Multi-valued responses of a nonlinear vibro-impact system excited by random narrow-band noise

We investigate the multi-valued responses of a non-linear vibro-impact oscillator with a one-sided barrier subject to random narrow-band excitation. The frequency response of the system is obtained using the Krylov-Bogoliubov averaging method. Meanwhile, the backbone curve and the critical equation of unstable region are also derived for the deterministic case. Then, the method of moment is applied to obtain the iterative calculation equation for the mean-square response amplitude under the stochastic case. Excitation frequency, nonlinearity intensity, damping parameters, especially the distance between the system's static equilibrium position and the barrier can lead to triple-valued response under certain case. In some conditions the impact system may have two or four steady-state solutions, which is an interesting phenomenon for impact system. The unstable region is one uniform part while under smaller nonlinearity intensity it is divided into two parts. Moreover, we also find that as random noise intensity increases, the pervasion of the phase trajectories strengthens, and then destroys the topological property of the phase trajectories.