# The Vibration Isolation Design of a Re-Entrant Negative Poisson's Ratio Metamaterial
## 4 Bandgap Properties of N-NPRM
### Effects of Design Parameters  
Introducing circular rings into the traditional T-NPRM structure can adjust the mode shapes and frequencies of the unit cell, thereby enabling bandgap control of the metamaterial. In this section, we present our research on the influence of the following design factors on the bandgap. The effects of the materials of assembly pins  
Based on the above analysis, it can be observed that to adjust the bandgap of the metamaterial structure, effective control can be achieved by varying the density of the core pins, which requires the use of different materials for the pins. This approach enables the manipulation of the dispersion curves, which are primarily governed by the wall-pin coupled vibration mode, thereby achieving bandgap tuning of the metamaterial. Figure 7 demonstrates the bandgap characteristics of the NPRM using different assembly pin materials, namely WL8000, aluminum, steel, and copper. The bandgaps of the N-NPRM with different pins are marked by blue areas. In the figure, it can be observed that the presence of the pins has an impact on the mode frequencies associated with the deformation of the connecting walls. In Figure 7a, where WL8000 is used as the core material, which is the same as the metamaterial, the corresponding mode frequencies of the wall deformation are slightly increased. In the case of Figure 7b-d, the modulus of the metal materials, being significantly higher than that of WL8000, allows the pins to behave as rigid bodies in the deformation mode of the connecting walls (mode B1 in Figure 6). Consequently, the upper frequency of the bandgap (mode B1) remains around 14,700 Hz and does not vary with changes in the pin material. On the other hand, the lower mode shape of the bandgap primarily involves a wall-pin coupled motion. As the density of the pins increases, the corresponding frequency decreases, resulting in an increased width of the bandgap. As the density difference between steel and copper is not significant, Figure 7c,d shows a minor increase in the width of the frequency band. However, the copper pins induce a decrease in frequency for the other modes, opening up a new bandgap with a frequency range of 8200 Hz to 8600 Hz. It is evident that by utilizing pins of different materials, the  
Figure 7: The dispersion curve of N-NPRM with different pins. (**a**) N-NPRM with plastic pins, (**b**) N-NPRM with aluminum pins, (**c**) N-NPRM with steel pins, (**d**) N-NPRM with copper pins. width of the bandgap can be effectively adjusted, potentially leading to the formation of new bandgaps. By using pins made of different materials, the width of the bandgap can be effectively controlled. This provides valuable insights for the bandgap design and optimization of metamaterials. These methods can be used in various applications, such as vibration control, wave manipulation, and energy harvesting, where precise control over bandgaps is desired. From the analysis of the vibration modes above, it can be determined that increasing the size of the pins, without altering the dimensions of the metamaterial unit, will have two main effects. Firstly, reducing the length of the connecting walls will increase the mode frequencies primarily associated with the vibration deformation of the connecting walls. Secondly, increasing the size of the pins will significantly increase the local mass, leading to a decrease in the mode frequencies dominated by the wall-pin coupled motion. The effect of the pin size on the bandgap is illustrated in Figure 8, using steel as the pin material. The bandgaps of N-NPRM with pins radii of 3 mm, 3.5 mm and 4 mm are marked by blue area, red area and yellow area, respectively. As the pin size increases, the upper limit frequency of the bandgap increases, while the lower limit frequency decreases, resulting in a significantly widened bandgap. Moreover, when the pin size reaches 4 mm, a new bandgap is opened up in the frequency range of 3800 Hz to 4000 Hz. It can be observed that increasing the pin size is advantageous for expanding the bandgap.