Application of Wavelets in Modeling Stochastic Dynamic Systems

This paper presents a wavelet based model for stochastic dynamic systems. In this model, the state variables and their variations are approximated using truncated linear sums of orthogonal polynomials, and a modified Hamilton’s law of varying action is used to reduce the integral equations representing dynamics of the system to a set of algebraic equations. For deterministic systems, the coefficients of the polynomials are constant, but for stochastic systems, the coefficients are random variables. The external forcing functions are treated as stationary Gaussian processes with specified mean and correlation functions. Using Karhunen-Loeve (K-L) expansion, the random input processes are represented in terms of linear sums of finite number of orthonormal eigenfunctions with uncorrelated random coefficients. A wavelet based technique is used to solve the integral eigenvalue problem. Application of wavelets and K-L expansion reduces the infinite dimensional input force vector to one with finite dimensions. Orthogonal properties of the polynomials and the wavelets are utilized to make the algebraic equations sparse and computationally efficient. A method to compute the mean and the variance functions for the state processes is developed. A single degree of freedom spring-mass-damper system subjected to a random forcing function is considered to show the feasibility and effectiveness of the formulation. Studies show that the results of this formulation agree well with those obtained using other schemes.