Stationary Response Probability Distribution of SDOF Nonlinear Stochastic Systems

Abstract

There has been no significant progress in developing new techniques for obtaining exact stationary probability density functions (PDFs) of nonlinear stochastic systems since the development of the method of generalized probability potential in 1990s. In this paper, a general technique is proposed for constructing approximate stationary PDF solutions of single degree of freedom (SDOF) nonlinear systems under external and parametric Gaussian white noise excitations. This technique consists of two novel components. The first one is the introduction of new trial solutions for the reduced Fokker–Planck–Kolmogorov (FPK) equation. The second one is the iterative method of weighted residuals to determine the unknown parameters in the trial solution. Numerical results of four challenging examples show that the proposed technique will converge to the exact solutions if they exist, or a highly accurate solution with a relatively low computational effort. Furthermore, the proposed technique can be extended to multi degree of freedom (MDOF) systems.