Dynamic Study Of A Two-Degree-of-Freedom Vibro-Impact System

In this dissertation, a two-degree-of-freedom vibro-impact system with damper is studied. At first, existence and stability of periodic motion of the above system are deduced. Then, we discuss the effect of system parameters on periodic motions through numerical simulations. Moreover, the method for calculating the spectrum of Lyapunov exponents of the impact-vibrating system is obtained. At last, using the technique of the damper coefficient feedback control, the vibro-impact system can be controlled from chaotic behavior to stable period-1 orbit and period-2 orbit. This research can rather completely reveals the complicated dynamic behaviors of the vibro-impact system aforementioned. The results and the main contributions of the dissertation are as following.Chapter 1 reviews some achievements of the study on vibro-impact systems, as well as the developments of chaos and chaos controlling. Furthermore the calculation methods for Lyapunov exponents of smooth dynamical system, and the recent research results of Lyapunov exponents of non-smooth dynamical system are presented. Finally, the mechanical model of the vibro-impact system and the equations of motion studied in this dissertation are deduced.In chapter 2, by using the Poincare map method, we get an existent criterion of single impact period-n motion in the vibro-impact system. It is shown that our conclusions are valid through numerical simulations. Moreover, the stability of periodic motions is considered and the corresponding Jacobian matrix is derived. After that, the local bifurcation and global bifurcation, and the influence of changing parameters on its periodic responses are investigated by numerical simulations. As a result, we find that more disciplinary periodic motions may appear in the vibro-impact system under strong damping, feeble harmonic excitation, small ratio of quality and quite big coefficient of restitution.Introducing the local maps, we obtain the method for calculating the spectrum of Lyapunov exponents of the two-degree-of-freedom vibro-impact system through the Gram-Schmidt orthogonalization and normalization method. It is shown that our conclusions are valid by numerical results. Making use of the method of damper coefficient feedback control, the chaotic behavior can be suppressed in the vibro-impact system. It is clear that selecting appropriate gain coefficient, the chaotic behavior can be controlled to stable period-1 orbit and period-2 orbit by means of simulation in chapter 3.