Stochastic Averaging of Strongly Nonlinear Oscillators under Poisson White Noise Excitation

2011 ◽  
pp. 147-155 ◽  
Author(s):  
Y. Zeng ◽  
W. Q. Zhu
Keyword(s):  
White Noise ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Poisson White Noise ◽  
Strongly Nonlinear ◽  
White Noise Excitation ◽  
Strongly Nonlinear Oscillators

Stochastic Averaging of Strongly Nonlinear Oscillators Under Combined Harmonic and Wide-Band Noise Excitations

2008 ◽  
Vol 130 (5) ◽  
Author(s):  
Y. J. Wu ◽  
W. Q. Zhu
Keyword(s):  
Probability Density ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

Physical and engineering systems are often subjected to combined harmonic and random excitations. The random excitation is often modeled as Gaussian white noise for mathematical tractability. However, in practice, the random excitation is nonwhite. This paper investigates the stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. By using generalized harmonic functions, a new stochastic averaging procedure for estimating stationary response probability density of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations is developed. The damping can be linear and (or) nonlinear and the excitations can be external and (or) parametric. After stochastic averaging, the system state is represented by two-dimensional time-homogeneous diffusive Markov processes. The method of reduced Fokker–Planck–Kolmogorov equation is used to investigate the stationary response of the vibration system. A nonlinearly damped Duffing oscillator is taken as an example to show the application and validity of the method. In the case of primary external resonance, based on the stationary joint probability density of amplitude and phase difference, the stochastic jump of the Duffing oscillator and P-bifurcation as the system parameters change are examined for the first time. The agreement between the analytical results and those from Monte Carlo simulation of original system shows that the proposed procedure works quite well.

scholarly journals Optimal Bounded Control for Stationary Response of Strongly Nonlinear Oscillators under Combined Harmonic and Wide-Band Noise Excitations

2011 ◽  
Vol 2011 ◽  
pp. 1-21 ◽  
Author(s):  
Yongjun Wu ◽  
Changshui Feng ◽  
Ronghua Huan
Keyword(s):  
Joint Probability ◽  
Wide Band ◽  
Nonlinear Oscillators ◽  
Stochastic Averaging ◽  
Bounded Control ◽  
Stationary Response ◽  
Strongly Nonlinear ◽  
Band Noise ◽  
Wide Band Noise ◽  
Strongly Nonlinear Oscillators

We study the stochastic optimal bounded control for minimizing the stationary response of strongly nonlinear oscillators under combined harmonic and wide-band noise excitations. The stochastic averaging method and the dynamical programming principle are combined to obtain the fully averaged Itô stochastic differential equations which describe the original controlled strongly nonlinear system approximately. The stationary joint probability density of the amplitude and phase difference of the optimally controlled systems is obtained from solving the corresponding reduced Fokker-Planck-Kolmogorov (FPK) equation. An example is given to illustrate the proposed procedure, and the theoretical results are verified by Monte Carlo simulation.

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.

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