# Co-Design of Mechanical and Vibration Properties of a Star Polygon-Coupled Honeycomb Metamaterial
## 4 Comparison and Analysis of the Bandgap Properties
### Comparison of Bandgap Properties between the IM and SPCHM  
The bandgaps of different metamaterials were simulated by the multi-physics software COMSOL Multiphysics 6.0 (COMSOL Inc., Stockholm, Sweden). The mesh size has a great influence on the calculation accuracy, and the maximum mesh is set to 0.3 mm. The finite element meshes are shown in Figure 4e-g. The cells of the IM, SPCHM, and SPCHM with metal pins during simulation and experiment are shown in Figure 4a-c, respectively. Figure 4d represents the Brillouin zone of the periodic lattice, along with four marked intersection points named O, 1, 2, and 3, accordingly. The software performs frequency scanning in the order of O-1-2-3-O and, finally, obtains the bandgap properties of the scanned metamaterials. The first 30 dispersion curves of the IM, SPCHM, and SPCHM with copper pins are drawn in Figure 5; the bandgaps are highlighted by red, the letters A-J represent the frequency ranges of different bandgaps, and the number 1 and 2 indicate the highest and lowest frequencies of corresponding bandgap, respectively. Figure 6 indicates the vibration mode shapes of the specific points marked in Figure 5. From the first 30 dispersion curves of the IM, the frequency band of the first bandgap (A2-A1) is from 1600 Hz to 2500 Hz. As shown in Figure 6, these kinds of mode shapes (A-E) have an overwhelming negative effect on the adjustment of the bandgaps, especially for the low-frequency bandgaps. However, changing the nodes to circular rings can change the overall mass, and the addition of the pins further increases the mass of the nodes, which changes the mass distribution and stiffness of the SPCHM-based structure, thus making its modal mode different from that of the IM and thereby changing the frequency of the bandgap. Based on Equation (6), the introduction of rings and metal pins can increase the local stiffness of the unit cell, making it possible to obtain low-frequency local resonant mode. Therefore, it is possible to create a bandgap by adjusting the design parameters. Figure 4: Diagrams for bandgap simulation: (**a**) unit cell of the IM; (**b**) unit cell of the SPCHM; (**c**) unit cell of the SPCHM with pins; (**d**) Brillouin zone of periodic lattices, (**e**) finite element mesh of the IM; (**f**) finite element mesh of the SPCHM; (**g**) finite element mesh of the SPCHM with copper pins. Figure 5: Dispersion curves of the metamaterials: (**a**) IM; (**b**) SPCHM; (**c**) SPCHM with copper pins. the letters A–J represent the frequency ranges of different bandgaps, and the number 1 and 2 indicate the highest and lowest frequencies of corresponding bandgap, respec-tively. The frequency of the first 30 dispersion curves of the SPCHM, which varies from 0 to about 6 kHz, is much lower than that of the IM. And the first bandgap of the SPCHM has higher frequency and narrower bandwidth compared to the IM, as marked by the red region from 3159 Hz to 3254 Hz. However, according to the modes of points F1 and F2 shown in Figure 6, the introduction of rings improves the local stiffness of the cell and results in more deformation forms between rods and rings, which makes the bandgap change internally and it indicates that the bandgap properties of the SPCHM can be regulated by changing the size of the rings or adding pins of different materials into the rings. Therefore, the factors, which could significantly influence the mode deformation of the unit cell, should be considered for generating specific bandgaps. The mass distribution of the cell changes dramatically when the metal pins are inserted into the rings. The change in mass generates new local resonant modes, which have better energy absorption capacity, and the width and frequency of the bandgaps are changed. As can be seen from Figure 5c, four more bandgaps are generated after the addition of copper pins, and the frequency ranges are from 989 Hz to 1343 Hz, 1853 Hz to 1947 Hz, 4538 Hz to 4950 Hz, and 5214 Hz to 5512 Hz, respectively.