Stochastic Dynamics of Impact Oscillators

Abstract

The purpose of this work is to develop an averaging approach to study the dynamics of a vibro-impact system excited by random perturbations. As a prototype, we consider a noisy single-degree-of-freedom equation with both positive and negative stiffness and achieve a model reduction, i.e., the development of rigorous methods to replace, in some asymptotic regime, a complicated system by a simpler one. To this end, we study the equations as a random perturbation of a two-dimensional weakly dissipative Hamiltonian system with either center type or saddle type fixed points. We achieve the model-reduction through stochastic averaging. Examination of the reduced Markov process on a graph yields mean exit times, probability density functions, and stochastic bifurcations. Transactions of the ASME Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org/ on 06/12/2015 Terms of Use: http://asme.org/termswhere the vector b and the matrix are given by Journal of Applied Mechanics NOVEMBER 2005, Vol. 72 / 863 Downloaded From: https://appliedmechanics.asmedigitalcollection.asme.org/ on 06/12/2015 Terms of Use: http://asme.org/terms

Random vibrations of Rayleigh vibroimpact oscillator under Parametric Poisson white noise

“…Rong et al [25] studied subharmonic response of a nonlinear vibro-impact system to a randomly disordered periodic excitation. Namachchivaya and Park [26] developed an averaging approach to explore dynamic behaviors of a vibro-impact system excited by random perturbations. However, most researches mentioned above are about Gaussian white noise.…”

Stochastic response of a vibro-impact Duffing system under external Poisson impulses

“…In these investigations, the stochastic averaging method was extensively adopted using different impact models. The model with instantaneous repetitive impacts was used in quite a few researches [13][14][15][16][17][18]. For this impact model, a restitution factor is used between impact and rebound velocities to describe the energy loss according to the Newton's law.…”

“…Several nonsmooth coordinate transformation techniques were also developed and studied for vibro-impact vibration [8][9][10][11][12]. In particular, the stochastic response of vibroimpact systems has received more and more attention in the past decades [13][14][15][16][17][18][19][20][21][22][23][24]. In these investigations, the stochastic averaging method was extensively adopted using different impact models.…”

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

“…Therefore, studies on the prediction of stochastic responses of vibroimpact systems are naturally important but difficult due to the presence of random excitations and nonsmooth factors. In recent years, some methods [22][23][24][25][26][27] have been developed to explore the response probability density function (PDF) which is an important characteristic of a stochastic system. For the classical impact model, Dimentberg et al [28,29] studied stochastic response of linear vibroimpact system under Gaussian white noise.…”