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Feasibility Determination of Reciprocal Configurations on Cylindrical Surface from Archimedean Pavings
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摘要: 互承结构由来已久，因节点简单、施工方便、造型美观等优点而受到学术界和工程界的关注。然而，互承结构中大量几何约束的存在，导致其构型困难，这成为了互承结构应用的一大障碍。为了寻找在给定曲面上生成互承构型的可行方案，探究了阿基米德铺砌的几何用于柱面互承构型的可行性。对于柱面网格生成互承构型的可行性判定，其判定流程为先进行平面预判定，然后对柱面网格进行直接转换，再通过三层次判定法或快捷判定法进行判定。11种阿基米德平面铺砌以不同的对称轴为纵轴弯曲，可生成21种阿基米德铺砌柱面网格，因此采用收缩法、旋转法和扩展平移法这3种直接转换法生成互承构型存在63种可能方案。
对这63种方案展开系统性的判定筛选分析。首先，通过平面预判定和三层次判定法中结点单元的物理判定可获得某些不可行方案的典型特征，根据这些特征对阿基米德铺砌柱面网格形式进行批量预处理，排除一部分不可行的方案。对剩下的方案，采用自编的MATLAB程序，对柱面网格用收缩法、旋转法和扩展平移法进行直接转换，之后，采用三层次判定法和快捷判定法进行逐一判断。研究结果表明：63种可能方案中，理论上满足互承构型要求的有6种，分别是A1②的收缩法和旋转法、A2②的扩展平移法、A3②的收缩法和旋转法以及A3①的收缩法，并同时给出了这6种可行方案的解析解。这6种构型的网格都具有单元几何形状单一、二度旋转对称且整体形态规则的特点。为实证研究结果，对其中5种可行方案，搭建了实体互承模型。Abstract: The reciprocal structure has a long history, which has attracted the attention of the academic and engineering circles due to its advantages of simple node, convenient construction and beautiful shape. However, the existence of a large number of geometric constraints in the reciprocal structure makes its configuration difficult, which becomes a big obstacle to the application of the reciprocal structure. In order to find feasible schemes for generating reciprocal configurations on a given surface, the feasibility of applying the geometries of Archimedean pavings to generate reciprocal configurations on the cylindrical surface is investigated. For the feasibility judgment of cylindrical grid generation of reciprocal configurations, the judgment process in this paper is to carry out the plane pre-judgment first, then directly transform the cylindrical grid, and then carry on the judgment through the three-level judgment method or the fast judgment method. 11 kinds of Archimedes plane pavements are curved with different axis of symmetry as the longitudinal axis, and 21 kinds of Archimedes cylindrical meshes can be generated. Therefore, there are 63 possible schemes for the generation of reciprocal configurations by three direct conversion methods:contraction method, element-rotating method and extended translation method.
These 63 schemes were systematically determined and screened in this paper. Firstly, the typical features of some infeasible schemes can be obtained through the plane pre-judgment and the physical judgment of nodal elements in the three-level judgment method. According to these characteristics, the grid form of Archimedes' paved cylinder can be preprocessed in batches to eliminate some infeasible schemes. For the rest of the schemes, through the developed MATLAB program based on geometric analytical solutions, transformations from Archimedean pavings to reciprocal configurations were conducted employing contraction method, element-rotating method and extended translation method, and furthermore judgment methods at three level and a fast judgment method were employed for determinations one by one. The results show that among the 63 possible schemes, there are only 6 schemes theoretically satisfy the requirements of reciprocal configuration, which are A1② by contraction method and element-rotating method, A2② by extended translation method, A3② by contraction method and element-rotating method, and A3① by contraction method. The analytical solutions of these 6 feasible schemes are given in this paper. The grids of these 6 configurations have the characteristics of single cell geometry, symmetrical rotation of the second degree and regular overall morphology. In order to validate the results of the study, practical entity reciprocal configuration models were successfully built for 5 feasible schemes. -
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