Nonlinear Dynamic Behaviour Of The Rotating Pendulum Linked A Clamped And Supported Spring

In this article,a kind of novel pendulum modle is proposed and established. Accoroding to the mechanics stucture of the modle, it is named Rotating Pendulum Linked a Clamped and Supported Spring. This kind of pendulum is a typical cylinder pendulum, and the system displays smooth and discontinuous dynamics behaviour depending on the value of a system parameter. The nonlinear dynamics behaviour is analized and computed from the pualitative and the puota which used the nonlinear dynamics theory method and the numerical method.The concrete arrangement of this article as follows:In the first chapter, the development of the pendulum, the research history and the present situation of the non-linear dynamics, and innovation spots are introduced.In the second chapter, the modle of the Rotating Pendulum Linked a Clamped and Supported Spring is established, at the same time, the dynamic motion differential equation is given, then using the Matlab, the diagram of the system's balance points, the analy of the resume strength fuction, the potential fuction, the phase diagram and the attractor basins are obtained. The conclution indicates that the system not only has the smooth dynamics behaviour but also contains the discontinual dynamics behaviour.In the third chapter, the puantitative analysis of the smooth Rotating Pendulum Linked a Clamped and Supported Spring is given by investigating the approximate system. The threshold value curve is obtained by the Melnikov function analysis forecast to the system of homoclinic orbits, the system of cylinder homoclinic orbits and the coexistence of two kinds of homoclinic orbits. In the end, the nrmerical simrlation is done by using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.In the forth chapter, the puantitative analysis of the discontinual Rotating Pendulum Linked a Clamped and Supported Spring is given. It contains the analysis of restoring nonlinear force, the potential function and the phase portraits. The conclution of the analysis indicates that the system has the complex dynamics behaviour of cylinder homoclinic-like orbits and homoclinic-like orbits. In the end, the nrmerical simrlation is done using the Dynamics software, then the existence of the cycle solution and the chaos solution is cinfirmed.In the fifth chapter, the cylinder approximate system under the periodic disturbances present Hopf bifurcation, two closed orbit bifurcations, homoclinic orbit bifurcation and cylindrical homoclinic orbit bifurcation.The bifurcation of the system is solved using the Melnikov function and Submelnikov function ,then we use the Matlab software to obtain global bifurcation diagram and Dynamics software for numerical validation.In the sixth chapter, this chapter is mainly focused on the study of discontinuous limit case, In the discontinuous limit regime, firstly, we obtained the solution of special orbits by using smart measurd; secondly, we make the Melnikov analysis to detect the discontinuous heteroclinic-orbits tangling under the perturbation of damping and driving.at last, the bifurcation of discontinuous system can be investigated under the periodic function.In the seventh chapter, the main work of this article is summarized, and then predicts the next step work of the rotating pendulum linked the clamped and supported spring. It can be researched on the theoretical analysis and the project application.Appendix1. The solutions of cylinder homoclinic orbits and homoclinic orbits;2. The solutions of cylinder homoclinic-like orbits;3. The solutions of cylinder heteroclinic-like orbits;4. The visual illustration procedure is compiled using the Matlab, the visible animation of the cycle solution and the chaos solution are vividly. The essence of the procedure lays in using the law of cosines and the corner formula.