多體機械系統動態模擬數值積分穩定法之研究

Abstract

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(kinematics)方程式及速度運動方程 式，亦 即積分的變數有相關性(dependent)。然而一般的數值積分法忽略了其中 的相關性 ，直接求解因而造成數值的偏差。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。本計畫將利用數位控制理論中的系統穩定度分析方法解決此問題 ，提供不同數值積分法在使用拘束穩定積分方法時係數的選擇，並以實例 驗證固定積分時距(stepsize)及變化積分時距之數值情形。

The objective of this paper is to resolve the stability problem for thenumerical integration of constrained multibody mechanical system. The dynamicequations of motion of the constrained multibody mechanical system is a mixeddifferential- algebraic equation(DAE) which contains external forces, constraintreaction forces as well as acceleration of the generalized coordinates of thesystem. In applying numerical integration methods to solve the mixeddifferential- algebraic equation, the constraint equation and its first andsecond derivatives must be satisfied. That is, the generalized coordinatesare dependent. Direct integration methods do not consider this dependency andconstraint violation occurs. To solve this problem, Baumgarte proposed aconstraint stabilization method in which a velocity term and a position termwere added in the second derivative of the constraint equation. The disadvan-tage of this method is that there is no known reliable method for selectingthe coefficients of the position and velocity term.Improper selection of thesecoefficients can lead to erroneous results.In this paper,we will use stabilityanalysis methods in digital control theory to resolve this problem. Correctchoice of the coefficients for different numerical integration methods arefound for both fixed and variable integration stepsize.