Stochastic Averaging Methods For Nonlinear Systems Subject To Non-Gaussian Random Excitation

Several stochastic averaging methods for nonlinear systems subject to non-Gaussian random excitations are investigated in the present dissertation.In chapter 1, the dynamical problems subject to non-Gaussian random excitations and recent development of stochastic averaging method are introduced.In chapter 2, Poisson white noise and its related concepts, stochastic differential rules and Monte Carlo simulation methods are introduced.In chapter 3, stochastic averaging method of coefficients of generalized Fokker-Planck- Kolmogorov (GFPK) equation for quasi linear systems subject to Poisson white noise is proposed. Approximate stationary solutions of the averaged GFPK equations are obtained by using the perturbation method for four typical quasi linear systems, i.e., van der Pol oscillator, Rayleigh oscillator, system with energy-dependent damping, and systems with power law damping. The reliability of the perturbation solution is assessed by performing appropriate Monte Carlo simulations. It is found that analytical and numerical results agree well and effect of non-Gaussianity of the excitation process is not negligible for predicting probability density of total energy of quasi linear system in most cases.In chapter 4, a stochastic averaging method is proposed for predicting the response of multi-degree-of-freedom quasi-noninegrable-Hamiltonian systems subject to Poisson white noises. A one-dimensional averaged GFPK equation for the transition probability density of the Hamiltonian is derived and the probability density of the stationary response of the system is obtained by using the perturbation method. Two examples, two linearly and nonlinearly coupled van der Pol oscillators and 2-degree-of-freedom vibro-impact system, are given to illustrate the application and validity of the proposed method.In chapter 5, the averaged GFPK equation for response of n-dimensional (n-d) nonlinear dynamical systems to non-Gaussian wide-band stationary random excitation is derived from the standard form of equation of motion. The explicit expressions for coefficients of the fourth-order approximation of the averaged GFPK equation are given in series form. Conditions for convergences of these series are pointed out. The averaged GFPK equation is then reduced to that for 1-d dynamical systems derived by Stratonovich and compared with the closed form of GFPK equation for n-d dynamical systems subject to Poisson white noise derived by Di Paola and Falsone. Finally, stochastic averaging method for quasi linear systems subject to non-Gaussian wide-band stationary random excitation is proposed by using the equation of motion generated from van der Pol transformations and its corresponding averaged GFPK equation. Stationary probability density for quasi linear system subject to filtered Poisson white noise is obtained. Theoretical results for the example are confirmed by using Monte-Carlo simulation for different parameter values.Finally, in chapter 6, conclusion remarks of the present paper are made and the direction of future research is pointed out.