Mobility of a class of perforated polyhedra
Mobility of a class of perforated polyhedra
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DOI：
10.1016/j.ijsolstr.2016.02.006
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发表时间：
2016-05
影响因子：
3.6
通讯作者：
P. Fowler;S. Guest;B. Schulze
中科院分区：
文献类型：
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作者：
P. Fowler;S. Guest;B. Schulze
描述了一类过支撑但典型的柔性体铰框架。它们基于具有刚性面的多面体，其中面的独立子集已被一组孔取代。描述物体（C的顶点）和它们的连接关节（C的边）的接触多面体C是通过对下面的立方多面体的边进行细分而导出的。对称性计算检测的灵活性不能单独计算。格吕布勒-库茨巴赫迁移率计数规则的一个通用的扩展版本解释了这种类型的无限族的净迁移率（基于棱柱、楔形、桶形和父多面体的一些一般膨胀的细分）。在细分构造下，所有面均为平的棱柱和所有桶形的棱柱都能生成柔性穿孔多面体。这项研究的灵感来自沃尔特·怀特利提出的一个问题，即一个穿孔多面体具有独特的机制，可以将八面体对称性降低到四面体对称性。事实证明，穿孔多面体与最高（O h）点群对称的基础上细分的立方体是机械等效的霍伯曼开关间距玩具。这两个对象表现出一个完全相似的机制，保留在有限范围内的T d子群对称性;这种机制在鲍勃·康奈利和芭芭拉·海斯提出的两种变体中生存，它们具有相同的接触图，但初始最大对称性较低。
A class of over-braced but typically flexible body-hinge frameworks is described. They are based on polyhedra with rigid faces where an independent subset of faces has been replaced by a set of holes. The contact polyhedron C describing the bodies (vertices of C) and their connecting joints (edges of C) is derived by subdivision of the edges of an underlying cubic polyhedron. Symmetry calculations detect flexibility not revealed by counting alone. A generic symmetry-extended version of the Grübler–Kutzbach mobility counting rule accounts for the net mobilities of infinite families of this type (based on subdivisions of prisms, wedges, barrels, and some general inflations of a parent polyhedron). The prisms with all faces even and all barrels are found to generate flexible perforated polyhedra under the subdivision construction. The investigation was inspired by a question raised by Walter Whiteley about a perforated polyhedron with a unique mechanism reducing octahedral to tetrahedral symmetry. It turns out that the perforated polyhedron with highest (O h) point-group symmetry based on subdivision of the cube is mechanically equivalent to the Hoberman Switch-Pitch toy. Both objects exhibit an exactly similar mechanism that preserves T d subgroup symmetry over a finite range; this mechanism survives in two variants suggested by Bob Connelly and Barbara Heys that have the same contact graph, but lower initial maximum symmetry.