Wave localization and conversion phenomena in multi-coupled multi-span beams

The linear dynamics of nearly periodic disordered multi-span beams resting on flexible supports are investigated. A wave transfer matrix methodology is chosen to examine the propagation of waves and the transmission of vibration along the structure. The spans are bi-coupled through the rotation and the transverse displacement at the supports and thus the beam motion is made up of two independent wave types. While for the ordered infinite beam there exists frequency passbands for which the free harmonic waves propagate without attenuation, the introduction of a slight disorder among the span lengths results in the localization of the vibration energy to few spans and in the conversion of the energy from one type of wave to the other. The energy conversion phenomenon renders the mechanism of localization much more complex than in mono-coupled periodic systems. The contribution of each type of wave to the global beam motion is analyzed in terms of frequency. It is observed that the spatial decay of each wave type is mainly governed by an exponential envelope. The corresponding exponential decay constants define a measure of localization for each wave and are found to be equal to the Lyapunov exponents of the product of random wave transfer matrices. It is also found that at frequencies which belong to a passband for both wave types, the decay rate of an incident wave vector is bounded by the two Lyapunov exponents, while at frequencies which belong to a passband for one wave type and a stopband for the other, localization effects are best predicted by the smallest of the two Lyapunov exponents.