CONTENTS
Preface = xv
1 Introduction = 1
1.1 History and Motivation = 1
1.2 Example = 4
1.3 Scope and Organization = 6
2 Random Variables = 9
2.1 Essential Probability = 9
2.1.1 Set Theory = 10
2.1.2 Axioms of Probability = 11
2.1.3 Conditional Probability and Independence = 12
2.2 Single Random Variables = 14
2.2.1 Discrete Random Variables and Probability Mass Function = 14
2.2.2 Continuous Random Variables and Probability Density Function = 15
2.2.3 Cumulative Distribution Functions = 19
2.2.4 Measures of Central Tendency and Dispersion = 21
2.2.5 Expected Values = 23
2.2.6 Moment-Generating Functions = 25
2.2.7 Gaussian (Normal) Random Variables = 26
2.3 Jointly Distributed Random Variables = 30
2.3.1 Joint and Marginal Distributions = 30
2.3.2 Conditional Distributions and Independence = 33
2.3.3 Expected Values, Covariance and Correlation Coefficient = 36
2.3.4 Linear Dependence = 38
2.3.5 Joint Normal Distribution = 41
2.4 Function of Random Variables = 43
2.4.1 Change of Variables by Cumulative Distribution Function = 43
2.4.2 Change of Variables by Probability Density Function = 45
2.4.3 Multidimensional Change of Variables = 49
2.4.4 Sums of Random Variables = 51
2.5 Central Limit Theorem and Distributions Related to Normal Distribution = 55
2.5.1 De Moivre-Laplace Approximation = 56
2.5.2 Central Limit Theorem = 57
2.5.3 Distribution of Sample Mean: Normal = 58
2.5.4 Distribution of Sample Variance: Chi Squared = 60
2.5.5 Distributions of Related Ratios of Random Variables: t and F = 64
2.5.6 Distribution of Extreme Values = 65
Problems = 68
3 Random Processes in Time Domain = 72
3.1 Random Process Definitions = 72
3.1.1 State Spaces and Index Sets = 72
3.1.2 Ensembles and Ensemble Averages = 73
3.1.3 Stationary Processes = 80
3.1.4 Ergodic Processes = 82
3.1.5 Gaussian Processes = 83
3.2 Correlation = 84
3.2.1 Characteristics of Autocorrelation Function = 84
3.2.2 Cross-Correlation Function and Linear Transformations = 86
3.2.3 Derivatives of Stationary Processes = 89
3.3 Some Essential Random Processes = 91
3.3.1 Harmonic Processes = 91
3.3.2 Poisson Process = 94
Problems = 96
4 Fourier Transforms = 100
4.1 Fourier Transform = 100
4.1.1 Fourier Series = 101
4.1.2 Fourier Transforms = 104
4.1.3 Symmetry = 107
4.1.4 Basic Theorems = 108
4.1.5 Correlation, Convolution, and Windowing = 111
4.2 Dirac Delta Function = 113
4.2,1 Definition and Properties of Delta Function = 113
4.2.2 Fourier Transform of Delta Function = 116
4.2.3 Transforms of Cosine and Sine Functions = 118
Problems = 119
5 Random Processes in Frequency Domain = 123
5,1 Spectral Density Function = 123
5.1.1 Definition = 124
5.1.2 Relationship with Fourier Transform of X(t) = 127
5.1.3 Practical Issues = 130
5.1.4 White Noise and Bandpass Filtered Spectra = 135
5.2 Cross-Spectral Density Function = 139
5.2.1 Definition = 139
5.2.2 Coherence Function and Linear Transformations = 140
5.2.3 Derivatives of Stationary Processes = 141
Problems = 143
6 Statistical Properties of Random Processes = 146
6.1 Level Crossings = 146
6.1.1 Preliminary Remarks = 146
6.1.2 Derivation of Expected Rate of Level Crossing = 150
6.1.3 Specializations = 152
6.1.4 Rice's Narrow-Band Envelope = 157
6.2 Distributions of Extrema = 162
6.2.1 Simple Approach to Distribution of Peak = 162
6.2.2 General Approach to Distribution of Peak = 164
6.2.3 Approximate Distribution for Height of Rise or Fall = 167
Problems = 171
