## 2 Model
### Layered-Beam Metamaterial Structure  
The general design approach for an n-layered metamaterial beam structure with r-number of local resonators per horizontal beam is shown in Figure 3. The local resonators are evenly distributed across each horizontal beam, and vertical beams are interlinking the horizontal beams. The design approach for an arbitrary number of layered metamaterial structures was studied in our previous work [19]. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Value** & **Unit** \\ \hline [PERSON]’s modulus of elasticity of the beam, \(E\) & 70 & GPa \\ \hline Density of beam, \(\rho\) & 2600 & Kg/m\({}^{3}\) \\ \hline Area of horizontal beam, \(A\) & \(1.062\times 10^{-4}\) & m\({}^{2}\) \\ \hline Area moment of inertia, \(I\) & \(5.968\times 10^{-9}\) & m\({}^{4}\) \\ \hline Modal damping ratio of horizontal beams, \(\zeta_{I}ii\) & 0.02 & \\ \hline Vibration force magnitude, \(f_{\mathrm{s}}\) & 10 & N \\ \hline Length of unit cell beam, \(L_{c}\) & 0.125 & m \\ \hline Local resonator stiffness, \(k_{r}\) & \(1.65\times 10^{5}\) & N/m \\ \hline Local resonator mass, \(m_{r}\) & 0.0437 & Kg \\ \hline Damping ratio of local resonators, \(\zeta_{r}\) & 0.01 & \\ \hline Length of vertical columns, \(l_{c}\) & 80 & mm \\ \hline Width of vertical column, \(b_{c}\) & 7 & mm \\ \hline Thickness of vertical columns, \(h_{c}\) & 5 & mm \\ \hline Damping ratio of vertical columns, \(\zeta_{c}\) & 0.02 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Model design parameters of the single-beam metamaterial structure and the layered-beam metamaterial structure. Figure 3: General design approach for an n-layered horizontal beam with r-number of local resonators and vertical beams interlinking the horizontal beams. In the present study, as a base, the behavior of a typical double-layer beam-type metamaterial structure is analyzed and compared to a single-beam metamaterial structure of similar material properties and specifications. The modeled layered metamaterial structure shown in Figure 4A has eight local resonators per horizontal beam, with all dimensions shown in Table 1. The horizontal beams are held together by vertical beams. In Figure 4B, the representation of the horizontal beams and their interaction with the interlinking vertical beams is shown. Localized spring and damping forces are used in representing the vertical beams, as shown in Figure 4B. The input vibration force is applied to the center of the lowest horizontal beam and the output displacement is taken from the center of the topmost horizontal beam. For the layered free-free beams in Figure 4A subject to external vibrations, the transmissibility is analyzed using the frequency response function (FRF). The transverse response of the beams with local resonators and an external vibration source applied to it are given by the Euler-Bernoulli equation below [18]:  
\[EI\frac{\partial^{4}U_{j}}{\partial x^{4}}+\rho A\frac{\partial^{2}U_{j}}{ \partial t^{2}}=f_{sh}(x,t)\delta\left(x-\frac{L}{2}\right)+\sum_{r=1}^{\mu}f (x_{r},t)\delta(x-x_{r}) \tag{5}\]  
In Equation (5), \(U_{j}\) represents the \(j\)th beam's transverse displacement. The positions of the local resonators are factored in as \(x_{r}\), while \(\delta\) and \(\mu\) are the Dirac delta function and the number of local resonators per horizontal beams, respectively. The transverse displacement in Equation (5) can be further evaluated using the [PERSON] approach and superposition of mode shapes to give the equation below:  
\[U(x,t)_{j=1,2,\cdot,\text{ n}}=\sum_{i=1}^{N}p_{ji}(t)\varnothing_{ji}(x) \tag{6}\]  
where \(p_{ji}\) and \(\varnothing_{ji}\) in Equation (6) represent the generalized displacement and the trial of the \(j\)th horizontal beam at the \(i\)th mode, respectively. Moreover, \(N\) is the number of modes considered in the model.