A direct probabilistic approach to solve state equations for nonlinear systems under random excitation

In this paper, a direct probabilistic approach（DPA） is presented to formulate and solve moment equations for nonlinear systems excited by environmental loads that can be either a stationary or nonstationary random process.The proposed method has the advantage of obtaining the response＇s moments directly from the initial conditions and statistical characteristics of the corresponding external excitations. First, the response＇s moment equations are directly derived based on a DPA, which is completely independent of the It？/filtering approach since no specific assumptions regarding the correlation structure of excitation are made.By solving them under Gaussian closure, the response＇s moments can be obtained. Subsequently, a multiscale algorithm for the numerical solution of moment equations is exploited to improve computational efficiency and avoid much wall-clock time. Finally, a comparison of the results with Monte Carlo（MC） simulation gives good agreement.Furthermore, the advantage of the multiscale algorithm in terms of efficiency is also demonstrated by an engineering example.