兩端受到插銷拘束的可拉伸彈性樑在步階負荷作用之動態跳躍式挫曲分析

本文探討兩端受插銷拘束、且具備拉伸性質的彈性樑，受到側向集中負荷之作用，分析動態系統之暫態響應行為。吾人著重於拉伸性質對於動態跳躍式挫曲的影響，應用高階克蘭克尼科爾森方法求解非線性的控制方程式，由於保守系統具備能量守恆的性質，茲定義為數值收斂性測試的標準。根據數值結果顯示，對於具備相同細長參數的可拉伸與不可拉伸之彈性樑而言，在特定的側向負荷作用之下，兩者分別以對稱與非對稱的形式發生動態跳躍式挫曲；對於不可拉伸的彈性樑而言，忽略拉伸性質的假設使得動態臨界負荷備受高估。此外，直接透過數值模擬預測動態臨界負荷的方法較為不切實際，故本文提出能量障壁的概念，藉以尋求絕對無法導致動態跳躍式挫曲的保守動態臨界負荷。最後，保守動態臨界負荷與端點初始角度呈正相關，不考慮拉伸性質的彈性樑理論在小變形範圍內顯得不堪使用。

關鍵字

可拉伸的彈性樑；步階負荷動態跳躍式挫曲；動態臨界負荷

並列摘要

In this paper we study the transient response of a hinged extensible elastica under a step load at the midpoint. Emphasis is placed on the effect of extensibility on the dynamic snapping load. A second-order Crank-Nicolson method associated with finite difference discretization is employed to solve for the dynamic response. A weak sense of convergence based on the conservation of total energy over time is proposed. It is possible that for the same load and slenderness ratio, inextensible model predicts unsymmetric dynamic snapping while the elastica model predicts a symmetric one. An inextensible elastica model ignoring the effect of axial extensibility tends to overestimate the dynamic critical load. It is impractical to determine the exact dynamic snapping load via direct simulation. Instead, the concept of energy barrier is used to find a conservative dynamic critical load, below which the elastica is guaranteed to be safe from snapping. The dynamic critical load grows as the initial end angle increases. Inextensible model is particularly inadequate in the small-deformation range.

並列關鍵字

Extensible elastica ； Step load ； Dynamic snapping ； Dynamic critical load

參考文獻

[1] Cazottes, P., Fernandes, A., Pouget, J., Hafez, M., 2009. Bistable buckled beam: Modeling of actuating force and experimental validations, ASME Journal of Mechanical Design, 131, 101001.

[2] Chen, J.-S., S.-Y. Hung, 2011a. Exact snapping loads of a buckled beam under a midpoint force. Appl. Math. Modell., in press.

[3] Chen, J.-S., Hung, S.-Y., 2011b. Snapping of an elastica under various loading mechanisms, Eur. J. Mech. A 30, 525-531.

[4] Chen, J.-S., Lin, W.-Z., June 2012. Dynamic snapping of a suddenly loaded elastica with fixed end slopes, International Journal of Non-Linear Mechanics, Volume 47, Issue 5, Pages 489-495.

[5] Chen, J.-S., Tsao, H.-W., Static Snapping Load of a Hinged Extensible Elastica. Appl. Math. Modell., in press.

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兩端受到插銷拘束的可拉伸彈性樑在步階負荷作用之動態跳躍式挫曲分析

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