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Analityczne określenie współrzędnych węzłów konstrukcji prętowej z nieregularnego romboedru

Mroczkowski J., Romaszkiewicz-Białas T.

Journal Biuletyn of Polish Society for Geometry and Engineering Graphics

2001

Vol. 12

35--43

In the study formulas have been derived. permitting to find the coordinates of joints of a space truss made by the partition of the walls of a rhombic dodecahedron /irregular rhombohedron/. Constructing the cover /ceiling, roof/ of buildig structures on the basis of a rhombic dodecahedron is possible in two cases: in horizontal projection. they have the contour of a square, or a regular hexagon /Fig. 1 and 2/. Dividing the rhombic walls of a dodecahedron /Fig.3/ into ..n" equal parts, we will obtain „n" of rhombi similar /homothetic/ to and congruent with the wall. If in this lattice of rhombi we introduce parallel diagonals as cross-braces, we will receiv a truss with two lengths of truss members: the length of sides of rhombi and the length of diagonals. The aim of the study is to define the coordinates of nodes of thus defined truss for both above-mentioned contours of buildings: square and hexagonal. For an assumed room covered with the abowe-mentioned truss the length of a square or hexahedral projection has been denoted „d“ and co-ordinate system „Oxyz” successively as in Fig. 1 and 2 assumed. Naming „i“ and “j” the divided sides of rhombic walls into „n” of equal parts as in Fig. 3, each truss joint has been denoted „Wij“ /i,j=0,…n/. After transformations and on analysis of the made assumptions the coordinates of truss joints finally equal to: [pat tern] 1) for the building of a square plan [pattern] 2) for the building of a hexagon plan [pattern]