1 Introduction 1.1 When Are Uncertainties Random? 1.2 Why Do We Study Random Vibration? 1.3 What Is Random Vibration About? 1.4 New Possibility with Random Vibration References 2 Theoretical Foundation 2.1 Probability 2.1.1 Random Variable 2.1.2 Probability Distribution 2.1.3 Function of Random Variable 2.2 Stochastic Process 2.2.1 Expectation of Stochastic Process 2.2.2 Characteristic and Cumulant Function 2.3 Spectral Analysis 2.4 Stochastic Calculus 2.4.1 Ito Calculus 2.4.2 Questions on Gaussian Property References 3 Simulation 3.1 Purpose of Simulation 3.2 Random Number 3.3 Random Process 3.4 Integration of Stochastic Differential Equation 3.5 A Note on Non-Gaussian Process References 4 Response Process 4.1 System with Random Uncertainty 4.2 Extended Sample Space 4.3 Nature of Response Process 4.4 Probability of Response Reference 5 Response Moment 5.1 Moment Equation 5.2 Cumulant Function 5.3 Cumulant-Neglect Closure References 6 Response Distribution 6.1 Governing Equation 6.1.1 Questions on FPK Equation 6.2 Backward Equation 6.3 Reliability Formulation 6.4 First-Passage Time Probability 6.5 Common Safe Domains References 7 Classical Random Vibration 7.1 Linear Random Vibration 7.2 Nonlinear Random Vibration 7.2.1 Short-Time Solution 7.2.2 Improvement of Short-Time Solution
7.2.3 Path Integral 7.2.4 Markov Chain Representation 7.2.5 Exact Solution of Stationary PDF 7.2.6 Detailed Balance 7.3 Equivalent Ito Equation References 8 Random Vibration with Neural Networks 8.1 Neural Networks 8.1.1 Sigmoid Function 8.1.2 Hyperbolic Tangent (Tanh) 8.1.3 Rectified Linear Unit (ReLU) 8.1.4 Leaky ReLU 8.1.5 Softmax 8.2 Layers of Neural Networks 8.2.1 Input Layer 8.2.2 Hidden Layer 8.2.3 Output Layer 8.3 Loss Function 8.4 Architecture of Neural Networks 8.5 Overfitting and Regularization 8.6 Radial Basis Function Neural Networks 8.7 RBFNN Solution of FPK Equation 8.8 Motivation for Gaussian Functions 8.9 Sampling Method for Integration 8.10 Response Moments 8.11 Examples of Application 8.11.1 First Order Nonlinear System 8.11.2 Van der Pol Oscillator 8.12 First-Passage Problem 8.12.1 RBFNN Solution 8.12.2 Solution in Null Space 8.13 First-Passage Examples 8.13.1 Linear Oscillator 8.13.2 Van der Pol Oscillator 8.13.3 Duffing System References 9 Method of Feynman-Kac 9.1 Background 9.2 Feynman-Kac Formula 9.3 Solution via Feynman-Kac Formula 9.3.1 RBFNN Solution 9.3.2 Loss Function 9.3.3 STGA Technique 9.3.4 Computation of Optimal Weights 9.4 Equation-Based RBFNN Method 9.5 Error Metric 9.6 Examples of Feynman-Kac 9.6.1 Van der Pol System 9.6.2 Nonlinear Vibro-Impact System 9.6.3 Two Coupled Duffing Oscillators
9.7 Comparison of Computational Time 9.8 A Summary References 10 Green's Operator 10.1 Boundary Value Problem 10.2 Algebraic Solution for Linear Problem 10.3 Neural Network Solution for Nonlinear Problem 10.4 Examples of Green's Operator 10.4.1 Linear Random Vibration Problem 10.4.2 Bouc-Wen System References 11 Data-Driven Random Vibration 11.1.1 What Can We Do with Data? 11.1.1 System Property from Data 11.1.2 Physical Laws in Data 11.1.3 Prediction Across Physical Domain 11.1.4 A Remark on Digital Twin 11.2 Response Moments from Data 11.3 Density Estimation 11.4 Model Identification with Data+Knowledge 11.4.1 Algebraic Method 11.4.2 Computation of Multiple Integration 11.4.3 Sparse Representation 11.4.4 Numerical Example 11.4.5 Experimental Example References 12 Data-Driven Generalized Cell Mapping 12.1 Generalized Cell Mapping 12.2 GCM from Data 12.2.1 Determination of Embedding Dimension 12.2.2 Autoencoder for Embedding Dimension 12.2.3 Determination of Mapping Time 12.2.4 Construction of Transition Probability Matrix 12.3 Example of Mathieu-Duffing Oscillator References 13 New Application of Simulation 13.1 Design of RBFNN 13.1.1 Neuron Performance Index 13.1.2 Statistical Properties of I(j) and w(j) 13.1.3 Optimal Selection Strategy 13.1.4 Van der Pol System 13.1.5 Lorenz System 13.2 Order Reduction 13.2.1 Equation for Marginal PDF 13.2.2 Computing Conditional Expectation References Index
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