Mechanical Vibrations and Shocks: Random Vibrations v. 3 (Mechanical vibration & shock)

365 Pages·2002·4.33 MB·English

by  Christian Lalanne

Random
Vibration
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MechanicalVibration
& Shock
Random
Vibration
Volume III
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 and licences issued
by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at
the undermentioned address:
Hermes Penton Science
120PentonvilleRoad
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 asserted 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 9605 8
Printed and bound in Great Britain by Biddies Ltd, Guildford and King's Lynn
www.biddles.co.uk
Contents
Introduction xiii
List of symbols xv
1 Statistical properties of a random process 1
1.1. Definitions 1
1.1.1. Random variable 1
1.1.2. Random process 1
1.2. Random vibration in real environments 2
1.3. Random vibration in laboratory tests 2
1.4. Methods of analysis of random vibration 3
1.5. Distribution of instantaneous values 4
1.5.1. Probability density 4
1.5.2. Distribution function 5
1.6. Gaussian random process 6
1.7. Rayleigh distribution 7
1.8. Ensemble averages: ''through the process' 8
1.8.1. n order average 8
1.8.2. Central moments 9
1.8.3. Variance 9
1.8.4. Standard deviation 10
1.8.5. Autocorrelation function 12
1.8.6. Cross-correlation function 12
1.8.7. Autocovariance 12
1.8.8. Covariance 13
1.8.9. Stationarity 13
1.9. Temporal averages: 'along the process' 15
1.9.1. Mean 15
1.9.2. Quadratic mean - rms value 16
1.9.3. Moments of order n 18
vi Random vibration
1.9.4. Variance - standard deviation 18
1.9.5. Skewness 18
1.9.6. Kurtosis 19
1.9.7. Temporal autocorrelation function 22
1.9.8. Properties of the autocorrelation function 25
1.9.9. Correlation duration 27
1.9.10. Cross-correlation 28
1.9.11. Cross-correlation coefficient 28
1.9.12. Ergodicity 29
1.10. Significance of the statistical analysis (ensemble or temporal) 30
1.11. Stationary and pseudo-stationary signals 30
1.12. Summary chart of main definitions (Table 1.2) 30.
2 Properties of random vibration in the frequency domain 33
2.1. Fourier transform 33
2.2. Power spectral density 35
2.2.1. Need 35
2.2.2. Definition 36
2.3. Cross-power spectral density 42
2.4. Power spectral density of a random process 43
2.5. Cross-power spectral density of two processes 43
2.6. Relation between PSD and correlation function of a process 44
2.7. Quadspectrum - cospectrum 45
2.8. Definitions 45
2.8.1. Broad-band process 45
2.8.2. White noise 46
2.8.3. Band-limited white noise 47
2.8.4. Narrow-band process 47
2.8.5. Pink noise 48
2.9. Autocorrelation function of white noise 48
2.10. Autocorrelation function of band-limited white noise 50
2.11. Peak factor 52
2.12. Standardized PSD/density of probability analogy 54
2.13. Spectral density as a function of time 54
2.14. Relation between PSD of excitation and response of a linear system 55
2.15. Relation between PSD of excitation and cross-power spectral
density of response of a linear system 57
2.16. Coherence function 58
2.17. Effects of truncation of peaks of acceleration signal 60
2.17.1. Acceleration signal selected for study 60
2.17.2. Power spectral densities obtained 61
Contents vii
3 Rms value of random vibration 63
3.1. Rms value of a signal as function of its PSD 63
3.2. Relations between PSD of acceleration, velocity and displacement 66
3.3. Graphical representation of PSD 69
3.4. Practical calculation of acceleration, velocity and
displacement rms values 70
3.4.1. General expressions 70
3.4.2. Constant PSD in frequency interval 71
3.4.3. PSD comprising several horizontal straight line segments 72
3.4.4. PSD defined by a linear segment of arbitrary slope 73
3.4.5. PSD comprising several segments of arbitrary slopes 81
3.5. Case: periodic signals 82
3.6. Case: periodic signal superimposed onto random noise 84
4 Practical calculation of power spectral density 87
4.1 Sampling of the signal 87
4.2. Calculation of PSD from rms value of filtered signal 90
4.3. Calculation of PSD starting from Fourier transform 91
4.3.1. Maximum frequency 92
4.3.2. Extraction of sample of duration T 92
4.3.3. Averaging 98
4.3.4. Addition of zeros 98
4.4. FFT 100
4.5. Particular case of a periodic excitation 101
4.6. Statistical error 102
4.6.1. Origin 102
4.6.2. Definition 103
4.7. Statistical error calculation 104
4.7.1. Distribution of measured PSD 104
