# Radial Gradient Seismic Metamaterials with Ultra-Low Frequency and Ultra-Wide Band Gap
## 4 Radial Gradient Seismic Metamaterial Structure after Optimization of Supercell
### The Band Gap Mechanism of OS-RGSM  
The supercell method is used to calculate the energy band of the OS-RGSM structure. Since the optimized supercell contains three unit cells, the Brillouin zone is three times that of the original unit cell. The Brillouin zone boundary of the triangular cross-sectional radial seismic metamaterial unit cell is from \(\Gamma\) (0, 0) to R (1, 0), and the Brillouin zone boundary of the optimized supercell is from \(\Gamma\) (0, 0) to R (3, 0). Therefore, one band of the unit cell is folded into three bands in the supercell, and the three bands in the supercell are called a set  
Figure 8: OS-RGSM barrier for attenuation of seismic Lamb wave. Figure 9: The band gap characteristics of the OS–RGSM structure; (**a**) the dispersion curve of the OS–RGSM structure; (**b**) the transmission spectrum curve of the OS–RGSM structure. of bands. As shown in Figure 10, the energy band diagram of the OS-RGSM structure is found to have an ultra-low broadband gap from 2.1-20 Hz. In addition, multiple localized flat bands can be observed in the band diagram. In the frequency range of 0-20 Hz, the relative bandwidth of the band gap generated by the OS-RGSM structure proposed in this paper is as high as 89.5%. In addition, this ultra-low-frequency characteristic is very important because the lower the frequency of the seismic waves, the greater the harm to buildings. Table 1 compares the Lamb wave band gaps of similar structures in the relevant literature. \begin{table}
\begin{tabular}{c c c c c} \hline
**Reference** & **Lattice Constant (m)** & **Optimal Band Gap Range (Hz)** & **Lowest Frequency (Hz)** & **Band Width Ratio** \\ \hline
[11] & 10 & 6.38–16.01, 16.39–19.39 & 6.38 & 62.9\% \\
[12] & 1.5 & 3.5–10 & 3.5 & 32.5\% \\
[13] & 10 & 8–20 & 8 & 60\% \\
[19] & 2.5 & 7.41–20 & 7.41 & 62.95\% \\ This work & 12 & 2.1–20 & 2.1 & 89.5\% \\ \hline \end{tabular}
\end{table} 
Table 1: Lamb band gap comparison of similar structures. Figure 10: Vibration modes at special points of OS-RGSM structure; (**a**) the vibration mode at point A1; (**b**) vibration mode at point A2; (**c**) vibration mode at point A3; (**d**) vibration mode at point A4; (**e**) vibration mode at point A5. In order to further study the band gap generation mechanism of the OS-RGSM structure, the vibration modes at five special points in Figure 9 are calculated. The three-dimensional and two-dimensional vibration modes are shown in Figure 10. The OS-RGSM structure can be seen as the soil acting as a connector, connecting different scatterers as a mass to stabilize, and acting as a spring between adjacent scatterers. As shown in Figure 10, the mode at point B1 shows that in the OS-RGSM structure, the steel ring in the second unit cell vibrates upward along the z-axis direction, and the other parts remain basically static; for the mode at the B2 point, the vibration and scattering occur on the scatterer steel ring in the first unit cell of the supercell, and the vibration direction is along the z-axis; for the mode at point B3, the vibration and scattering occur on the connecting body (soil) between the second unit cell and the third unit cell and vibrate downward along the axial direction. The mode at B4 is the axial vibration of the soil between the second and third unit cells along the R direction, and the scatterer is almost completely static. The mode at B5 shows the downward vibration of the connecting soil between the first and second cells along the z-axis. It can be found that the OS-RGSM structure can be equivalent to a multi-oscillator-spring structure. At different frequencies, different oscillators vibrate correspondingly. The design of this structure is beneficial to the practical application. The steel ring is used as the lower foundation of the building while attenuating the seismic Lamb wave, and the different soil material parameters can be used to flexibly control the band gap range, which greatly saves the economic cost.