Characteristics of band gaps of a metamaterial plate with membrane-type resonators based on the energy approach

Local resonance metamaterials have garnered substantial interest in recent years due to their potential to effectively control low-frequency noise and vibration. This study proposes a new metamaterial plate with double mass membrane-type resonators (DMMRs). Combining polynomial fitting and virtual spring method, the natural characteristics of the resonator and the dispersion surfaces of the metamaterial plate are obtained. Results show the proposed metamaterial plate exhibits multiple band gaps, resulting from the cancellation of shear forces in the plate by the reaction forces generated by the first two resonances of the resonators. Parametric analysis is performed to obtain the variation patterns of the band gaps, where approximate models of the resonator are adopted to explain the effects of the parameters. It is found that the occurrence and width of the two band gaps are closely related to the mass ratio between the masses on the resonator and the substrate plate. Furthermore, finite element analysis is conducted to verify the effectiveness of the theoretical derivation, as well as the suppression effect of the metamaterial plate. The findings demonstrate the advantages of the metamaterial plate in vibration control and band gap tuning. The study provides guidance for the design and analysis of elastic wave metamaterials.