On group-theoretic computation of natural frequencies for spring–mass dynamic systems with rectilinear motion

Abstract

In this article, group theory is employed to simplify the computation of natural circular frequencies for spring–mass dynamic systems with rectilinear motion. The systems are by themselves not physically symmetric (or they exhibit only very weak symmetry properties), but they can be transformed into graphs that preserve all the connectivities between masses and springs, while featuring the maximum possible symmetry. For physical dynamic systems exhibiting symmetry, a well-known group-theoretic approach involves the computation of the full stiffness matrix of the system first, followed by transformation of this matrix in order to cast it into block-diagonal form. The present approach involves the direct assembly of much smaller stiffness matrices within the decomposed subspaces of the problem, and is therefore computationally more efficient. Of particular focus in this study are transformed configurations belonging to ‘triangular’ symmetry groups, whose symmetries are difficult to exploit using conventional methods. It is shown how the repeating eigenvalues associated with the degenerate subspaces of such symmetry groups can easily be obtained. Copyright © 2007 John Wiley & Sons, Ltd.