Vibration mode structure and simplified modelling of cyclically symmetric or rotationally periodic systems

Abstract

This paper proves the modal vibration properties of general rotating, cyclically symmetric (or rotationally periodic) systems, including those with central components to which cyclically symmetric substructures are attached. This cyclic symmetry results in structured modal properties with only two possible mode types referred to as substructure and coupled modes. For systems with uncoupled central component translations and rotations, which is the usual case, all eigenvectors fall into one of three categories: substructure, translational and rotational modes. The properties of the system equations of motion resulting from the cyclic symmetry are discussed first. These properties are then used to prove the modal decomposition of general rotating, cyclically symmetric systems. The development leads to modelling and computational efficiencies. The vibration modes and natural frequencies for each mode type are determined from reduced eigenvalue problems that are much smaller than the full system eigenvalue problem. The full system matrices, although not needed for the current purposes, can be generated from much smaller matrices derived from simple subsystem models.