Random Vibrationand Spectral Analysisby

ANDR PREUMONTUniversit Libre de Bruxelles, Belgium

KLUWER ACADEMIC PUBLISHERSDORDRECHT / BOSTON / LONDON

TABLE OF CONTENTS

Preface xv

Introduction 11.1 Overview 1

1.1.1 Organization 41.1.2 Notations 4

1.2 The Fourier transform . . . 51.2.1 Differentiation theorem 5L2.2 Translation theorem 61.2.3 Parseval's theorem 61.2.4 Symmetry, change of scale, duality 61.2.5 Harmonic functions 7

1.3 Convolution, correlation 71.3.1 Convolution integral 71.3.2 Correlation integral 81.3.3 Example: The leakage 8

1.4 References 101.5 Problems 11

Random Variables 132.1 Axioms of probability theory 13

2.1.1 Bernoulli's law of large numbers 132.1.2 Alternative interpretation 132.1.3 Axioms 14

2.2 Theorems and definitions 152.3 Random variable 17

2.3.1 Discrete random variable 182.3.2 Continuous random variable 18

2.4 Jointly distributed random variables 202.5 Conditional distribution 212.6 Functions of Random variables 22

2.6.1 Function of one random variable 222.6.2 Function of two random variables 242.6.3 The sum of two independent random variables . . . 252.6.4 Rayleigh distribution 252.6.5 functions of random variables 26

2.7 Moments 27

viii Random Vibration and Spectral Analysis

2.7.1 Expected value 272.7.2 Moments 272.7.3 Schwarz inequality 282.7.4 Chebyshev's inequality 29

2.8 Characterstic function, Cumulants 292.8.1 Single random variable 292.8.2 Jointly distributed random variables 31

2.9 References 322.10 Problems 32

3 Random Processes 353.1 Introduction 353.2 Specification of a random process 36

3.2.1 Probability density functions 363.2.2 Characteristic function 373.2.3 Moment functions 383.2.4 Cumulant functions 383.2.5 Characteristic functional 39

3.3 Stationary random process 403.4 Properties of the correlation functions 413.5 Differentiation 43

3.5.1 Convergence 433.5.2 Continuity 433.5.3 Stochastic differentiation 44

3.6 Stochastic integrale, Ergodicity 453.6.1 Integration 453.6.2 Temporal mean 463.6.3 Ergodicity theorem 47

3.7 Spectral decomposition 483.7.1 Fourier transform 483.7.2 Power spectral density 48

3.8 Examples 503.8.1 White noise 503.8.2 Ideal low-pass process 503.8.3 Process with exponential correlation 513.8.4 Construction of a random process with specified po-

wer spectral density 513.9 Cross power spectral density 523.10 Periodic process 533.11 References 543.12 Problems 55

Contents ix

4 Gaussian Process, Poisson Process 574.1 Gaussian random variable 574.2 The central limit theorem 58

4.2.1 Example 1 594.2.2 Example 2: Binomial distribution 60

4.3 Jointly Gaussian random variables 624.3.1 Remark 64

4.4 Gaussian random vector 644.5 Gaussian random process 664.6 Poisson process 67

4.6.1 Counting process 674.6.2 Uniform Poisson process 684.6.3 Non-uniform Poisson process 70

4.7 Random pulses 704.8 Shot noise 724.9 References 724.10 Problems 73

5 Random Response of a Single Degree of Freedom Oscilla-tor 755.1 Response of a linear system 755.2 Single degree of freedom oscillator 765.3 Stationary response of a linear system 795.4 Stationary response of the linear oscillator. White noise app-

roximation 805.5 Transient response 82

5.5.1 Excitation applied from t = 0 825.5.2 Stationary excitation 83

5.6 Spectral moments 855.6.1 Definition 855.6.2 Computation for the linear oscillator 855.6.3 Rice formulae 88

5.7 Envelope of a narrow band process 905.7.1 Crandall & Mark's definition 905.7.2 Joint distribution of X and X 915.7.3 Probability distribution of the envelope 91

5.8 References 925.9 Problems 92

Random Vibration and Spectral Analysis

Random Response of Multi Degree of Freedom Systems 946.1 Some concepts of structural dynamics 94

6.1.1 Equation of motion 946.1.2 Input-output relationship 956.1.3 Modal decomposition 956.1.4 State variable form 976.1.5 Structural and hereditary damping 986.1.6 Remarks 100

6.2 Seismic excitation 1006.2.1 Equation of motion 1006.2.2 Effective modal mass 1026.2.3 Input-Output relationships in the frequency domain 104

6.3 Response to a stationary excitation 1076.4 Role of the cross-correlation 1086.5 Response to a stationary seismic excitation 1126.6 Continuous structures 113

6.6.1 Input-Output relationship 1136.6.2 Structure with normal modes 115

6.7 Co-spectrum 1186.8 Example: Boundary layer noise 1206.9 Discretization of the excitation 1226.10 Along-wind response of a tall building 123

6.10.1 Along-wind aerodynamic forces 1236.10.2 Mean wind 1246.10.3 Spectrum at a point 1246.10.4 Davenport spectrum 1256.10.5 Example 126

