Localized vibrations of nearly periodic structures with irregularities.

Abstract

Dynamic analysis of periodic assemblies is typically made under the assumption that the components making up the periodic structure are identical to one another. However, due to the inherent structural irregularities arising from unavoidable manufacturing and material tolerances, no structure can ever be made perfectly periodic. The effects of disorder on the dynamics of nearly periodic assemblies of monocoupled, multi-mode component systems are examined. Both the modal characteristics and the transmission of vibration from a local source of excitation are considered. Slight irregularities among the components of nearly periodic structures are found to have drastic effects on the dynamics of the entire assembly: the free vibration modes become localized to a small geometric region and the vibrational energy is confined near the source of excitation. Perturbation schemes are developed to predict the occurrence of localization and to evaluate the systems sensitivity to disorder. A statistical formulation is chosen that provides a comprehensive analytic description of localization effects. Since component mode analysis is used to model the nearly periodic structure, the theoretical results obtained are generic, and thus valid for many engineering periodic structures. The degree of localization is shown to depend on the disorder strength, the modal coupling, the location of the constraint between subsystems, and the excitation frequency. In general, the localization effects increase rapidly with frequency. In fact, for arbitrarily small disorders severe localization is unavoidable if the mode number is large enough. This has important consequences for periodic engineering structures such as mistuned bladed-disk assemblies.