含質點鉛垂振動旋轉圓管的規則與渾沌分析及渾沌控制

Abstract

本篇論文研究一含質點的鉛垂振動旋轉圓管系統之詳細規則與渾沌動力分析及渾沌控制. 旋轉圓管系統受到鉛垂簡諧週期振動及阻尼, 得到豐富的動力行為. 運用李亞普諾夫直接法, 得到系統相對平衡位置的穩定性. 藉中心流形定理, 一個餘維數的分歧分析應用於自治系統退化之後, 並發現系統的Hopf分歧行為. 應用多種數值分析方法, 如 相平面, 龐加萊映射, 時間響應, 功率譜法, 可觀察其週期解及渾沌行為. 參數的變化對系統的影響可以由分歧圖及參數圖來顯示. 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. 最後, 運用幾個控制方法來控制渾沌現象至穩定的週期行為, 如 外加定力矩, 外加週期力矩, 外加週期脈衝, 外力迴授控制, 延遲迴授控制, 適應控制, Bang-Bang控制, 及最佳控制.

The thesis is to present the detailed dynamic analysis of a particle in a vertically vibrating and rotating circular tube. By subjecting to a harmonic periodic vibration and damping force on this nonlinear system, enriched dynamics behaviors of the nonlinear system are presented. By applying the Lyapunov direct method, the conditions of stability or instability of relative equilibrium position can be determined. A codimension one bifurcation analysis for the autonomous system is carried out near the degenerate point, it is founded that Hopf bifurcation occurs in the system by center manifold theory. And by applying various numerical results, such as phase portrait, Poincare map, time history and power spectrum analysis and the behavior of the periodic and chaotic motion can be presented. The effects of the change of parameters in the system can be fond in the bifurcation diagrams and parameter diagram. Further, by using Lyapunov exponents and Lyapunov dimensions we can verify the chaotic behavior. Finally, eight methods, namely, the adding of constant torque, the adding of periodic torque, the adding of impulse, external force feedback control, delayed feedback control, adoptive control, bang-bang control, optimal control, are use to control chaos effectively to periodic orbit .