Bifurcation And Chaos Motion Of A Vibratory System With Clearance

In the field of engineering, there will be a collision or impact between the component of the mechanical system with clearance. This behaviour will have a serious influence on the mechanical system's dynamic characteristic, security, fatigue life and so on. According to the characteristic of the collision, people can also use it to manufacture mechanical equipment to achieve the purpose of production. Therefore, the research of the vibration system with clearance have the engineering practice significance.The actual mechanical system is used as the engineering background in this paper. By simplifying three vibration systems with clearance, the mechanical models are established. The analytical solution of the three models is obtained by using the analytical method. And the existence conditions of the periodic response are determined by the analysis. However, before solving the analytical solution, there is a need to use the modal analysis method to decouple the coupling system. According to the disturbed motion of the system, the Poincaré mapping and its Jacobi matrix are given. The dynamic behavior of each of the mechanical models is simulated by MATLAB software programming which is based on Poincaré mapping method. At the same time, the simulation results are analyzed, thus selecting the best parameters as the theoretical reference for the optimization of the mechanical system with clearance. In this paper, the research analysis of the three mechanical models is as following:Firstly, a two-degree-of-freedom bilateral vibro-impact system is set up, which can be regarded as the abstraction and simplification of the collision between kinematic pair elements and the wheel-rail lateral impact. The results of the numerical simulation show that there are torus-doubling and Hopf bifurcation in the mechanical model. In the torus doubling bifurcation, it is revealed that the evolution of 2T~1 torus routes to chaos motion. When the Hopf bifurcation appears in the system, there will be T~1 torus in the Poincaré section. By comparing the bifurcation diagrams under different parameters, it is found that the bifurcation characteristic of the system will have a great change with the change of some parameters, but some are almost no effect.Secondly, the vibrating compaction machine is used as the engineering background, and a three-degree-of-freedom vibro-impact system is established. The typical Hopf bifurcation, period doubling bifurcation and torus-doubling bifurcation exist in the system, and there are also codimension two bifurcation Hopf-flip and Hop-Hopf. When the system becomes unstable, there are different paths which can be evolved into multi-period motion, quasi-periodic motion and chaos motion. The evolutionary process shows the rich and wonderful dynamic behavior, such as T~2 torus, 4×T~1 torus, strange chaotic attractor, etc.Finally, a three-degree-of-freedom bilateral vibration system is established, which is based on the coupler and draft gear. The bifurcation behavior of the system includes Hopf-Hopf bifurcation, torus bifurcation and Hopf bifurcation of the periodic motion. In the process of the chaos evolution, the T~2 torus may be formed by torus bifurcation, and may be also caused by Hopf bifurcation of the multi-periodic motion. Meanwhile, the effect of two different parameters on the dynamic behavior of the system can be observed by means of the three-dimensional bifurcation diagram.