## 3 Cell Base Structure and Frequency Response Analysis
### Frequency Response Analysis of Negative Poisson's Ratio Honeycomb Base  
Proportioning mass blocks will change the natural frequency of the structure and produce a special resonance, and this will open the band gap of the structure to change the vibration isolation performance of the structure. So based on the re-entrant hexagonal cell structure, the vertical side of the cell structure is \(a~{}=~{}14\) cm, the hypotenuse side is \(b~{}=~{}10\) cm, and the side thickness is \(\tau~{}=~{}0.1\) cm and \(\tau~{}=~{}0.2\) cm, respectively, the cell structure with different Poisson's ratio is used to design the honeycomb base. As shown in the following schematic diagram, the honeycomb base is designed to be laid in 6 layers in the vertical direction. In the same way, the vertical coordinate of the frequency response curve graph represents the logarithm of the displacement ratio between the initial position and the position of the end point. The smaller the value is, the better the vibration isolation effect is. The larger the range is, the wider the vibration isolation bandwidth of the modified mechanism to external excitation is. From the frequency response curve of Figures 7-9, we can see that honeycomb base with re-entrant hexagonal structure has better isolation effect on external excitation. For example, it can be seen from Figure 7 that honeycomb base with re-entrant hexagonal structure with Poisson's ratio of \(v~{}=~{}-0.47\) and side thickness \(\tau~{}=~{}0.1\) cm has good isolation effect when the frequency is higher than 300 Hz and it has more effective isolation effect on frequency about 5000 Hz. From Figure 8, we can see that when the side thickness \(\tau~{}=~{}0.2\) cm with the same Poisson's ratio, honeycomb base provides better isolation at higher frequencies above 8000 Hz. From Figure 9, we can see when different cell structure are used in honeycomb, the base has isolation effect in wider frequency, when the side thickness is \(\tau~{}=~{}0.1\) cm, it can be seen that the cellular base has frequency vibration isolation when the frequency is higher than 300 Hz, and in the much more higher frequency, the base has better vibration isolation than Figures 7 and 8, and when the side thickness is \(\tau~{}=~{}0.2\) cm, there is significant vibration isolation when the frequency is higher than 8000 Hz, and better vibration isolation can be gotten when the frequency is lower, this is better than Figures 7 and 8. It can be seen from Figures 7-9 that when the re-entrant structure with different Poisson ratios is used for laying, it will have better vibration isolation effect at a wider frequency. Figure 7: The base diagram of re-entrant structure is given, (**a**) the diagram of honeycomb base, the Poisson’s ratio of the re-entrant hexagonal elements that make up the honeycomb base is \(\
u~{}=~{}-0.47\); (**b**) a fixed displacement 1 m upper plane of the base; (**c**) the side thickness of the re-entrant hexagonal elements that make up the honeycomb base is \(\tau~{}=~{}0.1\) cm; (**d**) the side thickness of the re-entrant hexagonal elements that make up the honeycomb base is \(\tau~{}=~{}0.2\) cm. Figure 8: (**a**) the diagram of honeycomb base, the Poisson’s ratio of the re-entrant hexagonal elements that make up the honeycomb base is \(v\ =\ -1.11\); (**b**) a fixed displacement 1 m upper plane of the base; (**c**) the side thickness of the re-entrant hexagonal elements that make up the honeycomb base is \(\tau\ =\ 0.1\) cm; (**d**) the side thickness of the re-entrant hexagonal elements that make up the honeycomb base is \(\tau\ =\ 0.2\) cm.