Probability density analysis of the stochastically excited bilateral vibro-impact system

This paper investigates the stochastic response of rigid bilateral vibro-impact systems. The abrupt change in velocity during impact events causes the state trajectory to exhibit discontinuous jumps in the phase space, which renders numerous conventional methods for obtaining the stochastic response inadequate. Additionally, practical systems are often subjected to harmonic excitation, which renders the problem inherently time-dependent and further complicates the analysis. To address this key issue, this paper introduces a non-smooth transformation technique based on coordinate mapping, which transfers the original discontinuous states into a continuous state space. A notable advantage of this approach is that it imposes no specific constraints on the positions of bilateral barriers or the coefficients of restitution. The transformed system is analyzed using the path integration method. Subsequently, the results are converted back into the probability density functions governed by the original state variables. Especially under periodic forcing, the transition probability density functions for multiple phases are computed within one period of external forcing, thereby revealing more detailed variations in the probability flow. The findings elucidate the influence mechanisms of various parameters, including the position of impact barriers, the restitution coefficient, and the intensity of periodic excitation, on the system motion. The validity of the proposed method is demonstrated through comparison with Monte Carlo simulations.