Bifurcation And Chaos Control Of Elastie Impacting Systems With Precewise-Linearity

In the kinds of mechanical systems, the system with piecewise-linearity always can be encountered and is a typical non-smooth dynamic system, so the analysis of its dynamics is the focus of the current research. It has very important significance for comprehending movement mechanism and analyzing the global properties by further investigating on the system with piecewise-linearity. The chaotic phenomenon only occurs in non-linear system. The chaotic vibration in the vibro-impact system can severely affect the performance and operational reliability of the mechanical system. So controlling the chaotic vibration of the vibro-impact system is of important significance in engineering project. In this thesis, the non-linear dynamical behavior of the system with piecewise-linearity is studied by integrating theoretical analysis and numerical simulations. Based on research of the non-linear dynamic system and understanding of the chaotic vibration, the chaotic control strategies of the non-smooth dynamic system are explored. The main content is organized in the following manner:1. Some research results, newest development trends and problems to be resolved is summarized and interpreted. The definitions and the analysis methods on non-smooth dynamical systems are introduced precisely, such as the movement properties of the system analyzed by the Poincare surface of sections; the properties of stability and chaotic motion by using Lyapunov exponents and Lyapunov dimension, the types of non-smooth bifurcations are distinguished by the eigenvalues distributed on the unite circle of the interfaces; the analysis on right-discontinuous systems based on differential conclusions theory, and so forth.2. The dynamics behaviors of transversal periodic motions of a single-degree-of-freedom elastic impacting system with piecewise-linearity is studied. The period-doubling bifurcation of periodic motions of the piecewise linear system is investigated by the salatio matrix and the Floquet theory. The period-doubling bifurcations and chaotic behaviors in the non-smooth system are further investigated by means of numerical simulations. The bifurcation and chaos of traversal periodic motions of a two-degree-of-freedom elastic impact system with piecewise-linearity is investigated. The salvation matrix is given out at the switching boundaries and the Neimark-Sacker bifurcation point of periodic motions of the system is investigated by the numerical calculation. Numerical results demonstrate the existence of Neimark-Sacker bifurcation and subharmonic bifurcation in the non-smooth system.3. The control of bifurcation and chaotic motion of the single-degree-of-freedom non-smooth system with piecewise-linearity is investigated by using two control strategies. Delayed feedback control strategy is the first control strategy which is proposed to control chaos. The bifurcation diagram and Poincare maps are used to analyze dynamic characters of the system. The obvious controlling effect is obtained while the delayed feedback control strategy is used to control chaos. It is shown that the delayed feedback control in fact is a perturbation terms, with suitable feedback controlling parameters R1, R2 and delayed timeτ, which can invert a chaotic motion to a regular periodic motion and make the chaos motion to be controlled.A periodical exciting force is investigated for controlling the chaotic motion of the vibro-impact system, and this second control strategy proposes to substitute the external sine force for the periodical exciting force. The periodical exciting force is produced and controlled easily in the engineering project. So this section proposes a method to suppress chaotic motion by applying an external periodical force to the single-degree non-smooth system. The strategy of the periodical exciting force weakens the chaotic motion of the system by means of the added control term. When periodical driving force and certain unstable periodic motion of the chaotic system bring the phenomenon of the sympathetic vibration, the certain limit cycle from unstable limit cycle of the system resonates to the external driving signal and it is stabled. So the chaotic motion is controlled.