## 3 Ultra-Low-Frequency Broadband Characteristics
### Mechanism of the Ultra-Low Broadband Characteristics  
In order to research the mechanism of the ultra-low-frequency broadband, the dispersion relation curves of conventional radial metamaterials are calculated under the condition that the structure and material parameters remain unchanged, and the results are presented in Figure 4. It can be seen that there is no ultra-low-frequency broadband in the range of the parsimonious frequency \(0<\Omega<0.50\). Figure 5 shows the calculated eigenmode displacement vector fields for the special points of the energy bands in the two states, respectively. Here, the black arrows indicate the directions of the vibration displacement and the colors indicate the magnitude of the vibration displacement. Figure 4: The unconstrained REM dispersion relation curve. Figure 3: Frequency response functions of the quasi –static REM. Referring to point \(A_{1}\) in Figure 2, the vibration energy is concentrated at the free end of the unit cell, whereas the whole represents a localized rotational vibration mode, similar to a \"pendulum\". This is due to the constraints imposed by \(Z\) on the base surface so that the vibration cannot be generated at the lower base surface. At point \(A_{2}\), since the upper and lower basal planes are free boundaries, the vibration is evenly distributed throughout the entire unit cell. This entails an overall vertical vibration mode of the axial direction. Due to the face constraints, the structural intrinsic eigenmode displacement vector field is changed, and the localized vibration of the metamaterials opens the ultra-low-frequency broadband. Furthermore, the vibration characteristics of the finite-period metamaterial structures are studied and the generation mechanism of the ultra-low-frequency broadband characteristics of the quasi-static radial metamaterials is explored. Figure 6 shows the displacement field along the radial direction at the center frequency of the first band gap of the quasi-static radial metamaterial for ten periods. Here, the center frequency \(\Omega\) of the first band gap is 0.10 in its quasi-static state. The incident wave is effectively reflected, and a localized vibration mode appears. Inside the metamaterial array, the vibration amplitude is the greatest at the first lattice. Afterwards, the amplitude decays exponentially towards the exit, where a low transmission is observed on the exit side. The vibration mode at the first lattice is symmetric along the axial direction, which is consistent with the experimental results [35]. However, in the radial displacement field outside the ten-period quasi-static radial metamaterial band gap range (at the parsimonious frequency \(\Omega=0.30\)), it is observed that the incident Lamb wave passes through the metamaterial structure almost unperturbed, as shown in Figure 7. Its displacement values can reach up to 2 mm due to the superposition of the reflections. Therefore, the structural local resonance is the main reason for the attenuation of the incident Lamb wave. According to the analysis of the above-mentioned quasi-static REM unit cell eigenmode displacement vector field and the radial vibration displacement field, the lower base surface of its local vibration mode is stationary, whereas the upper free end is equivalent to a single-degree-of-freedom system. In addition, its local resonance structure can be equivalent to a mass-spring system [35]. The cut-off frequency of its resonant band gap can be determined by the natural frequency of the local resonance, and it can be evaluated in terms of the equivalent mass (which provides the mass part) and the equivalent stiffness (which provides the elastic part) for this vibration mode:  
\[f=\sqrt{(K_{e}/M_{e})}/2\pi \tag{5}\]  
Figure 5: The eigenmodes and the displacement vector fields of the special points. The middle part of the I-shaped structure acts as a \(K_{e}\) providing equivalent stiffness, while the upper free end acts as an \(M_{e}\) providing equivalent mass.