Stochastic averaging of quasi-partially integrable Hamiltonian systems under combined Gaussian and Poisson white noise excitations

2014 ◽  
Vol 398 ◽  
pp. 125-144 ◽  
Author(s):  
Wantao Jia ◽  
Weiqiu Zhu
Keyword(s):  
White Noise ◽  
Hamiltonian Systems ◽  
Stochastic Averaging ◽  
Integrable Hamiltonian Systems ◽  
Poisson White Noise

Stochastic Averaging of Quasi-Integrable and Resonant Hamiltonian Systems Under Combined Gaussian and Poisson White Noise Excitations

2013 ◽  
Vol 81 (4) ◽  
Author(s):  
Wantao Jia ◽  
Weiqiu Zhu ◽  
Yong Xu ◽  
Weiyan Liu
Keyword(s):  
White Noise ◽  
Probability Density ◽  
Hamiltonian Systems ◽  
Transition Probability ◽  
Digital Simulation ◽  
Relaxation Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Transition Probability Density ◽  
Poisson White Noise

A stochastic averaging method for quasi-integrable and resonant Hamiltonian systems subject to combined Gaussian and Poisson white noise excitations is proposed. The case of resonance with α resonant relations is considered. An (n + α)-dimensional averaged Generalized Fokker–Plank–Kolmogorov (GFPK) equation for the transition probability density of n action variables and α combinations of phase angles is derived from the stochastic integrodifferential equations (SIDEs) of original quasi-integrable and resonant Hamiltonian systems by using the jump-diffusion chain rule. The reduced GFPK equation is solved by using finite difference method and the successive over relaxation method to obtain the stationary probability density of the system. An example of two nonlinearly damped oscillators under combined Gaussian and Poisson white noise excitations is given to illustrate the proposed method. The good agreement between the analytical results and those from digital simulation shows the validity of the proposed method.

STOCHASTIC HOPF BIFURCATION OF QUASI-INTEGRABLE HAMILTONIAN SYSTEMS WITH FRACTIONAL DERIVATIVE DAMPING

2012 ◽  
Vol 22 (04) ◽  
pp. 1250083 ◽  
Author(s):  
F. HU ◽  
W. Q. ZHU ◽  
L. C. CHEN
Keyword(s):  
Hopf Bifurcation ◽  
Fractional Derivative ◽  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Stochastic Averaging Method ◽  
Bifurcation Parameter ◽  
Integrable Hamiltonian Systems ◽  
Fractional Derivative Damping ◽  
Stochastic Hopf Bifurcation

The stochastic Hopf bifurcation of multi-degree-of-freedom (MDOF) quasi-integrable Hamiltonian systems with fractional derivative damping is investigated. First, the averaged Itô stochastic differential equations for n motion integrals are obtained by using the stochastic averaging method for quasi-integrable Hamiltonian systems. Then, an expression for the average bifurcation parameter of the averaged system is obtained and a criterion for determining the stochastic Hopf bifurcation of the system by using the average bifurcation parameter is proposed. An example is given to illustrate the proposed procedure in detail and the numerical results show the effect of fractional derivative order on the stochastic Hopf bifurcation.

Random Response of Wake-Oscillator Excited by Fluctuating Wind

2021 ◽  
pp. 1-33
Author(s):  
Mao Lin Deng ◽  
Genjin Mu ◽  
Weiqiu Zhu
Keyword(s):  
Hamiltonian Systems ◽  
Averaging Method ◽  
Stochastic Averaging ◽  
Random Excitation ◽  
Stochastic Averaging Method ◽  
Random Response ◽  
Integrable Hamiltonian Systems ◽  
Wake Oscillator ◽  
Fluctuating Wind ◽  
Oscillator Models

Abstract Many wake-oscillator models applied to study vortex-induced vibration (VIV) are assumed to be excited by ideal wind that is assumed to be uniform flow with constant velocity. While in the field of wind engineering, the real wind generally is described to be composed of mean wind and fluctuating wind. The wake-oscillator excited by fluctuating wind should be treated as a randomly excited and dissipated multi-degree of freedom (DOF) nonlinear system. The involved studies are very difficult and so far there are no exact solutions available. The present paper aims to carry out some study works on the stochastic dynamics of VIV. The stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation is applied to study the Hartlen-Currie wake-oscillator model and its modified model excited by fluctuating wind. The probability and statistics of the random response of wake-oscillator in resonant or lock-in case and in non-resonant case are analytically obtained, and the theoretical results are confirmed by using numerical simulation of original system. Finally, it is pointed out that the stochastic averaging method of quasi integrable Hamiltonian systems under wideband random excitation can also be applied to other wake-oscillator models, such as Skop-Griffin model and Krenk-Nielsen model excited by fluctuating wind.

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