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標題: 具PID型式之多體機械系統數值積分穩定法
The PID Type Constraint Stabilization Method for Numerical Integration of Multibody Mechanical
作者: 陳明仁
Chen, Ming-Ren
關鍵字: multibody mechanical;多體機械;DAE;constraint stabilization method;混合微分.代數方程式;拘束穩定法
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),不過此方法有一缺點在於無法改善使用大積分步距時的穩態誤差。本文將針對拘束穩定法這部份的研究成果,引入PID型式之數值積分穩定法,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,提出一個解決的方法。

The objective of this thesis 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 it can not correct the steady state error of the constraint equation when a large integration stepsize is used. In this thesis, we will use the so-called pseudo-integration equation to analyze the constraint stabilization method for numerical integration. The stability analysis methods in digital control theory will be used to solve this problem for the Adams predictor-corrector and Runge-Kutta method.
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