以向量式分析及剛體動力學分析平面機構運動

Abstract

本論文旨在使用向量式有限元與剛體動力學求解機構問題。不同於一般僅使用剛體或柔體的多體動力學，本文在計算運動對時使用剛體動力學，而其他柔體部分則使用向量式分析法。 剛體動力分析中，本文將不同運動對的約束方程式定出後，依拉格朗日乘數法帶入虛功方程式中，得出一個微分代數方程式，再透過時間積分及牛頓迭代法求解。在向量式有限元中，則以點值描述方式選取一組質點作為構件運動的描述方式，質點間以結構單元連接。質點的運動方程式則各別定義在每一組相連的途徑單元上；質點間的互制力則由桿件元的內力計算得到。為了求得桿件元的內力，透過逆向運動獲得純變形，並依此建立變形坐標計算應變，透過應力與應變關係及虛功等效關係與平衡關係後即可求得等效節點內力。利用集成，將內力帶入運動方程式後，即可透過時間積分逐步求解，獲得完整的運動分析。 為將剛體與柔體耦合，向量式有限元使用點值描述的特色可以直接與剛體連接，故僅需讓剛體與柔體連接的點共用一個質點參數，即可達到耦合效果。經過上述耦合方式，相較傳統有限元處理約束時，需建立完整的系統矩陣，使用剛體與柔體耦合處理運動對約束，將運動對分開處理後，能減少矩陣的階數且能夠降低計算困難。本文最後將以幾個例題對剛體與柔體耦合分析法進行驗證，比較結果後發現與文獻及商用軟體模擬結果相近，可以證明該方法的可行性及實用性。

The gist of this thesis is to analyze the motion of planar; by the vector form analysis and rigid body dynamics. Rather than using only rigid or flexible multi-body system dynamics, in this thesis, we solve the problem of kinematic pairs using the rigid body dynamics while the flexible parts using the vector form analysis. In rigid body dynamics, constraint equations for different kinematic pairs are firstly established. Then, the Lagrange multiplier is adopted into the formulation of principles of virtual work, and DAEs are obtained. Then, it could be solved by time integration and Newton-Raphson iteration method. In Vector Form Intrinsic Finite Element (VFIFE), a group of mass points to represent the components of mechanisms using point value description is established, then two neighboring mass points with a structural element are connected. The equations of motion could be defined by path elements, and the force between mass points is obtained by the inertia force of structure elements. Using the inverted motion to obtain the pure deformation and co-rotation coordinate for strain, then internal force could be obtained. The motion analysis of mechanisms can be obtained by the equations of motion and appropriate time integration. The method is verified by some example. The results are also compare with those in reference and by ANSYS.