Non-smooth Analysis In The Cantilever Beam System With Impacts

This paper analyzes and summarizes the research methods of non-smooth dynamical systems, and studies the non-smooth dynamical behavior of a cantilever beam collision system. Non-smooth dynamical systems are different from the general smooth systems, due to a large number of non-smooth factors (the impact of the collision, dry friction, etc.), will give non-smooth systems and smooth systems have similar kinetic the behavior, there are a large number of complex dynamical behaviors. For example, a variety of coexistence of attracting sets collision and electrical systems, and many border collision bifurcations (such as grazing, jump, switch, etc.). Because its movement is not continuous, the phenomenon causes great difficulties for the system. So, the paper directly uses the numerical simulation method to study the behavior of the dynamical behavior of the cantilever beam system with unilateral collision dynamical s of the cantilever system.The main research work is as follows:(1) To review the theory and its practical significance of the topics, to summarize the history and the present status of development of non-smooth dynamical systems, to point out the difficulty of this kind of problem the focus.(2) To summarize concepts and research methods of the non-smooth dynamical systems. Mainly summarize the research methods that scholars have proposed, such as the Poincare section method.(3) The paper carries on numerical simulation for the cantilever beam system with unilateral collision, and uses the phase diagram, bifurcation diagram, Poincare map in order to study above systems, and uses the mathematical software and fourth-order Runge-Kutta method so as to carry on numerical simulation for the original system, and found that the unilateral collision cantilever beam system in addition to having a smooth the dynamic behavior of the system, such as movement and the cycle of periodic motion, period doubling and chaos appear alternately and so on, as well as own unique and exotic kinetic behavior, such as jump.(4) For double-sided collision cantilever system, the paper uses of the chart, the phase diagram, Poincare sections, and bifurcation diagram to analysis of the dynamic behavior of the system, too.