隨機平均法及其應(yīng)用（上冊 英文版）

Contents
1 Introduction 1
References 8
2 Stochastic Processes 9
2.1 Fundamentals 9
2.1.1 Descriptions of Stochastic Processes 11
2.1.2 Stationarity and Ergodicity 13
2.1.3 Spectral Analysis 17
2.2 Gaussian Stochastic Processes 23
2.3 Markov Processes 24
2.3.1 Markov Processes and Chapman-Kolmogorov-Smoluwski Equation 24
2.3.2 Markov Diffusion Processes and Fokker–Planck-Kolmogorov (FPK) Equation 26
2.3.3 Wiener Processes and Gaussian White Noise 28
2.3.4 It？ Stochastic Differential Equations 31
2.3.5 Responses of Systems Under Gaussian White-Noise Excitations 34
2.4 PoissonWhite Noise Processes 38
2.4.1 Poisson Processes 38
2.4.2 PoissonWhite Noise 39
2.4.3 Stochastic Differential-Integral Equation and FPK Equation 42
2.5 Fractional Gaussian Processes 49
2.5.1 Fractional Calculus 49
2.5.2 Fractional Brownian Motion 50
2.5.3 Fractional Gaussian Noises 52
2.5.4 Stochastic Integration with Respect to Fractional Brownian Motion and Fractional Stochastic Differential Equations 54
2.5.5 Response of Linear Systems Excited by Fractional Gaussian Noises 57
2.6 Colored Noises 61
2.6.1 Noises Generated from Linear Filters 62
2.6.2 Noises Generated from Nonlinear Filters 64
2.6.3 Randomized Harmonic Process71
References 74
3 Nonlinear Stochastic Dynamical Systems 77
3.1 Modeling of Nonlinear Stochastic Dynamical Systems 77
3.2 Hamiltonian Systems and Their Classification 80
3.2.1 Hamilton Equation 80
3.2.2 Poisson Bracket 84
3.2.3 Phase Flow 86
3.2.4 Canonical Transformation 87
3.2.5 Completely Integrable Hamiltonian System 88
3.2.6 Non-Integrable Hamiltonian System 93
3.2.7 Partially Integrable Hamiltonian System 94
3.2.8 Ergodicity of Hamiltonian Systems 95
3.2.9 Stochastically Excited and Dissipated Hamiltonian Systems 96
3.3 The Generalized Hamiltonian System and its Classification 98
3.4 Forces with Genetic Effects 104
3.4.1 Hysteretic Forces 104
3.4.2 Visco-Elastic Force 114
3.4.3 Damping Force with Fractional Derivative 118
References 120
4 Stochastic Averaging Methods of Single-Degree-Of-Freedom Systems 123
4.1 Stochastic Averaging Principles 124
4.2 Stochastic Averaging Methods of SDOF Systems 130
4.2.1 Stochastic Averaging of Amplitude Envelope 131
4.2.2 Stochastic Averaging of Energy Envelope 134
4.3 Systems Under Gaussian White Noise Excitations 138
4.3.1 Linear Restoring Force 138
4.3.2 Nonlinear Restoring Force 142
4.4 Systems Under Broad-Band Random Excitations 145
4.4.1 Linear Restoring Force 146
4.4.2 A Primary-Secondary System 148
4.4.3 Energy-Dependent White-Noise Approximation 153
4.4.4 Fourier-Expansion Scheme 155
4.4.5 Residual Phase Procedure 159
4.5 Viscoelastic Systems Under Broad-Band Excitations 167
4.5.1 Linear Restoring Force 168
4.5.2 Nonlinear Restoring Force 173
4.6 A System with Double-Well Potential 180
4.6.1 Deterministic System with Double-Well Potential 181
4.6.2 Stochastic Averaging 184
4.7 Systems Under Combined Random and Harmonic Excitations 190
4.8 Systems Under Poisson White Noise Excitations 200
4.8.1 Amplitude Envelope 201
4.8.2 Energy Envelope 207
4.9 Systems Excited by Fractional Gaussian Noises 210
References 216
5 Stochastic Averaging Methods of Quasi-Hamiltonian Systems Under Gaussian White Noise Excitations. 219
5.1 Quasi-Non-Integrable Hamiltonian Systems 220
5.2 Quasi-Integrable Hamiltonian Systems 232
5.2.1 Non-Internal Resonant Case 234
5.2.2 Internal Resonant Case 242
5.3 Quasi-Partially Integrable Hamiltonian Systems 249
5.3.1 Noninternal Resonance Case 251
5.3.2 Internal Resonant Case 256
5.4 Stationary Response of 2-DOF Vibration-Impact System 266
5.4.1 Exact Stationary Solution 268
5.4.2 Application of Stochastic Averaging Method of Quasi-Non-Integrable Hamiltonian Systems 269
5.4.3 Application of Stochastic Averaging Method of Quasi-Integrable Hamiltonian Systems 274
5.4.4 Combined Application of Both Stochastic Averaging Methods of Quasi-Non-Integrable and Quasi-Integrable Hamiltonian Systems 281
5.5 Quasi-Non-Integrable Hamiltonian Systems with Markov Jump Parameters 284
5.5.1 Single-DOF Systems 286
5.5.2 Multi-DOF Systems 294
References 302
6 Stochastic Averaging Methods of Quasi-Hamiltonian Systems Excited by Gaussian and PoissonWhite Noises 303
6.1 Quasi-Hamiltonian Systems Excited by Gaussian and Poisson White Noises 303
6.2 Quasi-Non-Integrable Hamiltonian Systems 306
6.2.1 Combined Gaussian and Poisson White Noise Excitations306
6.2.2 PoissonWhite Noise Excitation 318
6.3 Quasi-Integrable Hamiltonian Systems 330
6.3.1 Non-Internal Resonant Case 332
6.3.2 Internal Resonant Case 340
6.4 Quasi-Partially Integrable Hamiltonian Systems 357
6.4.1 Non-Internal Resonant Case 361
6.4.2 Internal Resonant Case 367
References 387
7 Stochastic Averaging Methods