Anti-control Of Hopf Bifurcation In Nonlinear Double-mass Vibrator System And Its Chaos

Nonlinear vibrator is a very widely used class of machinery,such as industrial raw material of transmission machine,aggregate screening machine,etc.Therefore,the research on this kind of machinery is of great practical value.Whether the vibrating machine works well depends on whether the machine can be stable and for a long time in a vibrating state.In order to achieve this goal,we use the anti-control method to generate the limit cycle by Hopf bifurcation,and carry on the stability research and chaos control.The main work is as follows:1.The research status of nonlinear vibration machine,anti-control,chaos control and synchronization are reviewed at home and abroad.2.The dynamical equations of the system are established by the second Lagrange equation,and designed the washout-filter controller to control it,the system equation after the control is obtained.3.Explicit critical criterion of Hopf bifurcation,which don’t directly depend on the eigenvalue calculation,are used to obtain the linear control gain,which causes the system to generate Hopf bifurcation at a given parameter point.In order to generate a stable limit cycle,the first Lyapunov coefficient of the limit system on the central manifold at a given parameter point is calculated,and the nonlinear control gain of the control system is determined.4.The control effect is tested by Hopf bifurcation amplitude expression.The numerical simulation of the system using MATLAB software shows that the nonlinear control gain has a positive correlation with the control of the system amplitude.5.A periodic excitation force is applied to the original system,and the dynamical equation is established again.After the system parameters are simplified,the chaotic state of the system is obtained by period-doubling bifurcation.The chaotic state is described by programming the phase trajectory,displacement time history curve,power spectrum,Poincaré mapping and maximum Lyapunov exponent.The global bifurcation diagram and Lyapunov exponent spectrum of the system are obtained by MATLAB software.Through these two graphs,the complete process of system bifurcation is shown in detail.6.Chaos is controlled by the method of piecewise function x|x|and external constant force method and linear feedback method.The control bifurcation diagram and phase trajectory of the system are simulated by MATLAB.The numerical simulation results show that these control methods can well control the chaos and get better control results.It is proved that these control methods have the advantages of high control efficiency,simple process and convenient use.7.The chaotic system is synchronized by nonsingular TSM method.The chaotic system as a driving system,The sliding surface and the controller are designed for the response system.The results of MATLAB numerical simulation show that the phase trajectory of the response system and the phase trajectory of the driving system are completely consistent in a short time.This shows that the response system is quickly converged on the sliding surface by the designed controller,and finally the synchronization with the drive system is completed.