Wave Localization in Multi-Coupled Periodic Structures: Application to Truss Beams

Abstract

A tutorial and a review of recent developments in the area of localization in linear structural dynamics problems are presented. Particular emphasis is placed on multi-coupled nearly periodic structures, which carry more than one wave type. First, background on perfectly periodic structures is provided, including both the wave and modal descriptions of the dynamics. A wave transfer matrix formulation for disordered periodic structures is then presented, which is well suited to the analysis of localized dynamics. Next, stochastic analysis tools are introduced that allow one to quantify the degree of localization in an asymptotic sense. Means of calculating these localization factors as the Lyapunov exponents of the system wave transfer matrix are discussed. Finally, the general theory is illustrated on an example multi-coupled structure - a planar truss beam which carries four pairs of waves. The propagation of waves in the disordered structure is examined, and Lyapunov exponents are calculated. In addition to the localization of the incident wave, significant mixing of the various wave types occurs, causing the leakage of energy to the least localized waves, and enabling sustainment of motion according to the smallest Lyapunov exponent.