice An Engineering Handbook Douglas Thorby AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth...

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Structural Dynamics and Vibration in Practice

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Structural Dynamics and Vibration in Practice An Engineering Handbook Douglas Thorby

AMSTERDAM • BOSTON • HEIDELBERG • LONDON NEW YORK • OXFORD • PARIS • SAN DIEGO SAN FRANCISCO • SINGAPORE • SYDNEY • TOKYO Butterworth-Heinemann is an imprint of Elsevier

Butterworth-Heinemann is an imprint of Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 30 Corporate Drive, Suite 400, Burlington, MA 01803, USA First edition 2008 Copyright  2008 Elsevier Ltd. 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 written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Catalog Number: 2007941701 ISBN: 978-0-7506-8002-8

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To my wife, Marjory; our children, Chris and Anne; and our grandchildren, Tom, Jenny, and Rosa.

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Contents

xiii

Preface Acknowledgements Chapter 1 1.1 1.2 1.3

1.4

1.5

1.6 1.7

Statics, dynamics and structural dynamics Coordinates, displacement, velocity and acceleration Simple harmonic motion 1.3.1 Time history representation 1.3.2 Complex exponential representation Mass, stiffness and damping 1.4.1 Mass and inertia 1.4.2 Stiffness 1.4.3 Stiffness and flexibility matrices 1.4.4 Damping Energy methods in structural dynamics 1.5.1 Rayleigh’s energy method 1.5.2 The principle of virtual work 1.5.3 Lagrange’s equations Linear and non-linear systems Systems of units 1.7.1 Absolute and gravitational systems 1.7.2 Conversion between systems 1.7.3 The SI system References

Chapter 2 2.1

Basic Concepts

The Linear Single Degree of Freedom System: Classical Methods

Setting up the differential equation of motion 2.1.1 Single degree of freedom system with force input 2.1.2 Single degree of freedom system with base motion input 2.2 Free response of single-DOF systems by direct solution of the equation of motion 2.3 Forced response of the system by direct solution of the equation of motion

xv 1 1 1 2 3 5 7 7 10 12 14 16 17 19 21 23 23 24 26 27 28 29 29 29 33 34 38 vii

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Contents

Chapter 3 3.1

3.2

3.3

3.4 3.5

The Linear Single Degree of Freedom System: Response in the Time Domain

Exact analytical methods 3.1.1 The Laplace transform method 3.1.2 The convolution or Duhamel integral 3.1.3 Listings of standard responses ‘Semi-analytical’ methods 3.2.1 Impulse response method 3.2.2 Straight-line approximation to input function 3.2.3 Superposition of standard responses Step-by-step numerical methods using approximate derivatives 3.3.1 Euler method 3.3.2 Modified Euler method 3.3.3 Central difference method 3.3.4 The Runge–Kutta method 3.3.5 Discussion of the simpler finite difference methods Dynamic factors 3.4.1 Dynamic factor for a square step input Response spectra 3.5.1 Response spectrum for a rectangular pulse 3.5.2 Response spectrum for a sloping step References

Chapter 4

The Linear Single Degree of Freedom System: Response in the Frequency Domain

4.1

45 46 46 50 53 55 56 56 56 59 60 62 62 65 69 70 70 72 72 74 76 77

Response of a single degree of freedom system with applied force 4.1.1 Response expressed as amplitude and phase 4.1.2 Complex response functions 4.1.3 Frequency response functions 4.2 Single-DOF system excited by base motion 4.2.1 Base excitation, relative response 4.2.2 Base excitation: absolute response 4.3 Force transmissibility 4.4 Excitation by a rotating unbalance 4.4.1 Displacement response 4.4.2 Force transmitted to supports References

77 77 81 83 86 87 91 93 94 95 96 97

Chapter 5

99

5.1 5.2

Damping

Viscous and hysteretic damping models Damping as an energy loss 5.2.1 Energy loss per cycle – viscous model 5.2.2 Energy loss per cycle – hysteretic model 5.2.3 Graphical representation of energy loss 5.2.4 Specific damping capacity 5.3 Tests on damping materials

