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.

Some new advance on the research of stochastic non-smooth systems

“…[53] Zhu used the exponentialpolynomial closure method to get the probability density function for a stochastic vibro-impact system, and examined the effectiveness of this method by some examples. [54,55] Kumar et al used the finite element method to solve the FPK equation, then got the responses of the stochastic vibro-impact systems. [56] By defining an impact-to-impact mapping, a novel numerical scheme was used to obtain the probability density function of the stochastic vibro-impact systems based on the cell mapping method in Ref.…”

The determination of the activation energy for a vibro-impact system under multiple excitations

“…For the deterministic aspect, those studies mainly focus on the modeling of the Vibro-impact system, analyzing the bifurcation and the chaotic behavior, and studying the local stability [2][3][4][5][6][7][8]. For the stochastic aspect, because the PDF is very useful for us to analyze the stochastic system's stability, the researchers are dedicated to solving the response PDFs of the Vibro-impact system under various random excitations [9][10][11][12][13][14][15][16][17][18]. For example, Fen et al have studied the Vibro-impact Duffing oscillator's stationary responses excited by additive Gaussian white noise by using the quasi-conservative averaging method [13].…”

“…For example, Fen et al have studied the Vibro-impact Duffing oscillator's stationary responses excited by additive Gaussian white noise by using the quasi-conservative averaging method [13]. Zhu obtained the stationary PDFs of the Vibro-impact Duffing system under external Poisson impulses with the exponential-polynomial closure method [12]. And Dimentberg derived a closed-form solution for the stationary PDF of a single-mass Vibro-impact system to random white noise by the quasi-conservative approximation [11].…”

Stochastic Responses of Lightly Nonlinear Vibroimpact System with Inelastic Impact Subjected to External Poisson White Noise Excitation

“…The Poisson white noise is the most used model for non-Gaussian random excitations [37][38][39][40]. However, the research on stochastic responses of vibroimpact system subject to Poisson white noise has been rarely addressed [41] and is far from enough. This paper is devoted to presenting a procedure to predict the stochastic responses of lightly nonlinear vibroimpact system subject to external Poisson white noise excitation.…”