Flexural wave band gaps in locally resonant thin plates with periodically attached spring-mass resonators

The authors study the propagation of flexural waves in a locally resonant (LR) thin plate made of a two-dimensional periodic array of spring-mass resonators attached on a thin homogeneous plate. The well-known plane wave expansion method is extended to deal with such a plate system with a periodic array of lumped resonant elements. Explicit matrix formulations are developed for the calculation of complex band structures, in which the imaginary parts of Bloch wave vectors are displayed to quantify the wave attenuation performance of band gaps. It is found that resonance-type and Bragg-type band gaps coexist in the LR plate, and the bandwidth of these gaps can be dramatically affected by the resonant frequency of local resonators. In particular, a super-wide pseudo-directional gap can be formed by a combination of the resonance gap and Bragg gap; inside such a pseudo-gap, only a very narrow pass band exists. An explicit formula is further developed to facilitate the design of such a pseudo-gap. Finally, vibration transmission in finite LR plates is calculated using the finite element method. Vibration transmission gaps are observed, and the results are in good agreement with the band gap properties predicted by the complex band structures.