Abstract
An approximate analytical method based on complex fractional moments (CFMs) is proposed to obtain the transient response probability density function of a dynamic system with nonlinearity in both stiffness and damping excited by Gaussian white noise. The CFM is a new kind of statistical moment developed in recent years and is related to the Mellin transform of a probability density function. In the proposed method, first, the system response is expressed in the form of quasi-harmonic oscillation with amplitude and phase variables, and then, the equivalent natural frequency of the system, which is given as a function of the amplitude, is determined by the equivalent linearization method. Next, using the equivalent natural frequency and the stochastic averaging procedure, the Fokker-Planck equation governing the probability density of the response amplitude is derived. The Mellin transform of the Fokker-Planck equation yields the governing equations of the amplitude CFMs, which are given by simultaneous linear ordinary differential equations. Finally, the inverse Mellin transform of the CFMs obtained from the differential equations leads to the transient response probability density function. In numerical examples, two types of nonlinear stochastic oscillators are considered. The analytical results of the transient response probability densities obtained by the proposed method are in good agreement with the corresponding Monte Carlo simulation results, including their tail region.
