Stationary Solution of Duffing Oscillator Driven by Additive and Multiplicative Colored Noise Excitations

2017 ◽  
Vol 139 (2) ◽  
Author(s):  
Siu-Siu Guo ◽  
Qingxuan Shi
Keyword(s):  
White Noise ◽  
Colored Noise ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Approximate Solutions ◽  
Intensity Level ◽  
First Order ◽  
Fpk Equation ◽  
Polynomial Closure ◽  
Mcs Method

A bistable Duffing oscillator subjected to additive and multiplicative Ornstein–Uhlenbeck (OU) colored excitations is examined. It is modeled through a set of four first-order stochastic differential equations by representing the OU excitations as filtered Gaussian white noise excitations. Enlargement in the state-space vector leads to four-dimensional (4D) Fokker–Planck–Kolmogorov (FPK) equation. The exponential-polynomial closure (EPC) method, proposed previously for the case of white noise excitations, is further improved and developed to solve colored noise case, resulting in much more polynomial terms included in the approximate solution. Numerical results show that approximate solutions from the EPC method compare well with the predictions obtained via Monte Carlo simulation (MCS) method. Investigation is also carried out to examine the influence of intensity level on the probability distribution solutions of system responses.

Probabilistic Solution of Vibro-Impact Systems Under Additive Gaussian White Noise

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
H. T. Zhu
Keyword(s):  
White Noise ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Planck Equation ◽  
Solution Procedure ◽  
Additive Gaussian White Noise ◽  
Stationary Probability Density Function ◽  
Probabilistic Solution ◽  
Zero Offset ◽  
Polynomial Closure

This paper presents a solution procedure for the stationary probability density function (PDF) of the response of vibro-impact systems under additive Gaussian white noise. The constraint is a unilateral zero-offset barrier. The vibro-impact system is first converted into a system without barriers using the Zhuravlev nonsmooth coordinate transformation. The stationary PDF of the converted system is governed by the Fokker–Planck equation which is solved by the exponential-polynomial closure (EPC) method. A vibro-impact Duffing oscillator with either elastic or lightly inelastic impacts is considered in a numerical analysis. Meanwhile, the level of nonlinearity in displacement is also examined in this study as well as the case of negative linear stiffness. Comparison with the simulated results shows that the EPC method can present a satisfactory PDF for displacement and velocity when the polynomial order is taken as 4 in the investigated cases. The tail of the PDF also works well with the simulated result.

Probabilistic Solutions of Stochastic Oscillators Excited by Correlated External and Parametric White Noises

2014 ◽  
Vol 136 (3) ◽  
Author(s):  
Siu-Siu Guo
Keyword(s):  
Gaussian White Noise ◽  
Nonlinear Oscillators ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Linear And Nonlinear ◽  
Polynomial Closure ◽  
Stochastic Oscillators ◽  
Random Oscillators ◽  
White Noises ◽  
Probability Density Function Pdf

The stationary probability density function (PDF) solution of random oscillators with correlated additive and multiplicative Gaussian excitations is investigated in this paper. The correlation between additive and multiplicative Gaussian excitations is taken into account. As a result, the generalized Fokker-Planck-Kolmogorov (FPK) equation is expressed with the independent part and the correlated part, which can be solved by the exponential-polynomial closure (EPC) method. The linear and nonlinear oscillators under correlated additive and multiplicative Gaussian white noise excitations are investigated. Two cases of different correlated additive and multiplicative excitations are considered. Compared with the results in the case of independent external and parametric excitations, unsymmetrical PDFs and nonzero means of system responses can be obtained.

Extension of Nonlinear Stochastic Solution to Include Sinusoidal Excitation—Illustrated by Duffing Oscillator

2017 ◽  
Vol 12 (5) ◽  
Author(s):  
R. J. Chang
Keyword(s):  
White Noise ◽  
Correction Factor ◽  
Duffing Oscillator ◽  
Gaussian White Noise ◽  
Noise Spectrum ◽  
Linearization Method ◽  
Algebraic Equations ◽  
Stochastic Solution ◽  
Non Gaussian ◽  
Sinusoidal Excitation

A new non-Gaussian linearization method is developed for extending the analysis of Gaussian white-noise excited nonlinear oscillator to incorporate sinusoidal excitation. The non-Gaussian linearization method is developed through introducing a modulated correction factor on the linearization coefficient which is obtained by Gaussian linearization. The time average of cyclostationary response of variance and noise spectrum is analyzed through the correction factor. The validity of the present non-Gaussian approach in predicting the statistical response is supported by utilizing Monte Carlo simulations. The present non-Gaussian analysis, without imposing restrictive analytical conditions, can be obtained by solving nonlinear algebraic equations. The non-Gaussian solution effectively predicts accurate sinusoidal and noise response when the nonlinear system is subjected to both sinusoidal and white-noise excitations.

