Research On The Stochastic Dynamics In Typical Vibro-impact Systems And Real-power Vibration Isolation Systems

Complicated dynamical behaviors of nonlinear systems,together with the related problems,have been important subjects of scientific studies for years.As two kinds of typical nonlinear systems,vibro-impact systems and real-power vibration isolation systems have special nonlinear structure and complex dynamical behaviors.Thus,the research of these kinds of systems has become one of the hotspots and also attracted numerous researchers.Hence,the stochastic responses of vibro-impact systems are mainly considered under external excitation and parametric excitation,respectively.Then,the effects of the time delay on the dynamical behaviors of the systems are also discussed.Eespecially,complex dynamical properties of the elastic impact system and the vibration isolation system with real-power nonlinearities under delayed feedback control are investigated.The main results of this dissertation are as follows1.The multi-valued responses of a nonlinear rigid vibro-impact oscillator with a one-sided barrier subjected to random narrow-band excitation are studied.The dynamical behaviors of the system are analyzed by using the non-smooth variable transform and the KrylovBogoliubov averaging method.And the method of moment is applied to obtain the iterative calculation equation for the mean square response amplitude.Excitation amplitude,nonlinearity intensity,restitution coefficient,damping parameters,especially the distance between the system's static equilibrium position and the barrier can lead to the triple-valued responses under certain case.In some conditions,rigid impact system may have two or four steady-state solutions,which is an interesting phenomenon for the impact system.The unstable region is one uniform part,however,it is divided into two parts under smaller nonlinearity intensity.Moreover,we find that as random noise intensity increases,the pervasion of the phase trajectories will be strengthened,and finally it will destroy the topological property of the phase trajectories.2.This chapter is devoted to investigate the stochastic response of a nonlinear rigid vibroimpact system under parametric excitation.Based on the Krylov-Bogoliubov averaging method,the largest Lyapunov exponent which determines the almost sure stability of the trivial solution is derived,in which the modified Bessel function of the first kind is utilized.Results show that the largest Lyapunov exponent is different from the one in the system without impact.Meanwhile,the backbone curve and critical equation of unstable regions are also derived for the deterministic case.Then,the first and second order non-trivial steady-state moments of the system are considered.The frequency island phenomenon is found.Finally,the steady-state probability density is analyzed for parametric resonance via finite difference method.The basic jump phenomenon is found.3.Based on the method of multiple scales,the principal resonance response of a stochastically driven elastic impact system with time-delayed cubic velocity feedback is investigated.The steady-state response and its stability are analyzed both in deterministic and stochastic cases.It is shown that for the case of the multi-valued response with frequency island,only the smallest amplitude of the steady-state response is stable,which is different from the case of the traditional frequency response.Then,a design criterion is proposed to suppress the jump phenomenon.The effects of the feedback parameters on the steady-state responses,as well as the size,shape,and location of stability regions are studied.Furthermore,appropriate feedback gain and time delay are derived in order to suppress the amplitude peak and govern the resonance stability.4.The time-delayed cubic velocity feedback and the real-power form of restoring and damping forces are combined to improve the performance of the vibration isolation.The primary resonance,dynamical stability and energy transmissibility are studied for a forced vibration isolation system under base excitation.The equivalent damping is proposed to interpret the effect of feedback control on the dynamical behavior of the real-power isolation system.In this regard,the isolation system indicates the softening behavior for under-linear restoring force and hardening behavior for over-linear restoring force.And multi-valued responses,especially five valued responses,are found under under-linear restoring force.To verify this result,the stability boundaries are characterized and presented excellent agreement with the responses.Then,in order to avoid the jump phenomenon,an analytical criterion is derived and confirmed by the numerical simulation.Finally,the effects of the system parameters on the energy transmissibility are discussed.Results show that the strategy proposed in this chapter is practicable and feedback parameters are significant factors to improve the isolation effectiveness for the real-power isolation system.5.The principal resonance of a viscoelastic isolation system with time delay feedback and power-form restoring force subjected to narrow-band random parametric excitation is investigated.The largest Lyapunov exponent which determines the almost sure stability of the trivial solution is derived,in which the modified Bessel function of the first kind is utilized.Then,the first and second order non-trivial steady-state moments of the system are considered.Meanwhile,the analytical expressions of the corresponding critical bifurcation values are also derived.With the purpose of suppressing the peak amplitude and improving the vibration stability,suitable time delays are determined by the frequency response together with stability conditions.Finally,by solving the corresponding FPK equation,the phenomenon of stochastic jump is analyzed for principal parametric resonance.