Vibrations in Systems Possessing Geometric Symmetry: Effect of Asymmetry

Vibrations in mechanical systems with geometric symmetry have been studied. It has been shown that, in systems with a low level of asymmetry, stratification of multiple natural frequencies occurs, which leads to instability of forced vibrations in this frequency range, as well as to beats occurring in the course of natural vibrations. In the case of mechanical systems, symmetrically arranged structural elements usually have several degrees of freedom or represent separate subsystems. Therefore, block symmetry operators and basis vectors that characterize the interactions between these structural elements determined by the symmetry conditions have been introduced. It has been shown that, for systems with a symmetry hierarchy, the resulting symmetry operator represents the product of the operators corresponding to each symmetry group. It has been found that the basis vectors for a given type of symmetry remain the same for nonlinear systems with the same type of symmetry. Their use makes it possible to divide the initial equations with an odd nonlinearity function into independent nonlinear equations, each of which describes its own coordinate. The mathematical apparatus of the theory of group representation is used.