圓桿受端點扭矩下的後挫曲行為

本文研究桿件在兩端邊界為固定端，施予固定軸向力後再施予端點扭矩，找出其變形的趨勢，其中一固定端施予扭轉，另一固定端可滑動。本文將使用elastica模型來模擬彈性桿件的變形現象，其邊界值問題利用Shooting Mehhod來求解。求出靜態解後再利用振動法來決定各個平衡解的穩定性。文中將會提及兩種不同的軸向力以及兩種不同的材料參數設定，討論其不同的影響，並設計了一實驗機構驗證理論之靜態變形是否正確。而接著在探討桿件若未施予軸向力時其臨界扭矩，從elastica控制方程式開始進行理論推導，求出其曲率及位移的通解，將通解帶入邊界條件求出在不同邊界條件下的特徵方程是，在由數值方法求出其臨界扭矩，並考慮一些特殊的例子，本文包含兩種邊界條件：(1)Spherically-hinged (2)Clamped-clamped。影響臨界扭矩的參數包含桿件長寬比、剪力模數及初始扭率，分析各種參數對臨界扭矩的影響，其中將剪力模數假設為非常大時和前人的結果相符合。桿件有初始扭率其臨界扭矩會比沒有初始扭率來的大。若桿件其初始扭率相當大時，其臨界扭矩會趨近於一定值，和理論值相符合。

關鍵字

彈性圓桿；振動；臨界扭矩

並列摘要

In this paper we study post-buckling behavior of a rod with both ends clamped. One end is unrotatable and fixed, the other end is rotatable and allowed to slide. Edge thrust is fixed and the end rotation is varied. Elastica model is adopted to take into account exact geometry in large deformation. Vibration method is then employed to determine the stability of the equilibrium solution. We discuss to two different edge thrust and two diffenrent material constant, then point out the difference. Also, we derive the equation for the critical moment of a spherically-hinged pre-twisted rod under axial moment. It is found that in the case when the cross section has unequal principal moments of inertia the pre-rotation caused by end moment cannot be ignored in calculating the critical moment. On the other hand, if the two principal moments of inertia are equal, such as circular or square cross section, both the pre-twist and pre-rotation have no effect on the critical moment. The resulted equation for critical moment can be considered as the extension of the well-known Greenhill’s formula.

並列關鍵字

elastica ； large deformation ； vibration ； critical torque

參考文獻

[1] Nour-Omid, B., Rankin, C.C., 1991. Finite rotation analysis and consistent linearization using projectors. Computer Methods in Applied Mechanics and Engineering 93, 353-384.

[2] Miyazaki, Y., Kondo, K., 1997. Analytical solution of spatial elastica and its application to kinking problem. International Journal of Solids and Structures 34(27), 3619-3636.

[3] van der Heijden, G.H.M., Neukirch, S., Goss, V.G.A., Champneys, Thompson, J.M.T., 2003. Instability and self-contact phenomena in the writhing of clamped rods. Int. J. Mech. Sci. 45, 161-196.

[4] Goyal, A.G., Perkins, N.C., Lee, C. L., 2005. Nonlinear dynamics and loop formation in Kirchhoff rods with implications to the mechanics of DNA and cables Journal of Computational Physics 209, 371–389.

[5] Goyal, A.G., Perkins, N.C., Lee, C. L., 2008. Non-linear dynamic intertwining of rods with self-contact. International Journal of Non-Linear Mechanics 43, 65 – 73

國際替代計量

圓桿受端點扭矩下的後挫曲行為

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