兩端受插銷拘束之彈性樑受中點推力之動態反應及保守動態臨界負載預測

Transient Response of a Hinged Elastica Under a Step Load

林唯中(Wei-Zhong Lin)

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(2012年)

https://doi.org/10.6342/NTU.2012.02692

摘要

摘要

關於兩端受插銷(hinged)拘束之彈性樑之靜態變形與穩定性分析，前人已使用小變形理論得到。本文首次使用大變形理論彈性樑在任意點突然受一垂直推力之動態反應，同時使用能量法預測在中點受力時的動態臨界負載。文中使用Deformation Potentials將原本的六條統御方程式縮減為三條，使用有限差分法做離散化，且使用牛頓法使方程式的解達到收斂。本文將討論起始變形為兩端與x軸夾角為40度之彈性樑突然受一推力Q的動態反應；同時也考慮材料內阻尼c_m的效果。文中並比較推力Q由重物mg取代後產生的差異，及使用模擬得到非中點受力時40度、50度、80度的動態臨界負載。
當Q遠小於動態臨界負載時，彈性樑的動態反應在時間間隔k縮小時可以達到收斂。當Q遠大於動態臨界負載時，彈性樑發生動態挫曲的時間點可以準確預測，發生挫曲前之動態反應在k縮小時可以達到收斂，但彈性樑挫曲後之動態變形並無法由縮小k得到收斂。當Q靠近動態臨界負載時，我們發現數值方法無法依靠縮小k以達到準確預測發生動態挫曲的確切時間，因此我們必須放棄使用有限差分的方式來預測彈性樑突然受集中力作用時之臨界動態負載，而需仰賴能量法。以能量的觀點求得保守的臨界負載。以確保推力Q在小於此保守臨界負載一定不會發生動態挫曲。

關鍵字

彈性樑 ； 動態臨界負載 ； 折斷式挫曲

並列摘要

Abstract

The previous research has studied the static deformation and stability analysis of hinged shallow arch by small deformation theory. In this paper, large deformation theory is first applied to research the transient response of elastic beam (elastica) under a step load Q. The energy method is used to predict dynamical critical load when the step load applied on the middle point of elastica. The number of the governing equations is reduced from 6 to 3 by Deformation Potentials, and the governing equations are discretised by Finite Difference Method and solved by Newton Method. The angle between the end of the initial shape studied in this paper and x-axis is 40 degree. The internal damping of material c_m is considered. The difference of the transient response between force Q and weight mg is compared, and the dynamical critical load is obtained by numerical simulation when the load applied on the non middle point.
If the step load Q much less than critical load, the transient response can converge by fining time increment k. If the step load Q greater than critical load, the time happening snap-through can be predicted precisely, but the transient response after snap-through cannot converge by fining time increment k. If the step load Q similar with critical load, the time happening snap-through cannot be predicted precisely by fining time increment k. Therefore we use the energy method to obtain the conservative dynamical critical load instead of numerical simulation. It is assured that the elastica would not happen snap-through if the step load Q less than the conservative dynamical critical load.

並列關鍵字

Elastica ； Dynamical Critical Load ； Snap-through

參考文獻

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