Stochastic responses analysis of vibro-impact system via direct probability integral method

Abstract

Vibro-impact system, as an important type of non-smooth system in mechanical engineering, has received much research interest in the past few decades. The behaviors of vibro-impact system are complex and unusual, because it behaves as continuous dynamical system between two successive impacts and develops a discrete behavior when reaching a constraint. Note that vibro-impact system is inevitably subjected to random perturbations, and thus it is essential to develop powerful approach for stochastic dynamic analysis of vibro-impact system. This paper proposes a unified and efficient direct probability integral method (DPIM) to calculate the probability density functions of the unilateral vibro-impact system under random excitations, and explores its complicated dynamical behavior. The DPIM is a versatile approach with high accuracy and efficiency to facilitate stochastic response analysis for general systems, including linear and nonlinear systems, time independent and time dependent systems, and the systems with stochastic parameters under various random excitations. Firstly, based on the principle of probability conservation and Dirac delta function, the probability density integral equation (PDIE) for the unilateral vibro-impact system is derived. Secondly, the PDIE is solved by using the point selection technique based on generalized F discrepancy and the smoothing of Dirac delta function. Compared with Monte Carlo simulation, some numerical examples illustrate the merits of the proposed approach for the stochastic response of unilateral vibro-impact system under Gaussian white noise excitation. The influence of some parameters of the vibro-impact system on the stochastic dynamic responses is also scrutinized.