| This thesis concerns the analysis and passive control of waves that propagate in smooth elastic structures, such as beam and plates, with arrays of substructures attached at regularly spaced locations. Examples of such structures include aircraft fuselages, ship hulls, and a variety of composite materials. This thesis presents new analytical descriptions that illuminate the physics of existing structures and suggest new applications of periodic structures. A primary contribution is the derivation of Floquet wave dispersion relations that are hierarchical in form and thereby quantify the effects of individual arrays on the vibrational response of the structure. Analysis of these relations indicates when a subset of arrays may be approximated by modifying the parameters of the smooth structure and provides a method for choosing the modifications. This “partial homogenization” approach will reduce the computational burdens of numerically analyzing structural designs and will provide structural designers with a better understanding of the effects of array parameters on structural response. In contrast, an “inverse homogenization” approach is developed in which continuous portions of a structure are approximated by an array of attached substructures. This approach is useful in constructing finite element models of composite structures from vibrational response data. These analytical contributions are illustrated by two applications of current interest. In the first, a new class of high-Q MEMS resonators is proposed in which energy is localized by periodic structures that dynamically simulate rigid structures. In the second, the inverse homogenization method is used to construct a transient model of a beam damping treatment for inclusion in a time-domain finite element model. |
