向量式有限元素法於空間桁架之應用

Analysis of Space Truss Structure by The Vector Form Intrinsic Finite Element Method

陳世凱(Sui-Kai Chen)

指導教授 ： 莊清鏘

中原大學/工學院/土木工程研究所/碩士(2004年)

https://doi.org/10.6840/cycu200400391

摘要

本研究主要說明如何應用向量式有限元配合位移控制模擬空間桁架非線性行為。有別於傳統有限元素法以變分法為基本理論的方式，向量式有限元素法是以構件和節點為模擬基礎的物理模式，它將結構體定義成一群質點的組合，同時利用牛頓運動定律描述質點的運動，因此向量式有限元的計算變成一組向量方程式的計算。向量式有限元模擬問題時利用共轉座標分解剛體位移和變形位移，再藉用顯式時間積分法處理向量運動方程式，因此可以精確的計算多個連續體具有大剛體運動和大幾何變形行為。
由於原始的向量式有限元作用力已知，物體運動為代解量，對於某些問題，譬如：機構、自動控制裝置、機械裝置，預知的為物體位移歷時，作用力為代解量，此時整體結構的行為無法單純的只由節點載荷控制，部份節點為位移控制，本研究將探討向量式有限元配合位移控制的處理方式，並以桁架結構為例說明其可行性。
原始的向量式有限元利用動力平衡觀念建立運動方程式，本研究嘗試利用動量平衡觀念配合向量式有限元處理流程模擬空間桁架非線性行為。利用動量平衡觀念處理時，原始時間積分法不需改變，同時具有下面幾個特性：(1)外力經過一次積分具有平滑累加的效應，可以更有效的掌握外力急劇變化特性；(2)時間步長可以放大，減少誤差傳播效應。本文將探討向量有限元動力平衡和動量平衡的處理方式，並比較兩種方式模擬的結果。

關鍵字

向量式有限元 ； 材料非線性 ； 動量平衡方程式 ； 幾何非線性 ； 顯示時間積分 ； 共轉座標 ； 位移控制

並列摘要

In this paper, the vector form intrinsic finite element (VFIFE) method with displacement control is applied to simulate the geometrically nonlinear behaviors of space truss structures. Different from the conventional finite element method which is based on the variational theory, the VFIFE method is a physical model which models the analyzed domain to be composed by finite particles and Newton’s second law is applied to describe each particle’s motion. Thus, the calculation of VFIFE method becomes solving a set of uncoupled vector form equations. In the analysis of VFIFE method, the co-rotational coordinate system is used to separate the rigid body motion and pure deformation of the system. After combining it with explicit time integration scheme, the VFIFE method can effectively simulate the dynamic behaviors of multi-bodies system having large rigid body motion and deformation.
The forces of original VFIFE method are known and the motions of body are the results of the forces, but many problems like mechanism and automatic control device, the motion of body is prescribed, the applied forces are unknown, hence the behaviors of many structures can not simulate with force control only. In this study, the VFIFE method with displacement control is applied on the space structures to demonstrate feasibility of this strategy.
The governing equations of motion of original VFIFE method is dynamic equilibrium of force, in this study, VFIFE method with dynamic equilibrium of momentum is applied on the space structures. The time integration schemes in dynamic equilibrium of force and momentum are the same, and the advantages of dynamic equilibrium of momentum are: (1) a better capability in capturing the rapid changes of dynamic loading and (2) a less error propagation effect. The procedures and results of VFIFE method with dynamic equilibrium of force and momentum are discussed and compared on space structures.

並列關鍵字

displacement control ； momentum equations of motion ； explicit time integration ； geometrically nonlinear ； materially nonlinear ； VFIFE ； co-rotational approach

參考文獻

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