Low frequency resonant bandgaps in platonic crystals made of N-beam resonators

N-beam resonators are resonant structures of easy manufacturing with cutting machines. They consist of a circular hole containing a smaller plate connected to the background plate with N rectangular beams. We demonstrate that this type of resonators are effective in opening low frequency complete bandgaps for flexural waves. To begin with, we investigate the effect of the number of beams on the resonator fundamental mode, whose behaviour is analogous to a simple springmass oscillator. The analysis of the scattering cross-section for a single resonator reveals that the resonant frequency increases rapidly with the number of beams. Besides, the fundamental mode can only be excited in some specific directions when the resonator contains more than one beam. Therefore, a complete bandgap in the band structure is only obtained for lattices of 1-beam resonators. The response of the 1-beam resonators produces an extremely narrow frequency bandgap, but it can be substantially broadened by adding more resonators into the hole. As an example, we have shown that the width of the low frequency pseudo-complete bandgap is broadened by about one order of magnitude after introducing another three resonators into the hole. The results here reported have potential application for controlling low frequency vibration in plates.