## 4 Vibration Properties of the Proposed Metamaterials
### Bandgap Characteristics  
The bandgap properties of the metamaterials are figured by the multiphysics software COMSOL 6.0. The structures of the unit cells are shown in Figure 5a and the irreducible Brillouin region boundary (O\(\rightarrow\)M\(\rightarrow\)I\(\rightarrow\)O) is defined in Figure 5b. A series of characteristic frequencies are determined corresponding to the propagation of the wave vector \(\mathbf{k}\), and the band structures of the metamaterials are achieved for further analysis.  
Figure 6 shows the elastic wave dispersion curves of the DHM, CDCHM and CDCHM with the metal pins. There is a one-to-one correspondence between the scanning parameters 0\(\rightarrow\)1\(\rightarrow\)2\(\rightarrow\)3 and O\(\rightarrow\)M\(\rightarrow\)I\(\rightarrow\)O in the irreducible Brillouin region in Figure 6. The dispersion curves suggest that the DHM only has two narrow bandgaps within the range under 10,000 Hz, and its lowest bandgap is nearby at 4100 Hz, which indicates that the DHM has poor vibration-suppression performance at low frequencies. A1-A4 in Figure 7 illustrate the vibration modes corresponding to the marked frequencies in Figure 6b; we have noticed that the mode deformation is the global deformation of the cell, and changing the global mode deformation to the local mode deformation is an effective method to generate a local resonance bandgap.  
Figure 5: Structure diagram for bandgap analysis: (**a**) unit cells of the DHM, CDCHM and CDCHM with metal pins; (**b**) irreducible Brillouin zone of the periodic lattice.  
Figure 6: \(Cont.\)  
Figure 6: The simulation results of the metamaterials: (**a**,**b**) the dispersion curve of the DHM; (**c**,**d**) the dispersion curve of the CDCHM; (**e**,**f**) the dispersion curve of the CDCHM with the copper pins; (**g**) the VTL curves and bandgaps of the metamaterials.  
As explained in Figure 6c,d, with a chiral structure assembled into the diamond honeycomb structure, the bandgap properties of the CDCHM are greatly improved compared with the original DHM. The first bandgap of the CDCHM is nearby at 2700 Hz, which means that the modified metamaterial shows a better vibration-suppression effect at low frequencies. Moreover, the inferior limited frequency of the second bandgap of the CDCHM declines from about 9200 Hz to 5000 Hz, along with a wider bandwidth, and the third and fourth bandgaps are generated, and especially the width of the fourth bandgap is very wide, which is from 6187 Hz to 8584 Hz. According to A1-A4 and B1-B4, it is found that the node stiffness of the metamaterial is improved with the introduction of a chiral structure, and the local mode deformations of the cell are generated, which provide better bandgap properties. The metal pins further enhanced the local resonance and generated bandgaps in the lower-frequency range, as shown in Figure 6e,f. The corresponding modes are shown in Figure 7C1-C6.  
In order to further analyze the vibration-isolation performance of the metamaterials, the concept of vibration transmission loss (VTL) is introduced to evaluate the vibration-suppression ability, which can be expressed as:  
\[\text{VTL}=20\times\text{lg}(\frac{a_{0}}{a_{i}}) \tag{5}\]  
where \(a_{0}\) and \(a_{i}\) represent the acceleration amplitudes of the output and input of the metamaterials, respectively.  
The VTL simulation results of the metamaterials mentioned above are shown in Figure 6g. We found that the attenuation part of the VTL is in a good correspondence with the bandgap obtained by the COMSOL simulation (highlighted by corresponding colors in the background), so the effectiveness of the two simulation methods can be mutually verified. According to the above analysis, it can be inferred that the ring structure is a key factor affecting the generation of bandgaps, and changing the ring size of the CDCHM may obtain different bandgap characteristics.