Response analysis of Rayleigh–Van der Pol vibroimpact system with inelastic impact under two parametric white-noise excitations

In the present paper, the random vibration problem of Rayleigh–Van der Pol oscillator is considered by means of a single-degree-of-freedom system with a unilateral rigid barrier at its equilibrium position. The random excitations are two parametric Gaussian white noises. The nonsmooth transformation method is applied to convert the vibroimpact system to an equivalent system without velocity jump. The modified stochastic averaging technique for energy envelope is used to deal with the transformed system, and the averaging Itô stochastic differential equation is obtained. The analytical solution of the response of the original vibroimpact system is derived by solving the corresponding Fokker–Planck equation and using the inverse transformation of nonsmooth transformation. The validity of the analytical results is verified by those from Monte Carlo simulations based on original system. Effects of different system parameters and parametric white noises on the response of the system are examined. In addition, the critical condition of stochastic bifurcations is also explored.