Optimal mode localization in disordered, periodic structures

Abstract

Periodic structures which are slightly disordered undergo dramatic changes in mode shapes such that the responses go from being spatially extended to spatially localized. This phenomenon called mode localization, offers an excellent option for passive vibration isolation. In the first part of the thesis, we provide analytical prediction of modes exhibiting moderate localization using a newly developed Jordan Block Perturbation Method. We estimate and compare convergence zones of our newly developed method with perturbation techniques used to describe localized modes. In the second part of the thesis, we provide numerical evidence that complex branch points, which occur for complex disorder values in the mode-disorder relation, are responsible for modal sensitivity. We investigate the effects of the strength of the branch point and their location in the complex plane. In the third part of the thesis we perform an optimization study involving the selection of parameters which ensure a minimum level of localization of all modes. Optimal solutions were found to lie at maximum distances from the branch points, and the convergence basin of each optimum was demarcated by the branch point surface. The number of local optima were found to grow exponentially with the number of pendula. A statistical analysis showed that sampling of 10% provided an estimate that was within 2% of the global optimum, thereby reducing the computational effort for small to moderate systems of pendula. For larger systems of pendula, the problem of obtaining the global optimum in reasonable time still remains an open problem. In the fourth part of the thesis we propose an application for mode localization in vibration isolation. An oceanographic mooring with regularly spaced buoys is investigated for localization of inline elastic oscillations. Localization is found to be useful for confining the harmonics in deep water moorings of 1000 - 4000m.