Stability And Bifurcation Of A Delayed Vibration System With Time Delay State Feedback

A vibration system with delayed spring force and delayed state feedback is studied. Linear stability analyses of this system are considered and stability conditions of the trivial equilibrium are presented. The effect of the time delay on the system dynamics is also discussed. According to stability criteria and the bifurcation analysis, it is shown that the trivial equilibrium may lose stability via a pitchfork bifurcation, a Hopf bifurcation, or one of three types of codimension-two bifurcations. Results from theoretical analysis are examined using numerical experiments.Firs of fall, the background of a vibration system with delayed spring force and delayed state feedback is introduced. In the same time, the theses introduced some characteristics of the delay dynamic systems with time-delay state feedback and the research status of delay dynamic system. Then we study the dynamics of following delayed system on a Duffing oscillator thinking about "switch",where c≥0,k>0,0≤β<<1,s>0,A≥0 andτ≥0. Time delay s is called internal timedelay in this paper. This equation serves as a simplified mathematical model of many dynamical systems. For example, it can be used as the model of the nonlinear regenerative effects in metal cutting if considering nonlinear delayed spring force, and include the flexible structures with delayed spring force equipped and a displacement feedback to tune the natural frequency. We focus on the effect of time delay to the dynamics of this system. Simple stable criteria of trivial equilibrium are given according to pattern ofλ=iw curve in A-τplane. We developed an analytical approach to consider the stability and bifurcation of the system with the delayed spring force and delayed state feedback, and draw some conclusions at last.Simple criteria about the stability region for the trivial solution are presented. An analysis on the steady bifurcation, Hopf bifurcation and three types of codimension-two bifurcations of system is made when the stability of trivial solutions is lost, and it is demonstrated by numerical simulation.