Localization of Vibration Propagation in Randomly Disordered Weakly Coupled Two-Dimensional Cantilever-Spring Arrays

In this paper, the method of regular perturbation for a linear algebraic system is applied to study localization of vibration propagation in randomly disordered weakly coupled two-dimensional cantilever-spring arrays under external harmonic excitations. Iterative equations are obtained to express the displacement vector of the cantilevers on the Mth “layer” in the Mth-order perturbation in terms of those on the (M − 1)th “layer” in the (M − 1)th-order perturbation. Localization factors, which characterize the average exponential rates of decay of the amplitudes of vibration, are defined in terms of the angles of orientation. First-order approximate results of the localization factors are obtained using a combined analytical-numerical approach. The localization factors are symmetric about the horizontal and vertical axes passing through the cantilever that is being externally excited. For the systems under consideration, the direction in which vibration is dominant corresponds to the smallest localization factor; whereas the “diagonal” directions correspond to the largest localization factor.