Exact stationary solutions of the Fokker-Planck equation for nonlinear oscillators under stochastic parametric and external excitations

A systematic procedure is developed to obtain the stationary probability de nsity function for the response of general single-degree-of-freedom nonline ar oscillators under parametric and external Gaussian white noise excitatio ns. In a previous paper (Wang and Zhang 1998 J. Eng. Mech. ASCE 18 18-23) w e expressed a nonlinear function of oscillators using a polynomial formula. The nonlinear system described here has the following form: x + g(x, (x) o ver dot ) = k(1) xi(1) (t) + k(2)x xi(2) (t). where g(x, (x) over dot ) = S igma(i=0)(infinity) g(i) (x) (x) overdot(i) and xi 1, xi 2 are Gaussian whi te noise functions. Thus, this paper is a generalization of the results stu died in our previous paper. The stationary Fokker-Planck equation is employ ed to obtain the governing equation of the probability density function. Ba sed on this procedure, the exact stationary probability densities of many n onlinear stochastic oscillators are obtained and it is shown that some of t he exact stationary solutions described in the literature are only particul ar cases of the presented generalized results.