99 103 103 104 105 106 108

Contents

5.4

5.5 5.6

5.7

5.8

Quantifying linear damping 5.4.1 Quality factor, Q 5.4.2 Logarithmic decrement 5.4.3 Number of cycles to half amplitude 5.4.4 Summary table for linear damping Heat dissipated by damping Non-linear damping 5.6.1 Coulomb damping 5.6.2 Square law damping Equivalent linear dampers 5.7.1 Viscous equivalent for coulomb damping 5.7.2 Viscous equivalent for square law damping 5.7.3 Limit cycle oscillations with square-law damping Variation of damping and natural frequency in structures with amplitude and time

Chapter 6 6.1

6.2

6.3 6.4

6.5 6.6

Setting up the equations of motion for simple, undamped, multi-DOF systems 6.1.1 Equations of motion from Newton’s second law and d’Alembert’s principle 6.1.2 Equations of motion from the stiffness matrix 6.1.3 Equations of motion from Lagrange’s equations Matrix methods for multi-DOF systems 6.2.1 Mass and stiffness matrices: global coordinates 6.2.2 Modal coordinates 6.2.3 Transformation from global to modal coordinates Undamped normal modes 6.3.1 Introducing eigenvalues and eigenvectors Damping in multi-DOF systems 6.4.1 The damping matrix 6.4.2 Damped and undamped modes 6.4.3 Damping inserted from measurements 6.4.4 Proportional damping Response of multi-DOF systems by normal mode summation Response of multi-DOF systems by direct integration 6.6.1 Fourth-order Runge–Kutta method for multi-DOF systems

Chapter 7 7.1

Introduction to Multi-degree-of-freedom Systems

Eigenvalues and Eigenvectors

The eigenvalue problem in standard form 7.1.1 The modal matrix 7.2 Some basic methods for calculating real eigenvalues and eigenvectors 7.2.1 Eigenvalues from the roots of the characteristic equation and eigenvectors by Gaussian elimination 7.2.2 Matrix iteration 7.2.3 Jacobi diagonalization

ix

108 108 109 110 111 112 112 113 113 114 115 116 117 117 119 119 120 120 121 122 122 126 127 132 132 142 142 143 144 145 147 155 156 159 159 161 162 162 165 168

x

7.3 7.4 7.5

Contents

Choleski factorization More advanced methods for extracting real eigenvalues and eigenvectors Complex (damped) eigenvalues and eigenvectors References

Chapter 8

Vibration of Structures

8.1 8.2

177 178 179 180 181

A historical view of structural dynamics methods Continuous systems 8.2.1 Vibration of uniform beams in bending 8.2.2 The Rayleigh–Ritz method: classical and modern 8.3 Component mode methods 8.3.1 Component mode synthesis 8.3.2 The branch mode method 8.4 The finite element method 8.4.1 An overview 8.4.2 Equations of motion for individual elements 8.5 Symmetrical structures References

181 182 182 189 194 195 208 213 213 221 234 235

Chapter 9

237

Fourier Transformation and Related Topics

9.1

The Fourier series and its developments 9.1.1 Fourier series 9.1.2 Fourier coefficients in magnitude and phase form 9.1.3 The Fourier series in complex notation 9.1.4 The Fourier integral and Fourier transforms 9.2 The discrete Fourier transform 9.2.1 Derivation of the discrete Fourier transform 9.2.2 Proprietary DFT codes 9.2.3 The fast Fourier transform 9.3 Aliasing 9.4 Response of systems to periodic vibration 9.4.1 Response of a single-DOF system to a periodic input force References

237 237 243 245 246 247 248 255 256 256 260 261 265

Chapter 10

267

Random Vibration

10.1 Stationarity, ergodicity, expected and average values 10.2 Amplitude probability distribution and density functions 10.2.1 The Gaussian or normal distribution 10.3 The power spectrum 10.3.1 Power spectrum of a periodic waveform 10.3.2 The power spectrum of a random waveform 10.4 Response of a system to a single random input 10.4.1 The frequency response function 10.4.2 Response power spectrum in terms of the input power spectrum

