Nonlinear localization in mistuned periodic structures

A characteristic phenomenon encountered in weakly coupled repetitive systems (systems composed of identical, repetitive substructural elements) is mode localization. Such localized modes have been investigated widely in the literature, in which it has been shown that eigenvalue veering in mistuned linear systems and mode bifurcations in perfectly tuned nonlinear systems give rise to motions during which a system’s vibrational energy may be spatially confined to a small subset of its elements. In the present work, a systematic investigation of the combined effects of nonlinearities and structural mistunings is discussed. The method of multiple scales is utilized to compute localized modes for an n degree-of-freedom nonlinear cyclic system with structural mistunings. Strong and weak localized motions are computed for various structural parameters, and it is shown that the presence of nonlinearities and mistunings can enhance the localization effect for some modes, while simultaneously diminishing the effect for others. Sample calculations will be presented for systems composed of two, three, and four degrees of freedom. The implications of nonlinear mode localization for vibration isolation are also discussed.