Improved Form-Finding of Tensegrity Structures Using Blocks of Symmetry-Adapted Force Density Matrix

Abstract

Form-finding analysis is very important for developing innovative tensegrity structures. Nevertheless, it is frequently difficult to determine simultaneously the prestress mode and the exact nodal coordinates for a given geometry. This paper presents an improved symmetry method for analytical form-finding of tensegrity structures. The method is based on group representation theory and the force density method, and requires only the specified symmetry type and connectivity of a structure. The force densities of a d-dimensional tensegrity structure can be accurately determined using the zero determinant of d small-sized block matrices of a symmetry-adapted force density matrix. The nodal coordinate vectors can be obtained from the null spaces of the blocks through symmetry spaces for rigid-body translations along d directions. A number of geometries with cyclic symmetry, dihedral symmetry, or tetrahedral symmetry are investigated. Illustrative examples show that the proposed method allows significant simplification of the form-finding process. The analytical solutions offer all feasible stable or superstable tensegrity structures with expected symmetries. The obtained prestress modes for each independent structural configuration necessarily retain full symmetry.