A quite important topic of structural dynamics is deterministic mechanical systems subjected to stochastic dynamic actions, such as for wind or earthquake. EN such cases structural response have to be properly evaluated by a stochastic approach. Unfortunately for nonlinear mechanical systems only in a very few cases exact solutions are available, and usually simply approximate solutions should be used. A well known one is stochastic equivalent linearization, easy and simple from the conceptual point of view. Moreover it needs of specific numerical techniques to be properly implemented, whose complexity increases in case of non stationary conditions. In this paper a procedure to solve covariance analysis of stochastic linearized systems in case of non stationary excitation is proposed. The no stationary Lyapunov differential matrix covariance equation for the linearized system is solved by using a numerical algorithm that updates linearized system matrix coefficients step by step. This by a predictor-corrector procedure applied to a Euler-implicit integration scheme for the matrix covariance analysis. It is described in details to be simpler implemented by other researchers, and then applied to a typical and important applicative case, that is seismic response of a Bouc Wen Single Degree of Freedom (SDoF) system. Seismic input processes is modeled as linear filtered white noise non stationary separable process. Accuracy and computational costs are analyzed showing the efficiency of the proposed integrating procedure.
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