兩自由度振動支座物理擺非線性動力分析及渾沌控制

Abstract

本篇論文探討一兩自由度振動支座物理擺之動態行為。由解析和數值的方法可以得到系統的各種特性。相對平衡位置的穩定條件可由李亞普諾夫直接法得到。強非線性系統的穩定與不穩定週期解可由增量諧波平衡法得到。由許多數值分析的結果，如相平面圖、龐加菜映射、時間響應和功率譜法，可觀察其週其解及渾沌行為。參數化對系統的影響可以由分歧圖表現出來。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。以修正式內插包映射法對吸引區邊界及碎形結構作全局分析。本文進一步利用外加定力矩，外加週期外力，外加 dither 信號，延遲回授控制，適應控制及Bang-Bang控制法有效地改變渾沌現象。

The dynamic behaviors of a two-degrees of freedom physical pendulum with vibrating support are studied in this thesis. Both analytical and computational results are employed to obtain the characteristics of the systems. By using the Lyapunov direct method the conditions of stability of the relative equilibrium position can be determined .The incremental harmonic balance method (IHB) is used to find the stable and unstable perodic solutions for the strongly nonlinear system. By applying various numerical results such as phase plane, Pioncare map, time history and power spectrum analysis, a variety of periodic solutions and phenomena of the chaotic motion is presented. The effects of the changes of parameters in the system can be found in the bifurcation diagrams. Further, Chaotic behavior is verified by using Lyapunov exponents and Lyapunov dimensions. The modified interpolated cell mapping method (MICM)is used to study the basins of attraction of periodic attarctors and fractal structure. Besides, additions of a constant torque, addition of a period torque, addition of dither signals, delayed feedback control, adaptive control, and bang-bang control are to used controlling of chaos phenomena effectively.