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Mode Localization in a Class of Multidegree-of-Freedom Nonlinear Systems with Cyclic Symmetry

The free oscillations of n-degree-of-freedom (DOF) nonlinear systems with cyclic symmetry and weak coupling between substructures are examined. An asymptotic methodology is used to detect localized nonsimilar normal modes, i.e., free periodic motions spatially confined to only a limited number of substructures of the cyclic system. It is shown that nonlinear mode localization occurs in the perfectly symmetric, weakly coupled structure, in contrast to linear mode localization, which exists only in the presence of substructure “mistuning.” In addition to the localized modes, nonlocalized modes are also found in the weakly coupled system. The stability of the identified modes is investigated by means of an approximate two-timing averaging methodology, and the general theory is applied to the case of a cyclic system with three-DOF. The theoretical results are then verified by direct numerical integrations of the equations of motion.