Response, Bifurcation And Chaos And Their Control In Stochastic Structure And Stochastic Dynamical System

The problem of stochastic response and stochastic bifurcation and chaos,together with their control, in a stochastic structure or a stochastic dynamical systemis currently an interesting topic in mechanics, associated with a great scientificachievement in the past years. This is not only due to the challenging topic itself butalso due to its fruitful results in application. Based on available theories and methodsabout stochastic structures and stochastic dynamical systems, the dissertation isdevoted to exploring the response and its control in linear stochastic structuressubject to the evolutionary random excitation, and the bifurcation, chaos, chaoscontrol, chaos synchronization, flutter and flutter control in nonlinear stochasticdynamical systems by Gegenbauer polynomial approximation. The maincontributions of this dissertation are as follows.1) The developments of probabilistic and non-probabilistic methods aboutstochastic structures with uncertain parameters are reviewed briefly. The probabilitydensity function with respect of a parameterλ(λ-PDF for short), together withGegenbauer polynomials and Gegenbauer polynomial approximation is introduced.The probability density functions derived fromλ-PDF are investigated and it is foundthat theλ-PDF and its derived PDFs constitute a big family of probability densityfunctions for bounded random parameters in engineering including someunsymmetrical PDFs and Gauss truncated PDF.2) We study the response and its control of a linear stochastic structure subjectto the evolutionary random excitation first. The mass, damping and stiffness of thestructure are supposed to be a quadratic polynomial function of a random variablewithλ-PDF respectively. By Gegenbauer polynomial approximation, we transformthe stochastic structure into its equivalent deterministic system. Then the unifiedapproach applies to it to acquire the ensemble random evolutionary response, and theobtained results are in agreement with what we obtain with Monte Carlo method. Togain an insight into the effect of parameterλon the random responses, we define themean square difference between the response of stochastic structure and that of themean parameter structure (named nominal structure) as dispersion. Dispersion analysis shows that the smaller theλ, the more dispersive is the random response of arandom structure about that of the nominal structure. The dispersive analysis issupplementary to the stochastic structure response analysis.We use Tuned Mass Damper (TMD for short) to suppress the response of astochastic structure subject to an evolutionary random excitation model for the 1964Niigata earthquake. Then the relationship between the evolutionary random responseamplitude and the parameters of TMD is explicitly formulated by means of theresponse surface methodology, and the parameters of the TMD are obtained by amulti-step optimal strategy. Numerical results show that the optimized TMD caneffectively suppress the earthquake responses of the structure. Moreover, the TMDworks equally well for every sample system, so that in this sense the optimal TMDcontrol is robust.3) The bifurcation, chaos, chaos control and chaos synchronization instochastic Duffing system under the harmonic excitation are explored in detail. Theequivalent deterministic system for stochastic Duffing system is obtained viaGegenbauer polynomial approximation. Then the stochastic period-doublingbifurcations for T→2T and 2T→4T, stochastic chaos are investigated via theequivalent deterministic system. Numerical simulations show that similar to itscounterpart in deterministic form, the stochastic Duffing system possesses the aliketypical nonlinear dynamic behavior, such as period-doubling bifurcations and chaos,but in some local areas, it has its own specific features due to the parameterrandomness.Chaos control of the stochastic Duffing system is studied by adding anadjustable noisy phase to the harmonic excitation or by delayed feedback. The topLyapunov exponent of the equivalent system is calculated by Wolf's algorithm inorder to examine the effects of the noise level, feedback intensity and time delay onthe dynamic behavior of the stochastic Duffing system. It is shown that the chaoscontrol can be realized through adjusting the noise level or by proper choice of thefeedback intensity and time delay, which indicates that both the random phasecontrol strategy and delayed feedback strategy can be used to suppress stochasticchaos. Further, the strategies formerly suggested for deterministic chaos control canapply to stochastic chaos control as well.Chaos synchronization of two identical stochastic Duffing systems subject to harmonic excitations is probed by two feedback control strategies respectively. Toguarantee synchronization, the feedback control parameter zone, where the topLyapunov exponent of the controlled responding system is negative or the errorsystem is asymptotical stable, is discussed. Numerical analysis shows that thefeedback control strategy is an effective way to synchronize two identical stochasticDuffing systems with different initial conitions.4) The effects of the parameter uncertainty on the flutter characteristics of atwo-dimensional airfoil in an incompressible flow were investigated throughGegenbauer polynomial approximation. The uncertain parameters, such as the linearand cubic pitch stiffness coefficients are modeled as bounded random variables withλ-PDFs. With the aid of Gegenbauer polynomial approximation, the two-dimensionalstochastic airfoil system is transformed at first into its equivalent deterministic one,through which a Hopf-bifurcation point is then determined and the onset of the flutter,together with the flutter frequency against the probability density distributionparameter and the intensity of the random variable is explored. In addition, theranges of the peak pitch and plunge responses against the flight speed and the PDFsof the peak pitch response are obtained. Numerical results show how the probabilitydensity distribution parameters and the intensity of random variable parameters affectthe flutter onset speed, flutter angular frequency, the upper and lower boundaries ofpeak responses, and the PDFs of peak responses.Flutter suppression of the airfoil with uncertain pitch stiffness is explored byfeedback control of its trailing-edge surface. The influence of feedback control onsuppressing flutter of the stochastic airfoil system with different flight speeds isdiscussed. The controllability and validity of the feedback suppression strategy areinvestigated by means of the simulation analysis. Furthermore, the controllableparameter zone, able for suppressing flutter, is obtained for different airspeeds.Numerical results show that the critical flutter speed of a stochastic airfoil system canbe reasonably lifted by properly controlling its trailing-edge surface.