Complete bandgaps in three-dimensional holey phononic crystals with helmholtz resonators

In this paper, the bandgap properties of three-dimensional holey phononic crystals with Helmholtz resonators are investigated by using the finite element method. The resonators are periodically arranged cubic lumps in the cubic holes connected to the matrix by one narrow connector. In contrast to a system with cubic or spherical holes, which has no bandgaps, systems with Helmholtz resonators can exhibit complete bandgap, which is lower by an order of magnitude than the Bragg bandgap can be obtained. The vibration modes at the band edges of the lowest bandgap are analyzed in order to understand the mechanism of the bandgap generation. It is found that the emergence of the bandgap is due to the local resonance of the resonators. Spring-mass/ pendulum models are developed in order to evaluate the frequencies of the bandgap edges. The study in this paper is relevant to the optimal design of the bandgaps in light porous materials.