267 270 274 279 279 281 286 286 287

Contents

10.4.3

10.5

10.6

10.7

10.8

Response of a single-DOF system to a broadband random input 10.4.4 Response of a multi-DOF system to a single broad-band random input Correlation functions and cross-power spectral density functions 10.5.1 Statistical correlation 10.5.2 The autocorrelation function 10.5.3 The cross-correlation function 10.5.4 Relationships between correlation functions and power spectral density functions The response of structures to random inputs 10.6.1 The response of a structure to multiple random inputs 10.6.2 Measuring the dynamic properties of a structure Computing power spectra and correlation functions using the discrete Fourier transform 10.7.1 Computing spectral density functions 10.7.2 Computing correlation functions 10.7.3 Leakage and data windows 10.7.4 Accuracy of spectral estimates from random data Fatigue due to random vibration 10.8.1 The Rayleigh distribution 10.8.2 The S–N diagram References

xi

Chapter 11

Vibration Reduction

288 296 299 299 300 302 303 305 305 307 310 312 314 317 318 320 321 322 324 325

11.1 Vibration isolation 11.1.1 Isolation from high environmental vibration 11.1.2 Reducing the transmission of vibration forces 11.2 The dynamic absorber 11.2.1 The centrifugal pendulum dynamic absorber 11.3 The damped vibration absorber 11.3.1 The springless vibration absorber References

326 326 332 332 336 338 342 345

Chapter 12

347

Introduction to Self-Excited Systems

12.1 Friction-induced vibration 12.1.1 Small-amplitude behavior 12.1.2 Large-amplitude behavior 12.1.3 Friction-induced vibration in aircraft landing gear 12.2 Flutter 12.2.1 The bending-torsion flutter of a wing 12.2.2 Flutter equations 12.2.3 An aircraft flutter clearance program in practice 12.3 Landing gear shimmy References

347 347 349 350 353 354 358 360 362 366

xii

Chapter 13

Contents

Vibration testing

367

13.1 Modal testing 13.1.1 Theoretical basis 13.1.2 Modal testing applied to an aircraft 13.2 Environmental vibration testing 13.2.1 Vibration inputs 13.2.2 Functional tests and endurance tests 13.2.3 Test control strategies 13.3 Vibration fatigue testing in real time 13.4 Vibration testing equipment 13.4.1 Accelerometers 13.4.2 Force transducers 13.4.3 Exciters References

368 368 369 373 373 374 375 376 377 377 378 378 385

Appendix A

A Short Table of Laplace Transforms

387

Appendix B

Calculation of Flexibility Influence Coefficients

389

Appendix C

Acoustic Spectra

393

Index

397

Preface

This book is primarily intended as an introductory text for newly qualified graduates, and experienced engineers from other disciplines, entering the field of structural dynamics and vibration, in industry. It should also be found useful by test engineers and technicians working in this area, and by those studying the subject in universities, although it is not designed to meet the requirements of any particular course of study. No previous knowledge of structural dynamics is assumed, but the reader should be familiar with the elements of mechanical or structural engineering, and a basic knowledge of mathematics is also required. This should include calculus, complex numbers and matrices. Topics such as the solution of linear second-order differential equations, and eigenvalues and eigenvectors, are explained in the text. Each concept is explained in the simplest possible way, and the aim has been to give the reader a basic understanding of each topic, so that more specialized texts can be tackled with confidence. The book is largely based on the author’s experience in the aerospace industry, and this will inevitably show. However, most of the material presented is of completely general application, and it is hoped that the book will be found useful as an introduction to structural dynamics and vibration in all branches of engineering. Although the principles behind current computer software are explained, actual programs are not provided, or discussed in any detail, since this area is more than adequately covered elsewhere. It is assumed that the reader has access to a software  package such as MATLAB . A feature of the book is the relatively high proportion of space devoted to worked examples. These have been chosen to represent tasks that might be encountered in industry. It will be noticed that both SI and traditional ‘British’ units have been used in the examples. This is quite deliberate, and is intended to highlight the fact that in industry, at least, the changeover to the SI system is far from complete, and it is not unknown for young graduates, having used only the SI system, to have to learn the obsolete British system when starting out in industry. The author’s view is that, far from ignoring systems other than the SI, which is sometimes advocated, engineers must understand, and be comfortable with, all systems of units. It is hoped that the discussion of the subject presented in Chapter 1 will be useful in this respect. The book is organized as follows. After reviewing the basic concepts used in structural dynamics in Chapter 1, Chapters 2, 3 and 4 are all devoted to the response of the single degree of freedom system. Chapter 5 then looks at damping, including non-linear damping, in single degree of freedom systems. Multi-degree of freedom systems are introduced in Chapter 6, with a simple introduction to matrix methods, based on Lagrange’s equations, and the important concepts of modal coordinates and the normal mode summation method. Having briefly introduced eigenvalues and xiii

