Stochastic Vibro–Impact Dynamics

Introduction

The analysis of random excitation of vibro–impact systems is not a simple task. The difficulty arises due to the fact that impact loading introduces strong nonlinearity. The theory of nonlinear random vibration encompasses analytical techniques that can handle developed for weakly nonlinear systems. Thus it is imperative to recast the vibro–impact system into a form amenable for the traditional analytical techniques. A nice overview of vibro–impact dynamics under random excitation has been presented by Dimentberg and Iourtchenko [251]. The article addressed analytical approaches and some results pertaining to random excitation of systems with lumped parameters and “classical” impacts. Emphasis was given to special piecewise–linear transformation of state variables using Zhuravlev transformation. Exact analyses for stationary probability densities of the response to white–noise excitation were found in few cases, whereas the stochastic averaging method was applied in some other cases. The method of direct energy balance was also illustrated based on direct application of the stochastic differential calculus between impacts. The problem of random excitation of vibro-impact systems attracted the attention of several researchers in the former Soviet Union (see, e.g.,[56], [57], [58], [253], [254], [243], [244], [245], [247], [248], [252], [249], [250], [133], [448], [526], [527], [528], [529], [530], [61]). These studies reduced the modeling by using the stochastic averaging method and thus it was possible to estimate the statistical response characteristics of the vibro–impact motion. The response statistics revealed how the energy is transferred from the impacting mass to the secondary structure.