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標題: 混合微分代數方程式數值積分法之研究
Numerical Integration of Mixed Differential-Algebraic Equations
作者: 陳世昌
關鍵字: 混合微分代數方程式;Mixed Differential-Algebraic Equation;多體機械系統動態模擬;拘束穩定法;改良型可變結構控制法;Dynamic Simulation of Multibody Mechanical System;constraint stabilization method;modified variable structure control;Post-stabilization;CPU time
出版社: 機械工程學系
使用Lagrangian法推導拘束多體機械系統運動方程式,所得到的是一組index-3的混合微分-代數方程式(differential-algebraic equations, DAE)。使用數值方法求解此DAE時,通常必須先減少其index以進行數值求解。然而為了減少index而直接對DAE中的拘束方程式進行微分會造成數值積分的不穩定,因此學者們提出了許多不同的數值積分穩定法解決此問題。其中以Baumgarte提出的拘束穩定法(constraint stabilization method)最常被使用。此法雖然簡單,但缺點是其參數的選擇與積分步距以及所使用的積分法有關,難以求得最佳參數。不過在後進學者的努力下,現已發展出許多提供參數選擇的方法。
本文針對二種常用的多步數值積分法:Adams數值積分法以及Backward Differentiation Formulae數值積分法(BDF),配合三種數值積分穩定法:拘束穩定法、改良型可變結構控制法(modified variable structure control,MVSC)以及Post-stabilization法,比較其應用於多體機械系統動態模擬時的拘束誤差以及計算效率。比較結果證實了Post-stabilization法不但擁有拘束穩定法的簡易性,其拘束誤差以及計算效率亦十分出色,較適合用於需要高精確度的多體機械系統數值模擬。

An index-3 mixed differential-algebraic equation (DAE) is obtained when using Lagrangian method to derive the equation of motion of constrained multi-body system. When solving this DAE numerically, some index reduction techniques have to be used before numerical discretization can be applied. However, direct differentiation of the constraints introduces instability. To cope with this problem, researchers have proposed many stabilization methods. Among these methods, constraint stabilization method proposed by Baumgarte is considered the most popular. The disadvantage of Baumgarte's technique is that choice of parameter relies on step size and the discretization method, it is difficult to find optimal parameter. Fortunately, many methods on the choice of parameter of Baumgarte's technique have been proposed.
In this thesis, two kinds of commonly used multi-step numerical integration methods: Adams and Backward Differentiation Formulae (BDF) method have been used. With three different stabilization methods: constraint stabilization, modified variable structure control and Post-stabilization method, we have compared the constraint error and efficiency when they are applied to computer simulation of constrained multi-body system. The result shows that Post-stabilization method is much more suitable for high accuracy simulation of constrained multi-body system because of it's simplicity and efficiency.
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