Nonlinear localization in a flexible system of beams

In this work a system of two linearly coupled, identical cantilevered beams with geometric nonlinearities is considered. Using a Galerkin expansion, the system is discretized and the dynamics of the first three linearized modes are examined. Localized motions are detected during which motion is confined to only one of the two beams. Since the first linearized natural frequency is incommensurable with higher frequencies, the dynamics of the first linearized mode can be decoupled from those of higher modes. Localized motion of the first mode exists only for small values of the stiffness parameter (the ratio between the coupling stiffness coefficient and the beam stiffness). The second and third modes must be considered together due to the 1:3 relationship between the second and third linearized natural frequencies. The character of localization in the presence of this internal resonance is found to differ greatly from the first case where the single mode is considered. Localization of the second and third modes is found to occur for higher values of the stiffness parameter, a result which agrees with linear localization theory.