## 4 Transmissibility  
In this section, the transfer function linking the outer mass displacement of the last unit cell with the input excitation force is derived for the finite-chain hierarchical metamaterial comprising \(k\) unit cells. Applying this transfer function, the outer mass displacement transmissibility and the normalized total power output of the last unit cell are evaluated. The evaluation of displacement transmissibility and normalized total power output serves two main purposes. First, it validates the band structure identified in the infinite chain. Second, it facilitates the optimization of vibration control and energy harvesting in electromechanical hierarchical metamaterials. The finite chain transfer function is derived using fixed-free boundary conditions. An input excitation force in the form of \(A_{0}^{(0)}e^{i\omega t}\) is applied to the fixed wall attached to the first unit cell. After performing the necessary mathematical rearrangements on Equations (1)-(7), applying the boundary conditions and  
Figure 10: Comparison of the total band gap width to last passband upper limit frequency ratio for the weakly coupled hierarchical metamaterials. taking the Laplace transform (\(s=i\omega\)), the resulting finite chain equations for the outward and inward configurations of the first-order hierarchy are provided in Equation (20) and Equation (21), respectively. \[ \begin{cases}\Big{(}s^{2}m_{0}+2k_{0}+k_{1}+k_{2}\Big{)}A_{0}^{(j)}-k_{0}A_{0}^{( j-1)}-k_{0}A_{0}^{(j+1)}\\ \quad-k_{1}A_{1}^{(j)}-k_{2}A_{2}^{(j)}-\theta_{1}B_{1}^{(j)}-\theta_{2}B_{2}^{ (j)}=0,\\ \Big{(}s^{2}m_{0}+k_{0}+k_{1}+k_{2}\Big{)}A_{0}^{(k)}-k_{0}A_{0}^{(k-1)}-k_{1}A_ {1}^{(k)}\\ \quad-k_{2}A_{2}^{(k)}-\theta_{1}B_{1}^{(k)}-\theta_{2}B_{2}^{(k)}=0,\\ \Big{(}s^{2}m_{1}+k_{1}\Big{)}A_{1}^{(j)}-k_{1}A_{0}^{(j)}+\theta_{1}B_{1}^{( j)}=0,\\ \Big{(}s^{2}m_{2}+k_{2}\Big{)}A_{2}^{(j)}-k_{2}A_{0}^{(j)}+\theta_{2}B_{2}^{( j)}=0,\\ (sR_{1}C_{1}+1)B_{1}^{(j)}+sR_{1}\theta_{1}A_{0}^{(j)}-sR_{1}\theta_{1}A_{1}^{ (j)}=0,\\ (sR_{2}C_{2}+1)B_{2}^{(j)}+sR_{2}\theta_{2}A_{0}^{(j)}-sR_{2}\theta_{2}A_{1}^{ (j)}=0.\end{cases}  \tag{20}\]  
\[ \begin{cases}\Big{(}s^{2}m_{0}+2k_{0}+k_{1}\Big{)}A_{0}^{(j)}-k_{0}A_{0}^{(j-1 )}-k_{0}A_{0}^{(j+1)}\\ \quad-k_{1}A_{1}^{(j)}-\theta_{1}B_{1}^{(j)}=0,\\ \Big{(}s^{2}m_{0}+k_{0}+k_{1}\Big{)}A_{0}^{(k)}-k_{0}A_{0}^{(k-1)}-k_{1}A_{1}^ {(k)}-\theta_{1}B_{1}^{(k)}=0,\\ \Big{(}s^{2}m_{1}+k_{1}+k_{2}\Big{)}A_{1}^{(j)}-k_{1}A_{0}^{(j)}-k_{2}A_{2}^{ (j)}+\theta_{1}B_{1}^{(j)}-\theta_{2}B_{2}^{(j)}=0,\\ \Big{(}s^{2}m_{2}+k_{2}\Big{)}A_{2}^{(j)}-k_{1}A_{1}^{(j)}-\theta_{2}B_{2}^{ (j)}=0,\\ (sR_{1}C_{1}+1)B_{1}^{(j)}+sR_{1}\theta_{1}A_{0}^{(j)}-sR_{1}\theta_{1}A_{1}^ {(j)}=0,\\ (sR_{2}C_{2}+1)B_{2}^{(j)}+sR_{2}\theta_{2}A_{0}^{(j)}-sR_{2}\theta_{2}A_{1}^ {(j)}=0.\end{cases}  \tag{21}\]  
In Equations (20) and (21), the first equation corresponds to the \(j^{\text{th}}\) unit cell, while the second equation, identified by the superscript \(k\), pertains to the last unit cell. The remaining equations describe the resonators and shunt circuits. By applying the back substitution method [39], the displacement amplitudes can be expressed as Equation (22)  
\[ \begin{cases}A_{0}^{(j)}=T_{1}\Big{(}A_{0}^{(j-1)}+A_{0}^{(j+1)}\Big{)},\\ A_{0}^{(k)}=T_{k}A_{0}^{(k-1)},\\ B_{n}^{(k)}=b_{n}A_{0}^{(k)}.\end{cases}  \tag{22}\]  
The coefficients \(T_{1}\) and \(T_{k}\) are transfer functions that connect the \(j^{\text{th}}\) unit cell to its neighboring unit cells and the last unit cell with the preceding unit cell, respectively, and \(b_{n}\) is a coefficient that connects the voltage output of the \(n^{\text{th}}\) shunt circuit to the displacement amplitude of the last unit cell. \(T_{1}\) and \(T_{k}\) are distinct for the first and last unit cells and expressed as  
\[ \begin{cases}T_{1}=\dfrac{c_{1}}{d_{1}},\\ T_{k}=\dfrac{c_{k}}{d_{k}}.\end{cases}  \tag{23}\]  
Refer to Appendix C for the details of the constants \(c_{1}\), \(c_{k}\), \(d_{1}\), \(d_{k}\), and \(b_{n}\).