Dynamics of periodic and near-periodic structures

The purpose of this work is to initiate the study of effects of imperfections on dynamics of continuous one- or two-dimensional periodic systems. First, longitudinal wave motions and localized natural modes in a rod-like structure with periodically-alternating material properties are investigated. A simple finite element method is developed to simulate a wave field within the periodic rods. A new transfer matrix method is proposed to get closed-form wave and natural mode solutions. The effects of a single disorder on localized modes are investigated. Subsequently, the natural frequency structure and mode localization of multiply-disordered periodic rods are investigated. The Monte Carlo and the perturbation methods are applied to study some intricate effects of material parameter uncertainties on natural modes and frequencies of randomly-disordered periodic rods. It is shown that multiple and/or random disorders lead to more localized modes in or near stopbands in a more complex way. Next, longitudinal wave localization in layers of a materially periodic laminated media is investigated. The effect of material parameter imperfections on wave localization is studied statistically using Monte Carlo simulation. Wave localization is shown to be a fundamental feature of high frequency dynamic behavior of randomly-disordered continuous periodic systems. Finally, the vibration of multi-coupled 2-D and 3-D lattices with imperfections are parametrically investigated. It is shown that both free and forced vibration localization due to random mass disorder exist, but are otherwise abated with the increase in the dimension of the lattices. Disorder also tends to increase energy leakage from one degree-of-freedom to another degree-of-freedom at the vibration localization sites.