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

Abstract

The paper is to present the detailed dynamic analysis of a vertically vibrating and rotating elliptic tube containing a particle. By subjecting to an external periodic excitation, it has shown that the system exhibits both regular and chaotic motions. By using the Lyapunov direct method and Chetaev's theorem, the stability and instability of the relative equilibrium position of the particle in the tube can be determined. The center manifold theorem is applied to verify the conditions of stability when system is under the critical case. The effects of the changes of parameters in the system can be found in the bifurcation and parametric diagrams. By applying various numerical results such as phase plane, Poincare map and power spectrum analysis, a variety of the periodic solutions and the phenomena of the chaotic motion can be presented Further, chaotic behavior can be verified by using Lyapunov exponents and Lyapunov dimensions.