EXACT STATIONARY RESPONSE SOLUTIONS OF 6 CLASSES OF NONLINEAR STOCHASTIC-SYSTEMS UNDER STOCHASTIC PARAMETRIC AND EXTERNAL EXCITATIONS

A systematic procedure is developed to obtain the stationary probabili ty density function for the response of a general nonlinear system und er parametric and external excitations of Gaussian white noises. The n onlinear system described here has the following form: (x) DOUBLE OVER DOT + g(0)(x) + g(1)(x)(x) OVER DOT + g(2)(x)(x) OVER DOT (2) + g(3)( x)(x) OVER DOT (3) = k(1) xi(1)(t) + k(2)x xi(2)(t) + k(3)(x) OVER DOT xi(3)(t), where xi(i)(t) = 1, 2, 3 are Gaussian white noise. The redu ced Fokker-Planck equation is employed to get the governing equation o f the probability density function. Based on this procedure, the prima ry focus of this paper is to find an undetermined function h(x, (x) OV ER DOT), which can satisfy the general FPK equation, so that the solut ion of the FPK equation can be found for each class of the nonlinear s tochastic systems. By doing the parameter studies, we get the exact st ationary response solutions of six classes of nonlinear stochastic sys tems. This paper will illustrate that previous certain classes of exac t steady-state solutions available up until now have been special case s of exact stationary response solutions under specific conditions of stochastic parametric and external excitations.