Analytical amd experimental aspects of nonlinear normal modes and nonlinear mode localization.

摘要

In this work, several aspects of nonlinear normal modes (NNMs) and free and forced nonlinear mode localization are investigated. In the first phase of the work, new asymptotic methodologies based on the concept of NNM are developed in order to compute synchronous responses for two classes of conservative nonlinear systems; namely (i) periodic systems composed of repeated sub-structural elements and (ii) systems possessing internal resonances. For coupled repetitive systems, NNMs are shown to provide an excellent framework by which to study the nonlinear mode localization phenomenon, wherein vibrational energy is spatially confined to a limited number of sub-structures, and criteria are established in order to detect these localized normal modes. Internal resonances have provided somewhat of an obstacle in existing NNM methodologies, thus modifications of the energy-based NNM methodology are also developed in order to appropriately account for resonant interactions. Several applications of these methodologies are considered. The second phase of this thesis is concerned with analytical and experimental studies of localization in cyclic and non-cyclic periodic systems. Examples of such systems encountered in engineering practice include bladed disk assemblies, large space structures composed of repeated bays and periodically stiffened plates and shells such as those found in airplane fuselages. Extensive work in the literature has focused on localization in linear systems, from which two necessary ingredients for the existence of localized modes have been identified: weak sub-structure coupling and weak structural mistunings. In the present work, free and forced mode localization are investigated in periodic systems whose sub-structures are weakly nonlinear. In contrast to linear theory, nonlinear mode localization is shown to exist in the absence of any structural mistunings (i.e. in perfectly periodic systems). In an effort to verify some of the analytical localization results, an experimental investigation of steady-state localization in a flexible system with active stiffness nonlinearities is performed. It is hoped that this work provides a first step toward the development of new approaches to passive and/or active vibration and shock isolation designs based on the concepts of NNMs and nonlinear mode localization.