A non-Gaussian stochastic linearization method

Abstract

A new method for the stochastic analysis of non-linear structural systems under white noise excitation is presented. It is based on the well-known stochastic linearization technique, which substitutes the original non-linear system with an equivalent linear one, whose coefficients, evaluated by minimizing the mean square error, depend on the statistics of the response process. In contrast with the Gaussian stochastic linearization, which is based on the Gaussian assumption of the response process in order to approximate the coefficients of the equivalent linear system, the proposed method allows us to take into account the non-Gaussian character of the response. This goal is pursued by assuming a modified A-type Gram–Charlier series approximation of the probability density function. The proposed non-Gaussian stochastic linearization method allows improving the results derived from the Gaussian stochastic linearization conducted at a first stage, by simply solving, at a second stage, a set of linear equations, whose unknowns are the series coefficients. The linear system is obtained by taking advantage from the markovianity of the response process and by using the reduced Fokker–Planck–Kolmogorov equation associated with the stochastic dynamic problem. Four examples show the performance of the method.