FINITE SYMMETRICAL SYSTEMS AND THEIR ANALYSIS

Symmetry of linear mechanical systems permits one to substantially reduce the effort in their analysis. Classification and structural analysis of the finite linear symmetric systems (models) are studied in this paper. There are two variants of the symmetry approach: mechanical and algebraic. In accordance with the former, a symmetric system is replaced by one or several small nonsymmetric subsystems which are subjected to special loads obtained from the initial set. The total response of the original symmetric system is found by special superposition of partial responses of these subsystems. The algebraic approach is based on the explicit block diagonal decomposition of the matrix equation corresponding to a symmetric system. While both approaches have the same efficiency the latter is easier to implement and describe. It is presented here. The conditions under which the symmetry technique may be utilized do not include symmetry of the applied loads (specifically, symmetry of those loads which form the right side of the associated equations). Nevertheless, if the loads are symmetric, the efficiency of the symmetry approach substantially increases. Group theory, which is widely used in this paper, is the mathematical tool for the study of symmetry, and all necessary notions are introduced.