## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures
\[\lim_{N_{r}\rightarrow\infty}\sum_{r=1}^{N_{r}/2}m(x_{2r-1})\phi_{m}(x_{2r-1} )\phi_{p}(x_{2r-1})dx_{2r-1}=\frac{1}{2}\delta_{mp},\quad m,p=1,2,\ldots \tag{31}\]  
\[\lim_{N_{r}\rightarrow\infty}\sum_{r=1}^{N_{r}/2}m(x_{2r})\phi_{m}(x_{2r}) \phi_{p}(x_{2r})dx_{2r}=\frac{1}{2}\delta_{mp},\quad m,p=1,2,\ldots \tag{32}\]  
Taking the Laplace transform of Equations (27) and (28), and applying the orthogonality of the mode shapes, as demonstrated in Equations (31) and (32), results in the derivation of the transfer function for a metastructure with internally coupled resonators, as represented in Equation (33). \[\frac{Z_{m}(s)}{Q_{b_{m}}(s)}=\frac{1}{s^{2}\Bigl{(}1+\frac{\mu\omega_{2r-1}^{ 2}}{s^{2}+\omega_{2r-1}^{2}}\Bigr{)}\Bigl{(}1+\frac{\frac{\kappa}{2}\omega_{2r }^{2}}{s^{2}+\omega_{2r}^{2}}\Bigr{)}+\omega_{m}^{2}}\quad m=1,2,\ldots,N_{m} \tag{33}\]  
The transfer function presented in Equation (33) incorporates coupling effects through the \(\kappa\) parameter, allowing for the interaction between multiple resonators, denoted as \(\omega_{2r-1}\) and \(\omega_{2r}\). This interaction can lead to complex dynamic behavior, including the potential for multiple bandgaps or more pronounced resonant effects. The integration of damping elements for both the plain structure and the resonators can be conveniently executed at this stage. In distributed parameter systems, such as beams, the resonators are two-degree-of-freedom (2 DOF) systems. It can be proven that \(\omega_{2r-1}=\omega_{r}\), where \(\omega_{r}\) is the natural frequency of a resonator when it is not coupled with its adjacent resonator. Additionally, \(\omega_{2r}=\sqrt{\omega_{2r-1}^{2}+\frac{2\kappa}{m_{r}}}\), where \(\kappa\) is the mechanical coupling coefficient and \(m_{r}\) is the mass of the resonator. This framework leads to the formation of secondary bandgaps in metastructures with internally coupled resonators. These bandgaps are associated with a 180-degree phase shift in the resonators. Consequently, such metastructures exhibit both primary and secondary bandgaps, a distinct feature compared to traditional structures. The condition of no stretching in the coupling spring essentially renders its influence negligible. Consequently, this scenario simplifies the equation, reducing it to a form that corresponds to the conventional metastructure dynamics, as established in Equation (22). This simplification allows for a more straightforward analysis of the metastructure by reverting to a more basic, yet fundamental, form of the equation. On the other hand, if the resonators differ in frequency or have the same frequency but with a phase difference, the parameter \(\kappa\) experiences stretching. This results in an additional pole and zero, creating an extra bandgap. Now, leveraging the transfer function method enables the utilization of the well-known root locus analysis. By considering the modal response as the closed-loop transfer function of a negative feedback system, which incorporates a proportional feedback gain of \(\omega_{m}^{2}\), and defining the feedforward transfer function as \(G(s)\), as specified in Equation (34), one can observe this interpretation. \[G(s)=\frac{s^{2}+\omega_{2r-1}^{2}}{s^{2}\big{(}s^{2}+\omega_{2r-1}^{2}+\mu \omega_{2r-1}^{2}\big{)}}\frac{s^{2}+\omega_{2r-1}^{2}+\frac{2\kappa}{m_{r}}}{ \big{(}s^{2}+(1+(\frac{\mu}{4}))(\omega_{2r-1}^{2}+\frac{2\kappa}{m_{r}})\big{)}} \tag{34}\]  
The first transfer function accurately represents what is found in a conventional metastructure. This function has two poles at the origin, characteristic of a system's inherent response dynamics. It includes an additional pole at \(\sqrt{1+\mu}\omega_{2r-1}\), influenced by the mass of the resonators. This pole is responsible for creating a bandgap with a length of \(\sqrt{1+\mu}\), indicative of the system's frequency-selective behavior. The internal coupling of resonators introduces additional dynamics, particularly influencing the system's behavior near resonant frequencies.