## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures
It illustrates how resonator displacements are influenced by the modes of the structure:  
\[\frac{\Psi_{m}(s)}{Q_{b_{m}}(s)}=\frac{-s^{2}}{\left[s^{2}+2\zeta_{r}\omega_{r }s+\omega_{r}^{2}\right]\left[s^{2}\left(1+\frac{\mu(\omega_{r}^{2}+2\zeta_{r} \omega_{r}s)}{s^{2}+2\zeta_{r}\omega_{r}s+\omega_{r}^{2}}\right)+2\zeta_{m} \omega_{m}s+\omega_{m}^{2}\right]},\quad m=1,2,\ldots,N_{m} \tag{25}\]  
The transfer function in Equation (25) clarifies how the input forces are transformed into modal responses. The function's poles, indicative of the system's natural frequencies, are the points at which the system exhibits peak responses. By introducing an internal linear coupling term, \(\kappa\), within the resonators illustrated in Figure 2, the system evolves into internally coupled resonators. This transformation creates an environment where the displacements of the resonators are no longer independent but are coupled. Specifically, the displacement of one resonator influences the displacement of the other, establishing a dynamic interaction. The energy associated with this coupling is quantified by the coupling potential energy, in which each pair of resonators (1 and 2, 3 and 4, 5 and 6, etc.) forms a system with two degrees of freedom: \(V_{c}(t)=\frac{1}{2}\kappa(z_{r_{1}}(t)-z_{r_{2}}(t)+w(x_{r_{1}},t)-w(x_{r_{2}}, t))^{2}\). The equations for the coupled oscillator system can be formulated as follows:  
\[ \begin{split}&\ddot{z}_{m}(t)+2\zeta_{m}\omega_{m}\dot{z}_{m}(t)+ \omega_{m}^{2}z_{m}(t)-\sum_{r=1}^{N_{r}/2}\left(m_{2r-1}\omega_{2r-1}^{2}z_{ 2r-1}(t)\phi_{m}(x_{2r-1})+\right.\\ &\left.m_{2r}\omega_{2r}^{2}z_{2r}(t)\phi_{m}(x_{2r})\right)= \mathcal{Q}_{b_{m}},\quad m=1,2,\ldots,N_{m},\text{ and }N_{r}\in 2\mathbb{N}\end{split}  \tag{26}\]  
Meanwhile, the equation for the resonators is given by the following:  
\[ \begin{split}&\ddot{z}_{2r-1}(t)+2\zeta_{2r-1}\omega_{2r-1} \dot{z}_{2r-1}(t)+\omega_{2r-1}^{2}z_{2r-1}(t)+\sum_{m=1}^{N_{m}}\dot{z}_{m}( t)\phi_{m}(x_{2r-1})+\\ &\beta\frac{\kappa}{m_{2r-1}}=\mathcal{Q}_{b_{2r-1}},\quad r=1,2,\ldots,N_{r}/2\end{split}  \tag{27}\]  
\[ \begin{split}&\ddot{z}_{2r}(t)+2\zeta_{2r}\omega_{2r}\dot{z}_{2r }(t)+\omega_{2r}^{2}z_{2r}(t)+\sum_{m=1}^{N_{m}}\dot{z}_{m}(t)\phi_{m}(x_{2r}) \\ &\quad-\beta\frac{\kappa}{m_{2r}}=\mathcal{Q}_{b_{2r}},\quad r=1, 2,\ldots,N_{r}/2\end{split}  \tag{28}\]  
Figure 2: Locally resonant metastructures with internally coupled resonators. Each pair of resonators forms one unit cell, with \(m\) representing the mass of the resonators, \(c\) the damping, \(k\) the stiffness of the resonators, and \(\kappa\) the internal coupling stiffness between them. where  
\[\beta=z_{2r-1}(t)-z_{2r}(t)+\sum_{m=1}^{N_{m}}z_{m}(t)\phi_{m}(x_{2r-1})-\sum_{m=1 }^{N_{m}}z_{m}(t)\phi_{m}(x_{2r}) \tag{29}\]  
These equations characterize the underlying dynamics of both the beam and the internally linear coupled resonator system. The Laplace transform of Equation (26) with zero initial conditions results in Equation (30). \[ \begin{split}&(s^{2}+2\zeta_{m}\omega_{m}s+\omega_{m}^{2})Z_{m}(s) -\sum_{r=1}^{N_{r}/2}\Bigl{(}m_{2r-1}\omega_{2r-1}^{2}Z_{2r-1}(s)\phi_{m}(x_{2r -1})+\\ & m_{2r}\omega_{2r}^{2}Z_{2r}(s)\phi_{m}(x_{2r})\Bigr{)}=Q_{b_{m} }(s)\end{split}  \tag{30}\]  
Given that the forces \(\mathcal{Q}_{b_{2r-1}}\) and \(\mathcal{Q}_{b_{2r}}\) are equal to \(-\tilde{w}_{b}(t)\), and considering that \(m_{2r-1}=m_{2r}\) or identical mass \(m_{r}\) for all resonators, along with the distribution of numerous resonators along a beam, it is assumed that the derivative of position within each unit cell is the same, indicated by \(dx_{2r-1}=dx_{2r}\). This reflects a uniform position derivative across all resonators. Furthermore, the following relationships are established: \(m_{2r-1}=\mu m(x_{2r-1})dx_{2r-1}\), and \(m_{2r}=\mu m(x_{2r})dx_{2r}\).