Vibration localization in one-dimensional linear and nonlinear lattices: discrete and continuum models

The phenomenon of vibration localization plays an important role in the dynamics of inhomogeneous and nonlinear materials and structures. The vibration localization can occur in the case of inhomogeneity under the following conditions: (i) the frequency spectrum of the periodic structure includes stopbands, (ii) a perturbation of periodicity is present, and (iii) the eigenfrequency of the perturbed element falls into a stopband. Under these conditions, the energy can be spatially localized in the vicinity of the defect with an exponential decay in the infinity. The influence of nonlinearity can shift frequency into the stopband zone. In the present paper, the localization of vibrations in one-dimensional linear and nonlinear lattices is investigated. The localization frequencies are determined and the attenuation factors are calculated. Discrete and continuum models are developed and compared. The limits of the applicability of the continuum models are established. Analysis of the linear problem has allowed a better understanding of specifics of the nonlinear problem and has led to developing a new approach for the analysis of nonlinear lattices alternative to the method of continualization.