Stochastic Dynamics Of Vibro-Impact Systems Coupled With Visco-Elastic Damping And Time-Delayed Feedback Control

The phenomenon of vibration is widespread,and in most cases it will cause harm.Therefore,it is particularly important and meaningful to apply vibration reduction thinking to structural systems existing in such as nature,human social life and production,and engineering practice.For the control of vibration,it is possible to take measures to suppress,and it is more popular to be able to utilize vibration.Visco-elastic damper made of visco-elastic material is an effective passive control device for energy dissipation and vibration reduction in structural vibration control of engineering practice.Time-delayed feedback control(TDFC)for the structural vibration reduction is an emerging technology and belongs to semi-active control of vibration.As a typical non-smooth vibration system,the dynamic behavior of vibro-impact(VI)system is complex and abundant in performance due to its special nonlinear structure,which has become a hotspot and focus in scientific research.In this dissertation,the author tries to take these factors and phenomena into account properly in the structural vibration system,and explores the dynamic behavior of the structural vibration system and pursues the essential characteristics of these behavioral representations in order to obtain some application enlightenment in the practical situation.Firstly,stationary responses have been exhibited to characterize the vibration law of the stochastic visco-elastic system with right nonzero offset barrier impacts.The visco-elastic damping(VED)is replaced by a combination of equivalent effect terms.The motions of free VI systems are sorted into periodic motion without impact and quasi-periodic motion with impact based on the levels of system energy.Then,the impacts condition for the displacement and velocity is transformed into the system energy description.The stationary probability density functions(PDFs)of system are then obtained by utilizing the stochastic averaging of energy envelope(SAEE)based upon the assumption of lightly damping and weakly random perturbation,and the accuracy of the method is substantiated by comparing the theoretical results with those from Monte Carlo(MC)simulations.Through the presented approach,stochastic P-bifurcation is explored.It is interesting that the change of elastic modulus from negative to zero and then to positive instead of just the negative value in previous considerations has witnessed the evolution process of stochastic P-bifurcation.From the vicinity of the usual value to a wider range,the relaxation time induces the stochastic P-bifurcation in two interval schemes.Secondly,this chapter is mainly dealing with the stochastic responses of nonlinear VI system coupled with visco-elastic force excited by colored noise.By the aid of visco-elastic force treatment,the original system is approximately equivalent to a simplified system without visco-elastic force.Then,the non-smooth transformation is applied to convert the non-smooth system into a conventional smooth system,and the approximated solutions of which are derived by the stochastic averaging method.The validity of the analytical method has been verified by using MC simulation results for a biquadratic Van der Pol VI oscillator with visco-elastic behavior in detail.In addition,the stationary responses are also used to explore the stochastic P-bifurcation of this Van der Pol VI system,and the bifurcation forms of the system have been clearly exhibited in two different angles.Amplitude discussion results give bifurcation diagrams of the considered system in the parameter planes,and the corresponding stationary PDFs for the amplitude.Simultaneously,the joint stationary PDFs displayed in the dissertation state clearly that two important visco-elastic parameters,the restitution coefficient can induce the emergence of stochastic P-bifurcation as well as the structural instability of the system.Thirdly,an investigation is presented for primary resonance and internal resonance of two-degree-of-freedom(TDOF)visco-elastic system with some complex nonlinear coupled terms under randomly disordered periodic excitations,where the visco-elastic damping is described by a convolution integral.Considering such a complex system,exact solutions of visco-elastic damping and randomly disordered periodic excitation are not optimistic through the theoretical analysis method.The multiple scales method coupled with linearization techniques are absorbed to derive the approximate steady-state solutions in deterministic and stochastic resonance cases,and then,the means of numerical simulation is also applied to verify the accuracy and reliability of the analysis method previously mentioned.It is clearly informed that(random)jump,(random)saturation,and double-jumping phenomena exist in such primary resonance with internal resonance dynamical system though the description of steady-state moments as changing excitation amplitudes and frequency.In addition,the visco-elastic parameters have a greater impact on the steady-state response of the TDOF system,especially the magnitudes can accelerate or delay the occurrence of random jump and saturation,and the influence for the steady-state responses of external resonance mode is quite different from that of internal resonance mode with the changing visco-elastic parameters.We also provide a discussion for the effect of the intensity of random excitation on the TDOF system.Finally,the maximal Lyapunov exponent(MLE)and steady-state moments(SSM)are considered to investigate a Rayleigh VI system based upon TDFC and VED under bounded random excitations.By the pretreatment of both,the TDFC and VED,and the adoption of the mirror image transformation,we get an approximate equivalent system,which is further processed by the Krylov-Bogoliubov(KB)averaging method in order to seeking explicit the asymptotic formulas for MLE of the trivial steady-state amplitude solution by means of the first kind of a modified Bessel function.Meanwhile,SSM of the nontrivial solutions are calculated according to the moment method and Ito's calculus.The almost certainly stability and critical situations(bifurcation point)of the trivial solution are described in detail by combining qualitative and quantitative analysis.In addition,conclusions including the merging of the general frequency response curve and the frequency island,the analysis of conditions satisfied by the unstable boundary,and both the appearance and disappearance of the time-delayed island are observed based upon SSM of the system nontrivial solution.Stochastic jump and bifurcation are observed for the stationary joint transition probability density function(TPDF)of the system's trivial and nontrivial solutions when vary the elastic modulus from positive to negative and the displacement feedback control gain from positive to negative.