Localization of vibration in disordered multi-span beams with damping

The combined effects of disorder and structural damping on the dynamics of a multi-span beam with slight randomness in the spacing between supports are investigated. A wave transfer matrix approach is chosen to calculate the free and forced harmonic responses of this nearly periodic structure. It is shown that both harmonic waves and normal modes of vibration that extend throughout the ordered, undamped beam become spatially attenuated if either small damping or small disorder is present in the system. The physical mechanism which causes the attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder. The corresponding rates of spatial exponential decay are approximated by applying statistical perturbation methods. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong inter-span coupling, but interact less trivially in the weak coupling case. Furthermore, the effect of disorder is found to be small relative to that of damping in the case of strong inter-span coupling, but of comparable magnitude for weak coupling between spans. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.