Stochastic response of a vibro-impact Duffing system under external Poisson impulses

Abstract

This paper studies the stationary probability density function (PDF) of a vibro-impact Duffing system under external Poisson impulses. A one-sided constraint is located at the equilibrium position of the system, and the system collides with the constraint by instantaneous repetitive impacts. A recently proposed solution procedure is extended to the case of Poisson impulses including three steps. First, the Zhuravlev non-smooth coordinate transformation is utilized to make the original Duffing system and impact condition be integrated into one equation. An additional impulsive damping term is introduced in the new equation. Second, the PDF of the new system is obtained with the exponential–polynomial closure method by solving the generalized Fokker–Planck–Kolmogorov equation. Last, the PDF of the original system is established following the methodology on seeking the PDF of a function of random variables. In numerical analysis, different levels of nonlinearity degree and excitation intensity are considered in four illustrative examples to show the effectiveness of the proposed solution procedure. The numerical results show that when the polynomial order is taken as six in the proposed solution procedure, it can present a satisfactory PDF solution compared with the simulated result. The tail region of the PDF solution is also approximated well for both displacement and velocity.