Linearization Methods for Stochastic Dynamic Systems

For most cases of interest, exact solutions to nonlinear equations describing stochastic dynamical systems are not available. This book details the relatively simple and popular linearization techniques available, covering theory as well as application. It examines models with continuous external and parametric excitations, those that cover the majority of known approaches. Introduction References Mathematical Preliminaries Moments and Cumulants of Random Variables Gaussian and Non-Gaussian Distributions Moments and Cumulants of Stochastic Processes Second-Order Stochastic Processes Gaussian and Non-Gaussian Processes Markov Processes Diffusion Processes and Kolmogorov Equations Wiener Process White and Colored Noise Integration and Differentiation Formulas of Diffusion Processes Stochastic Differential Equations Stochastic Stability The Method of Fokker--Planck--Kolmogorov Equations Bibliography Notes References Moment Equations for Linear Stochastic Dynamic Systems (LSDS) Gaussian White Noise External Excitation Gaussian White Noise External and Parametric Excitation Gaussian Colored Noise External and Parametric Excitation Nonstationary Gaussian External Excitation Spectral Method Non-Gaussian External Excitation Bibliography Notes References Moment Equations for Nonlinear Stochastic Dynamic Systems (NSDS) Moment Equations for Polynomial SDS Under Parametric and External Gaussian Excitation Simple Closure Techniques Non-Gaussian Closure Techniques Bibliography Notes References Statistical Linearization of Stochastic Dynamic Systems Under External Excitations Moment Criteria Criteria in Probability Density Functions Space Stationary Gaussian Excitations Nonstationary Gaussian Excitations Non-Gaussian Excitations Bibliography Notes References Equivalent Linearization of Stochastic Dynamic Systems Under External Excitation Introduction Moment Criteria Criteria in Probability Density Space Criteria in Spectral Density Space Multi-criterial Linearization Methods Special Linearization Methods Bibliography Notes References Nonlinearization Methods Introduction Moment Criteria Probability Density Criteria Application of the Generalized Stationary Potential Approach Application of Stochastic Averaging Approach Application of Volterra Functional Series Approach Bibliography Notes References Linearization of Dynamic Systems with Stochastic Parametric Excitations Introduction Statistical Linearization Equivalent Linearization Bibliography Notes References Applications of Linearization Methods in Vibration Analysis of Stochastic Mechanical Structures Introduction Applications in Hysteretic Systems Vibrations of Structures under Earthquake Excitations Vibrations of Structures Under Wave Excitations Vibrations of Structures Under Wind Excitations Applications in Control Problems Bibliography Notes References Accuracy of Linearization Methods Theoretical Study of the Accuracy of Linearization Methods Comparison of Linearized and Exact Response Characteristics Comparison of Linearized and Simulated Response Characteristics Validation of Linearization Method by Experiments Limitations of Applicability of Linearization Methods Bibliography Notes References Index For most cases of interest, exact solutions to nonlinear equations describing stochastic dynamical systems are not available. The aim of this book is to give a systematic introduction to and overview of the relatively simple and popular linearization methods available. The scope is limited to models with continuous external and parametric excitations, yet these cover the majority of known approaches. The book contains an application chapter with emphasis on vibration analysis of stochastic mechanical structures as well as a chapter devoted to the assessment of the accuracy of the theoretical methods presented, both with respect to numerical and to experimental studies. The material derives partly from graduate course notes developed by the author for courses and seminars over the past 20 years.