Non-stationary semi-analytical solution of vibro-impact system with multiplicative and external random stimulations

This article aims to investigate the non-stationary semi-analytical solution of the vibro-impact (VI) system with multiplicative and external random stimulations. Firstly, the original VI system is replaced by an equivalent nonlinear system without collisional barriers in the new phase space using the Zhuravlev transformation. Afterward, the Fokker–Planck (FP) equation for the equivalent nonlinear system is formulated and solved using the evolutionary EPC method. The transitional probability density functions (TPDFs) of responses of the equivalent nonlinear system are then achieved. Subsequently, by using the variable relationships between the original VI system and the transformed nonlinear system, the TPDFs of responses of the original VI system are achieved at various time instants. Finally, three VI systems with various nonlinearity driven by both multiplicative and external random stimulations are investigated and studied by the presented procedure. The results show that the responses of the VI systems are strongly non-Gaussian due to the influence of the barrier, the nonlinear terms and the correlated random stimulations. Moreover, the accuracy of the obtained results and computational efficiency of the presented procedure are assessed by comparison with the simulated results.