Response localization in disordered structures governed by the Sturm-Liouville differential equation. (Review)

Abstract

The review is dedicated to the relatively new problem in structural engineering: localization of the response by structural irregularities. This review is aimed to outline all relevant discoveries in the response localization in mechanical problems (vibration, buckling) from the perspective of the common mathematical representation through Sturm-Liouville problem. Two possible approaches to analyze the influence of the disorder are discussed: exact dynamic stiffness formulation of the mistuned structure and the perturbation of the eigen solution of the tuned structure. Both approaches shown to lead to the same localization phenomena end exponential decay of the eigenvector from the source of disorder. In the section dedicated to the buckling mode localization the approach to analyze localization of the randomly disordered multi-span beam based on the Furstenberg’s theorem in presented. The examples of the localization phenomena in the real engineering structures are given.