The closed-form stationary probability distribution of the stochastically excited vibro-impact oscillators

The response of vibro-impact oscillators under random excitations has been studied using various techniques in the last three decades. However, the results available were smooth approximations of the intrinsically non-smooth response of vibroimpact oscillators. This paper proposes a new procedure for the closed-form stationary probability density function (PDF) of the response of single-degree-of-freedom (SDOF) vibro-impact system under Gaussian white noise excitation. First, the Zhuravlev-Ivanov transformation is adopted to convert a vibro-impact oscillator with one-side barrier into an oscillator without barrier. The probabilistic description of the system is subsequently defined through the corresponding Fokker-Planck-Kolmogorov (FPK) equation. The closed-form stationary PDF of the response is obtained by solving the reduced FPK equation using the iterative method of weighted residue together with the concepts of the circulatory probability flow and the potential probability flow. Two examples are examined to illustrate the effectiveness of the proposed procedure. Good agreements are found between the analytical solutions and the simulated results of the PDFs and logarithmic PDFs of the examples. The reported studies also show that the proposed procedure can deal with case of small restitution factor. In particular, the closed-form solution for the case of small restitution factor is firstly obtained in literature, and can be used as the benchmark problem to examine the other method in random vibration. Furthermore, the approximate solutions obtained with the proposed procedure are piece-wise form, reflecting the true nature of the intrinsically discontinuous vibro-impact oscillators.