Probabilistic Solutions of a Nonlinear Plate Excited by Gaussian White Noise Fully Correlated in Space

2017 ◽  
Vol 17 (09) ◽  
pp. 1750097 ◽  
Author(s):  
G. K. Er ◽  
V. P. Iu
Keyword(s):  
White Noise ◽  
Random Vibration ◽  
Gaussian White Noise ◽  
Computational Effort ◽  
Degree Of Freedom ◽  
Equivalent Linearization ◽  
Nonlinear Random Vibration ◽  
Split Method ◽  
Probabilistic Solution ◽  
Polynomial Closure

This paper addresses the nonlinear random vibration of a rectangular von Kármán plate excited by uniformly distributed Gaussian white noise which is fully correlated in space. The state-space-split method and exponential polynomial closure method are jointly utilized to analyze the probabilistic solutions of the plate. The computational efficiency and numerical accuracy of the methodology for analyzing the nonlinear random vibration of the plate are verified by comparing the computational effort and numerical results with those obtained by Monte Carlo simulation and equivalent linearization, respectively. Meanwhile, the convergence of the probabilistic solution in the sense of Galerkin’s approximation is examined by analyzing the plate modeled as single-degree-of-freedom and multi-degree-of-freedom systems. Some phenomena are discussed after numerically studying the behaviors of probabilistic solutions of the deflection at different locations of the plate.

scholarly journals The influence of second order narrow-band colored noises on non-linear random vibrations

1999 ◽  
Vol 21 (2) ◽  
pp. 65-74
Author(s):  
Nguyen Dong Anh ◽  
Nguyen Duc Tinh
Keyword(s):  
Colored Noise ◽  
Averaging Method ◽  
Narrow Band ◽  
Duffing Oscillator ◽  
Stochastic Averaging ◽  
Random Processes ◽  
Stochastic Averaging Method ◽  
Second Order ◽  
External Excitation ◽  
First Order

Since the effect of some nonlinear terms is lost during the first order averaging procedure, the higher order stochastic averaging method is developed to predict approximately the response of linear and lightly nonlinear systems subject to weakly external excitation of second order colored noise random processes. Application to Duffing oscillator is considered.

Probabilistic Solutions of the In-Plane Nonlinear Random Vibrations of Shallow Cables Under Filtered Gaussian White Noise

2018 ◽  
Vol 18 (04) ◽  
pp. 1850062 ◽  
Author(s):  
G. K. Er ◽  
K. Wang ◽  
V. P. Iu
Keyword(s):  
White Noise ◽  
Computational Efficiency ◽  
Gaussian White Noise ◽  
Peak Frequency ◽  
Seismic Intensity ◽  
Linearization Method ◽  
Equivalent Linearization ◽  
Seismic Force ◽  
Cable Systems ◽  
Polynomial Closure

The probabilistic solutions of the responses of shallow cable are studied when the cable is excited by filtered Gaussian white noise. The nonlinear multi-degree-of-freedom system is formulated which governs the random vibration of cable. The state-space-split (SSS) method and exponential polynomial closure (EPC) method are adopted to analyze the probabilistic solutions of cable systems in order to study the effectiveness and computational efficiency of SSS-EPC procedure in analyzing the probabilistic solutions of the cable systems under the excitation of filtered Gaussian white noise. Numerical results obtained by SSS-EPC method, Monte Carlo simulation, and equivalent linearization method are compared to examine the computational efficiency and numerical accuracy of SSS-EPC method in this case. Thereafter, the behaviors of probabilistic solutions of the cable systems are studied with different values of peak frequency and seismic intensity of excitation when the cable is excited by Kanai–Tajimi seismic force. Some observations and discussions are given by introducing a probabilistic quantity to show the influence of excitations on the probabilistic solutions.

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