## 2 Model
### Layered-Beam Metamaterial Structure
Applying the effects of the vertical beams connecting the two  
Figure 4: Schematic representation of the model approach to a layered-beam metamaterial structure: **(A)** Double-beam metamaterial structure with 8 local resonators and vertical beams connecting the horizontal beam; **(B)** representative diagram of the interlinking vertical beams, the damping and stiffness applied in the modeling approach for the dynamic displacement of the layered-beam metamaterial structure. horizontal beams to Equation (5), a new set of equations for the two horizontal beams is given below:  
\[\left\{ \begin{array}{ll}EI\frac{\partial^{4}U_{1}}{\partial x^{4}}+k_{v}(U_{1}-U_ {2})+r_{v}(\dot{U_{1}}-\dot{U_{2}})+\ \rho A\frac{\partial^{2}U_{1}}{\partial t^{2}}=\ f_{sh}(x,t)\delta(x-\frac{L}{2 })+\ \sum\limits_{r=1}^{\mathrm{H}}f(x_{r},t)\delta(x-x_{r})\\ E\frac{\partial^{4}U_{2}}{\partial x^{4}}+k_{v}(U_{2}-U_{1})+r_{v}(\dot{U_{2}}- \dot{U_{1}})+\ \rho A\frac{\partial^{2}U_{2}}{\partial t^{2}}=\ \sum\limits_{r=1}^{ \mathrm{H}}f(x_{r},t)\delta(x-x_{r})\end{array} \right. \tag{7}\]  
The force exerted on the lowest beam by the external vibration source is still given by the sinusoidal input \(f_{sh}=f_{s}\cos(\omega t)\), where \(f_{s}\) is the amplitude of the vibration force. The expression in Equation (7) can be rearranged as forces to show the effects of the vertical beams on specific locations (\(x_{v}\)) on the horizontal beams in the format given below:  
\[\left\{ \begin{array}{ll}EI\frac{\partial^{4}U_{1}}{\partial x^{4}}+\rho A \frac{\partial^{2}U_{2}}{\partial t^{2}}=\ f_{sh}(x,t)\delta(x-\frac{L}{2})+ \ \sum\limits_{r=1}^{n}f(x_{r},t)\delta(x-x_{r})+\sum\limits_{v=1}^{q}f_{1}(x_{v },t)\delta(x-x_{v})\\ E\frac{\partial^{4}U_{2}}{\partial x^{4}}+\rho A\frac{\partial^{2}U_{2}}{ \partial t^{2}}=\ \sum\limits_{r=1}^{n}f(x_{r},t)\delta(x-x_{r})+\sum \limits_{v=1}^{q}f_{2}(x_{v},t)\delta(x-x_{v})\end{array} \right. \tag{8}\]  
where \(f_{1}\) and \(f_{2}\) are the forces from the vertical beams acting on the lowest and topmost horizontal beams, respectively. They are expressed fully below:  
\[\left\{ \begin{array}{ll}f_{1}(x_{v},t)=-[k_{v}(U_{1}-U_{2})+r_{v}\big{(} \dot{U_{1}}-\dot{U_{2}}\big{)}]\\ f_{2}(x_{v},t)=-[k_{v}(U_{2}-U_{1})+r_{v}\big{(}\dot{U_{2}}-\dot{U_{1}}\big{)} ]\end{array} \right. \tag{9}\]  
Furthermore, \(k_{v}\) and \(r_{v}\) in Equation (9) represent the stiffness and the damping coefficient of the vertical beams and are given by Equation (10), respectively [2; 31]:  
\[\left\{ \begin{array}{ll}k_{v}=\frac{32\ast E\ast b_{v}\ast h_{v}\ast h_{v} \ast}{l_{v}}\\ r_{v}=2\xi_{v}\sqrt{m_{v}k_{v}}\\ m_{v}=0.33\ast\rho\ast A\ast l_{v}\end{array} \right. \tag{10}\]  
where \(m_{v}\) in Equation (10) is the effective mass of the vertical beams. Applying the damping effect of the resonators to Equation (8) transforms the equation into the form below:  
\[\left\{\begin{array}{ll}EI\frac{\partial^{4}U_{1}}{\partial x^{4}}+\rho A \frac{\partial^{2}U_{2}}{\partial t^{2}}=\ f_{sh}(x,t)\delta(x-\frac{L}{2})+ \ \sum\limits_{r=1}^{\mathrm{H}}f(x_{r},t)\delta(x-x_{r})+\sum\limits_{r=1}^{ \mathrm{H}}f_{d}(x_{r},t)\delta(x-x_{r})+\sum\limits_{v=1}^{q}f_{1}(x_{v},tWhen Equation (14) is multiplied by the trial function \(\varnothing_{\#}(x)\) it is stretched further into Equation (15) below:  
\[\left\{