xiv

Preface

eigenvectors in Chapter 6, some of the simpler procedures for their extraction are described in Chapter 7. Methods for dealing with larger structures, from the original Ritz method of 1909, to today’s finite element method, are believed to be explained most clearly by considering them from a historical viewpoint, and this approach is used in Chapter 8. Chapter 9 then introduces the classical Fourier series, and its digital development, the Discrete Fourier Transform (DFT), still the mainstay of practical digital vibration analysis. Chapter 10 is a simple introduction to random vibration, and vibration isolation and absorption are discussed in Chapter 11. In Chapter 12, some of the more commonly encountered self-excited phenomena are introduced, including vibration induced by friction, a brief introduction to the important subject of aircraft flutter, and the phenomenon of shimmy in aircraft landing gear. Finally, Chapter 13 gives an overview of vibration testing, introducing modal testing, environmental testing and vibration fatigue testing in real time. Douglas Thorby

Acknowledgements

The author would like to acknowledge the assistance of his former colleague, Mike Child, in checking the draft of this book, and pointing out numerous errors. Thanks are also due to the staff at Elsevier for their help and encouragement, and good humor at all times.

xv

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1 Basic Concepts Contents 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Statics, dynamics and structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Coordinates, displacement, velocity and acceleration . . . . . . . . . . . . . . . . . . . . . . Simple harmonic motion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mass, stiffness and damping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Energy methods in structural dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear and non-linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of units . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 7 16 23 23 28

This introductory chapter discusses some of the basic concepts in the fascinating subject of structural dynamics.

1.1 Statics, dynamics and structural dynamics Statics deals with the effect of forces on bodies at rest. Dynamics deals with the motion of nominally rigid bodies. The two aspects of dynamics are kinematics and kinetics. Kinematics is concerned only with the motion of bodies with geometric constraints, irrespective of the forces acting. So, for example, a body connected by a link so that it can only rotate about a fixed point is constrained by its kinematics to move in a circular path, irrespective of any forces that may be acting. On the other hand, in kinetics, the path of a particle may vary as a result of the applied forces. The term structural dynamics implies that, in addition to having motion, the bodies are non-rigid, i.e. ‘elastic’. ‘Structural dynamics’ is slightly wider in meaning than ‘vibration’, which implies only oscillatory behavior.

1.2 Coordinates, displacement, velocity and acceleration The word coordinate acquires a slightly different, additional meaning in structural dynamics. We are used to using coordinates, x, y and z, say, when describing the location of a point in a structure. These are Cartesian coordinates (named after Rene´ Descartes), sometimes also known as ‘rectangular’ coordinates. However, the same word ‘coordinate’ can be used to mean the movement of a point on a structure from some standard configuration. As an example, the positions of the grid points chosen for the analysis of a structure could be specified as x, y and z coordinates from some fixed point. However, the displacements of those points, when the structure is loaded in some way, are often also referred to as coordinates. 1

2

Structural dynamics and vibration in practice

y1

r1

θ1 m1

x1

θ2

y2

r2

m2 x2

Fig. 1.1 Double pendulum illustrating generalized coordinates.

Cartesian coordinates of this kind are not always suitable for defining the vibrati...