Exact stationary probability density functions for non-linear systems under Poisson white noise excitation

2015

Nonlinear Dyn

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.

Section: Introductionmentioning

confidence: 99%

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

2015

Nonlinear Dyn

“…Zhu ( )single-degree-of-freedom (SDOF) systems. Even in the case of the SDOF systems, only a few stationary PDF solutions have been obtained in some special cases by solving the associated Fokker-Planck-Kolmogorov (FPK) equation (Caughey and Ma 1982;Dimentberg 1982;Lin and Cai 1988;Vasta 1995;Proppe 2002aProppe , 2003. Most problems need some approximation methods, such as perturbation method (Roberts 1972;Cai and Lin 1992), finite element method (Langley 1985), PetrovGalerkin method (Köylüoglu et al 1994), cell-to-cell mapping (path integration) technique (Köylüoglu et al 1995;Di Paola and Santoro 2008), and finite difference approach (Wojtkiewicz et al 1999).…”

Section: Introductionmentioning

confidence: 99%

Approximate Probability Density Function Solution of Multi-Degree-of-Freedom Coupled Systems Under Poisson Impulses

Numerical Methods for Reliability and Safety Assessment

2014

A solution procedure is proposed to approximate the probability density function (PDF) solution of high-dimensional non-linear systems under Poisson impulses. The PDF solution yields the generalized Fokker-Planck-Kolmogorov (FPK) equation. First a state-space-split method is proposed to reduce the highdimensional generalized FPK equation to a low dimensional equation. After that, the exponential-polynomial closure method is further adopted to solve the reduced FPK equation for the PDF solution. In order to show the effectiveness of the proposed solution procedure, a two-degree-of-freedom coupled pitch-roll ship motion system and a 10-degree-of-freedom mass-spring-damper system are investigated, respectively. Compared to the simulated results, the proposed solution procedure is effective to obtain the PDF solution, especially in the tail region which is very important for reliability analysis.

“…One is in the form of a partial integrodifferential equation and the other is in the form of partial differential equation containing an infinite number of derivate higher-order moments. Only some exact stationary solutions were solved in some special cases for the generalized FPK equation [16,17]. In these solvable cases, an inverse solution procedure is developed.…”

Section: Introductionmentioning

confidence: 99%

Probability density function solution to nonlinear ship roll motion excited by external Poisson white noise

Sci. China Technol. Sci.

et al. 2011

The stationary probability density function (PDF) solution to nonlinear ship roll motion excited by Poisson white noise is analyzed. Subjected to such random excitation, the joint PDF solution to the roll angle and angular velocity is governed by the generalized Fokker-Planck-Kolmogorov (FPK) equation. To solve this equation, the exponential-polynomial closure (EPC) method is adopted. With the EPC method, the PDF solution is assumed to be an exponential-polynomial function of state variables. Special measure is taken such that the generalized FPK equation is satisfied in the average sense of integration with the assumed PDF. The problem of determining the unknown parameters in the approximate PDF finally results in solving simultaneous nonlinear algebraic equations. Both slight and high nonlinearities are considered in the illustrative examples. The analysis shows that when a second-order polynomial is taken, the result of the EPC method is the same as the one given by the equivalent linearization (EQL) method. The EQL results differ significantly from the simulated results in the case of high nonlinearity. When a fourth-order or sixth-order polynomial is taken, the results of the EPC method agree well with the simulated ones, especially in the tail regions of the PDF. This agreement is observed in the cases of both slight and high nonlinearities. probability density function, ship roll motion, Poisson white noise, stochastic process, nonlinearity Citation:Er G K, Zhu H T, Iu V P, et al. Probability density function solution to nonlinear ship roll motion excited by external Poisson white noise.