## 2 Model and Formulations
### Band Structure of Infinite Systems  
The resonant frequencies of the QZS resonators can be calculated by  
\[f_{1,2}=\frac{1}{4\pi}\left[\left(\frac{k_{1}+k_{Q}}{m_{1}}+\frac{k_{Q}}{m_{2}} \right)\mp\sqrt{\left(\frac{k_{1}}{m_{1}}-\frac{k_{Q}}{m_{2}}\right)^{2}+2\frac {k_{Q}}{m_{1}}\left(\frac{k_{1}}{m_{1}}+\frac{k_{Q}}{m_{2}}\right)+\left(\frac {k_{Q}}{m_{1}}\right)^{2}}\right] \tag{18}\]  
According to Equation (18) and the parameter settings in Table 1, the resonant frequencies of the QZS resonators are 45 Hz and 200 Hz. The band structures of the LR plate with QZS resonators can be acquired by calculating the dispersion curves in its irreducible region, surrounded by the edges of TX, XM, and MT. By sweeping the Bloch wave vector \(\kappa=(\kappa_{x},\kappa_{y})\) in \(\Gamma\)X direction (\(\kappa_{x}:0\rightarrow\pi/a_{1}\), \(\kappa_{y}:0\)), XM direction (\(\kappa_{x}:\pi/a_{1}\), \(\kappa_{y}:0\rightarrow\pi/a_{2}\)), and MT direction (\(\kappa_{x}:\pi/a_{1}\to 0\), \(\kappa_{y}:\pi/a_{2}\to 0\)), the dispersion curves between frequency and wave number are obtained, from which the bandgap can then be identified, as shown in Figure 2a. It can be seen that two bandgaps exist around the two resonant frequencies, i.e., 45-49 Hz and 200-217 Hz, where the first bandgap is generated by the resonance of mass \(m_{2}\) and the second bandgap is generated by the resonance of mass \(m_{1}\). To verify the correctness of the bandgap theory, a finite plate periodically attached with QZS resonators was modeled by FEM to obtain its vibration transmittance. The FEM model was established with COMSOL Multiphysics software (Version 5.0). For clamped or simply supported boundary conditions, the forced vibration response at the position near the corner points is usually very small, which is inappropriate for assessing the vibration transmittance characteristics of the periodic plate. Therefore, in the simulation model, the four edges of the plate were all set in free boundary conditions. The size of the finite plate is \(1.5\) m \(\times\) 0.8 m \(\times\) 0.005 m with steel material property. \(15\times 8\) QZS resonators are periodically distributed on the baseplate with the parameters given in Table 1. The quadratic element type was used in the model. The maximum mesh size is set as 10 mm, which can provide adequate precision for the upper analytical \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Plate Parameters**} & \multicolumn{2}{c}{**QZS Resonator Parameters**} \\ \hline Density & \(\rho=7850\) kg/m\({}^{3}\) & Mass 1 & \(m_{1}=78.5\) g \\ Young’s modulus & \(E=216\) GPa & Mass 2 & \(m_{2}=78.5\) g \\ Poisson’s ratio & \(\
u=0.28\) & Spring stiffness \(k_{1}\) & \(k_{1}=1.17\times 10^{5}\) N/m \\ Plate thickness & \(h=5\) mm & Spring stiffness \(k_{p}\) & \(k_{p}=5\times 10^{4}\) N/m \\ Lattice constant & \(a_{1}=a_{2}=0.1\) m & Spring stiffness \(k_{n}\) & \(k_{n}=4.33\times 10^{4}\) N/m \\  & & Compression ratio & \(\gamma=0.5\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the QZS LR plate. Figure 2: Comparison of (**a**) the band structure of the LR plate with QZS resonators and (**b**) the vibration transmittance computed by FEM. frequency of 300 Hz. In order to simplify the model, the QZS resonator is equivalent to a 2-DOF resonator based on the theory in Section 2, as shown in Figure 3. A transverse point force with a harmonic form is applied at one corner of the plate, represented by the red arrow in the figure. The label \"Response\" represents the position on the board where the transverse vibration is of interest. Assuming that the acceleration responses of the force point and response point are, respectively, represented by \(a_{\text{F}}\) and \(a_{\text{r}}\), the vibration transmittance can be defined by \(TL=20\log_{10}\lvert a_{\text{r}}/a_{\text{F}}\rvert\). The vibration transmittance of the finite LR plate is drawn in Figure 2b.