Exact stationary response of SDOF nonlinear stochastic oscillators

In this paper, a systematic procedure is developed to obtain the stationary probability density function for the response of a general single-degree-of-freedom (SDOF) nonlinear oscillators under parametric and external Gaussian white-noise excitations. Wang and Zhang (1998) expressed the nonlinear function of oscillators by a polynomial formula. The nonlinear system described here has the following form: $\text{x}\text{̈}\text{+g(x,}\text{x}\text{̇}\text{)=k}{}_1\text{ξ}{}_1\text{(t)+k}{}_2\text{xξ}{}_2\text{(t)}$ , where $\text{g(x,}\text{x}\text{̇}\text{)=}\text{∑}\text{i=0}\text{∞}\text{g}{}_{\mathrm{i}}\text{(x)}\text{x}\text{̇}{}^{\mathrm{i}}$ and ξ₁,ξ₂ are Gaussian white noises. Thus, this paper is a generalization for the results obtained by Wang and Zhang (1998). 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 oscillators 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.