受點拘束之彈性板條的變形和穩定性分析

Abstract

本文研究一個受固定於空間上的拘束點所限制之彈性板條(elastica)的挫曲變形和穩定性，其中包含了理論的分析和實驗量測。彈性板條的一端為完全夾持(clamp)，另一端允許彈性板條經由沒有摩擦力和空隙的導引通道而滑動，使範圍內的彈性板條長度增加或縮短。拘束點固定於夾持平面的特定高度上方。一般而言，當推力達到臨界負荷時彈性板條會挫曲，跳躍至以對稱的變形接觸拘束點。當推力持續增加，達到一個對稱分裂(symmetry-breaking)的分支點(bifurcation)時，彈性板條將變成非對稱的變形。非對稱變形的彈性板條受到持續增加的推力會發生折斷式挫曲(snap-through buckling)，跳躍到自我接觸的平衡位置。我們將以Eulerian的描述方法來替代Lagrangian描述式，以此為基礎來研究受拘束之彈性板條的振動分析。同時我們也利用實驗驗證了理論上所預測的兩次跳躍和對稱分裂的分支點。

This paper studies, both theoretically and experimentally, the deformation and stability of a buckled elastic strip (also known as an elastica) constrained by a space-fixed point in the middle. One end of the elastica is fully clamped, while the other end is allowed to slide without friction and clearance inside a rigid channel. The point constraint is located at a specified height above the clamping plane. The elastic strip buckles when the pushing force reaches the conventional buckling load. At this buckling load, the elastica jumps to a symmetric configuration in contact with the point constraint. As the pushing force increases, a symmetry-breaking bifurcation occurs and the elastica evolves to one of a pair of asymmetric deformations. As the pushing force continues to increase the asymmetric deformation experiences a second jump to a self-contact configuration. A vibration analysis of the constrained elastica based on an Eulerian description is described, and the stability of the calculated equilibrium configuration is determined. The experiment confirms the two jumps and the symmetry-breaking bifurcation predicted theoretically.