Suppression For Chaos And Basin Erosion Of A Nonlinear Relative Rotation System By Delayed Position Feedback

Shaft rotation is a common motion transmission form in daily life and industrial production. It is also a typical power transmission form in the mechanical systems. However, because of physical properties of the shaft system and external factors, especially the nonlinear factors influencing the system, there are nonlinear relative correlations between input and output of transmission system which is called nonlinear relative rotation system. The complex dynamics of a classical nonlinear relative rotation system and the suppression for chaos and basin erosion of the system by delayed position feedback control is investigated in the thesis.First, the kinetic model of two rotators relative rotation system with nonlinear stiffness and nonlinear damping is established, also the kinetic model of the time delayed control system with time delayed position feedback control is established. The two models are the foundation of further study.Secondly, the excitation amplitude threshold of chaotic motion and the basin erosion of the uncontrolled relative rotation system is obtained by the Melnikov method. Then, Melnikov method is applied in time delayed control systems. The condition of Hopf bifurcation of time delay control system is discussed for estimating the available range of control parameters for the Melnikov method. Then the necessary condition of the global bifurcation of time delayed control system is obtained.Finally, chaotic motion and the basin erosion of the uncontrolled relative rotation system and the evolutions of dynamical behavior in the delayed control system with control parameters are presented numerically by 4th order Runge-Kutta approach and point-to-point mapping method, which verifies the validity of delayed position feedback control.It is found that chaotic motion and basin erosion can be effectively suppressed and effective area of the safe basin can be expanded by time delay position feedback, when the gain is positive and time delay is short. While simple motion of relative vibration system can be transformed to chaos when the gain is negative or time delay is long. It indicates that time delay position feedback control in the original dynamical system has two-direction control effect to dynamical behavior.