Band gap extending of locally resonant phononic crystal with outward hierarchical structure

Locally resonant phononic crystals (LRPC) are able to give low-frequency band gap to attenuate the vibration in a specific frequency range; however, the local resonance band gaps (LRBP) obtained with conventional structures are too narrow for practical applications. In this paper, the outward hierarchical locally resonant phononic crystal (OHLRPC) structure is constructed to extend the band gap. To this end, the band gap range is determined by band structure calculation, and the band gap mechanism is discussed by modal analysis. The band structure indicates OHLRPC can extend the locally resonant band gap width, which is proven by frequency response function. In addition, the modal analysis shows that the multiple oscillator vibration caused by OHLRPC results in strong coupling response, which produces the more and wider band gap. The band gap determination formula based on effective mass theory is derived to obtain quickly the band gap range. The discussions of parameter analysis demonstrate that the band gap location and width can be optimized to obtain low-frequency and broadband simultaneously by configurating reasonably the geometric parameter, such as the primitive cell size, the ratio of coating layer radius to scatterer radius and lattice constant. Furthermore, the MOHLRPC with multiple oscillator system is proposed to expand the investigation, which has wider band gaps and is easy to fabricate. The investigation can provide references for the design of two-dimensional local resonance structure to tailor the attenuation frequency range and effect according to the actual vibration environment.