Generalized Eigenvalue Analysis of Symmetric Prestressed Structures Using Group Theory

As conventional approaches for calculating natural frequencies do not make full use of the inherent symmetry of a structure, the rising degree of freedoms often leads to significant increase in computational demand. In this study, a simplified technique for analyzing dynamic characteristics of symmetric prestressed structures is described using group theory. First, the generalized eigenvalue equation of a prestressed structure based on tangent stiffness matrix and lumped mass matrix is built to get natural frequencies and the corresponding vibration shapes in which the contribution of initial prestresses is considered. A symmetry-adapted coordinate system for the structure is adopted to block-diagonalize the stiffness and mass matrices. The complexity of generalized eigenvalue equation is reduced by solving the mutually independent subspaces, and thus natural frequencies and the corresponding vibration modes could be obtained. Illustrative examples point out the general procedure, and show the superiority. Compared with numerical results, it has been proven that the novel symmetry method using group theory is accurate and very efficient.