Scattering reduction and resonant trapping of flexural waves: Two rings to rule them

In this paper, we discuss two problems concerning scattering and localisation of flexural waves in structured elastic plates. Firstly, we compare the scattering amplitudes of waves in a thin plate, generated by a point source, due to a single mass and to a large number of smaller masses, having the same equivalent mass and located around a circle. We show that in the second case, the scattering can be reduced, in particular in the medium-and high-frequency regimes. Secondly, we develop a homogenised model for a double-ring cluster of spring-mass resonators, connected to an elastic thin plate. We determine the conditions for which the plate exhibits vibration modes trapped between the two rings. Further, we show that the frequencies of the localised modes can be tuned by varying the geometry of the two rings and the characteristics of the resonators. The analytical results are corroborated by numerical simulations performed with independent finite element models.