## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures
This process is expressed in Equation (16), where \(\mathcal{H}_{b_{m}}(x,t)\) is determined by integrating the effects of external forces, base motion, and damping into a net external  
The Laplace transform is applied to the system of equations, assuming zero initial conditions, to transition the analysis to the frequency domain, as seen in Equations (18) and (19). In these equations, \(Z_{m}(s)\) and \(Z_{r}(s)\) represent the Laplace transforms of the modal and resonator displacements, respectively. This transformation simplifies the algebraic manipulation and analysis of the system's dynamics. \[Z_{r}(s)=\frac{\mathcal{Q}_{b_{r}}(s)-\sum_{m=1}^{N_{m}}s^{2}Z_{m}(s)\phi_{m}( x_{r})}{s^{2}+2\zeta_{r}\omega_{r}s+\omega_{r}^{2}},\quad r=1,2,\ldots,N_{r} \tag{18}\]  
\[ \begin{split} s^{2}Z_{m}(s)+2\zeta_{m}\omega_{m}sZ_{m}(s)+\omega_ {m}^{2}Z_{m}(s)+\sum_{r=1}^{N_{r}}m_{r}\phi_{m}(x_{r})\sum_{p=1}^{N_{m}}s^{2}Z _{p}(s)\phi_{p}(x_{r})+\\ \sum_{r=1}^{N_{r}}m_{r}s^{2}Z_{r}(s)\phi_{m}(x_{r})+2\sum_{r=1}^{ N_{r}}m_{r}\phi_{m}(x_{r})\zeta_{r}\omega_{r}sZ_{r}(s)=H_{b_{m}}(s),\quad m=1,2, \ldots,N_{m}\end{split}  \tag{19}\]  
For a deeper analytical understanding, applying Equation (18) to the Laplace transform of Equation (14) yields the following expression in Equation (20). \[ \begin{split}(s^{2}+2\zeta_{m}\omega_{m}s+\omega_{m}^{2})Z_{m}(s )-\sum_{r=1}^{N_{r}}m_{r}\omega_{r}^{2}\left(\frac{\mathcal{Q}_{b_{r}}(s)- \sum_{p=1}^{N_{m}}s^{2}Z_{p}(s)\phi_{p}(x_{r})}{s^{2}+2\zeta_{r}\omega_{r}s+ \omega_{r}^{2}}\right)\phi_{m}(x_{r})=\mathcal{Q}_{b_{m}}(s),\\ m=1,2,\ldots,N_{m}\end{split}  \tag{20}\]  
The analysis focuses on the transfer function \(\frac{Z_{m}}{Q_{b_{m}}}\), particularly when the effect of \(Q_{b_{r}}\) is ignored. Here, the mass ratio \(\mu\), a dimensionless quantity, relates the mass of each resonator to a differential mass element of the system and is defined as \(\mu=\frac{m_{r}}{m(x_{r})dx_{r}}\), where \(m(x_{r})\) represents the mass per unit length at \(x_{r}\) and \(dx_{r}\) is an infinitesimal segment length at this point. To simplify the system of equations, it is assumed that an infinite number of resonators are distributed throughout the entire domain of \(x\), and the regions represented by \(x_{r}\) become infinitesimally small. \[\lim_{N_{r}\rightarrow\infty}\sum_{r=1}^{N_{r}}m(x_{r})\phi_{m}(x_{r})\phi_{ p}(x_{r})dx_{r}\approx\int_{0}^{L}m(x)\phi_{m}(x)\phi_{p}(x)dx=\delta_{mp}, \quad m,p=1,2,\ldots \tag{21}\]  
Applying these assumptions results in the following expression:  
\[ \begin{split}\frac{Z_{m}(s)}{Q_{b_{m}}(s)}=\frac{1}{s^{2}\Big{(} 1+\frac{\mu(2\zeta_{r}\omega_{r}s+\omega_{r}^{2})}{s^{2}+2\zeta_{r}\omega_{r} s+\omega_{r}^{2}}\Big{)}+2\zeta_{m}\omega_{m}s+\omega_{m}^{2}}\\ m=1,2,\ldots,N_{m}\end{split}  \tag{22}\]  
Equation (22) indicates that resonators add a frequency-dependent mass to the system. With the assumption of an infinite resonator distribution, leading to continuous spatial displacements, similar reductions apply to the resonator displacements. By substituting Equations (22) into (18) and transitioning from the discrete \(x_{r}\) to a continuous spatial domain \(x\), a simplified equation emerges as presented in Equation (23). \[Z_{r}(x,s)=-\frac{s^{2}}{s^{2}+2\zeta_{r}\omega_{r}s+\omega_{r}^{2}}\sum_{m=1 }^{N_{m}}\frac{Q_{b_{m}}(s)\phi_{m}(x)}{s^{2}\Big{(}1+\frac{\mu(\omega_{r}^{2} +2\zeta_{r}\omega_{r}s)}{s^{2}+2\zeta_{r}\omega_{r}s+\omega_{r}^{2}}\Big{)}+2 \zeta_{m}\omega_{m}s+\omega_{m}^{2}} \tag{23}\]Equation (23) defines the motion \(Z_{r}(x,s)\) of resonators along the beam in the Laplace domain, influenced by modal forces \(Q_{b_{m}}(s)\). The displacement of the resonators is presented as a weighted sum of the beam's modal shapes:  
\[Z_{r}(x,s)=\sum_{m=1}^{N_{m}}\Psi_{m}(s)\phi_{m}(x) \tag{24}\]  
Equation (25) expresses the relationship between the modal force \(Q_{b_{m}}(s)\) and the modal coordinate \(\Psi_{m}(s)\) in the Laplace domain.