A technique note of exact stationary solutions on nonlinear systems under stochastic parametric excitations

Abstract

In this paper, a systematic procedure is developed to obtain the stationary probability density function for the response of a general nonlinear system under parametric and external Gaussian white noise excitations. In Ref. [11], nonlinear function of system was expressed to the polynomial formula. The nonlinear system described here has the following form: x¨ + g(x, x˙) = k1ξ1(t)+ k2xξ2(t), where g(x,x˙) = ∑ i = 0∞g i (x)x˙ i and ξ1,ξ2 are Gaussian white noises. Thus, this paper is a generalization for the results studied in Ref. [11]. The reduced Fokker–Planck (FP) equation is employed to get the governing equation of the probability density function. Based on this procedure, the exact stationary probability densities of many nonlinear stochastic systems are obtained, and it is shown that some of the exact stationary solutions described in the literature are only particular cases of the presented generalized results.