Decomposition of symmetric mass–spring vibrating systems using groups, graphs and linear algebra

Abstract

SUMMARYThe main objective of this article is to develop a methodology for an efficient calculation of the eigenvalues for symmetric mass-spring systems in order to reduce the size of the eigenproblem involved. This is achieved using group-theoretical method, whereby the model of a symmetric mass-spring system is decomposed into appropriate submodels. The eigenvalues of the entire system is then obtained by calculating the eigenvalues of its submodels. The results are compared to those of the existing methods based on graph theory and linear algebra. Examples are provided to illustrate the simplicity and efficiency of the present method.

Group‐theoretic exploitations of symmetry in computational solid and structural mechanics

“…Certain configurations of spring-mass systems, while themselves not physically symmetric, may be transformed into equivalent symmetric models that preserve all the graphical connectivities between masses and springs, allowing group theory to be employed to simplify the extraction of eigenvalues for the equivalent system. In more recent considerations of the vibration of such systems, Kaveh and Nikbakht [43] successfully bring together techniques of graphs and group theory, and employ the group-theoretic approach to simplify the calculation of eigenvalues. Their procedure involves assembly of the full stiffness matrix of the entire system, which is then blockdiagonalized via a transformation matrix.…”

Symmetry Exploitation in the Natural Vibrations of Rod Systems

“…This approach has been successfully applied to mechanical systems with a finite number of degrees of freedom [4] (molecular vibrations [5], mass-spring models of mechanical systems [6,7], finite element and finite difference models [8][9][10] etc.). In this case the symmetry group is a finite group, usually represented by spatial symmetry of the system, and the irreducible representations and their respective projectors can be easily found with the tables of characters of the finite groups irreducible representations [11].…”

A self-regularized approach for deriving the free–free flexibility and stiffness matrices

“…Related works can be found in the four references [14][15][16][17]. However, our approach introduces a slack variable as well as a corresponding constraint to deal with rank-deficient matrices.…”