Vibration of general symmetric systems using group theory

Abstract

This work presents group-theory-based vibration analysis to determine the highly structured, symmetry-related modal properties of symmetric systems and classify them into specific types. No eigenvalue problem solutions are required. In fact, the properties are obtained without a mathematical model. They derive only from symmetry and the degrees of freedom chosen to describe the deformation. Dynamic or static response is similarly decomposed into the sum of multiple response components associated with the mode types. One knows a priori, without any numerical computation, which mode types can or can not be excited by a given input excitation. Group theory divides the full system equations of motion and eigenvalue problem into multiple, smaller, decoupled problems associated with each mode type, which is computationally efficient. The method applies for general three-dimensional systems having any type of symmetry where components of the system are modeled using any mixture of lumped-parameter (including finite element) and elastic continuum models. Only basic elements of group theory are required, and they are introduced. Among three examples of the method, one derives the modal properties for general cyclically symmetric systems having any number of central components and substructures that can be modeled as any of rigid bodies, finite element meshes, and elastic continua.