Regular and chaotic dynamic analysis for a vibratically vibrating and rotating elliptic tube containing a particle

The paper is to present the detailed dynamic analysis of a vertically vibra ting and rotating elliptic tube containing a particle. By subjecting to an external periodic excitation, it has shown that the system exhibits both re gular and chaotic motions. By using the Lyapunov direct method and Chetaev' s theorem, the stability and instability of the relative equilibrium positi on of the particle in the tube can be determined. The center manifold theor em is applied to verify the conditions of stability when system is under th e critical case. The effects of the changes of parameters in the system can be found in the bifurcation and parametric diagrams. By applying various n umerical results such as phase plane, Poincare map and power spectrum analy sis, a variety of the periodic solutions and the phenomena of the chaotic m otion can be presented Further, chaotic behavior can be verified by using L yapunov exponents and Lyapunov dimensions.