Probability Density Function Solution of Nonlinear Oscillators Subjected to Multiplicative Poisson Pulse Excitation on Velocity

2010 ◽  
Vol 77 (3) ◽  
Author(s):  
H. T. Zhu ◽  
G. K. Er ◽  
V. P. Iu ◽  
K. P. Kou
Keyword(s):  
Probability Density Function ◽  
White Noise ◽  
Probability Density ◽  
Density Function ◽  
Nonlinear Oscillators ◽  
Solution Procedure ◽  
Pulse Excitation ◽  
Poisson White Noise ◽  
White Noise Excitation ◽  
Fpk Equation

The stationary probability density function (PDF) solution of the stochastic responses is derived for nonlinear oscillators subjected to both additive and multiplicative Poisson white noises. The PDF solution is governed by the generalized Fokker–Planck–Kolmogorov (FPK) equation and obtained with the exponential-polynomial closure (EPC) method, which was originally proposed for solving the FPK equation. The extended EPC solution procedure is presented for the case of Poisson pulses in this paper. In order to evaluate the effectiveness of the solution procedure, nonlinear oscillators are investigated under multiplicative Poisson white noise excitation on velocity and additive Poisson white noise excitation. Both weakly and strongly nonlinear oscillators are considered, respectively. In the numerical analysis, both the unimodal and bimodal stationary PDFs of oscillator responses are obtained with the EPC method and Monte Carlo simulation. Compared with the simulation results, good agreement is achieved with the presented solution procedure in the case of the polynomial degree being 6, especially in the tail regions of the PDFs of the system responses.

scholarly journals Dynamics of a Nonlinear Business Cycle Model Under Poisson White Noise Excitation

2015 ◽  
Vol 3 (2) ◽  
pp. 176-183 ◽  
Author(s):  
Jiaorui Li ◽  
Shuang Li
Keyword(s):  
Business Cycle ◽  
White Noise ◽  
Economic System ◽  
Probability Density ◽  
Stationary Probability ◽  
Cycle Model ◽  
Poisson White Noise ◽  
Stationary Probability Density ◽  
White Noise Excitation ◽  
Business Cycle Model

AbstractSeveral observations in real economic systems have shown the evidence of non-Gaussianity behavior, and one of mathematical models to describe these behaviors is Poisson noise. In this paper, stationary probability density of a nonlinear business cycle model under Poisson white noise excitation has been studied analytically. By using the stochastic averaged method, the approximate stationary probability density of the averaged generalized FPK equations are obtained analytically. The results show that the economic system occurs jump and bifurcation when there is a Poisson impulse existing in the periodic economic system. Furthermore, the numerical solutions are presented to show the effectiveness of the obtained analytical solutions.

scholarly journals Analysis of exceedance probability of displacement response of randomly nonlinear structures

2000 ◽  
Vol 22 (4) ◽  
pp. 212-224 ◽  
Author(s):  
Luu Xuan Hung
Keyword(s):  
Probability Density Function ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Density Function ◽  
Exceedance Probability ◽  
Nonlinear Structures ◽  
Exact Probability ◽  
Displacement Response ◽  
Linearization Techniques

The paper presents the estimation of the exact exceedance probability (EEP) of stationary responses of some white noise-randomly excited nonlinear systems whose exact probability density function can be known. Consequently, the approximate exceedance probabilities (AEPs) are evaluated based on the analysis of equivalent linearized systems using the traditional Caughey method and the extension technique of LOMSEC. Comparisons of the AEPs versus the EEP are demonstrated. The obtained results indicate important characters of the exceedance probability (EP) as well as the influence of nonlinearity over EP. The evaluation of the applied possibility of the proposed linearization techniques for estimating AEPs are made.

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

2018 ◽  
Vol 2018 ◽  
pp. 1-12
Author(s):  
Guidong Yang ◽  
Wei Xu ◽  
Dongmei Huang ◽  
Mengli Hao
Keyword(s):  
White Noise ◽  
Coordinate Transformation ◽  
Perturbation Technique ◽  
Inverse Transformation ◽  
Poisson White Noise ◽  
Dirac Delta ◽  
White Noise Excitation ◽  
Inelastic Impact ◽  
Fpk Equation ◽  
Vibroimpact System

A procedure for analyzing stationary responses of lightly nonlinear vibroimpact system with inelastic impact subjected to external Poisson white noise excitation is proposed. First, the original vibroimpact system is transformed to a new system without velocity jump in terms of the Zhuravlev nonsmooth coordinate transformation and the Dirac delta function. Second, the averaged generalized Fokker-Planck-Kolmogorov (FPK) equation for transformed system under parametric excitation of Poisson white noise is derived by stochastic averaging method. Third, the averaged generalized FPK equation is solved by using the perturbation technique and inverse transformation of the Zhuravlev nonsmooth coordinate transformation to obtain the approximately stationary solutions for response probability density functions of original vibroimpact system. Last, analytical and numerical results for two typical lightly nonlinear vibroimpact systems are presented to assess the effectiveness of the proposed method. It is found that they are in good agreement and the proposed method is quite effective.

The Probabilistic Solutions to Nonlinear Random Vibrations of Multi-Degree-of-Freedom Systems

1999 ◽  
Vol 67 (2) ◽  
pp. 355-359 ◽  
Author(s):  
G.-K. Er
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Degree Of Freedom ◽  
State Variables ◽  
Algebraic Equations ◽  
Nonlinear Random Vibration ◽  
Highly Nonlinear ◽  
Fpk Equation ◽  
Two Degree Of Freedom

The probability density function of the responses of nonlinear random vibration of a multi-degree-of-freedom system is formulated in the defined domain as an exponential function of polynomials in state variables. The probability density function is assumed to be governed by Fokker-Planck-Kolmogorov (FPK) equation. Special measure is taken to satisfy the FPK equation in the average sense of integration with the assumed function and quadratic algebraic equations are obtained for determining the unknown probability density function. Two-degree-of-freedom systems are analyzed with the proposed method to validate the method for nonlinear multi-degree-of-freedom systems. The probability density functions obtained with the proposed method are compared with the obtainable exact and simulated ones. Numerical results showed that the probability density function solutions obtained with the presented method are much closer to the exact and simulated solutions even for highly nonlinear systems with both external and parametric excitations. [S0021-8936(00)01602-0]

Stochastic path summation with memory

2015 ◽  
Vol 36 ◽  
pp. 1560006
Author(s):  
Christopher C. Bernido ◽  
M. Victoria Carpio-Bernido
Keyword(s):  
Probability Density Function ◽  
Diffusion Equation ◽  
Stochastic Processes ◽  
White Noise ◽  
Probability Density ◽  
Density Function ◽  
Path Integral ◽  
Different Types ◽  
With Memory

Some classes of stochastic processes with memory properties are investigated by evaluating the probability density function as a white noise path integral. The corresponding modified diffusion equation for different types of memory behavior is then discussed.

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