多連體動力分析的拘束穩定技巧之研究

Abstract

在本論文中，我們從微分代數方程式的觀點來發展多連體動力/機動分析的數值工具。首先，我們利用輸入-輸出回授線性化與微分幾何的理論證明了拘束分離準則(Constraint Separation Principle)，這個準則間接證明了所有輸出回授型的拘束發散穩定(Constraint ViolationStabilization)技巧都是穩定的。利用此準則與控制理論我們發展出三種新的拘束發散穩定技巧：(1) Baumgarte 技巧的最佳化; (2) 改良式可變結構控制技巧; (3) 速度校正技巧。 在多連體動力的數值分析中，數值積分器是不可或缺的工具，因此，本論文的第二個工作就是根據傳統的 Adams-Bashforth-Moulton 估測-校正法，我們提出了改良式的Adams-Bashforth-Moulton 估測-校正法，來提高數值積分器的準確度，理論與數值模擬皆證明了新的積分器比舊的積分器提高了將近一個次數(order)的準度，同時，根據數值計算，新的積分器比舊的積分器更具穩健性。 本論文的第三個成果是提出一個拘束守恆積分器來解決在解尤拉四參數(quaternion equations)時所遭遇到的拘束發散現象。此拘束守恆積分器不需另外的校正技巧，就能自動且準確地使 '尤拉參數的範值(norm)等於1'。這將使模擬的結果更準確也更符合物理意義。 最後，我們利用一些數值模擬的例子來證明本論文所提出之方法的可行性，所有的模擬都是利用INTEL PENTINUM的個人電腦所完成的。這些例子的模擬結果證明了本論文提出方法的可行性與優越性。

Computational Procedures for kinematic and dynamic analysis of three-dimensional multibody dynamic (MBD) analysis are developed from differential algebraic equations (DAEs) viewpoint. Firstly, based on the input-output feedback linearization theory, a global constraint separation principle is proposed to guarantee the stability of the output-feedback-like constraint violation stabilization schemes. Furthermore, by using this proposed principle and control theory, three new constraint stabilization technique, i.e., optimal Baumgarte technique, modified variable structure control (M.V.S.C.) technique, velocity correction method (V.C.M), are developed to increase the accuracy for MBD analysis. Secondly, based on the traditional Adams-Bashforth-Moulton predictor-corrector (ABMPC) method, a modified Adams-Bashforth-Moulton predictor-corrector (MABMPC) method is developed to increase the accuracy and robustness of the numerical integrators that are needed in MBD analysis. This proposed numerical integrator is accurate about one order of magnitude than the ABMPC method by only making a small modification. Moreover, we conclude this proposed integrator provide a larger stability region for MBD systems from numerical analysis. Thirdly, we have developed a family of constraint preserving integrator for solving quaternion equations that is need to solve in spatial dynamic/ kinematic analysis. The inherent constraint, summation of norm of Euler parameters equal unity, involved in quaternion equations can satisfy exactly without any additional correction procedure. Therefore, the accuracy can be improved and the computational costs can be reduced in the multibody dynamic/kinematic analysis. To evaluate the computational procedures developed in the present work, many numerical examples are implemented in INTEL PENTIUM personal computer. These test examples demonstrate the robustness and efficiency of the proposed constraint stabilization techniques and numerical integrators.