LOCALIZATION OF LONGITUDINAL WAVES IN BI-PERIODIC ELASTIC STRUCTURES WITH DISORDER

Abstract

A formulation for studying the effect of random variation in the transfer matrix on the attenuation behavior of disordered one-dimensional bi-periodic layered structures is developed. This formulation, however, can be used for both stochastically and deterministically disordered systems. The mean and variance of localization factors for the disordered systems are numerically evaluated under the assumption that the source of disorder is the variance in the Young's modulus of the first layer of a set, and this variation is modelled by a random variable with uniform probability density function. In the calculations for the mean localization factor and its variance for two layered systems with different elasticity properties are considered for four different levels of disorder. The results presented in the current work is used to explain how the attenuation zones of a perfect system expand into adjacent propagation zones due to disorder. It is found that while the existence of disorder affects the structures of all propagation zones, its significance is most predominant in the first propagation zones of both systems. However, as the frequency increases, the disorder level and the effect of elastic coupling become less significant. Therefore, the wave components corresponding to higher propagation zones penetrate deeper into a structure. It is observed that the right boundary of the propagation zones is the mean localization factor asymptote. The behavior of the mean Lyapunov curves near the right propagation zone boundaries explains how the disorder level governs the expansion of attenuation zones into propagation zones. This expansion is strongest in the first propagation zone. Since the power density of motion induced by an external force is high in the first propagation zone, the structure and size of first propagation zone are of a particular interest in transient analysis. The structures of the variance curves exhibit similar behavior to those for the mean localization factor.