旋轉載具上帶回饋之速率陀螺儀的渾沌與非線性動力分析

Abstract

本論文將對一裝置於飛行器上的單軸速率陀螺儀作詳細的動力分析。首先，探討一飛行器相對於輸出軸作穩態轉動且分別在輸入軸與自轉軸作簡諧轉動之兩種情況下，單軸速率陀螺儀的動力分析。這兩個系統皆為參數激勵強非線性耗散系統。分別採用具有快速伽遼金處理之諧波平衡方法與增量諧波平衡方法及福洛開理論對系統周期解的穩定性與分歧行為作分析。並進一步對系統研究，以相軌跡，龐加萊截面，平均功率譜與李雅普諾夫指數等數值方法來觀察其各種分歧與渾沌等行為。並利用改良式內插胞映射方法研究周期解之吸引區域及碎形結構作全局分析。 更進一步研究帶線性迴餽單軸速率陀螺儀之三維非線性系統。利用李雅普諾夫直接法得出該飛行器相對於自轉軸作不確定轉動ωz(t)且輸出軸作穩態轉動時之穩定條件。當飛行器相對於自轉軸作穩態轉動時，系統為自治系統，利用中心流形理論，正規形理論與分歧理論分析在兩個餘維數退化點附近之定性行為且發現系統存在數種不同的分歧行為如Hopf與pitchfork等分歧，於全局分歧分析可得鞍點連結，並以數值方法驗證分析結果。另外也以Melnikov方法在理論上證實系統渾沌運動的存在。並在適當頻率下對參數微擾而能有效地抑制渾沌運動。

In this dissertation, a detailed analysis is presented of a single axis rate gyro mounted on a space vehicle. First, the dynamics is analyzed of a single axis rate gyro mounted on a space vehicle undergoing harmonic motion about its input or spin axis and steady angular velocity about output axis. This is a strongly nonlinear damped system subjected to parametric excitation. The harmonic balance method with fast Galerkin procedure and the incremental harmonic balance method with the multivariable Floquet theory are applied to analyze the stability of periodic attractors and the behavior of bifurcation. Besides, phase portraits, Poincare maps, average power spectra, bifurcation diagrams, parametric diagrams, Lyapunov exponents and fractal dimensions are presented to observe Hopf bifurcation, symmetry breaking bifurcation, period doubling bifurcation, interior crisis and chaotic behavior. The modified interpolated cell mapping technique (MICM) is also used to study the basins of attraction of periodic attractors and fractal structure. Further, an analysis is presented for a single axis rate gyro subjected to feedback control mounted on a space vehicle that is spinning with uncertain angular velocity ωz(t) about its spin axis of the gyro. The stability of the nonlinear nonautonomous system is investigated by Lyapunov stability and instability theorems. When ωz is steady, the system is autonomous. The dynamics of the resulting system is examined on the center manifold near the double zero degenerate point by using center manifold and normal form methods. There exist a few kinds of bifurcations such as pitchfork and Hopf bifurcation for local bifurcation analyses, a saddle connection bifurcation for global analyses. The numerical simulations are performed to verify the analytical results. The criteria for the existence of chaos by using the Melnikov technique are given also and chaotic motions are suppressed by a small parameter perturbation of suitable frequency.