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]

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 On the theory of advective effects on biological dynamics in the sea. III. The role of turbulence in biological–physical interactions

2007 ◽  
Vol 464 (2091) ◽  
pp. 555-572 ◽  
Author(s):  
Louis Goodman ◽  
Allan R Robinson
Keyword(s):  
Probability Density Function ◽  
Probability Density ◽  
Density Function ◽  
Euphotic Zone ◽  
Biological Interaction ◽  
State Variables ◽  
Turbulent Layer ◽  
Wide Range ◽  
The Mean ◽  
Interaction Term

A nonlinear model for biological and physical dynamical interactions in a laminar upwelling flow field in parts I and II of this study is extended to turbulent flow. In the previous studies, a prescription for obtaining quadrature solutions to the fundamental biodynamical equations was developed. In this study, we use a probability density function approach on these solutions to obtain statistics of the biodynamical state variables and their self-interaction for the case of turbulent advection. To illustrate the theory, a simple nutrient ( N ), phytoplankton ( P ) problem is considered, that of upwelling into a surface turbulent layer. Biological interaction is modelled as bilinear, representing the uptake of N by P in a uniform light euphotic zone. A random walk model is used to obtain the appropriate probability density function for the advective turbulent field. The mean quantities, , , as well as the biological interaction term are calculated. The term has two contributions, , and the turbulence-induced interaction term, . It is shown that the often neglected turbulence-induced coupling term is of the order and opposite in sign. This results in, over a wide range of Peclet numbers, the mean interaction term being significantly smaller than either of its constituent terms, and .

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