Feasibility Determination of Reciprocal Configurations on Cylindrical Surface from Archimedean Pavings

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.