7 Vibration of Single-Degree-of-Freedom Systems = 174
7.1 Free Vibration of Single-Degree-of-Freedom System = 174
7.1.1 Background = 174
7.1.2 Harmonic Motion = 174
7.1.3 Free Vibration of Undamped Single-Degree-of-Freedom System = 177
7.1.4 Damped Free Vibration of Single-Degree-of-Freedom System = 179
7.2 Forced Vibration = 180
7.2.1 Force-Excited System: Harmonic Excitation = 180
7.2.2 Base-Excited System: Absolute Motion = 183
7.2.3 Base-Excited System: Relative Motion = 184
7.3 Background for Response of Single-Degree-of-Freedom System to Random Forces = 190
7.3.1 Response of Single-Degree-of-Freedom System to Impulsive Forces = 190
7.3,2 Response of Single-Degree-of-Freedom System to Arbitrary Loading = 191
7.3.3 Relationship Between h(t) and H(w) = 192
7.3.4 Relationship Between X(w) and F(w) = 193
Problems = 194
8 Response of Single-Degree-of-Freedom Linear Systems to Random Environments = 197
8.1 Response to Stationary Random Forces = 197
8.1.1 Preliminary Remarks = 197
8.1.2 Mean of Response Process = 198
8.1.3 Autocorrelation of Response Process = 199
8.1.4 Spectral Density of Response Process = 202
8.1.5 Distribution of Response Process = 203
8.2 White-Noise Process as Model for Force = 203
8.2.1 Definition of Process = 203
8.2.2 Response of Force-Excited System to White Noise = 204
8.2.3 Engineering Significance of White-Noise Process = 206
8.3 Examples of Response of Single-Degree-of- Freedom Systems to Random Forces = 207
Problems = 213
9 Random Vibration of Multi-Degree-of-Freedom Systems = 218
9.1 Equations of Motion for Multi-Degree-of-Freedom System = 218
9.1.1 Introduction = 218
9.1.2 Equations of Motion = 220
9.1.3 Transfer Function and Impulse Response Functions for System = 221
9.2 Direct Model for Determining Response of MDOF System = 223
9.2.1 Expression for Response = 223
9.2.2 Mean Value of Response = 224
9.2.3 Autocorrelation Function = 225
9.2.4 Spectral Density Function of Response = 226
9.2.5 Single Response Variable: Special Cases = 227
9.3 Normal Mode Method = 234
9.3.1 Preliminary Remarks = 234
9.3.2 The Eigenvalue Problem: Free-Vibration Solution to Equations of Motion = 234
9.3.3 Example of Eigenvalue Problem = 235
9.3.4 Normal Mode Method: Orthogonality Conditions = 237
9.3.5 Normal Mode Method: Equations of Motion = 241
9.3.6 Big Payoff for Using Normal Mode Method = 242
9.3.7 Introduction of Damping into Equations of Motion = 243
9.3.8 Normal Mode Method: Mode Combination Problem-Some Methods = 246
9.3.9 Example of Mode Combination Algorithm Results = 248
9.4 Computer Codes for Analyzing MDOF Structural Systems = 249
Problems = 250
10 Design to Avoid Structural Failures Due to Random Vibration = 254
10.1 Three-Sigma Design = 254
10.1.1 Basic Design Criterion = 254
10.1.2 More General Statement of Three Sigma Criterion = 256
10.2 First-Passage Failure = 257
10.2.1 Introductory Remarks = 257
10.2.2 Basic Formulation of First-Passage Problem = 259
10.2.3 First-Passage Failure: Exact Distribution of Largest in Sample of Size n Independent Peaks = 260
10.2.4 First-Passage Failure: Use of Extreme-Value Distribution to Model Distribution of Largest Peak = 263
10.2.5 First-Passage Failure: Determination of Design Value Using Concept of Return Period = 264
10.3 Fatigue: An Introduction = 266
10.3.1 Physical Process of Fatigue = 266
10.3.2 Fatigue Strength Models = 267
10.3.3 Miner's Rule = 271
10.3.4 Models of Fatigue Damage Under Narrow-Band Random Stress = 274
