Stochastic Averaging Methods For Quasi Integrable And Partially Integrable Hamiltonian Systems Under Fractional Gaussian Noise Excitation

In the past research of nonlinear stochastic dynamics,most excitations of dynamical systems are Gaussian white noise or wide-band stochastic process,which has short correlation time.However,in nature and engineering,many stochastic excitations have long correlation time.Fractional Gaussian noise(fGn)has long correlation time and can be used to model these excitations.For researching the response of multi-dgree-of-freedom(MDOF)strongly nonlinear system to fGn,the stochastic averaging methods of quasi integrable and partially integrable Hamiltonian systems under fGn excitation are developed in this dissertation.Firstly,the recent development of dynamical system excited by fGn and the stochastic averaging method are briefly reviewed.The definations of fractional calculus,the definitions and the basic characteristics of fractional Brownian motion(fBm)and fGn,the auto-correlation and power spectral density of fGn,the simulation algorithm for fBm and fGn,the three fractional pathwise integrals(symmetric pathwise integral,forward pathwise integral and backward pathwise integral)for fBm and the fractional stochastic differential rule for fBm are introduced.Secondly,the stochastic averaging method of quasi integrable Hamilonian systems excited by fGn is developed.The averaging principle for dynamical systems under fGn excitation and the Hamiltonian systems and their classification are briefly introduced.Based on the resonant relation of Hamiltonian system,quasi integrable Hamilonian systems are divided into two cases:quasi integrable and non-resonant and quasi integrable and resonant.Using the fractional stochastic differential rule and stochastic averaging principle,the averaged fractional stochastic differential equations(SDEs)for two cases of quasi integrable Hamiltonian systems are derived,respectively.The stochastic averaging method of quasi integrable and non-resonant Hamiltonian systems is applied to two examples:linearly and nonlinearly coupled two linear oscillators subjected to external fGn excitations,a van der Pol oscillator nonlinearly coupled with a Duffing oscillator subject to external excitations of fGn.The stochastic averaging method of quasi integrable and resonant Hamiltonian systems is applied to a 2-DOF damping coupled system.The approximate stationary responses of example systems are obtained from simulating the averaged fractional SDEs and compared with the results obtained from original systems.The results show that this method can largely reduce the dimension of original system and significantly improve computational efficiency.Thirdly,the stochastic averaging method of quasi partially integrable Hamilonian system excited by fGn is developed for both non-resonant and resonant cases.The averaged fractional SDEs for both cases are derived,respectively.The proposed method in non-resonant case is applied to a 4-DOF and a 3-DOF strongly nonlinear system.The proposed method in resonant case is applied to another 4-DOF strongly nonlinear system.The approximate stationary responses are obtained from simulating the averaged fractional SDEs and compared with the results obtained from original systems.The results of examples show that this method can largely reduce the dimensions of original system and significantly improve computational efficiency.Finally,conclusion remarks are made and the innovations of present dissertation and problems needed to be further researched are pointed out.