Response Of SDOF Nonlinear Stochastic Systems

In this paper, a new procedure is proposed to obtain the exact stationary solutions of some classes of the single-degree-of-freedom(SDOF) strongly nonlinear stochastic systems subjected to parametric and/or external Gaussian white noises. The main idea of the procedure is to find the exact stationary solution of the equivalent nonlinear stochastic system. The conditions for the existence of the exact stationary solution are described, and the forms of the exact stationary solutions are obtained. It is shown that the known exact stationary solutions which are dependent of energy are only the special cases of solutions presented in this paper, while some other classes of solutions all possess the property that the solutions are independent of energy. Thus, the exact stationary solutions obtained in this paper are those belonging to the most general class of the SDOF strongly nonlinear stochastic systems so far.The exact stationary solutions for Fokker-Planck-Kolmogorov(FPK) equation are usually obtained under strict conditions. The FPK equation is usually solved numerically. Several methods such as path integral method, finite difference method et.al, have been employed to numerically solve the FPK equation. The finite difference method is applied to solve the issue of spatial discretization in order to transform the FPK equation to a system of ordinary differential equations. Then the nonstationary and stationary solutions are obtained by solving the set of ordinary differential equations. Several examples are given to illustrate the application of proposed procedures. It is shown that this method is a very good procedure for numerical solution of FPK equation and the method can be futher applied to higher dimensional FPK equation.