兩端斜率固定之彈性圓桿的變形與振動分析

本論文中，討論兩端夾持之彈性圓桿在空間中的變形與振動頻率，彈性圓桿其中一端固定夾持於空間中，另一端同樣夾持並連接一滑塊，使其可以沿線性軌道作無摩擦滑動。本研究主要強調彈性圓桿端點受到推力時，兩端夾持角度的變化對彈性圓桿變形和穩定性的影響。在理論方面，以elastica模型模擬三度空間中的大變形彈性圓桿，並以數值方法shooting method求解彈性圓桿的靜態變形和振動頻率。為驗證理論預測的正確性，設計一組實驗量測彈性圓桿在不同夾持角度差下的受力和位移關係，以及其振動頻率。當兩夾持端的斜率差小於時，可以找到兩條不同的受力－變形曲線，一條為三維的空間變形曲線，一條為二維的平面變形曲線。在控制軸向推力的情況下，若推力持續增加或減少，彈性圓桿會在平面變形和空間變形間發生挫曲現象，而在控制端點位移的情況下，則不會發生此現象。當兩夾持端的斜率差大於時，彈性圓桿只存在平面變形。最後研究夾持角度差為零的直桿，由於直圓桿具軸對稱的特性，因此存在一零自然頻率，本文進一步探討橢圓桿以證實此論點。

關鍵字

彈性圓桿；端點斜率固定；振動；大變形理論； shooting method

並列摘要

In this paper we study the static deformations and vibration frequencies of a clamped-clamped spatial elastica. One clamp is fixed in space, and the other is attached to a slider which is allowed to slide without friction on a linear track. Emphasis is placed on the effect of the slope difference between the two clamps on the response of the elastica when the slider is under the action of an edge thrust. The equations of motion of the elastica are formulated within the framework of director theory, and the static deformations and vibration frequencies are calculated by shooting method. An experimental set-up is designed to examine the theoretical predictions. When the slope difference between the two clamps is smaller than , two separate load-deflection curves can be found; one for spatial deformation and the other for purely planar deformation. In force control, the elastica-slider assembly may jump between planar and spatial deformations as the edge thrust increases or decreases monotonically. In displacement control, on the other hand, no jump will occur. In the case when the slope difference is greater than , only planar deformation exists. For the case when the slope difference is zero, there exists a zero nature frequency for the circular rod because of the axial symmetry. We further study elliptic rods to confirm this point.

並列關鍵字

Spatial elastica ； fixed end slope ； vibration ； director theory ； shooting method

參考文獻

[1] Howell, L.L., 2001. Compliant Mechanisms. John Wiley and Sons, New York.

[2] Shoup, T.E., McLarnan, C.W., 1971. On the use of a doubly clamped flexible strip as a nonlinear spring. ASME J. Appl. Mech. 38, 559–560.

[4] Chen, J.-S., Lin, Y.-Z., 2008. Snapping of a planar elastica with fixed end slopes. ASME J. Appl. Mech. 75, 041024.

[5] Atanackovic, T.M., Glavardanov, V.B., 2002. Buckling of a twisted and compressed rod. Int. J. Solids Struct. 39, 2987–2999.

[6] Glavardanov, V., Maretic, R.B., 2009. Stability of a twisted and compressed clamped rod, Acta Mech. 202, 17-33.

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兩端斜率固定之彈性圓桿的變形與振動分析

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