Analytical Solutions of a Class of Nonlinear Single Degree-of-Freedom Systems to Nonstationary Random Excitations

Abstract

An analytical method for responses of a class of nonlinear single degree-of-freedom (DOF) systems under nonstationary random excitations (NSREs) is presented. It is based on the transformation of the nonlinear stochastic differential equations that govern the vibrations of nonlinear systems under NSREs into a polynomial of which the exact roots are available, and therefore solutions without approximation of the nonlinear system can be evaluated. For demonstration purposes, the van der Pol-Duffing oscillator under a time-modulated zero mean Gaussian white noise process is considered. Representative solutions are verified by the Monte Carlo simulation (MCS) technique. The main conclusions are as follows. First, a simple and relatively straightforward method has been developed to transform the nonlinear stochastic differential equations of a class of nonlinear single DOF systems into polynomials for which exact solutions are available. The analytical solution developed in this paper is for the deterministic equation of motion for a specific realization of the input excitation function