## 2 Theoretical Derivation and Energy Band Analysis of Mechanical Properties of the Cell
### Energy Band Analysis of Re-Entrant Hexagonal Cell and Traditional Cell Structure  
The energy band and the natural frequency analysis of a single re-entrant hexagonal cell is carried out with the help of phononic crystal knowledge. Firstly, the selection of the size of a single structure is summarized after reading many papers and do some research, the size of a single cell is determined, as shown in Figure 3.  
Figure 3: Schematic diagram of re-entrant hexagonal honeycomb cell is given.  
The vertical edge \(h=10\) cm and side thickness \(\tau=0.1\) cm or \(\tau=0.2\) cm, \(\theta=-15^{\circ}\) or \(\theta=-30^{\circ}\) are selected in the experiment, as shown in Figure 3 Single cell \(a~{}=~{}14\) cm, \(b~{}=~{}10\) cm. As is shown in formula, range from Equations (1)-(9), Poisson ratio is corresponding to different angles \(\theta~{}=~{}-15^{\circ}\) and \(\theta~{}=~{}-30^{\circ}\) are \(v~{}=~{}-0.47\) and \(v~{}=~{}-1.11\) respectively.  
The meta-material vibration isolation base is made of high strength steel, its elastic modulus \(E~{}=~{}210\) Gpa, and its Poisson's ratio \(v~{}=~{}0.30\) and density \(\rho~{}=~{}7800\) kg/m\({}^{3}\). The irreducible Brillouin region of the reduced Brillouin region of the cell can be regarded as a periodic structure. The natural frequencies of the irreducible Brillouin region of the cell can be analyzed by parameter scanning, and the band diagram can be obtained.  
According to the band diagrams of cell structure from different angles, there is a band gap in the vertical direction of cell structure, that is, there is a vibration isolation effect in the vertical direction. The cell structure with different Poisson's ratio has more obvious isolation effect at different frequencies. For example, from Figure 4, when side thickness \(\tau~{}=~{}0.1\)cm, we can see that when the Poisson's ratio \(v~{}=~{}-1.11\), there is an obvious band gap between cell structure and frequency range 300-600 Hz. 900-1400 Hz. 2600-2900 Hz. We can see that when the Poisson's ratio \(v~{}=~{}-0.47\), there is an obvious band gap between cell structure and frequency range 210-410 Hz. 790-1100 Hz. 1800-2200 Hz; From Figure 5, when the side thickness \(\tau~{}=~{}0.2\)cm, we can see that when the Poisson's ratio \(v~{}=~{}-0.47\) there is an obvious band gap between cell structure and frequency range 510-1200 Hz. 1900-3000 Hz. 3600-4500 Hz. We can see that when the Poisson's ratio \(v~{}=~{}-1.11\), there is an obvious band gap between cell structure and frequency range 500-900 Hz. 1000-1300 Hz. 3600-4400 Hz.  
Figure 4: Tenth order natural frequency band diagram of a single cell structure is given, (**a**) the single cell’s side thickness: \(\tau~{}=~{}0.1\) cm, its angle \(\theta~{}=~{}-15^{\circ}\), thus, its Poisson’s ratio is deduced: \(v~{}=~{}-0.47\); (**b**) the single cell’s side thickness: \(\tau~{}=~{}0.1\) cm, its angle \(\theta~{}=~{}-30^{\circ}\), thus, its Poisson’s ratio is deduced: \(v~{}=~{}-1.11\).  
The experimental results show that the re-entrant hexagonal structure does have the effect of vibration isolation, and the re-entrant hexagonal structure with different Poisson's ratio and different side thickness has the different frequency band gap, it means that the cell structure with different Poisson's ratio can isolate the excitation in different frequency bands.