Probabilistic solution of vibro-impact stochastic Duffing systems with a unilateral non-zero offset barrier

Abstract

This paper proposes a solution procedure for obtaining an approximate probability density function (PDF) of the response of vibro-impact Duffing systems with a unilateral non-zero offset barrier. The impact model is used with classical impacts with instantaneous velocity jumps and the excitation is modelled by a zero mean stationary Gaussian white noise. First the original vibro-impact system is converted into a system without barriers by adopting the Zhuravlev non-smooth coordinate transformation. Second the PDF of the converted system is obtained by solving the Fokker–Planck (FP) equation with the exponential–polynomial closure (EPC) method. Last the PDF of the original system is achieved using the well-established methodology on seeking the PDF distribution of a function of a random variable. A further parametric study is also conducted to evaluate the effectiveness of the proposed solution procedure. Comparison with the simulated result shows that when lightly inelastic impacts occur (i.e, the restitution coefficient is very close to 1.0), the technique of the Zhuravlev non-smooth coordinate transformation is feasible. The transformed system can adequately represent the original system. Consequently the EPC method can provide a good approximate PDF solution. The tail region of the PDF is also approximated well. The analysis further shows that the non-zero offset significantly affects the shape of the PDF distribution of displacement.