Numerical computation of wave localization in large disordered beam-like lattice trusses

Abstract

Wave localization in randomly disordered multiwave structures is investigated. The smallest localization factor is of particular interest and is related to the smallest positive Lyapunov exponent. The numerical algorithm by Wolf et al. is modified to determine all of the Lyapunov exponents. The wave localization in a large beamlike lattice truss modeled as an equivalent continuous Timoshenko beam is studied. ANY engineering structures are designed to be composed of identically constructed elements assembled end to end to form a spatially periodic structure, such as long space antennae or large beamlike lattice trusses used in space solar power stations. Periodic structures behave like bandpass filters when propagating stress waves. If damping is neglected, waves are classified as traveling waves that propagate without attenuation and nontraveling or attenuating waves whose amplitudes attenuate as the waves propagate. For single-wave periodic structures, the frequency axis is divided into alternating passbands and stopbands. In the frequency passbands, the wave is a traveling wave, whereas in the frequency stopbands, the wave is an attenuating wave. For multiwave structures, in a given frequency band, some of the waves may be traveling waves and the others nontraveling waves. However, due to defects in manufacture and assembly, no structure designed as a periodic structure can be perfectly periodic. Disorder can occur in the geometry of configuration s and material properties of the structure. In a disordered periodic structure, wave amplitudes of all waves will be attenuated, even those that are traveling waves in the perfectly periodic counterpart. This means that the vibrational energy imparted to the structure by an external source cannot propagate to arbitrarily long distances but is instead substantially confined to a region close to the source. This phenomenon is known as wave localization. In disordered periodic structures, it is therefore of great practical importance to study the localization behavior and evaluate the so-called localization factors, namely, the exponential rates at which the amplitudes of the waves propagating in the structures decay. The reciprocal of the localization factor gives the localization length, which characterizes the distance that the propagating wave extends in the structure. As an application, the range of damage that is spread in a structure due to impact at some location can be estimated. There are d waves in a d-wave disordered periodic structure; each wave attenuates at a certain exponential rate or corresponds to a certain localization factor, which implies that each wave will extend to a certain localization length. The smallest localization factor or the largest localization length is of particular interest for multiwave structures, since it corresponds to the wave that has potentially the least amount of decay or that carries energy along the multiwave structure farther than any other waves. The study of the localization phenomenon has been and remains an active research area in solid state physics after the celebrated work of Anderson. 1 Hodges2 was the first to recognize the relevance of localization theory to dynamical behavior of periodic engineering structures. Since then, there have been several studies on the localization of vibration of structures. In a series of publications, Pierre and Dowell,3 Pierre,4 and Cha and Pierre5 studied the localization phenomenon for monocoupled disordered structural systems. Cai and Lin6 developed a perturbation scheme to calculate the localization factor based on a generic periodic structure. Kissel7 and Ariaratnam and Xie8'9 used a traveling wave approach to investigate the localization effects in one-dimension al periodic engineering structures. A transfer matrix formulation, including wave transfer matrices, was used to model disordered periodic structures. Furstenberg's theorem10 for products of random matrices was applied to calculate the localization effect as a function of frequency. The localization factor was related to the largest Lyapunov exponent. In these publications, only one-dimensional or monocoupled periodic structures were considered, in which the transfer matrices are of dimension 2. Few studies have been done on the localization behavior of multiwave periodic structures. Kissel11 derived the localization factor as a function of the transmission matrix for multiwave disordered systems using the multiplicative ergodic theorem of Oseledec. 12 The difficulty in the study of multiwave periodic structures is that the localization factor is related to the smallest positive Lyapunov exponent and not the largest Lyapunov exponent as in the singlewave case. The numerical algorithms used previously for evaluating the largest Lyapunov exponent for single-wave periodic structures cannot be employed for multiwave structures. In this paper, a modification of an algorithm due to Wolf et al.13 is used to determine all of the Lyapunov exponents for randomly disordered multiwave periodic structures. The localization factor of a large beamlike lattice truss, modeled as an equivalent continuous Timoshenko beam, is evaluated.