Stochastic Responses of Lightly Nonlinear Vibroimpact System with Inelastic Impact Subjected to External Poisson White Noise Excitation

Guidong Yang,Wei Xu,Dongmei Huang,Mengli Hao

doi:10.1155/2018/3627195

Abstract

A procedure for analyzing stationary responses of lightly nonlinear vibroimpact system with inelastic impact subjected to external Poisson white noise excitation is proposed. First, the original vibroimpact system is transformed to a new system without velocity jump in terms of the Zhuravlev nonsmooth coordinate transformation and the Dirac delta function. Second, the averaged generalized Fokker-Planck-Kolmogorov (FPK) equation for transformed system under parametric excitation of Poisson white noise is derived by stochastic averaging method. Third, the averaged generalized FPK equation is solved by using the perturbation technique and inverse transformation of the Zhuravlev nonsmooth coordinate transformation to obtain the approximately stationary solutions for response probability density functions of original vibroimpact system. Last, analytical and numerical results for two typical lightly nonlinear vibroimpact systems are presented to assess the effectiveness of the proposed method. It is found that they are in good agreement and the proposed method is quite effective.

Highlights

Vibroimpact system, as a class of typical nonsmooth system [1], has attracted much attention in the past few decades due to its existence in various engineering applications [2, 3]
The classical impact model consists of two parts: the ordinary motion described by a differential equation and the impact condition
The paper is devoted to presenting a solution procedure for predicting stationary responses of single-degree-of-freedom lightly nonlinear vibroimpact systems with inelastic impact subjected to external Poisson white noise excitations

Summary

Introduction

Vibroimpact system, as a class of typical nonsmooth system [1], has attracted much attention in the past few decades due to its existence in various engineering applications [2, 3]. For the classical impact model, Dimentberg et al [28, 29] studied stochastic response of linear vibroimpact system under Gaussian white noise. The Poisson white noise is the most used model for non-Gaussian random excitations [37,38,39,40]. This paper is devoted to presenting a procedure to predict the stochastic responses of lightly nonlinear vibroimpact system subject to external Poisson white noise excitation. The Zhuravlev nonsmooth coordinate transformation [22], the stochastic averaging method [42], and the perturbation technique [43] are applied in succession to obtain the stationary solutions of response PDFs. Two examples are presented to assess the effectiveness of the procedure.

Problem Statement

Stochastic Responses

Illustrative Examples

Conclusions