Stochastic Response of a Vibro-Impact System via a New Impact-to-Impact Mapping

In this paper, a new impact-to-impact mapping is constructed to investigate the stochastic response of a nonautonomous vibro-impact system. The significant feature lies in the choice of Poincaré section, which consists of impact surface and codimensional time. Firstly, we construct a new impact-to-impact mapping to calculate the one-step transition probability matrix from a given impact to the next. Then, according to the matrix, we can investigate the stochastic responses of a nonautonomous vibro-impact system at the impact instants. The new impact-to-impact mapping is smooth and it effectively overcomes the nondifferentiability caused by the impact. A linear and a nonlinear nonautonomous vibro-impact systems are analyzed to verify the effectiveness of the strategy. The stochastic P-bifurcations induced by the noise intensity and system parameters are studied at the impact instants. Compared with Monte Carlo simulations, the new impact-to-impact strategy is accurate for nonautonomous vibro-impact systems with arbitrary restitution coefficients.