Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states
Multiple equilibrium states of a curved-sided hexagram: Part I—stability of states
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曲边六边形的多重平衡状态：第一部分——状态稳定性
DOI：
10.1016/j.jmps.2023.105406
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发表时间：
2023
影响因子：
5.3
通讯作者：
Hutchinson, John W.
中科院分区：
文献类型：
--
作者：
Lu, Lu;Dai, Jize;Leanza, Sophie;Zhao, Ruike Renee;Hutchinson, John W.
研究了六曲边六角环多重平衡态的稳定性。六个段中的每一个都是具有相同长度和均匀自然曲率的杆。这些杆在六边形的平面内均匀弯曲成120 °或240 °的相等弧，并在其末端相遇的尖点处连接，形成一个单回路平面环。由120 °或240 °弧形成的1-环圈是彼此的反转，并且它们又可以折叠成3-环圈直线构型或3-环圈，其中每个环圈呈“8”形。这四个平衡状态中的每一个都有一个均匀的弯矩。另外两个有趣的平面形状，6圆的六边形，平衡状态，也是均匀弯曲，确定和分析的稳定性。当自然曲率福尔斯落在上限和下限之外时，稳定性丧失，其形式为涉及杆段的耦合面外偏转和扭转的分叉模式。稳定性的条件或缺乏稳定性的条件取决于杆横截面的几何形状以及其自然曲率。将使用基尔霍夫棒理论的特殊形式分析圆形和矩形截面的棒，并详细说明所有四种感兴趣的状态相互稳定的特性。实验演示的各种状态和它们的稳定性。第二部分提出了数值模拟的状态之间的转换，使用杆理论和三维有限元公式，包括确认第一部分中建立的稳定性限制，并提出了额外的实验演示和验证。
The stability of the multiple equilibrium states of a hexagram ring with six curved sides is investigated. Each of the six segments is a rod having the same length and uniform natural curvature. These rods are bent uniformly in the plane of the hexagram into equal arcs of 120oor 240oand joined at a cusp where their ends meet to form a 1-loop planar ring. The 1-loop rings formed from 120oor 240oarcs are inversions of one another and they, in turn, can be folded into a 3-loop straight line configuration or a 3-loop ring with each loop in an “8” shape. Each of these four equilibrium states has a uniform bending moment. Two additional intriguing planar shapes, 6-circle hexagrams, with equilibrium states that are also uniform bending, are identified and analyzed for stability. Stability is lost when the natural curvature falls outside the upper and lower limits in the form of a bifurcation mode involving coupled out-of-plane deflection and torsion of the rod segments. Conditions for stability, or lack thereof, depend on the geometry of the rod cross-section as well as its natural curvature. Rods with circular and rectangular cross-sections will be analyzed using a specialized form of Kirchhoff rod theory, and properties will be detailed such that all four of the states of interest are mutually stable. Experimental demonstrations of the various states and their stability are presented. Part II presents numerical simulations of transitions between states using both rod theory and a three-dimensional finite element formulation, includes confirmation of the stability limits established in Part I, and presents additional experimental demonstrations and verifications.