臺灣博碩士論文加值系統

將一初始直樑彎曲成圓形，在端點給定反向且相等的旋轉角度，持續旋轉使板條自我接觸最終發生折斷式挫曲現象(snap)，本文將研究此過程中的變形情況與穩定性分析。穩定性的部分我們利用歐拉公式(Eulerian formulation)進行振動分析，該理論可以將振動時表面接觸點的滑動或滾動運動納入考慮。理論結果預測彎曲樑在轉到分歧點時會發生側向的折斷式挫曲現象。然而在實驗中，自我接觸樑的折斷式挫曲卻在超過分歧點的地方發生，並以將自身擠壓向兩端點中心線且維持對稱的方式翻至另一側。通過觀察在分歧點附近不穩定的模態形狀，可以合理判斷此結果是因接觸表面之間的滑動摩擦力阻止了理論上應發生的側向折斷式挫曲，其中摩擦力並未考慮進理論分析中。另外，比對求解的模態與頻率結果顯示，實驗中彎曲樑的折斷式挫曲現象更像是以第二種不穩定的模態發生。於是，我們提出一非對稱分析，使彎曲樑在初始情況下向一側傾斜一個小角度，而不再是完美對稱的情況。在傾斜角為3°的實驗中，變形結果與路徑確實與非對稱模型的力與變形圖吻合，並在分歧點附近產生側向傾斜的情況。該實驗結果可以驗證完美對稱模型中確實存在分歧點及相應的側向折斷式挫曲現象。

An initially straight beam is first bent into a circular configuration. Equal rotation angles of opposite direction are then prescribed at the two ends to bend the beam into self contact and eventually make it snap. We conduct a vibration analysis based on an Eulerian formulation, which can take into account the sliding or rolling motions between the contact surfaces during vibration. The theory predicts that the rotated beam snaps sideway at the bifurcation point. In experiment, however, the self-contacted beam passes through the bifurcation point and snaps symmetrically by squeezing itself past the center of the two end points to the other side. By looking into the mode shapes of the unstable mode at the bifurcation point, it is believed that the predicted sideway snapping may be prevented by the sliding friction between the contact surfaces, which is not included in the theory. Instead, the beam snaps in a second unstable mode which involves rolling between contact surfaces. We then propose an imperfection analysis in which the bent beam is slanted to one side by a small angle in its initial configuration. In an experiment with a slant angle of 3°, the deformation follows the load-deflection curve of the imperfect model and snaps sideway near the limit point. This experimental result may be considered as an auxuliary evidence of the existence of the bifurcation point and the associated sideway snapping phenomenon in the perfect model.

口試委員審定書 i
致謝 ii
摘要 iii
Abstract iv
圖目錄 vii
表目錄 ix
第一章 導論 1
第二章 理論模型與統御方程式 3
第三章 靜態變形分析 9
3.1 未接觸分析 9
3.2 自我接觸分析 10
3.3 數值方法 12
3.4 數值結果 14
第四章 振動與穩定性分析 22
4.1 未接觸分析 22
4.2 自我接觸分析 24
4.3 數值方法 28
4.4 數值結果 31
第五章 實驗量測 35
5.1 實驗設備 35
5.2 實驗結果 37
第六章 非對稱分析 43
第七章 結果與討論 47
參考文獻 49
附錄一 未接觸分析線性方程式推導 50
附錄二 自我接觸分析線性方程式推導 56
附錄三 位移控制的邊界條件推導 65
附錄四 座標轉換推導 71
附錄五 接觸點條件推導 72

[1] Plaut, R., "Optimal Arch Form for Stability Under End Moments," Proc. Developments in Mechanics, Proceedings of the 18th Midwestern Mechanics Conference, pp. 16-18.
[2] Chen , J.-S., and Lin , J.-S., 2005, "Exact Critical Loads for a Pinned Half-Sine Arch Under End Couples," Journal of Applied Mechanics, 72(1), pp. 147-148.
[3] Plaut, R. H., 2009, "Snap-Through of Shallow Elastic Arches Under End Moments," Journal of Applied Mechanics, 76(1).
[4] Chen, J.-S., and Ro, W.-C., 2009, "Dynamic response of a shallow arch under end moments," Journal of Sound and Vibration, 326(1-2), pp. 321-331.
[5] Plaut, R., and Virgin, L., 2009, "Vibration and snap-through of bent elastica strips subjected to end rotations," Journal of Applied Mechanics, 76(4).
[6] Chen, J.-S., and Ro, W.-C., 2010, "Deformations and Stability of an Elastica Subjected to an Off-Axis Point Constraint," Journal of Applied Mechanics, 77(3).
[7] Chen, J.-S., and Lin, Y.-C., 2013, "Vibration and stability of a long heavy elastica on rigid foundation," International Journal of Non-Linear Mechanics, 50, pp. 11-18.