Mechanical Vibrations and Shocks: Fatigue Damage v. 4 (Mechanical vibration & shock)

369 Pages·2002·5.61 MB·English

by  Christian Lalanne

Fatigue
Damage
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Mechanical Vibration
& Shock
Fatigue
Damage
Volume IV
Christian Lalanne
HPS
HERMES PENTON SCIENCE
First published in 1999 by Hermes Science Publications, Paris
First published in English in 2002 by Hermes Penton Ltd
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 of licences issued by the CLA. Enquiries concerning repro-
duction outside these terms should be sent to the publishers at the undermentioned
address:
Hermes Penton Science
120 Pentonville Road
London Nl 9JN
© Hermes Science Publications, 1999
© English language edition Hermes Penton Ltd, 2002
The right of Christian Lalanne to be identified as the author of this work has been assert-
ed by him in accordance with the Copyright, Designs and Patents Act 1988.
British Library Cataloguing in Publication Data
A CIP record for this book is available from the British Library.
ISBN 1 9039 9606 6
Printed and bound in Great Britain by Biddies Ltd, www.biddles.co.uk
Contents
Introduction xi
List of symbols xiii
1 Response of a linear one degree-of-freedom linear system
to random vibration 1
1.1. Average value of response of a linear system 1
1.2. Response of perfect bandpass filter to random vibration 2
1.3. PSD of response of a single-degree-of-freedom linear system 4
1.4. Rms value of response to white noise 5
1.5. Rms value of response of a linear one-degree-of-freedom
system subjected to bands of random noise 11
1.5.1. Case: excitation is a power spectral density defined
by a straight line segment in logarithmic scales 11
1.5.2. Case: vibration has a power spectral density defined
by a straight line segment of arbitrary slope in linear scales 17
1.5.3. Case: vibration has a constant power spectral
density between two frequencies 19
1.5.3.1. Power spectral density defined in a frequency
interval of arbitrary width 19
1.5.3.2. Case: narrow band noise of width Af = ^fo/Q 22
1.5.4. Excitation defined by an absolute displacement 25
1.5.5. Case: excitation defined by power spectral density
comprising n straight line segments 27
1.6. Rms value of the absolute acceleration of the response 30
1.7. Transitory response of dynamic system under stationary
random excitation 31
1.8. Transitory response of dynamic system under amplitude
modulated white noise excitation 38
vi Fatigue damage
2 Characteristics of the response of a one-degree-of-freedom
linear system to random vibration 41
2.1. Moments of response of a one-degree-of-freedom linear system:
irregularity factor of response 41
2.1.1. Moments 41
2.1.2. Irregularity factor of response to noise of constant PSD 45
2.1.3. Characteristics of irregularity factor of response 47
2.1.4. Case: band-limited noise 58
2.2. Autocorrelation function of response displacement 60
2.3. Average numbers of maxima and minima per second 60
2.4. Equivalence between transfer functions of bandpass filter
and one-degree-of-freedom linear system 63
2.4.1. Equivalence suggested by D.M. Aspinwall 63
2.4.2. Equivalence suggested by K.W. Smith 66
2.4.3. Rms value of signal filtered by the equivalent bandpass filter 67
3 Concepts of material fatigue 69
3.1. Introduction 69
3.2. Damage arising from fatigue 70
3.3. Characterization of endurance of materials 73
3.3.1.S-Ncurve 73
3.3.2. Statistical aspect 76
3.3.3. Distribution laws of endurance 77
3.3.4. Distribution laws of fatigue strength 81
3.3.5. Relation between fatigue limit and static properties of materials 82
3.3.6. Analytical representations of S-N curve 85
3.3.6.1. Wohler relation (1870) 85
3.3.6.2. Basquin relation 86
3.3.6.3. Some other laws 92
3.4. Factors of influence 94
3.4.1. General 94
3.4.2. Scale 95
3.4.3. Overloads 95
3.4.4. Frequency of stresses 97
3.4.5. Types of stresses 98
3.4.6. Non zero mean stress 98
3.5. Other representations of S-N curves 100
3.5.1.Haighdiagram 100
3.5.2. Statistical representation of Haigh diagram 107
3.6. Prediction of fatigue life of complex structures 108
3.7. Fatigue in composite materials 109
Contents vii
4 Accumulation of fatigue damage Ill
4.1. Evolution of fatigue damage Ill
4.2. Classification of various laws of accumulation 112
4.3. Miner's method 113
4.3.1. Miner's rule 113
4.3.2. Scatter of damage to failure as evaluated by Miner 117
