Random vibration of SDOF vibro-impact oscillators with restitution factor related to velocity under wide-band noise excitations

Hertz contact model has not included the impact of inelastic, while the modified Hertz contact models are too complicated to handle. The classical impact model only uses a material-related constant to describe the relationship between the velocity before and after a collision, without considering the energy dissipation mechanism in the impact process. On the other hand, many recent experiments have shown that the restitution factor was a function of impact velocity. Such impact velocity dependent restitution factor offers a comprehensive and reasonable impact model. In the present paper, the random vibration of single-degree-of-freedom (SDOF) vibro-impact oscillators with the restitution factor related to velocity under wide-band noise excitations is investigated. The given stochastic vibro-impact oscillator is firstly transformed into a nonlinear system without impact by a non-smooth transformation. Then, a Markov approximation is applied to obtain the Fokker-Plank-Kolmogorov (FPK) equation governing the evolution of the probability density function (PDF) of the energy envelope. The stationary PDFs of the energy envelope and amplitude envelope are solvable in a closed-form. Finally, two examples are given to verify the effectiveness of the proposed method. The results showed that the classical impact model can simplify the calculations to some extent. However, the error caused by this method cannot be ignored. The revised model presented in this paper is more in line with the actual working conditions. For complex dynamical system, especially for multi-point impact vibration systems, the work of this paper is helpful to the optimization of the system. (C) 2020 Elsevier Ltd. All rights reserved.