# The Vibration Isolation Design of a Re-Entrant Negative Poisson's Ratio Metamaterial
## 4 Bandgap Properties of N-NPRM
### Bandgap Analysis of the NPRM  
In this section, the bandgap performance of the metamaterials is calculated using the multi-physics software COMSOL 5.6. To ensure computational accuracy, the minimum wavelength is generally required to contain 10 elements, and thus, the maximum element size was set as 0.2 mm. Three periodic boundaries were set; the unit cell can be seen in Figure 4a. The properties of the materials were set according to Table 2.  
The bandgap performances of both T-NPRM and N-NPRM were analyzed to find the factors influencing the distribution and the width of the bandgap. This work will further help in establishing methods for bandgap tuning. As shown in Figure 4a, the unit cell of the NPRM was used for the bandgap calculations. The Brillouin zone of the periodic lattices, which represents the reciprocal space of the unit cell, is shown in Figure 4b, and the four intersection points are marked by letters O, A, Band C, respectively. The scanning direction follows the path of O-A-B-C-O, corresponding to the scanning parameters 0-1-2-3-4 in COMSOL 5.6. As shown in Figure 5, the dispersion curves of the traditional T-NPRM, N-NPRM, and N-NPRM with aluminum pins are depicted. Figure 6 illustrates the corresponding mode shapes at specific points. Combining Figures 5 and 6, it can be observed that the T-NPRM structure exhibited a higher bandgap frequency. The dominant mode determining the bandgap width was the deformation of the connecting walls, resulting in a higher starting frequency, such as A1\(\sim\)A4 in Figure 6. Such vibration modes make it challenging to adjust the bandgap width, as changes in the material properties will cause the upper and lower mode frequencies to shift in the same direction. And B1-E4 are the modes corresponding to the marks in Figure 5c. It can be seen that the mode shapes are changed, as the stiffness and mass distribution of the unit cell is changed by the introduction of the rings and pins. The darker the color of the mode shape indicates that the greater the deformation of the corresponding position, and this is not the true deformation amplitude and is only used to represent the deformation shape. Figure 4: Bandgap analysis of the metamaterial. (**a**) Unit cells of the T-NPRM (dotted blue box) and N-NPRM with pins (dotted red box). (**b**) Brillouin zone of periodic lattices. Figure 5: The dispersion curves of three honeycomb structures. (**a**) T-NPRM, (**b**) N-NPRM, (**c**) N-NPRM with aluminum pins. For the newly proposed N-NPRM structure, as shown in Figure 5b, the circular rings increased the local stiffness of the unit structure, leading to more coupled modes between the walls and rings. Within the frequency range of the first 20 modes, no bandgap was formed. Introducing the ring at the node mainly increased the stiffness of the node, and the mass change also increased. For the initial cell, the stiffness of the node was larger than that of the thin wall, and as the introduction of the ring resulted in a larger rigid and mass, a slight increase in the mode frequencies was generated. This small and universal change of the dispersion curves did not easily enable them to separate from each other and form bandgaps. As can be seen in Figure 5b, compared to the initial cell, the number of bandgaps reduced from 1 to 0. Therefore, in the design of cell structure, the factors that can bring significant changes to the cell mode must be considered to form the bandgap. When metal pins were installed in the circular rings, the mass distribution of the N-NPRM cell had a noticeable change. Figure 5c shows that two bandgaps were formed, located near 7000 Hz and 14,000 Hz, respectively. By examining Figure 6, it can be observed that the mode frequencies of D1 and E1 determined the width of the first bandgap. However, both modes are the coupled motion between the walls and pins, and altering the mass of the pins will have an impact on both modes. Therefore, adjusting the material of the assembly pins may not effectively broaden the bandgap width.