## 4 Time Domain Analysis of the Ultra-Low-Frequency Broadband Characteristics  
In order to further explore the attenuation characteristics of the quasi-static metamaterials for the elastic waves in the ultra-low frequency range, a three-dimensional full-scale radial metamaterial structure with finite period is designed. It contains in-plane and out-of-plane modes, as shown in Figure 11. The distance between the vibration source and the center of the quasistatic metamaterial finite periodic structure is 15a, the cross-sectional width is 19a, and its structure height is small enough to make only Lamb waves propagate in the quasi-static metamaterial.  
In addition, the finite period model of the quasi-static metamaterial used for the calculation has the same fixed bottom boundary. The tetrahedral mesh adopts the quadratic element size \(\lambda_{s}/5\) (\(\lambda_{s}\) is the wavelength of the surface wave), and its transient calculation time step is set to \(1/60\)\(f_{max}\) (\(f_{max}\) is the maximum expected frequency) to ensure the accuracy of the calculation results.  
At point A, the oscillation source in the X direction on the free surface is used to introduce Lamb waves, and the low-reflection boundaries are set on the surrounding sides of the structure to reduce any unwanted reflections and to make the results more realistic and reliable. In this section, two different input signals are investigated. The excitation signals are modulated by the sum of 7 and 21 sinusoidal cycles with a Hanning window  
Figure 10: Effect of the material parameters on the band gap characteristics.  
At point A (see Figure 10), two excited signals of 7/21 sinusoidal cycles modulated by Hanning at 580 Hz are entered. The instantaneous transient display responses of 0.05 s duration are performed respectively, ensuring that the excitation signal emitted by the emission point A can reach and pass through the quasi-static metamaterial structure. To avoid reflections from the boundary around the model, the response signals are picked up at point B, as shown in Figures 14 and 15. As shown in Figures 14a and 15a, due to the existence of the quasi-static metamaterials, the propagation of Lamb waves is effectively suppressed, and its amplitude attenuation can exceed 70% (the blue solid line represents the amplitude signal of the Lamb wave without the metamaterial, and the red dotted line represents the amplitude signal of the Lamb wave with the metamaterials). Subsequently, the Fourier transform of the response signal is performed, and its frequency contents are compared, as shown in Figures 14b and 15b.  
These results show that the quasi-static metamaterial can attenuate the Lamb waves finitely in the band gap range, and the amplitude of the attenuation can exceed 80% of the maximum value. Figure 16 shows the Von [PERSON] stress plot under a Hanning-modulated seven-sinusoidal-cycle excitation signal with a center frequency of 580 Hz. As shown in Figure 16a, when the Lamb wave passes through the metamaterial-free region, the energy does not suffer a loss, and it can carry the energy to travel a long distance. In Figure 16b, when the incident Lamb wave propagates in the metamaterial, the energy is obviously effectively absorbed. The propagation of the Lamb wave is attenuated and blocked, and the inner region of the metamaterial is hardly affected.  
Figure 14: 580 Hz, 21-cycle excitation signal response. (**a**) Time–history curve of response signal; (**b**) Fourier spectrum of response signal.  
Figure 15: 580 Hz, 21-cycle excitation signal response. (**a**) Time–history curve of response signal and (**b**) Fourier spectrum of response signal.