Stochastic response and stability of system with friction and a rigid barrier

In the present paper, considering the significant nonsmooth factors, friction and impact, the responses and asymptotic stability with probability one of a system are investigated for the cases of additive and multiplicative Gaussian white noise random excitations. First, the original system which contains impact and viscoelastic terms can be approximated as an equivalent system by using nonsmooth transformation and converting the viscoelastic force to the sum of stiffness and damping terms. Then, the stationary probability density functions are analytically obtained by utilizing the stochastic averaging method. And the equation of the Top Lyapunov Exponent is also obtained with the help of combining the stochastic averaging method and Khasminskii method, which is used to discuss the effects of various parameters on the system's stability. The validity of the analytical results derived from the proposed procedures is verified by comparing with directly numerical simulation. In particular, under certain conditions, there are examples about how to control the change of stability of the system by adjusting the variation of the restitution coefficient and the friction from the mathematical standpoint.