Platonic crystal with low-frequency locally-resonant spiral structures: wave trapping, transmission amplification, shielding and edge waves.

We propose a new type of platonic crystal, which includes spiral resonators with low- frequency resonant vibrations. The special dynamic effects of the resonators are high- lighted by a comparative analysis of dispersion properties of homogeneous and perforated plates. Analytical and numerical estimates of classes of standing waves are given and the analysis of a macrocell shows the possibility to obtain localization, wave trapping and edge waves. Applications include transmission amplification within two plates separated by a small ligament and a novel design to suppress low-frequency flexural vibrations in an elongated plate implementing a bypass system that re-routes waves within the mechanical system. Finally, we show the possibility to obtain Konenkov-Bloch edge waves for general boundary conditions.