Stochastic stationary responses of a viscoelastic system with impacts under additive Gaussian white noise excitation

•The viscoelastic force is replaced by a combination of stiffness and damping terms.•The stationary responses of the system are obtained analytically.•The occurrence of stochastic P-bifurcation is investigated.•The theoretical analyses are verified by numerical results.

This paper is devoted to investigate the stochastic stationary responses of a viscoelastic system with impacts under additive Gaussian white noise excitation. First, the viscoelastic force is approximated as equivalent stiffness and damping terms, and the original system is replaced by a system without viscoelasticity. Then, the stationary responses of the system are obtained analytically by using non-smooth transformation and stochastic averaging method. The theoretical analyses are verified by numerical results from Monte Carlo simulations. Effects of the noise intensity and the nonlinear damping coefficient on the system responses are investigated. At last, it is worthy noting that the restitution coefficient, the linear damping coefficient and the viscoelastic parameters can induce the occurrence of stochastic P-bifurcation.