Stochastic vibration analysis of a nonlinear oscillator with symmetric viscoelastic impact protection under wide-band noise excitations

To reduce the damage to the structure of mechanical vibration devices caused by the vibration, the end stops impact protection structure is often introduced in the design of the devices, which will inevitably occur nonclassical impact phenomenon. In this manuscript, the stochastic responses of a class of nonlinear impact protection oscillators are subjected to the excitation of wide-band noise. To better describe the changes before and after the impact protection, a stochastic differential equation with an indicator function is used to describe those changes. Then the Fourier series expansion of the indicator function is used to derive the Fokker-Planck-Kolmogorov (FPK) equation for the total energy of the system based on the Markov approximation. Then we obtain the stationary probability density functions (PDFs) of the total energy and the state variable. Finally, two different wide-band noises driven nonlinear Duffing-typed oscillations with end-stop impact protection blue are given as examples to verify all the proposed methods. The results show that the introduction of viscoelastic impact protection will increase the probability of oscillator vibration near its stable point, which means that the maximum displacement of the system will be effectively controlled. The results of this paper have some valuable references to guide the optimization designs, systemic integration, and vibration reliability of the Vibro-impact dynamical systems, especially the symmetric viscoelastic impact system.