4.3.3. Validity of Miner's law of accumulation of damage
in case of random stress 121
4.4. Modified Miner's theory 122
4.5. Henry's method 127
4.6. Modified Henry's method 128
4.7. H. Corten and T. Dolan's method 129
4.8. Other theories 131
5 Counting methods for analysing random time history 135
5.1. General 135
5.2. Peak count method 139
5.2.1. Presentation of method 139
5.2.2. Derived methods 141
5.2.3. Range-restricted peak count method 142
5.2.4. Level-restricted peak count method 143
5.3. Peak between mean-crossing count method 145
5.3.1. Presentation of method 145
5.3.2. Elimination of small variations 147
5.4. Range count method 148
5.4.1. Presentation of method 148
5.4.2. Elimination of small variations 150
5.5. Range-mean count method 151
5.5.1. Presentation of method 151
5.5.2. Elimination of small variations 154
5.6. Range-pair count method 156
5.7. Hayes counting method 160
5.8. Ordered overall range counting method 162
5.9. Level-crossing count method 164
5.10. Peak valley peak counting method 168
5.11. Fatigue-meter counting method 173
5.12. Rainflow counting method 175
5.12.1. Principle of method 175
5.12.2. Subroutine for rainflow counting 181
5.13. NRL counting method (National Luchtvaart Laboratorium) 184
5.14. Evaluation of time spent at given level 187
5.15. Influence of levels of load below fatigue limit on fatigue life 188
5.16. Test acceleration 188
5.17. Presentation of fatigue curves determined by random vibration tests 190
viii Fatigue damage
6 Damage by fatigue undergone by a one-degree-of-freedom
mechanical system 193
6.1. Introduction 193
6.2. Calculation of fatigue damage due to signal versus time 194
6.3. Calculation of fatigue damage due to acceleration spectral density 196
6.3.1. General case 196
6.3.2. Particular case of a wide-band response, e.g., at the limit, r = 0 200
6.3.3. Particular case of narrow band response 201
6.3.3.1. Expression for expected damage 201
6.3.3.2. Notes 205
6.3.3.3. Calculation of gamma function 210
6.3.4. Rms response to narrow band noise G of width Af when
0
GO Af = constant 212
6.4. Equivalent narrow band noise 213
6.4.1. Use of relation established for narrow band response 213
6.4.2. Alternative: use of mean number of maxima per second 216
6.4.3. P.H. Wirsching's approach 217
6.4.4. G.K. Chaudhury and W.D. Dover's approach 220
6.4.5. Approximation to real maxima distribution using a modified
Rayleigh distribution 224
6.5. Comparison of S-N curves established under sinusoidal
and random loads 228
6.6. Comparison of theory and experiment 232
6.7. Influence of shape of power spectral density and value
of irregularity factor 237
6.8. Effects of peak truncation 237
6.9. Truncation of stress peaks 238
7 Standard deviation of fatigue damage 253
7.1. Calculation of standard deviation of damage: J.S. Bendat's method 253
7.2. Calculation of standard deviation of damage: S.H. Crandall,
W.D. Mark and G.R. Khabbaz method 258
7.3. Comparison of W.D. Mark and J.S. Bendat's results 263
7.4. Statistical S-N curves 269
7.4.1. Definition of statistical curves 269
7.4.2. J.S. Bendat's formulation 270
7.4.3. W.D. Mark's formulation 273
8 Fatigue damage using other assumptions for calculation 277
8.1. S-N curve represented by two segments of a straight line
on logarithmic scales (taking into account fatigue limit) 277
8.2. S-N curve defined by two segments of straight line on log-lin scales 280
8.3. Hypothesis of non-linear accumulation of damage 283
Contents ix
8.3.1. Corten-Dolan's accumulation law 283
8.3.2. J.D. Morrow's accumulation model 284
8.4. Random vibration with non zero mean: use of modified
Goodman diagram 287
8.5. Non Gaussian distribution of instantaneous values of signal 289
8.5.1. Influence of distribution law of instantaneous values 289
8.5.2. Influence of peak distribution 290
8.5.3. Calculation of damage using Weibull distribution 291
8.5.4. Comparison of Rayleigh assumption/peak counting 294
8.6. Non-linear mechanical system 295
Appendices 297
Al. Gamma function 297
A 1.1. Definition 297
A1.2. Properties 297
A1.3. Approximations for arbitrary x 300
A2. Incomplete gamma function 301
A2.1. Definition 301
A2.2. Relation between complete gamma function and
incomplete gamma function 302
A2.3. Pearson form of incomplete gamma function 303
A3. Various integrals 303
Bibliography 323
Index 347
Synopsis of five volume series 351