Dynamic Response Analysis For Several Classes Of Typical Stochastic Systems

With the development of science and society, new phenomena and innovationsare emerging in a large number from various fields of applications. Thecomprehension of these new problems requires advanced mathematical theories in theprocesses of modeling and analyzing, and the introduction of stochastic elements alsoforms a trend.In this thesis, methods of dynamic response analysis are provided for stochasticsystems with fractional derivatives and multiple time delays. Main results andachievements contain:Pragmatic numerical schemes are proposed for different representations offractional derivatives. For stochastic systems with multiple fractional derivativesadopting Caputo definition, based on the concept of Boundary Element Method, agroup of linear and uncoupled stochastic fractional-order analog equations areintroduced. Laplace transform is conducted to solve the original system iteratively.For Grünwald-Letnikov-defined fractional derivative, GL coefficient sequence for aspecific fractional order is obtained, and a proper truncation is applied to weaken thedependence of fractional derivative on historical data. Classical numerical methodshelp to fulfill a computational procedure.Response power spectral density is formulated and analyzed as a function of thederivative order for a linear system with fractional derivative subjected to Gaussianwhite noise. Decomposition of fractional derivative derives a linear combination ofdamping and stiffness, and the coefficients are discussed elaborately for the impact onstochastic response with the change of the fractional order. Stochastic averaging isemployed to obtain Markov approximation for the response of equivalent system.Based on the modified statistical linearization theory, an equivalent linear systemfor a Duffing oscillator comprising fractional derivative element excited by Gaussian white noise is obtained utilizing generalized harmonic balance technique. Conditionalpower spectral density and stationary probability density function derived bystochastic averaging complete an estimation of response power spectral density. Thevariation of the power spectral density with respect to the derivative orderdemonstrates the effects of the fractional derivative on the response, which alsoindicates that the nonlinear property of the Duffing system is reserved after thelinearization procedure.Steady state analysis for a class of stochastic systems with multiple time delays isimplemented employing functional calculation. With the assistance of Novikov’stheorem, a delay FPK equation governing the evolution of probability densityfunction for the response is obtained. Utilizing small delay approximation technique,the stationary solution is derived, for which detailed analysis of parameters isexecuted. Indications of different effects of noise correlation strength and time delayfeedbacks on the response are received.