Vibration of multi-stage systems with arbitrary symmetry of stages: A group theory approach

Abstract

This work develops a group-theory-based method to identify the highly structured, symmetry related modal properties of multi-stage systems with symmetric component stages. The symmetry of each component stage is arbitrary and can differ among the stages. The full multi-stage system, however, is not symmetric. Without solving any eigenvalue problems, all modes are classified into mode types and the modal properties of each mode type are obtained. These mode types and properties are determined from the symmetry of the component stages and the coupling between the stages. In particular, full system mathematical models are unnecessary to determine the modal properties; they are needed only when numerical computation of modes is required. All vibration modes and natural frequencies are solved from multiple, smaller, decoupled eigenvalue problems associated with each mode type. These smaller problems are computationally efficient. Dynamic and static response are similarly analyzed using reduced problems with smaller matrices associated with each mode type. For general multi-stage systems with arbitrary stage symmetries, all vibration modes are categorized into single-stage substructure modes and overall modes. In single-stage substructure modes, the modal deflections occur in the substructures of one component stage, while all the central components and the substructures in the other stages are non-vibrating. In overall modes, all stage substructures and central components vibrate. More detailed conclusions are possible for specific stage symmetries.