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

• Published in: Nonlinear dynamics Vol. 76; no. 2; pp. 1271 - 1289
• Main Authors: Jia, Wantao, Zhu, Weiqiu
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.04.2014; Springer Nature B.V
• Subjects: Automotive Engineering; Classical Mechanics; Control; Damping; Density; Differential equations; Dynamical Systems; Engineering; Equations of motion; Excitation; Gaussian; Hamiltonian functions; Independent variables; Linear damping; Mathematical analysis; Mechanical Engineering; Noise; Noise prediction; Nonlinear dynamics; Original Paper; Oscillators; Perturbation methods; Probability theory; Stationary response; Stochasticity; Transition probabilities; Vibration; White noise; Perturbation method; Stochastic averaging method; Quasi-integrable and non-resonant Hamiltonian system; Combined Gaussian and Poisson white noise excitations
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-013-1209-9

A stochastic averaging method for predicting the response of quasi-integrable and non-resonant Hamiltonian systems to combined Gaussian and Poisson white noise excitations is proposed. First, the motion equations of a quasi-integrable and non-resonant Hamiltonian system subject to combined Gaussian and Poisson white noise excitations is transformed into stochastic integro-differential equations (SIDEs). Then n -dimensional averaged SIDEs and generalized Fokker–Plank–Kolmogrov (GFPK) equations for the transition probability densities of n action variables and n - independent integrals of motion are derived by using stochastic jump–diffusion chain rule and stochastic averaging principle. The probability density of the stationary response is obtained by solving the averaged GFPK equation using the perturbation method. Finally, as an example, two coupled non-linear damping oscillators under both external and parametric excitations of combined Gaussian and Poisson white noises are worked out in detail to illustrate the application and validity of the proposed stochastic averaging method.