10.3.5 Models of Fatigue Damage Under Wide-Band Random Stresses = 283
10.3.6 Quality of Miner's Rule = 288
Problems = 290
11 Introduction to Parameter Estimation = 296
11.1 Estimation and Analysis of Mean = 297
11.1.1 Maximum Likelihood Estimation = 297
11.1.2 Bias and Consistency of Mean Estimator = 301
11.1.3 Sampling Distribution of Mean Estimator and Confidence Intervals on Mean = 303
11.1.4 Bias in Variance Estimator = 307
11.2 Other Important Problems in Parameter Estimation = 308
11.2.1 Sampling Distribution for Variance = 308
11.2.2 Confidence Intervals for Mean with Unknown Variance = 309
11.3 Closure = 311
Problems = 312
12 Time-Domain Estimation of Random Process Parameters = 315
12.1 Random Process Parameter Estimation via Ensemble Average = 316
12,1.1 Mean, Variance, and Standard Deviation Function = 316
12.1.2 Correlation = 318
12.1.3 Autocorrelation Function = 320
12.1.4 Cross-Correlation Function = 324
12.1.5 Covariance and Correlation Coefficient Function = 324
12.2 Stationary Random Process Parameter Estimation via Temporal Averaging = 325
12.3 Other Methods for Nonstationary Random Process Analysis = 330
12.3.1 Direct Analysis of Nonstationary Random Processes = 330
12.3.2 Indirect Analysis of Nonstationary Random Processes: Method of Shock Response Spectra = 333
Problems = 338
13 Discrete Fourier Transform = 341
13.1 Definition of DFT and Fundamental Characteristics = 341
13.2 Periodicity of DFT; DFT of Real Signal = 344
13.3 DFT of Continuous-Time Signals: Aliasing and Leakage = 350
13.4 Data Windows = 356
13.5 Fast Fourier Transform = 361
13.6 Generation of Real-Valued, Finite-Duration, Sampled Realizations of Stationary, Normal Random Processes = 365
Problems = 368
14 Frequency-Domain Estimation of Random Processes = 370
14.1 Fundamental Concepts in Estimation of Autospectral Density = 370
14.1.1 Direct Estimation of Autospectral Density = 371
14.1.2 Maximum Likelihood Estimation = 375
14.1.3 Bias, Consistency, and Sampling Distribution of Autospectral Density Estimator = 378
14.2 Practical Aspects of Estimation of Autospectral Density, '= 381
14.3 Real-Time Estimation of Spectral Density = 386
14.4 Estimation of Cross-Spectral Density and Ordinary Coherence = 389
14.4.1 Cross-Spectral Density Estimation = 389
14.4.2 Coherence Estimation = 390
14.4.3 Bias and Consistency of Cross-Spectral Density Estimator = 391
14.5 Estimation of Frequency Response Function = 392
14.5.1 Frequency Response in SISO Case = 392
14.5.2 Sampling Distribution and Variance of Frequency Response Function Estimator = 396
14.5.3 Frequency Response Function in MISO Case = 398
Problems = 401
Appendix A Convergence of Random Processes = 404
A.1 Introductory Comments = 404
A.2 Modes of Convergence = 405
A.3 Mean-Square Continuity = 407
A.4 Mean-Square Differentiability = 407
A.5 Mean-Square Integrability = 409
A.6 Ergodicity = 412
A.7 Central Limit Theorem = 413
Appendix B Integrals of Transfer Functions = 416
Appendix C Formulas for Approximate Evaluation of Some Integrals Useful in Probabili ty = 417
Appendix D Some Fast Fourier Transform Programs = 419
D.1 C Language = 419
D.2 Fortran = 421
Appendix E Tables = 423
E.1 Table of Cumulative Distribution of Standard Normal Random Variable = 423
E.2 Table of Percentage Points of X² Distribution = 426
E.3 Table of Percentage Points of t Distribution = 429
E.4 Table of Percentage Points of the F Distribution = 430
References = 433
Index = 441
