Stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations

• The viscoelastic force is approximately equivalent to stiffness and damping forces.
• The stationary responses of the system are obtained analytical.
• The validity of the theoretical results is tested by numerical results.
• The occurrence of stochastic P-bifurcation is discussed.

This paper deals with the stochastic responses of a viscoelastic-impact system under additive and multiplicative random excitations. The viscoelastic force is replaced by a combination of stiffness and damping terms. The non-smooth transformation of the state variables is utilized to transform the original system to a new system without the impact term. The stochastic averaging method is applied to yield the stationary probability density functions. The validity of the analytical method is verified by comparing the analytical results with the numerical results. It is invaluable to note that the restitution coefficient, the viscoelastic parameters and the damping coefficients can induce the occurrence of stochastic P-bifurcation. Furthermore, the joint stationary probability density functions with three peaks are explored.