Response of a vibro-impact Duffing system with a randomly varying damping term

Abstract

This paper proposes a solution procedure for the probability density function (PDF) solution of a vibro-impact Duffing system with a randomly varying damping term. The study considers the one-sided barrier located at the equilibrium of the system. The classical model with instantaneous impacts is used to model the colliding between the system and the barrier. First, the Zhuravlev non-smooth coordinate transformation is employed to convert the original vibro-impact system into a new system without any barrier by introducing an additional damping term. Second, the PDF of the new system is governed by the Fokker–Planck equation which is solved by the exponential–polynomial closure method. Last, the PDF of the original system is formulated in terms of the methodology on seeking the PDF of a function of random variables. Six illustrative examples are examined to show the effectiveness of the proposed solution procedure. The effects of the parameters, namely the non-linearity in displacement, the parametric excitation intensity, the negative linear stiffness and the restitution factor, are further investigated on the PDF distribution of the vibro-impact systems. Comparison with the simulated result shows that the proposed solution procedure can provide a satisfactory PDF solution for the examined examples. The tail region of the PDF is also approximated well.