圓桿受端點扭矩與推力下的後挫曲行為

Post-Buckling Behavior of a Rod under End Torque and Axial Load

謝承勳(Cheng-Xun Sie)

指導教授 ： 陳振山

國立臺灣大學/工學院/機械工程學研究所/碩士(2016年)

https://doi.org/10.6342/NTU201603323

摘要

本文研究彈性直桿於兩端邊界箝制下，受端點力作用時之變形行為，我們考慮兩種不同施力方法：(1)先縮短兩端點水平距離，再於一端點施加扭矩。(2)先於一端點預設扭轉角，再縮短兩端點水平距離。本文將使用elastica模型來模擬彈性桿件的變形現象，利用Shooting Method求解其邊界值問題，並以牛頓法來幫助收斂。文中首先討論將彈性直桿兩端點水平距離縮短至一半桿長，再於端點施加扭矩，桿件變形從無自我接觸跳躍到一點自我接觸，再跳躍至兩點自我接觸，接著捲曲為三點自我接觸，最後變形成點線點接觸。相反地，若先於一端點預設扭轉角，再縮短兩端點水平距離，桿件變形順序同樣為無自我接觸、一點自我接觸、兩點自我接觸、三點自我接觸、點線點接觸，而兩端點水平距離縮短至非常靠近時，桿件的最後變形與預設轉角有關，預設轉角 桿件變形為點線點接觸，而預設轉角 、 、 桿件變形分別為三點自我接觸、兩點自我接觸、一點自我接觸。文中設計一組實驗機構，以一條具有彈性的鎳鈦合金線、軸承以及夾具來逼近以elastica為基礎的受力模型來驗證理論解的正確性。然而，在理論分析上，材料參數是線性的，代表材料性質不論受到多大的施力仍能保持固定，但實驗桿件的材料參數在承受高扭力下會改變。因此，雖然在本文實驗中無法準確地驗證高受力時的變形，但可驗證低受力時之理論變形是否正確。

關鍵字

彈性圓桿 ； 後挫曲 ； 大變形理論

並列摘要

We study the deformation of a clamped-clamped rod under combined edge thrust and twist. Two types of loadingcombinations are considered, i.e., twist-shortening and shortening-twist. In twist-shortening, the rod is pre-rotated a specified angle and then the two ends are brought closer quasi-statically. Shortening-twist is the other way around. Continuation scheme based on shooting method is employed to study the rod deformation as the end twist and shortening vary. We first examine the deformation sequence when the two ends are brought to one half of the initial distance and then the twist angle is increased quasi-statically. The deformation jumps from free-of-contact to one-point contact, then jumps again to two-point, then evolves smoothly to three-point and then to point-line-point writhing. In the case of twist-shortening, a typical deformation sequence also includes free of self-contact, one-point, two-point, three-point, and point-line-point contact. The final stage of deformation when the two ends approach each other depends on the pre-rotation. For instance, when the rod is pre-rotated and beyond, the final deformation as the two ends approach each other is point-line-point self-contact. For pre-rotations , , and , the final deformations are three-point, two-point, and one-point self-contacts, respectively.

並列關鍵字

elastica ； self-contact ； line-contact

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