4.7.2. Variance of measured PSD 106
4.7.3. Statistical error 107
4.7.4. Relationship between number of degrees of freedom,
duration and bandwidth of analysis 108
4.7.5. Confidence interval 111
4.7.6. Expression for statistical error in decibels 124
4.7.7. Statistical error calculation from digitized signal 126
4.8. Overlapping 128
4.8.1. Utility 128
4.8.2. Influence on number of degrees of freedom 129
4.8.3. Influence on statistical error 130
4.8.4. Choice of overlapping rate 132
4.9. Calculation of PSD for given statistical error 134
4.9.1. Case: digitalization of signal is to be carried out 134
4.9.2. Case: only one sample of an already digitized signal is available 135
viii Random vibration
4.10. Choice of filter bandwidth 137
4.10.1. Rules 137
4.10.2. Bias error 138
4.10.3. Maximum statistical error 143
4.10.4. Optimum bandwidth 145
4.11. Probability that the measured PSD lies between ± one
standard deviation 149
4.12. Statistical error: other quantities 150
4.13. Generation of random signal of given PSD 154
4.13.1. Method 154
4.13.2. Expression for phase 155
4.13.2. I.Gaussian law 155
4.13.2.2. Other laws 155
5 Properties of random vibration in the time domain 159
5.1. Averages 159
5.1.1. Mean value 159
5.1.2. Mean quadratic value; rms value 160
5.2. Statistical properties of instantaneous values of random signal 162
5.2.1. Distribution of instantaneous values 162
5.2.2. Properties of derivative process 163
5.2.3. Number of threshold crossings per unit time 167
5.2.4. Average frequency 171
5.2.5. Threshold level crossing curves 174
5.3. Moments : 181
5.4. Average frequency of PSD defined by straight line segments 184
5.4.1. Linear-linear scales 184
5.4.2. Linear-logarithmic scales 186
5.4.3. Logarithmic-linear scales , 187
5.4.4. Logarithmic-logarithmic scales 188
5.5. Fourth moment of PSD defined by straight line segments 189
5.5.1. Linear-linear scales 189
5.5.2. Linear-logarithmic scales 190
5.5.3. Logarithmic-linear scales 191
5.5.4. Logarithmic-logarithmic scales 192
5.6. Generalization; moment of order n 193
5.6.1. Linear-linear scales 193
5.6.2. Linear-logarithmic scales 193
5.6.3. Logarithmic-linear scales 193
5.6.4. Logarithmic-logarithmic scales 194
Contents ix
6 Probability distribution of maxima of random vibration 195
6.1. Probability density of maxima 195
6.2. Expected number of maxima per unit time 203
6.3. Average time interval between two successive maxima 206
6.4. Average correlation between two successive maxima 207
6.5. Properties of the irregularity factor 208
6.5.1. Variation interval 208
6.5.2. Calculation of irregularity factor for band-limited white noise 212
6.5.3. Calculation of irregularity factor for noise of form G = Const, f b.... 215
6.5.4. Study: variations of irregularity factor for two
narrow band signals 218
6.6. Error related to the use of Rayleigh's law instead of complete
probability density function 220
6.7. Peak distribution function 222
6.7.1. General case 222
6.7.2. Particular case of narrow band Gaussian process 224
6.8. Mean number of maxima greater than given threshold (by unit time) 226
6.9. Mean number of maxima above given threshold between two times 230
6.10. Mean time interval between two successive maxima 230
6.11. Mean number of maxima above given level reached by
signal excursion above this threshold 231
6.12. Time during which signal is above a given value 234
6.13. Probability that a maximum is positive or negative 235
6.14. Probability density of positive maxima 236
6.15. Probability that positive maxima is lower than given threshold 236
6.16. Average number of positive maxima per unit time 236
6.17. Average amplitude jump between two successive extrema 237
7 Statistics of extreme values 241
7.1. Probability density of maxima greater than given value 241
7.2. Return period 242
7.3. Peak £ expected among N peaks 242
p p
7.4. Logarithmic rise 243
7.5. Average maximum of N peaks 243
p
7.6. Variance of maximum 243
7.7. Mode (most probable maximum value) 244
7.8. Maximum value exceeded with risk a 244
7.9. Application to case of centred narrow band normal process 244
7.9.1. Distribution function of largest peaks over duration T 244
7.9.2. Probability that one peak at least exceeds a given threshold 247
7.9.3. Probability density of the largest maxima over duration T 247
7.9.4. Average of highest peaks 250
7.9.5. Standard deviation of highest peaks 252