FREQUENCY COALESCENCE AND MODE LOCALIZATION PHENOMENA - A GEOMETRIC-THEORY

Physical systems, the natural frequencies of which depend parametrically on certain quantities, may exhibit the phenomenon of frequency coalescence, when two or more natural frequencies become equal; or the phenomenon of avoided crossings, when two or more frequencies approach rapidly and then avoid crossing each other. In the process, the natural modes undergo significant variations. It is shown that both phenomena can be described in a unified manner from the singularities of the mapping from the complex parameter plane into the complex frequency plane. In particular, the formation of a saddle point in the frequency plane and a branch point in the parameter space is the basis for explaining the properties associated with these phenomena. The question of sensitivity of the system response near such singularities is subsequently examined. It is shown that the development of a branch point singularity in the complex parameter plane can cause large or even infinite sensitivity of the system about this point, depending on the distance of the branch point from the real axis in the parameter space. It is further shown that mode localization is also associated with the appearance of a branch point near the real axis of the complex parameter plane, which causes the large sensitivity that characterizes the phenomenon. The theory is applied to examples of conservative and dissipative systems undergoing free vibrations. © 1991.