Probabilistic response and analysis for a vibro-impact system driven by real noise

In this paper, a new stochastic averaging method is proposed to derive the stationary probabilistic response of vibro-impact system subject to real noise excitation, and the effects of real noise and the restitution coefficient on the response are discussed in detail. According to the size relationship between the system energy and the potential energy of the impact position , the motions of the unperturbed vibro-impact system are divided into two types: the free motion with no energy loss and the impact motion with energy loss. Based on those relationships, a new method that combines a transformation and the Fourier expansion scheme by introducing, the mean drift and diffusion coefficients of the averaged It stochastic differential equation for the two types of motions are obtained, which are depended on the bandwidth and the noise density of real noise. The probability density function of stationary response of vibro-impact system is derived through solving the associated Fokker-Planck-Kolmogorov equation. Finally, the effects of the restitution coefficient and the noise intensity of real noise in different bandwidths are detailedly analyzed through an example. The main results on probabilistic response analysis of vibro-impact Duffing system are obtained to illustrate the proposed stochastic averaging method, and Monte Carlo simulation method is also conducted to show the effectiveness of the proposed method.