兩端銷接的空間彈性桿件受中點集中力負載之變形與穩定性分析

本文研究一個能夠在空間中振動與變形彈性桿件，中點受集中力作用下其變形過程以及其挫曲的模式，其中包含了理論分析與實驗模擬。彈性桿件的兩端為銷接頭，只允許對特定軸的旋轉，而不能在銷接頭上滑動。彈性桿件只有中點處施加一個中點向下的集中力，其他部分則不受外力影響。本文的主題將探討彈性桿件的挫曲現象，其中有平面以及空間的挫曲形式。利用elastica模型來模擬彈性桿件的變形現象，以shooting method來求解。就如猜測的一樣，受中點力下的彈性桿件存在許多種靜態解，然而利用振動法來決定各種平衡解的穩定性。本文中的彈性桿件之變形模式可以由兩個參數來決定，一個為兩端銷接的距離；另一個則是彈性桿件截面的特性─截面主軸抗彎剛度的比值。在本文中彈性桿件的變形模式可以由四個特徵來分類出十種模式：挫曲前的變形、挫曲後的變形、挫曲發生時的臨界點類型以及發生挫曲時其挫曲形式。實驗的部分將以一條具有彈性的金屬線、軸承以及鋁合金夾具來逼近以elastica為基礎的受力模型並且驗證其模擬分析的結果。若想要設計兩端任意長的elastica並且希望桿件的變形模式皆在平面上變化的話，則截面主軸抗彎剛度的比值就必須大於28.24。

關鍵字

空間；中點力；折斷式挫曲； elastica

並列摘要

In this paper we study the deformation and stability of a pinned-pinned buckled beam under the action of a concentrated force at the midpoint. Focus is placed on the snapping-through phenomenon, which may take place in a plane or three-dimensionally. We first find the equilibrium configurations by using shooting method. Elastica model is adopted to take into account exact geometry in large deformation. As expected, multiple solutions may exist for a specified set of loading parameters. Vibration method is then employed to determine the stability of the equilibrium solutions. Through these analyses the deformation sequence as the midpoint force increases quasi-statically can be predicted. It is found that the deformation sequence of the elastica is determined by two parameters; (1) the distance between the two ends of the buckled beam, and (2) the bending stiffness ratio of the cross section. Ten different deformation patterns can be identified according to four characteristics; the deformations before, after, and during the jump, and the type of critical point at the jump. A metallic wire with circular cross section is used to verify the predicted deformation sequence. It is concluded that if one wishes to design an elastica capable of only plane deformation in all range of end distance, then the bending stiffness ratio has to be greater than 28.24.

並列關鍵字

Spatial elastica ； midpoint force ； snap-through

參考文獻

[1] Maurini, C., Pouget, J., Vidoli, S., 2007. Distributed piezoelectric actuation of a bistable buckled beam. European Journal of Mechanics –A/Solids 26, 837-853.

[3] Krylov, S., Ilic, B.R., Schreiber, D., Seretensky, S., and Craighead, H., 2008. The pull-in behavior of electrostatically actuated bistable microstructures J. Michromech. Microeng. 18, 055026.

[5] Schreyer, H.L., 1972. The effect of imperfections on the buckling load of shallow circular arches. Journal of Applied Mechanics 445-450

[6] Plaut, R.H., 1978. Influence of load position on the stability of shallow arches. Journal of Applied Mathematics and Physics (ZAMP) 30 548-552.

[7] Chen, J.-S., Ro, W.-C., Lin, J.-S., 2009. Exact static and dynamic critical loads of a shallow arch under a point Force at the midpoint, International Journal of Non-Linear Mechanics 44, pp. 66-70.

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兩端銷接的空間彈性桿件受中點集中力負載之變形與穩定性分析

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