Numerical algorithm for non-stationary covariance analysis of nonlinear mechanical system using equivalent stochastic linearization

When deterministic mechanical systems are subjected to dynamic actions whose nature is stochastic, response must be evaluated by a stochastic approach. Unfortunately, only in few nonlinear mechanical cases, exact solutions are available and therefore approximate solutions should be adopted. A typical solution, extremely easy and simple, is based on stochastic equivalent linearization. Moreover, it needs specific numerical approaches and algorithms to be properly implemented. The complexity increases because of non-stationary conditions and there is no exhaustive literature about. In this paper, a numerical procedure to solve covariance analysis of stochastic linearized systems in presence of non-stationary excitation is proposed. The non-stationary Lyapunov differential matrix covariance equation for the linearized system is solved by using a numerical algorithm that updates linearized system matrix coefficients with a step by step procedure. In particular, a predictor-corrector procedure is applied to an Euler-implicit integration scheme for the matrix covariance analysis.