# Co-Design of Mechanical and Vibration Properties of a Star Polygon-Coupled Honeycomb Metamaterial
## 5 Evaluation of Vibration Suppression Performance by Simulation and Experiment  
In this section, several simulations and experiments are conducted to verify the vibration suppression performance of the proposed metamaterials. The vibration suppression performance is assessed through the vibration transmission loss (VTL), which is calculated by:  
\[\text{VTL}=20\times\log_{10}(\frac{a_{1}}{a_{2}}) \tag{7}\]  
in which \(a_{1}\) and \(a_{2}\) represent the measured values of the output and input acceleration sensors, respectively, and the installation positions of the sensors are shown in Figure 10. It can be concluded from Equation (7) that the proposed metamaterial suppresses vibrations if the VTL value is negative. The finite element analysis software Abaqus CAE 2018 was used to calculate the VTLs of the metamaterials. In the simulation, the upper and lower aluminum blocks are set as rigid bodies, the output result is the acceleration of the upper aluminum block, and the simulation frequency range is from 1 Hz to 5000 Hz. To ensure the accuracy of the calculation, while ensuring that there are at least 10 elements in the shortest wavelength, the maximum mesh size of the SPCHM is 0.3 mm. Figure 11 reveals that the frequency histories of the VTLs generated from the IM, SPCHM, SPCHM with copper pins, and SPCHM with tungsten pins. All the four metamaterials could effectively restrain vibrations but exhibit different performances. Through Figure 11b-d, the vibration suppression ability of the SPCHM is well enhanced after adding pins into the rings, especially in the low frequency range (within 1000 Hz). As illustrated in Figure 11c,d, the maximum VTL of the SPCHM with copper pins is close to \(-120\) dB, while that of the SPCHM with tungsten pins is about \(-160\) dB; these results indicate that the vibration suppression ability of the SPCHM is able to be regulated by changing the mass distribution inside the rings, which is in accordance with the bandgap characteristic analyzed above. The green marked areas in Figure 11 represent the frequency ranges of bandgaps; the vibration isolation performance of metamaterials is improved within the bandgaps, which implies that the VTL simulation results are in good agreement with the bandgaps. Figure 10: The experimental system. With the increase in frequency, the vibration modes of the SPCHM show certain characteristics, as shown in Figure 12. The selected k value is 2.5 and the frequencies of Figure 12a-d are 94 Hz, 986 Hz, 3390 Hz, and 5214 Hz, respectively. It can be demonstrated from the figure that, when the frequency is low, the deformation of the unit cell of the SPCHM is manifested as moving and, as the frequency increases, the unit cell is gradually manifested as rotating. When the frequency is large enough, the local deformation behaviors occur on both the left and right sides of the unit cell. Figure 11: VTL simulation results: (**a**) IM; (**b**) SPCHM; (**c**) SPCHM with copper pins; (**d**) SPCHM with tungsten pins. The green marked areas in Figure 11 represents the frequency ranges of bandgaps. Figure 12: The deformation varies with frequency. As can be seen from Figure 12, the rings move evidently in the low frequency range. In order to further strengthen the suppression ability of vibration excitation outside the bandgap frequency and realize continuous adjustment of the bandgap, metal pipes with particle damping were added inside the rings to replace the metal pins and experiments were carried out to verify its effects. Figure 10 illustrates the composition of the experimental system. A sweep signal generator is adopted to produce white noise as a system excitation signal, and the frequency automatically sweeps from 2 Hz to 5000 Hz. An electrodynamic vibration shaker is used to realize the excitation signal from the generator. Two acceleration sensors with sensitivities of 1.075 \(\text{mv}/(\text{m}\cdot\text{s}^{-2})\) and 1.067 \(\text{mv}/(\text{m}\cdot\text{s}^{-2})\) are used to sample the input and output acceleration signals at a rate of 10 kHz.