Approximate Analysis of Nonlinear Systems Driven by Gaussian White Noise

1992 ◽  
Author(s):  
Dingyu Xue ◽  
D.P. Atherton
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Gaussian White Noise ◽  
Approximate Analysis

The Response Statistics, Jump and Bifurcation of Nonlinear Dynamical Systems Subjected to White Noise and Combined Sinusoidal and White Noise Excitation

2007 ◽  
Author(s):  
Pankaj Kumar ◽  
S. Narayanan
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Nonlinear Dynamical Systems ◽  
Gaussian White Noise ◽  
Nonlinear Dynamical ◽  
White Noise Excitation ◽  
Bladed Disk Assembly ◽  
Noise Frequency ◽  
System Nonlinearity ◽  
Non Gaussian

The prediction of the response of nonlinear systems subjected to stochastic parametric, narrowband and wideband or coloured external excitation is of importance in the field of structural and rotor dynamics. The transitional probability density function (pdf) for the random response of nonlinear systems under white or coloured noise excitation (delta-correlated) is governed by both the forward Fokker-Planck (FP) and backward Kolmogorov equations. This paper presents efficient numerical solution of the FP equation for the pdf of response for general nonlinear systems subjected to external white noise and combined sinusoidal and white noise excitation. The effect of intensity of white noise, frequency and amplitude of sinusoidal excitation and level of system nonlinearity on the non-Gaussian nature of response caused by the system nonlinearity are investigated. Stochastic behaviours like stability, jump, bifurcation are examined as the system parameters change. The finite element (FE) scheme is used to solve the FP equation and obtain the statistics of a two degree-of-freedom linear system representative of the vibration of gas turbine tip-shrouded bladed disk assembly subjected to Gaussian white noise excitation as an illustrative example.

Exact Stationary-Response Solution for Second Order Nonlinear Systems Under Parametric and External White-Noise Excitations

1987 ◽  
Vol 54 (2) ◽  
pp. 414-418 ◽  
Author(s):  
Y. Yong ◽  
Y. K. Lin
Keyword(s):  
Nonlinear Systems ◽  
White Noise ◽  
Gaussian White Noise ◽  
Detailed Balance ◽  
Stationary Solutions ◽  
Second Order ◽  
Unexpected Result ◽  
Sporadic Cases ◽  
Suitable Combination ◽  
Parametric And External Excitations

Except for rare sporadic cases, exact stationary solutions for second order nonlinear systems under Gaussian white-noise excitations are known only for certain types of systems and only when the excitations are purely external (additive). Yet, in many engineering problems, random excitations may also be parametric (multiplicative). It is shown in this paper that the method of detailed balance developed by the physicists can be applied to obtain systematically the stationary solutions for a large class of nonlinear systems under either external random excitations or parametric random excitations, or both. Examples are given for those cases where solutions have been given previously in the literature as well as other cases where solutions are new. An unexpected result is revealed in one of the new solutions, namely, under suitable combination of the parametric and external excitations of Gaussian white noises, the response of a nonlinear system can be Gaussian.

Equivalent Nonlinear System Method for Stochastically Excited Hamiltonian Systems

1994 ◽  
Vol 61 (3) ◽  
pp. 618-623 ◽  
Author(s):  
W. Q. Zhu ◽  
T. T. Soong ◽  
Y. Lei
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Gaussian White Noise ◽  
Energy Balancing ◽  
Dissipation Energy ◽  
System Method ◽  
Approximate Probability

An equivalent nonlinear system method is presented to obtain the approximate probability density for the stationary response of multi-degree-of-freedom nonlinear Hamiltonian systems to Gaussian white noise parametric and/or external excitations. The equivalent nonlinear systems are obtained on the basis of one of the following three criteria: least mean-squared deficiency of damping forces, dissipation energy balancing, and least mean-squared deficiency of dissipation energies. An example is given to illustrate the application and validity of the method and the differences in the three equivalence criteria.

scholarly journals Stochastic P-bifurcation in a bistable Van der Pol oscillator with fractional time-delay feedback under Gaussian white noise excitation

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Yajie Li ◽  
Zhiqiang Wu ◽  
Guoqi Zhang ◽  
Feng Wang ◽  
Yuancen Wang
Keyword(s):  
Time Delay ◽  
White Noise ◽  
Gaussian White Noise ◽  
Singularity Theory ◽  
Van Der Pol Oscillator ◽  
Order System ◽  
Damping Force ◽  
Van Der Pol ◽  
Restoring Force ◽  
White Noise Excitation

Abstract The stochastic P-bifurcation behavior of a bistable Van der Pol system with fractional time-delay feedback under Gaussian white noise excitation is studied. Firstly, based on the minimal mean square error principle, the fractional derivative term is found to be equivalent to the linear combination of damping force and restoring force, and the original system is further simplified to an equivalent integer order system. Secondly, the stationary Probability Density Function (PDF) of system amplitude is obtained by stochastic averaging, and the critical parametric conditions for stochastic P-bifurcation of system amplitude are determined according to the singularity theory. Finally, the types of stationary PDF curves of system amplitude are qualitatively analyzed by choosing the corresponding parameters in each area divided by the transition set curves. The consistency between the analytical solutions and Monte Carlo simulation results verifies the theoretical analysis in this paper.

Exact Stationary Response Solution for Second Order Nonlinear Systems Under Parametric and External White Noise Excitations: Part II

1988 ◽  
Vol 55 (3) ◽  
pp. 702-705 ◽  
Author(s):  
Y. K. Lin ◽  
Guoqiang Cai
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Probability Distribution ◽  
White Noise ◽  
Probability Density ◽  
Detailed Balance ◽  
Stationary Probability ◽  
Probability Flow ◽  
Parametric And External Excitations ◽  
White Noises

A systematic procedure is developed to obtain the stationary probability density for the response of a nonlinear system under parametric and external excitations of Gaussian white noises. The procedure is devised by separating the circulatory portion of the probability flow from the noncirculatory flow, thus obtaining two sets of equations that must be satisfied by the probability potential. It is shown that these equations are identical to two of the conditions established previously under the assumption of detailed balance; therefore, one remaining condition for detailed balance is superfluous. Three examples are given for illustration, one of which is capable of exhibiting limit cycle and bifurcation behaviors, while another is selected to show that two different systems under two differents sets of excitations may result in the same probability distribution for their responses.

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