Vibration mode localization in two-dimensional systems

The method of regular perturbation is applied to study vibration mode localization in randomly disordered weakly coupled two-dimensional cantilever-spring arrays. Localization factors, which characterize the average exponential rates of decay or growth of the amplitudes of vibration, are defined in terms of the angles of orientation. First-order approximate results of the localization factor are obtained using a combined analytical-numerical approach. The localization factors are symmetric about the cantilever and the horizontal and vertical axes passing through the cantilever at which vibration is originated. For the systems under consideration, the direction in which vibration is originated corresponds to the smallest localization factor; whereas the diagonal directions correspond to the largest rate of decay or growth of the amplitudes of vibration. When plotted in the logarithmic scale, the vibration modes are of a hill shape with the amplitudes of vibration decaying linearly away from the cantilever at which vibration is originated.