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.

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.

scholarly journals The probabilistic solution of stochastic oscillators with even nonlinearity under poisson excitation

Open Physics ◽  
2012 ◽  
Vol 10 (3) ◽  
Author(s):  
Siu-Siu Guo ◽  
Guo-Kang Er
Keyword(s):  
Monte Carlo Simulation ◽  
Monte Carlo ◽  
Numerical Analysis ◽  
Kolmogorov Equation ◽  
Poisson White Noise ◽  
Stationary Probability Density Function ◽  
Probabilistic Solution ◽  
The Mean ◽  
Stochastic Oscillators ◽  
Probability Density Function Pdf

AbstractThe probabilistic solutions of nonlinear stochastic oscillators with even nonlinearity driven by Poisson white noise are investigated in this paper. The stationary probability density function (PDF) of the oscillator responses governed by the reduced Fokker-Planck-Kolmogorov equation is obtained with exponentialpolynomial closure (EPC) method. Different types of nonlinear oscillators are considered. Monte Carlo simulation is conducted to examine the effectiveness and accuracy of the EPC method in this case. It is found that the PDF solutions obtained with EPC agree well with those obtained with Monte Carlo simulation, especially in the tail regions of the PDFs of oscillator responses. Numerical analysis shows that the mean of displacement is nonzero and the PDF of displacement is nonsymmetric about its mean when there is even nonlinearity in displacement in the oscillator. Numerical analysis further shows that the mean of velocity always equals zero and the PDF of velocity is symmetrically distributed about its mean.

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 Analysis of Random Structural Intensity for Structural Members under Stochastic Loadings

2015 ◽  
Vol 12 (03) ◽  
pp. 1550013 ◽  
Author(s):  
Siu-Siu Guo ◽  
Dongfang Wang ◽  
Zishun Liu
Keyword(s):  
Energy Flow ◽  
Early Stage ◽  
Structural Members ◽  
Structural Intensity ◽  
Dynamic Loadings ◽  
Arbitrary Section ◽  
Stationary Loading ◽  
Fpk Equation ◽  
Definition Of ◽  
Probability Density Function Pdf

The concept of structural intensity (SI) is extended to the random domain by introducing a physical quantity denominated random structural intensity (RSI). This quantity is formulated for mechanical systems whose dynamical responses are stochastic due to random excitations. In order to fully characterize the stochastic behavior of a system under random loadings, it is imperative to obtain the probability density function (PDF) of RSI. Based on the elastic theory and the definition of SI, RSI is expressed as functions of system responses. In general, the PDF of system responses is governed by Fokker–Planck–Kolmogorov (FPK) equation under the assumption that random dynamic loadings are idealized as white noise excitations. Therefore, the PDF of RSI is derived with the joint PDF of system responses. In the present study, four demonstrating cases of beams and plates under separately concentrated and uniform random loadings are studied to investigate the properties of RSI. Stationary and non-stationary PDFs of RSI at arbitrary section of beam and plate are obtained. Numerical results show that the PDF of RSI is transient at early stage of stationary loading and then converges to the exact stationary ones as time increases. With the obtained PDFs of RSI, energy transmission path over the beam and plate can be determined, which is guided from the locations with lower probabilities of RSI to the ones with higher probabilities of RSI. Furthermore, virtual energy flow sinks on the plate and beam can be found, which are identified by the locations with the maximum probabilities of RSI.

The Effects of Correlated Noise on Bifurcation in Birhythmicity Driven by Delay

2018 ◽  
Vol 28 (10) ◽  
pp. 1850127 ◽  
Author(s):  
Lijuan Ning ◽  
Zhidan Ma
Keyword(s):  
Averaging Method ◽  
Harmonic Approximation ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Self Control ◽  
Correlated Noise ◽  
State Variables ◽  
Stationary Probability Density Function ◽  
Fpk Equation ◽  
Control Feedback

We consider bifurcation regulations under the effects of correlated noise and delay self-control feedback excitation in a birhythmic model. Firstly, the term of delay self-control feedback is transferred into state variables without delay by harmonic approximation. Secondly, FPK equation and stationary probability density function (SPDF) for amplitude can be theoretically mapped with stochastic averaging method. Thirdly, the intriguing effects on bifurcation regulations in a birhythmic model induced by delay and correlated noise are observed, which suggest the violent dependence of bifurcation in this model on delay and correlated noise. Particularly, the inner limit cycle (LC) is always standing due to noise. Lastly, the validity of analytical results was confirmed by Monte Carlo simulation for the dynamics.

scholarly journals Calculus on Gaussian and Poisson white noises

1988 ◽  
Vol 111 ◽  
pp. 41-84 ◽  
Author(s):  
Yoshifusa Ito ◽  
Izumi Kubo
Keyword(s):  
White Noise ◽  
Gaussian White Noise ◽  
Exceptional Point ◽  
Stochastic Integrals ◽  
Generalized Poisson ◽  
Gaussian Case ◽  
White Noises

Recently one of the authors has introduced the concept of generalized Poisson functionals and discussed the differentiation, renormalization, stochastic integrals etc. ([8], [9]), analogously to the works of T. Hida ([3], [4], [5]). Here we introduce a transformation for Poisson fnnctionals with the idea as in the case of Gaussian white noise (cf. [10], [11], [12], [13]). Then we can discuss the differentiation, renormalization, multiple Wiener integrals etc. in a way completely parallel with the Gaussian case. The only one exceptional point, which is most significant, is that the multiplications are described by for the Gaussian case, for the Poisson case,as will be stated in Section 5. Conversely, those formulae characterize the types of white noises.

Sign in / Sign up

Export Citation Format

Share Document