## 2 Modal Analysis and Bandgap Formation in Mechanical Metastructures
The second transfer function introduces terms that model the added poles and zeros in the metastructure due to the internal coupling of resonators. In this function, the roots progress from zero at \(\omega_{2r}\) to a pole at \(\sqrt{1+\frac{\mu}{4}}\omega_{2r}\), creating a bandgap with a length of \(\sqrt{1+\frac{\mu}{4}}\). The comparative analysis of the root locus plots for a conventional metamaterial and a metamaterial with internal resonator coupling, as depicted in Figures 3 and 4, clearly indicate the influence of the coupling term \(\kappa\) on the system dynamics. Figure 3 illustrates the resonance characteristics and bandgap frequencies of a metastructure, as indicated by the poles of its transfer function. The system's resonances correspond to the imaginary components of these poles. Modal responses of the plain beam are modeled as a closed-loop transfer function with proportional feedback. Bandgap edge frequencies are identified using root locus analysis, with specific zeros and poles on the imaginary axis determining these frequencies. Notably, within the bandgap defined by \(\omega_{2r-1}<\omega<\omega_{2r-1}\sqrt{1+\mu}\), and \(\omega_{2r}<\omega<\omega_{2r}\sqrt{1+\frac{\mu}{4}}\), no poles are present. Root locus analysis is advantageous for evaluating general linear attachments and facilitating the creation of multiple bandgaps. Figure 3: (**Left**) Conventional metastructure root locus with \(\omega_{m}=0.5\), \(\omega_{2r-1}=\omega_{r}=1\), \(k_{r}=1\), \(m_{r}=1\), and \(\mu=0.5\). (**Right**) Metastructure with internal coupling, showing narrow bandgap with \(\omega_{m}=0.5\), \(\omega_{2r}=1.7\), \(\kappa=k_{r}\), and \(\mu=0.5\). Internal coupling’s impact on system dynamics is highlighted by the additional bandgap in the right plot. As mentioned earlier, the introduction of \(\kappa\) in the coupled system leads to additional zeros and poles, as evidenced by the second root locus in Figure 4. This modification is characterized by the resonant frequencies \(\omega_{2r-1}=1\ (\mathrm{rad}/\mathrm{s})\) and \(\omega_{2r}=1.7\ (\mathrm{rad}/\mathrm{s})\), suggesting the emergence of a second bandgap. Moreover, the root locus plot in Figure 4 (left) indicates that the internal coupling parameter influences the system's pole trajectories. Conversely, the Bode plot (right) reveals a pronounced resonant peak, suggesting an increased selective sensitivity to certain frequencies. While the phase response indicates the overall system's stability under the new coupling condition, it must be carefully evaluated to ensure robustness, especially in control applications where stability is critical. Dispersion Analysis and Model Validation of Internally Coupled Resonators by Plane Wave Expansion Method  
The plane wave expansion (PWE) method is commonly used for analyzing the propagation of waves in periodic structures, and provides valuable insights into the behavior of these waves, facilitating the design and optimization of these periodic structures for a wide range of applications, such as vibration suppression and energy harvesting [12; 13]. The transverse displacement of a metastructure with linearly internally coupled resonators in absolute coordinates is defined as \(W_{t}(x,t)=\hat{W}_{t}e^{i(\hat{G}_{n}x-\omega t)}\) for the beam, \(z_{2r-1}(t)=\hat{z}_{2r-1}e^{i(\omega t)}\) for the first resonator, and \(z_{2r}(t)=\hat{z}_{2r}e^{i(\omega t)}\) for the second resonator. The dispersion relation emerges from applying periodic boundary conditions to find nontrivial solutions. The relationship between frequency \(\omega\) and wavevector \(G_{n}\) in one-directional transformation is established by multiplying the variable's amplitude with \(\exp(i(G_{n}x-\omega t))\). With both \(k_{r_{1}}=k_{r_{2}}=k_{r}\) and \(m_{r_{1}}=m_{r_{2}}=m_{r}\), the equation simplifies as follows:  
\[C_{1}\omega^{6}+C_{2}\omega^{4}+C_{3}\omega^{2}+C_{4}=0 \tag{35}\]  
where:  
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