Use of symmetry groups for generation of complex space grids and group-theoretic vibration analysis of triple-layer grids

Abstract

Space grids often exhibit a highly regular structure which is formed by repeated structural units. In the past, various algorithms have been developed to generate more complex structural assemblies using simple commands. In this paper, we explore the innovative use of symmetry groups to create various configurations of space grids, ranging from rectangular grids and their variants, to triangular and hexagonal grids, as well as grid forms of more complex shape. The advantage of this approach is that the generated space grid has pre-determined symmetry properties; this represents a shift in current design philosophy. A second objective is to show how group theory may be used to simplify the vibration analysis of layered space grids. The procedure is applied to the transverse vibrations of triple-layer space grids of D2h and D4h symmetries, modelled as discrete-parameter systems. A study of the symmetry of subspace basis vectors allows some important predictions to be made on the pattern of the motions and the location of stationary nodes. It is found that the lowest and highest frequencies of space grids of this type occur in the first subspace of the system, a finding that has significant computational importance.