## 3 Examples and Discussion
### Band-Gap Property and Vibration Attenuation
With the exception of this point, the bandwidth in the TRIEPE LR plate stays between the bandwidths in both ORIEPE LR plates below 500 Hz. Figure 4: The total bandwidths of the band gaps in the three locally resonant plates in Figure 3.  
Figure 3: The effect of \(f_{\rm R2}/f_{\rm R1}\) on the band-gap ranges in the locally resonant plate with two resonators in each periodic element (TRIEPE LR plate) with \((m_{\rm R1}\), \(k_{\rm R1})\) and \((m_{\rm R2}\), \(k_{\rm R2})\), and two corresponding locally resonant plates with one resonator in each periodic element (ORIEPE LR plates). It can be seen that when aiming to achieve a new low-frequency band gap, the introduction of the new periodic resonators might make the total bandwidth decrease in the TRIEPE LR plate. Appropriate spring-mass parameters for both kinds of resonators could make the total bandwidth not be greatly affected. Hence, the chosen parameters of the periodic element of the TRIEPE LR plate in this paper are given in Table 1, together with the details of the finite base plate. In the following analysis, the parameters are kept unchanged unless otherwise stated. As illustrated in Section 2.1, the four boundaries of the plate are supported by a linear spring with stiffness \(k\) and a torsional spring with stiffness \(K\). The free, simply supported, and clamped boundary conditions can be simulated by using specific stiffness values of \(k\) and \(K\): the free boundary condition can be expressed when \(k\ =\ 0\) and \(K\ =\ 0\); the simply supported boundary condition can be expressed when \(k\) tends to infinity and \(K\ =\ 0\); and the clamped boundary condition can be expressed when both \(k\) and \(K\) tend to infinity. The variation of the boundary conditions can result in the difference in the vibration response and acoustic performance [29; 30]. In this paper, three kinds of different boundary conditions are applied to the plate, including fully free boundary conditions ('FFFF'), fully simply supported boundary conditions ('SSSS'), and fully clamped boundary conditions ('CCCC'). Before the analysis of the vibration attenuation property of the LR plate, taking the SSSS boundary condition as an example, the theory to simulate the vibration response in this paper is examined by using the FEM software of Ansys. The comparisons are shown in Figure 5a-c, respectively for the bare plate, the mass-loaded (ML) plate, and the LR plate. The ML plate indicates the plate periodically attached with point masses, for which the plate, the lattice constant, and the loaded masses are all identical to those of the LR plate. The above theory is also appropriate for the ML plate via replacing Equation (21) by \(w_{Rist}\ =\ w(x_{is},y_{t})\). Good agreements can be found between the root-mean-square velocities (RMSVs) averaged over the whole surface calculated by the theory and FEM, which validates the correctness of the theory in this paper. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline \(m_{R1}\) & Mass of the resonator \((m_{R1},\ k_{R1})\) & 0.020 kg \\ \(k_{R1}\) & Spring stiffness of the resonator \((m_{R1},\ k_{R1})\) & \(3.2\times 10^{4}\) N/m \\ \(m_{R2}\) & Mass of the resonator \((m_{R2},\ k_{R2})\) & 0.014 kg \\ \(k_{R2}\) & Spring stiffness of the resonator \((m_{R2},\ k_{R2})\) & \(5\times 10^{4}\) N/m \\ \(a_{L}\) & Lattice constant & 0.05 m \\ \(h_{p}\) & Thickness of the base plate & 0.003 m \\ \(a\) & Length of the finite base plate & 0.8 m \\ \(b\) & Width of the finite base plate & 0.5 m \\ \(F_{0}\) & Amplitude of excitation force & 1 N \\ \((x_{0},y_{0})\) & Force position on the finite base plate & \((0.1\,\mathrm{m},0.1\,\mathrm{m})\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters of the periodic element and the details of the base plate in the finite locally resonant plate with two resonators in each periodic element.