6.11 Earthquake 1286.11.1 Response spectrum 1286.11.2 Cascade analysis 130

6.12 Remark on sound pressure level 1316.13 References 1326.14 Problems 133

Input-Output Relationship for Physical Systems 1357.1 Estimation of frequency response functions 1357.2 Coherence function 1367.3 Effect of measurement noise 1377.4 Example 1397.5 Remark 1417.6 References 141

Contents xi

8 Spectral Description of Non-stationary Random Processesl428.1 Introduction 142

8.1.1 Stationary random process 1428.1.2 Non-stationary random process 1438.1.3 Objectives of a spectral description 144

8.2 Instantaneous power spectrum 1458.3 Mark's Physical Spectrum 146

8.3.1 Definition and properties 1468.3.2 Duality, uncertainty principle 1488.3.3 Relation to the PSD of a stationary process 1508.3.4 Example: Structural response to a sweep sine . . . . 151

8.4 Priestley's Evolutionary Spectrum 1528.4.1 Generalized harmonic analysis 1528.4.2 Evolutionary spectrum 1548.4.3 Vector process 1558.4.4 Input-output relationship 1568.4.5 State variable form 1578.4.6 Remarks 158

8.5 Applications 1588.5.1 Structural response to a sweep sine 1588.5.2 Transient response of an oscillator 1598.5.3 Earthquake records 159

8.6 Summary 1608.7 References 1618.8 Problems 162

9 Markov Process 1649.1 Conditional piobability 1649.2 Classification of random processes 1659.3 Smoluchoweki equation 1669.4 Process with independent increments 166

9.4.1 Random Walk 1679.4.2 Wiener process 167

9.5 Markov process and state variables 1699.6 Gaussian Markov process 171

9.6.1 Covariance matrix 1719.6.2 Wide sense Markov process 1739.6.3 Power spectral density matrix 173

9.7 Random walk and diffusion equation 1759.7.1 Random walk of a free particle 1759.7.2 Random walk of an elastically bound particle . . . . 176

9.8 One-dimensional Fokker-Planck equation 177

xii Random Vibration and Spectral Analysis

9.8.1 Derivation of the Fokker-Planck equation 1799.8.2 Kolmogorov equation 180

9.9 Multi-dimensional Fokker-Planck equation 1819.10 The Brownian motion of an oscillator 1829.11 Replacement of an actual process by a Markov process . . . 184

9.11.1 One-dimensional process 1849.11.2 Stochastically equivalent systems 1859.11.3 Multi-dimensional process 186

9.12 References 1869.13 Problems 187

10 Threshold Crossings, Maxima, Envelope and Peak Factor 18810.1 Introduction 18810.2 Threshold crossings 189

10.2.1 Up-crossings of a level b 18910.2.2 Central frequency 190

10.3 Maxima 19110.4 Envelope 194

10.4.1 Crandall & Mark's definition 19410.4.2 Rice's definition 19410.4.3 The Hilbert transform 19610.4.4 Cramer & Leadbetter's definition 19710.4.5 Discussion 19710.4.6 Second order joint distribution of the envelope . . . 19910.4.7 Threshold crossings 20010.4.8 Clump size 202

10.5 First-crossing problem 20610.5.1 Introduction 20610.5.2 Independent crossings 20710.5.3 Independent envelope crossings 20810.5.4 Approach based on the dump size 20810.5.5 Vanmarcke's model 20810.5.6 Extreme point process 209

10.6 First-passage problem and Fokker-Planck equation 21110.6.1 Multidimensional Markov process 21110.6.2 Fokker-Planck equation of the envelope 21110.6.3 Kolmogorov equation of the reliability 212

10.7 Peak factor 21310.7.1 Extreme value probability 21310.7.2 Formulae for the peak factor 214

10.8 References 21610.9 Problems 217

Contents xiii

11 Random fatigue 22011.1 Introduction 22011.2 Uniaxial loading with zero mean 22111.3 Biaxial loading with zero mean 22211.4 Finite element formulation 22411.5 Fluctuating stresses 22411.6 Recommended procedure 22511.7 Example 22611.8 References 22711.9 Problems 228

12 The Discrete Fourier Transform 22912.1 Introduction 22912.2 Consequences of the convolution theorem 231

12.2.1 Periodic continuation 23112.2.2 Sampling 232

12.3 Shannon's theorem, Aliasing 23312.4 Fourier series 235

12.4.1 Orthogonal functions 23512.4.2 Fourier series 23612.4.3 Gibbs phenomenon 23712.4.4 Relation to the Fourier transform 238

12.5 Graphical development of the DFT 23912.6 Analytical development of the DFT 24112.7 Definition and properties of the DFT 243

12.7.1 Definition of the DFT ma IDFT 24312.7.2 Properties of the DFT 243

12.8 Leakage reduction 24612.9 Power spectrum estimation 24912.l0 Convolution and correlation via FFT 250

12.10.1 Periodic convolution and correlation 25012.10.2 Approximation of the continuous convolution . . . . 25212.10.3 Sectioning Overlap-save 25412.10.4 Sectioning Overlap-add 254

12.11 FFT simulation of Gaussian processes with prescribed PSD 25412.12 References 25712.13Problems 258

Bibliography 261

Index 269