Wave localization and conversion phenomena in disordered multi-span beams: Theory and experiment

The combined effects of periodicity-destroying irregularities and structural damping on the dynamics of multi-span beams are investigated. 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 this attenuation, however, is one of energy dissipation in the case of damping but one of energy confinement in the case of disorder.; The confinement phenomena are studied over a wide frequency range for beams resting on simple supports and with variable interspan coupling. Approximations of the localization factor (the average rate of spatial exponential decay of the vibration amplitude) are derived by statistical perturbation methods and validated by Monte Carlo simulations. If the interspan coupling is strong, localization effects are weak and of no practical significance for most engineering structures. On the other hand, localization is severe for small interspan coupling. It is found that the effects of damping and disorder simply superpose for a multi-span beam with strong interspan coupling, but interact less trivially in the weak coupling case. The adequacy of the statistical analysis to predict accurately localization in finite disordered beams with boundary conditions is also examined.; An experimental investigation of the effects of span length disorder on the dynamics of a twelve-span beam specimen is performed. Experimental results demonstrate that the transmission of vibration which takes place within the frequency passbands of the periodic beam is greatly hindered when span length randomness is introduced. The spatial localization of both the mode shapes and the steady-state harmonic response to an end excitation is observed. This study reports one of the first systematic experiments carried out to demonstrate the occurrence of vibration localization in nearly periodic structures.; The dynamics of ordered and disordered multi-span beams on flexible supports are also investigated. This multi-coupled structure is characterized by two pairs of waves. While for the ordered infinite structure there exist frequency passbands for which the free harmonic waves propagate without attenuation, the introduction of disorder results in a wave localization to a few of the spans and an energy conversion from one type of wave to the other. It is shown that the spatial decay of each wave type is governed by an exponential envelope. These exponential decay constants define a measure of localization and are found to be equal to the Lyapunov exponents of the product of the random wave transfer matrices of the spans.