## 2 Model  
### Single-Beam Metamaterial Structure  
A representative design of a single-beam metamaterial structure is shown in Figure 1. The structure is comprised of a single horizontal beam and various local resonators attachments, as shown in Figure 1. The spacing between unit cells, \(L_{c}\), is regarded as the length of the unit cell; the effect of the number of unit cells per beam has been studied for different design forms in the literature [18]. In Figure 1, \(M\) is the mass of the local resonators, while \(Z_{ir}\) is the displacement of the local resonators when an external vibration is applied to the metamaterial structure. In principle, when external vibration is incident on the single-beam metamaterial structure in Figure 1, at frequencies near the resonant frequency of the unit cell, the vibrational energies are trapped in the local resonators. This trapping of vibrational energy generates the local resonator bandgaps seen in the overall transmissibility of the metamaterial structure. The local resonators of the single-beam metamaterial structure in Figure 1 are further represented in the form of stiffness and damping shown in Figure 2. The symbols \(U(x,t)\), \(\xi_{r}\), and \(k_{r}\) represent the displacement of the beam, the damping coefficient of the local resonator, and the stiffness, respectively. Figure 1: Schematic representation of a single-beam metamaterial structure. M represents the mass of the local resonators, while \(Z_{ir}\) is the displacement of the local resonator. The displacement of the single-beam metamaterial structure in Figure 1 can be modeled with the standard Euler-Bernoulli beam's equation in Equation (1) below [30]:  
\[EI\frac{\partial^{4}U}{\partial x^{4}}+\rho A\frac{\partial^{2}U}{\partial t^{2} }=f_{sh}(x,t)\delta\bigg{(}x-\frac{L}{2}\bigg{)}+\sum_{r=1}^{n}f(x_{r},t) \delta(x-x_{r}) \tag{1}\]  
where \(L\), \(E\), and \(I\) represent, respectively, the total length of the horizontal beam (\(L=\gamma*L_{c}\)), [PERSON]'s elastic modulus, and the cross-sectional moment of inertia of the horizontal beam. In addition, \(f_{sh}(x,t)\) and \(f(x_{r},t)\) in Equation (1) represent the force from the vibration source on the beam and the force from the local resonators, respectively. The transverse displacement of the horizontal beam in Equation (1), \(U(x,t)\), can be analyzed using the [PERSON] approach and superposition of mode shapes given in Equation (2) below [18]:  
\[U(x,t)=\sum_{i=1}^{N}p_{i}(t)\varnothing_{i}(x) \tag{2}\]  
where \(p_{i}\) and \(\varnothing_{i}\) in Equation (2) represent the generalized displacement and the trial of the horizontal beam at the \(i_{th}\) mode, respectively. Moreover, \(N\) is the number of modes considered in the modeling effort. Furthermore, the displacement of the local resonators is shown in Equation (3) below:  
\[m_{r}\tilde{Z_{ir}}(t)+f_{d}(x_{r},t)+f(x_{r},t)=0 \tag{3}\]  
The expression \(f_{d}(x_{r},t)\) in Equation (3) represents the damping force of the local resonator, while \(m_{r}\) is the mass of the local resonator. In addition, \(Z_{ir}\) is the displacement of the \(r\)th local resonator in the \(i\)th mode shape. The expressions of the forces in Equations (1) and (2) are given below:  
\[ \begin{cases}\begin{aligned} f(x_{r},t)=k_{r}\big{[}Z_{r}(t)-U_{j}( x_{r},t)\big{]}\\ f_{d}(x_{r},t)=2\xi_{r}\sqrt{m_{r}k_{r}}\ \Big{[}\dot{Z}_{r}(t)-\dot{U}_{j}(x_{r},t) \Big{]}\\ f_{sh}=f_{s}\ \cos(\omega t)\end{aligned}\end{cases}  \tag{4}\]  
The expression \(f_{s}\) in Equation (4) represents the amplitude of the external vibration force. Here, the same material parameters used by [PERSON] et al. [5] and [PERSON] et al. [18], shown in Table 1, were used in this work. The dimensions and parameters reported by [PERSON] et al. [5] and [PERSON] et al. [18] were used because the results of the modeling approach were thoroughly verified with experiments in the literature.