Band-gap dynamics and programming for low-frequency broadband elastic metamaterial

Local resonant elastic metamaterials are broadly studied as sound barriers, vibration isolators and energy harvesting walls. For real applications of these devices in the environment with low frequency and broadband sound waves, band-gap frequency and width of the elastic metamaterial should be programmed. This paper proposes an effective dynamic method to estimate the band-gap edge frequencies of the local resonant elastic metamaterials with four representative resonators. Particularly, uniform resonator, tapered resonator, double-decked resonator and necked resonator are examined. According to Bloch's theorem, periodic boundary conditions are introduced to transform vibration study of the elastic metamaterial into vibration analysis of its unit cell. Wave equations of the axial vibrations of the unit cells are established. The frequency functions are deduced for predicting the lower and upper edge frequencies of the band-gap. Finite element analyses with Bloch periodic conditions are performed, and the band structures and eigenmodes of the four configurations verify the mathematical model and frequency functions. The resonator dimension is programmed for realizing the maximum normalized band-gap width with respect to the lower edge frequency and unit cell volume considering space limitation. The elastic metamaterial with the necked resonators is demonstrated to have the most superior low-frequency and broadband performances.