含質點鉛垂振動旋轉橢圓管的規則與渾沌動力分析及渾沌控制

Abstract

本篇論文討論一含質點旋轉且受鉛垂振動橢圓管的詳細動力分析。經由系統本身受到週期性的振動因而展現出規則與渾沌運動。運用李亞普諾夫直接法和切達耶夫定理可以得到系統相對平衡位置的穩定性和不穩定性。藉中心流形定理可以得到系統在臨界情況下的穩定條件。參數變化對系統的影響可以經由分歧圖與參數圖表現出來。由許多數值分析的結果，例如相平面圖、龐加萊映射、時間響應和功率譜法，可以觀察其週期與渾沌行為。利用李亞普諾夫指數和李亞普諾夫維度可驗證系統之渾沌現象。除此之外，本文更進一步利用開迴路控制，例如外加定力矩、外加週期力矩、外加週期脈衝，與閉迴路控制，例如延遲迴授控制、bang-bang控制、適應控制、最佳化控制，來控制系統從渾沌現象回到週期現象。

The thesis 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, time history 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. On the other hand, open-loop strategies such as using constant torque input as control torque, using periodic torque input as control torque, using periodic impulse input as control torque, and close-loop feedback methods such as delayed feedback control, bang-bang control, model referenced adaptive control, optimal control, are used to control our system from chaos to order.