On group-theoretic eigenvalue vibration analysis of structural systems with C6v symmetry

In considerations of the linear vibration of symmetric systems, group theory allows the space of the eigenvalue problem to be decomposed into independent subspaces that are spanned by symmetry-adapted freedoms. These problems usually feature one or more degenerate subspaces (i.e. subspaces that contain repeating solutions). For such subspaces, the associated idempotents, as calculated from the character table of the symmetry group, are not capable of full decomposition of the subspace. In this paper, and based on group theory, simple algebraic operators that fully decompose the two degenerate subspaces of structural problems belonging to the C6v symmetry group are proposed. The operators are applied to the vibration of a spring-mass system, for which the results for natural frequencies are found to agree exactly with results from the literature. Their application to the vibration of a hexagonal plane grid reveals new insights on the character of the modes of degenerate subspaces. The overall conclusion is that, for problems belonging to the C6v symmetry group, the proposed operators allow the mixed modes of degenerate subspaces to be separated into two distinct symmetry categories, and are very effective in simplifying the actual computation of the repeating eigenvalues of these subspaces.