A novel method for solving response of stochastic vibro-impact systems with two stoppers

The research of stochastic steady-state probability density response of vibration systems including bilateral impacts and gaps is of great theoretical and practical significance. Usually, the non-smooth transformation is introduced in the response calculation, while eliminates the non-smooth characteristics of the systems. Therefore, this paper propose a new method to preserve the velocity jump under the bilateral impact without non-smooth transformation. Based on the generalized cell mapping theory, a complete process is regarded as the one-step transition probability that the stochastic trajectories start from one stopper, and return to the original stopper after impacting with another stopper. Then, the intervals of response are established based on the initial stopper to ensure the continuity of the state space. We analyzed the probability density response of a bilateral Rayleigh vibro-impact oscillator and a piezoelectric energy harvesting device with two symmetric stoppers by utilizing the proposed method, respectively. Comparing with Monte Carlo simulations, it is fully demonstrated the effectiveness of this method. At the same time, the cases using our proposed method are found to have wide applicability under different position of stoppers, restitution coefficient, noise excitation and system parameters.