# Ultra-Wide Bandgap in Two-Dimensional Metamaterial Embedded with Acoustic Black Hole Structures
## 3 Results and Discussion
The first eigenfrequency is significantly reduced, but the latter eigenfrequency will be less affected, resulting in the wider attenuation band. Figure 5 illustrates the average decrease in lattice eigenfrequency and the width of the first bandgap as the parameter \(m\) increases. As \(m=2\), the frequencies of the shear state S3 (S4) and spinning state S2 are very close, which is accidental. As \(m\) increases to 10, the connecting part (ABH structures) is approximately equivalent to a uniform beam, and the bending stiffness decreases greatly, resulting in the lower rotational eigenstates (487 Hz and 2245 Hz) and transverse eigenstates (5097 Hz and 9790 Hz), which indicates that the first bandgap width changes to 5097 Hz-9790 Hz. It is worth noting that the influence of the variation of \(m\) on different eigenstates is different: the first rotation state (S1) and the first transverse eigenstates (S3, S4) are almost unaffected, and the second rotation state (S2) and the second transverse eigenstates (S5, S6) are significantly affected, which are reasonable. The increase of m leads to the decrease of the width at both ends of the ABHs, resulting in the reduction of bending stiffness at the ends. Different from state S2, state S1 is more sensitive to the lowest width because the deformation part is in the middle of the ABH structure. The deformation of the lumped masses in State S5 (S6) is much smaller than that in ABHs, and the vibration mode is represented by the local resonance of ABHs. Hence, state S5 (S6) is significantly affected. To show the attenuation ability and the bandgaps of the finite periodic structure, simulations based on FEM and experiments have been carried out. Figure 6a,b shows the finite structure model (1 m \(\times\) 1 m) consisting of 8 \(\times\) 8 unit cells in the frequency domain. A 5 cm thick perfect matching layer (PML) marked with blue areas surrounds the finite structure. The 0.8 m load line was excited by a harmonic force of 1 N. Figure 6c shows the experimental prototype made of Polycarbonate, which was processed by the high-precision milling machine. The parameters of the lattice are consistent with the lattice parameters in Figure 3a. Six PZT-5H patches (25 mm \(\times\) 20 mm \(\times\) 1 mm) were pasted at three points  
Figure 5: Average decrease in lattice eigenfrequency as the parameter \(m\) increases. on the left side of the prototype and driven by a power amplifier with a sweep signal source. During the experiment, the prototype was suspended by two strings to simulate the free boundary (see Figure S1 in the Supplementary Materials). The perforated structure on the plate is conducive to the measurement of the vibration response in the face. The velocity signals (the input side and output side) were measured by two laser vibrometers (Polytec PSV-400-3D). A good agreement between experimental results (red dotted line) and simulation results (blue dotted line) with PML in Figure 7 shows wide bandgaps. The transmission spectrum, in dB, is defined as 20 log10(\(A_{\text{out}}\)/\(A_{\text{in}}\)), \(A_{\text{out}}\) and \(A_{\text{in}}\) are the output and input of acceleration amplitude. The damping effect of the material is not introduced in the aforementioned numerical calculations. However, the materials used in this case often have damping characteristics that can't be ignored. As material loss factor is set to 0.01 and the PML is removed, the transmission spectrum is shown as the dark solid line in Figure 7, which is more in line with the experimental results. Compared with simulation results without damping, the peaks around 15 kHz in experimental results are attenuated to \(-20\) dB due to the damping feature of materials, resulting in an ultra-wide bandgap from 4.5 kHz to 21 kHz. This design can be applied in the requirements, such as the battery package, which have periodicity. This structure can provide support for the battery units and has good vibration damping performance. Figure 6: The simulation configuration and the experimental setup. The finite structure model.