# The Vibration Isolation Design of a Re-Entrant Negative Poisson's Ratio Metamaterial
## 4 Bandgap Properties of N-NPRM
### Effects of Design Parameters
This shows that by adjusting the size of the pins, it is possible to modify the mode frequencies and broaden the bandgap without altering the dimensions of the metamaterial unit. This work provides guidance for the design and optimization of metamaterials with  
Figure 8: The effect of pin size on the bandgaps of metamaterials. (**a**) The pin radius is 3 mm, (**b**) The pin radius is 3.5 mm, (**c**) The pin radius is 4 mm. desired bandgap properties, enabling enhanced vibration control and wave manipulation capabilities. The effect of particle damping  
In engineering applications, low-frequency vibration sources often are common. Therefore, low-frequency and wide-band vibration control methods have been a significant research focus. For instance, in traditional industries, like the automotive sector, vibrations below 2000 Hz are of utmost concern. In the context of vibration control in metamaterial structures, bandgap manipulation is an important design approach. In the proposed N-NPRM structure, the addition of circular rings shifts the dispersion curves of the metamaterial towards lower frequencies. However, it remains challenging to form effective bandgaps below 2000 Hz. To expand the application range of metamaterials, new methods must be introduced to suppress vibrations in the low-frequency range. For the traditional T-NPRM structure, its mode shapes are combinations of the deformations of the walls that make up the structure. In the case of N-NPRM, the addition of circular rings leads to significant changes in the stiffness and mass distribution within the unit cell. The structural stiffness is noticeably increased, which is beneficial for enhancing its load-bearing capacity, as shown in Figure 3. At the same time, the mode shapes of N-NPRM exhibit specific characteristics as the frequency increases, as depicted in Figure 9. The darker the color of the mode shape indicates that the greater the deformation of the corresponding position. For the circular rings, as the frequency increases, they exhibit translational, rotational, and deformation motions, represented by M1 to M3 in Figure 9. At very high frequencies, the mode shapes resemble localized deformations of the walls, as shown by M4 in Figure 9. In the above analysis, it was found that increasing the weight at the location of the circular rings can effectively lower the frequencies of modes, such as M1 to M3. Based on the analysis of the mode shapes of the metamaterial structure shown in Figure 9, it was observed that in the low-frequency range, the circular rings undergo significant motion. Therefore, by combining traditional damping vibration control methods, the introduction of particle damping within the circular rings becomes a possibility for the broad-spectrum suppression of low-frequency vibrations. In the simulation, the circular rings and particle damping were treated as an integrated system. The increase in weight due to particle damping can be achieved by adjusting the density of the circular rings. To prevent particle leakage, covers were added at both ends of the circular rings, which enhanced the stiffness at the location of the circular rings. The effect on the stiffness can be adjusted by modifying the elastic modulus of the circular rings, disregarding the impact when it is negligible and ensuring equal weight. Figure 10c shows the dispersion diagram of the metamaterial when the weight of the circular rings was equal to the weight of the steel pins. It can be observed in the figure that when particle  
Figure 9: Mode shapes changing with frequency. damping was added within the circular rings, the mode frequencies were generally lower compared to the metamaterial with inserted steel pins since the increase in stiffness due to particle damping was not significant. As shown in Figure 10b,c, the first 20 mode shapes occurred within the frequency ranges of 0-16,000 Hz and 0-9000 Hz, respectively. Although the addition of particles created a bandgap at 6500 Hz, it was still ineffective for controlling the vibrations at even lower frequencies.