圓管內的彈性圓桿於端點受扭矩作用下之研究

On an Elastic Rod inside a Tube under End Twisting Moment

李弘基(Hong-Chi Li)

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(2010年)

https://doi.org/10.6342/NTU.2010.01389

摘要

在本文中，我們研究一圓管內的細長彈性圓桿於端點施加扭矩作用下的變形分析。圓桿的端點位於圓管的中心點並於側向受到夾持。在前人的研究中，他們考慮整段的圓桿與管壁間均為線接觸變形。與前人不同的地方在於我們將無因次扭矩Mx由0開始增加，並針對圓桿的變形作完整的分析。我們發現初始形狀為直線的圓桿在Mx達到8.987 時發生挫曲並形成螺旋狀的變形，此時圓桿中點與管壁接觸，為一點接觸變形。當Mx增至11.472時接觸點由中點一分為二，圓桿的中點與管壁分離，形成兩點接觸變形。當Mx增至13.022，圓桿中點再次與管壁接觸，其變形發展為三點接觸變形。從Mx=13.098開始，圓桿中點的接觸型態由點接觸發展為線接觸。隨著Mx繼續增加，圓桿變形均以點-線-點接觸分佈，同時線接觸部分的長度隨著Mx增加而增加。在線接觸變形模型中，當線接觸部分夠長時，我們可以預測其變形的解析解。我們發現數值結果與解析解十分符合。最後，我們建立一套實驗設備觀察圓管內的圓桿於端點受扭矩作用下的變形發展。

關鍵字

彈性圓桿 ； 圓管 ； 挫曲 ； 扭矩 ； 夾持 ； 變形

並列摘要

In this paper we study the deformation of a thin elastic rod constrained inside a cylindrical tube and under the action of an end twisting moment. The ends of the rod are clamped in the lateral direction at the centers of the tube. Unlike the previous works of others, in which only the fully-developed line-contact spiral was considered, we present a complete analysis on the deformation when the dimensionless twisting moment Mx is increases from zero. It is found that the straight rod buckles into a spiral shape and touches the inner wall of the tube at the midpoint when Mx reaches 8.987. As Mx increases to 11.472 the contact point in the middle splits into two, leaving the midpoint floating in the air. As Mx increases to 13.022, the midpoint returns to touch the tube wall and the two-point-contact deformation evolves to a three-point-contact deformation. Starting from Mx=13.098, the point contact in the middle evolves to a line contact, and the deformation becomes a point-line-point contact configuration and remains so thereafter. In the case when the line-contact pattern is fully developed, it is possible to predict the spiral shape analytically. The numerical results are found to agree very well with those predicted analytically. Finally, an experimental set-up is constructed to observe the deformation evolution of the constrained rod under end twist.

並列關鍵字

elastic rod ； tube ； buckle ； torque ； clamp ； deformation

參考文獻

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