測速器的非線性動力學及渾沌控制

Abstract

本篇論文探討一帶阻尼的旋轉測速器受到鉛垂振動所產生的動態行為。經由不同的解析和數值的方法可以得到系統的各種特性。由李亞普諾夫直接法可求得系統平衡位置的穩定條件。用中心流形理論可得到系統平衡位置處於臨界情況的穩定條件。應用不同的數值分析如：相位圖、功率譜法、龐加萊映射法，可觀察其週期解及渾沌現象。參數變化對系統的影響可以由分歧圖和參數圖表現出來。同時由李亞普諾夫指數和李亞普諾夫維度可驗證系統的週期和渾沌現象。最後利用外加的定值力矩、週期外力矩、延遲回授控制、適應控制、最佳控制、Bang-Bang 控制和外加週期脈衝控制方法有效的改變渾沌現象。

The dynamic behaviors of a rotational tachometer with vibrating support are studied in the thesis. Both analytical and computational results are used to obtain the characteristics of the system. The Lyapunov direct method is applied to obtain the conditions of stability of the equilibrium position of the system. The center manifold theorem determine the conditions of stability while the system in critical case. By applying various numerical analyses such as phase plane, Poincare map, time history and power spectrum analysis, a variety of periodic solutions and phenomena of the chaotic motion is observed. The effects of the changes of parameters in the system can be found in the bifurcation diagrams and parametric diagrams. By using Lyapunov exponents and Lyapunov dimensions of the periodic and chaotic behaviors are verified. Finally, various methods, such as addition of a constant torque, addition of a periodic torque, delayed feedback control, adaptive control, Bang-Bang control, optimal control and addition of a periodic impulse are used to control chaos effectively