Stationary response analysis for a linear system under non-Gaussian random excitation by the equivalent non-Gaussian excitation method and the Hermite moment model

Abstract

An approximate analytical method is developed to obtain the stationary response probability density and mean upcrossing rate for a linear system subjected to non-Gaussian random excitation. The random excitation is a stationary stochastic process prescribed by an arbitrary first-order probability density and a class of power spectral density with a bandwidth parameter, and is described by an Itô stochastic differential equation. In the proposed method, the equivalent non-Gaussian excitation method is first utilized to derive a closed set of moment equations for the system response. The exact solutions of response moments up to the fourth order are calculated from the moment equations. Next, using the response moments, the Hermite moment model is built for the stationary displacement response, and its probability density function is obtained through the model. Finally, the mean upcrossing rate of the displacement is determined from the response probability distribution by making two approximations that the displacement and velocity of the system are independent and the velocity is Gaussian. The proposed method can be used for any excitation probability density and bandwidth parameter value as long as the skewness and kurtosis of the response are within the applicable range of the Hermite moment model. In illustrative examples, the proposed method is applied to linear systems under random excitations with two-types of non-Gaussian probability densities. The effectiveness of the method is demonstrated by comparing the analytical results with the pertinent Monte Carlo simulation results while widely varying the bandwidth ratio between the excitation spectral density and the frequency response function of the system. Moreover, the effect of excitation non-Gaussianity on the mean upcrossing rate is discussed. It is shown that for small bandwidth ratio, the crossing rate differs considerably depending on the excitation probability density. The difference due to the excitation distribution decreases monotonically as the bandwidth ratio increases, and the crossing rate becomes almost the same as that in the case of Gaussian excitation when the bandwidth ratio is large.