MODELLING OF STOCHASTIC DYNAMIC SYSTEMS USING HAMILTON'S LAW OF VARYING ACTION

Abstract

This paper presents a model based on Hamilton's law of varying action for stochastic dynamic systems. In this model, the state variables are approximated as a linear sum of orthogonal polynomials. For deterministic systems, the coefficients of the polynomials are constant, but for stochastic systems, the coefficients are random variables. The initial conditions are treated as Gaussian random variables with specified mean values and covariance matrix, and the external forcing functions (input processes) are treated as stationary Gaussian processes with specified mean and correlation functions. Using the Karhunen-Loeve expansion, the random input processes are represented in terms of truncated linear sums of orthonormal eigenfunctions with uncorrelated random variables as coefficients. This method reduces the infinite dimensional input force vector to one with finite dimensions. A method to compute the mean and the variance functions for both the singular and the regular components of the state processes is developed. A single-degree-of-freedom spring-mass-damper systems subjected to random initial conditions and a random forcing function is considered to show the feasibility and effectiveness of the formulation. The response is obtained without any reference to the differential equations associated with the dynamics of the system. The study shows that the results of this formulation agree well with those obtained using other schemes.