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標題: 多體機械系統使用拘束穩定數值積分法時參數選擇之研究
Parameters Selection of the Constraint Stabilization Method for Numerical Integration of Multibody Mechanical Systems
作者: 黃健楠
Huang, Jiann-Nan
關鍵字: 多體機械系統;multibody mechanical systems;數值積分;混合微分;代數方程式;拘束穩定法;假積分;numerical integration;differential-algebraic equation;constraint stabilization method;pseudo-integration
出版社: 機械工程學系
本文的目的在解決受拘束(constraint)條件下,多體(multibody)機械系統數值積分的穩定性問題。多體系統之運動方程式是一個包含外力、拘束力、加速度的混合微分、代數方程式(mixed differential-algebraic equations,簡稱DAE),在使用數值積分法解開微分方程式的同時,其位置及速度必須滿足拘束運動方程式及速度運動方程式,也就是說積分的變數是相關的。然而一般的數值積分法並不考慮其中的相關性,直接求解因此造成數值的偏差。
為了解決此問題,Baumgarte嘗試將加速度方程式加入位置項及速度項,當適當的選擇兩項之係數後,數值積分將獲得正確的解,稱為拘束穩定方法(constraint stabilization method),不過此方法唯一缺點在於係數之選擇,並無規則可循。本文將針對拘束穩定法這部份的研究成果,引入假積分(pseudo-integration)的觀念,並利用數位控制理論中的系統穩定度分析,針對數值積分法中最常用的兩種方法-Predictor-Corrector及Runge-Kutta法,提出一個正確選擇係數的方法。

The objective of this project is to resolve the stability problem for the numerical integration of constrained multibody mechanical systems. The dynamic equations of motion of the constrained multibody mechanical system is a mixed differential-algebraic equation(DAE) which contains external forces, constraint reaction forces as well as acceleration of the generalized coordinates of the system. In applying numerical integration methods to solve the mixed differential-algebraic equation, the constraint equation and its first and second derivatives must be satisfied simultaneously. That is, the generalized coordinates are dependent. Direct integration methods do not consider this dependency and constraint violation occurs.
To solve this problem, Baumgarte proposed a constraint stabilization method in which a velocity term and a position term were added in the second derivative of the constraint equation. The disadvantage of this method is that there is no known reliable method for selecting the coefficients of the position and velocity term. Improper selection of these coefficients can lead to erroneous results. In this paper, we will use the so-called pseudo-integration equation to analyse the constraint stabilization method for numerical integration. The stability analysis methods in digital control theory will be used to give correct choice of the coefficients for the Adams predictor-corrector and Runge-Kutta method.
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