An improved non-smooth coordinate transformation for analyzing bilateral vibro-impact systems with stochastic excitations

Vibro-impact systems exhibit non-smooth characteristics and pose significant challenges for analysis. Non-smooth coordinate transformations are widely recognized for their ability to convert vibro-impact systems into systems with continuous trajectories, thereby enabling the application of some classical methods. This paper introduces an improved non-smooth coordinate transformation method [Su et al., Chaos 32, 043118 (2022)], developed from the Zhuravlev and Ivanov transformations, and extends it to the analysis of bilateral vibro-impact systems with stochastic excitations. We provide a detailed derivation of the transformation, which allows the conversion of the original non-smooth system into a form with continuous and periodic trajectories. According to two typical examples, the effectiveness of the proposed method is validated by solving the corresponding Fokker-Planck equation and comparing the stationary probability density functions obtained from this approach with results from Monte-Carlo simulations. The good agreement demonstrates that the improved transformation method, which can be directly applied to vibro-impact systems with asymmetric bilateral barriers accompanied with distinct restitution coefficients or a unilateral barrier, offers an effective tool for studying stochastic responses and bifurcations of such complex systems.