## 3 Band Structure of Hierarchical Metamaterials
### Effect of Electromechanical Coupling Coefficients and Shunt Circuit Resistances  
The effect of the strong and weak electromechanical coupling coefficients is shown in Figure 2, generated at a shunt circuit resistance of \(10^{4}\) ohms. According to [PERSON] et al. [50], strong electromechanical coupling is for an electromechanical coupling coefficient that is greater than \(10^{-3}\) N/V, while the electromechanical coupling coefficient between \(10^{-10}\) N/V and \(10^{-3}\) N/V is considered as weak electromechanical coupling. In this study, a transverse electromechanical coupling system was selected [55]. In transverse \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties** & \(\mathbf{m_{0}}\) & \(\mathbf{\omega_{0}}\) & \(\mathbf{C_{1},C_{2}}\) \\ \hline Value & 0.125 kg & \(2\times 10^{5}\) rad/s & \(1.13\times 10^{-10}\) F \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Material parameters used in the analysis. coupling, the mechanical and electrical domains are coupled such that the electric field polarization aligns with the thickness direction of the piezoelectric material, and strain is applied along the axial direction. To quantify the effects of hierarchy, electromechanical coupling, and shunt circuits, attention will be given to changes to the band gaps and bandwidths in the dispersion relations. A band gap is the frequency range between the upper and lower limits of consecutive passbands. On the other hand, the bandwidth of the passbands is the range of frequencies within which wave propagation is allowed. The dispersion of the zero-order hierarchy has two passbands and a single local resonance bandgap. In contrast, the dispersion of the first-order hierarchy, which is distinct for the outward and inward configurations, generates three passbands and two local resonance bandgaps. A dispersion relation at a given wavenumber corresponds to the natural frequencies of the masses. This explains why the number of passbands and local resonance bandgaps increases with higher-order hierarchies. This relationship is affected by the electromechanical couplings and the resistances of the shunt circuit. The parameters used to generate Figure 2a-c are selected based on the order of magnitude of the strong electromechanical coupling coefficient, ranging from \(10^{-2}\) N/V to \(10^{-1}\) N/V, as \(10^{-3}\) N/V represents the lower boundary. For lower electromechanical coupling coefficients, such as \(10^{-2}\) N/V and \(3\times 10^{-1}\) N/V, the influence on the band structure and band gap width is minimal, resulting in only subtle shifts on the curves. This highlights that at the lower end of the strong coupling regime, the system's dynamic response is not highly sensitive to variations in the electromechanical coupling coefficient. However, as the electromechanical coupling coefficient approaches higher values, \(9\times 10^{-1}\) N/V, a more noticeable widening of the band gap occurs. Strong electromechanical coupling coefficients shift the location of the local resonance band gaps upward. Increasing the strong electromechanical coupling coefficients, as shown in Figure 2a-c, increases the  
Figure 2: The effect of strong electromechanical coupling coefficients on the band structure of (**a**) zero-order, (**b**) outward, and (**c**) inward configurations. The effect of weak electromechanical coupling coefficients on the band structure of (**d**) zero-order, (**e**) outward, and (**f**) inward configurations at \(\eta=0.75\) and \(\beta_{1}=0.5\). width of the local resonance band gaps and the bandwidths. The band gap width and bandwidth changes are greater in the first-order hierarchy compared to the zero-order hierarchy, with the inward configuration showing the most significant increment. Due to strong electromechanical coupling, the band gap width increases by 11.52% for the zero-order configuration, and by 13.49% and 7.51% in the first band gap for the outward and inward configurations, respectively.