## 4 Transmissibility
By substituting the second equation of Equation (22) into the first, the transfer function of the \(j^{th}\) unit cell, \(Q_{j}\), can be expressed in terms of the transfer function of the last unit cell, \(Q_{k}\), as  
\[ \begin{cases}Q_{k}=\frac{A_{0}^{(k)}}{A_{0}^{(k-1)}}=T_{k},\\ \vdots\\ Q_{j}=\frac{A_{0}^{(j)}}{A_{0}^{(j-1)}}=\frac{T_{1}}{1-T_{1}Q_{j}},\\ \vdots\\ Q_{1}=\frac{A_{0}^{(1)}}{A_{0}^{(0)}}=\frac{T_{1}}{1-T_{1}Q_{2}}.\end{cases}  \tag{24}\]  
The outer mass displacement transmissibility of the last unit cell relative to the input excitation is [56]  
\[T=\frac{A_{0}^{(k)}}{A_{0}^{(0)}}=\prod_{j=1}^{k}Q_{j} \tag{25}\]  
where \(T\) is the transfer function that links the last unit cell outer mass displacement to the input excitation amplitude, \(A_{0}^{(0)}\). The power output, \(P_{n}^{(k)}\), of the \(n^{\text{th}}\) shunt circuit in the last unit cell is given by Equation (26). \[P_{n}^{(k)}=\frac{\left(B_{n}^{(k)}\right)^{2}}{R_{n}^{(k)}} \tag{26}\]  
The power output from each shunt circuit can be expressed in a non-dimensional form by normalizing it with respect to the resonators input power, \(\frac{m_{n}\left(\omega^{2}A_{0}^{(0)}\right)^{2}}{\omega_{n}^{2}}\)[39]. The normalized total power output of the last unit cell is obtained by summing the dimensionless power harvested from all the shunt circuits within the unit cell. This is expressed in Equation (27). \[\left|P_{norm}\right|=\sum_{n=1}^{N}\left|\frac{P_{n}^{(k)}}{\frac{m_{n}\left( \omega^{2}A_{0}^{(0)}\right)^{2}}{\omega_{n}^{2}}}\right| \tag{27}\]  
The outer mass displacement transmissibility and the normalized total power output of the last unit cell relative to the input excitation for the finite chain consisting of 100 unit cells are examined for the zero-order and first-order hierarchies. Figure 11 demonstrates the outer mass displacement transmissibility of the last unit cell for weakly and strongly coupled hierarchical metamaterials at different \(\beta_{1}\) values, respectively. The band structure identified in the dispersion is in good agreement with the transmissibility results, which confirms the consistency of the results. The outer mass displacement transmissibility of the weakly coupled first-order hierarchy shows that the vibration attenuation frequency range depends on the value of \(\beta_{1}\), as illustrated in Figure 11a-c. In contrast, the outer mass displacement transmissibility of the zero-order hierarchy exhibits a fixed frequency range. The weakly coupled zero-order hierarchy exhibits higher local displacement transmissibility peaks compared to the first-order hierarchy, particularly at lower frequencies. The vibration attenuation depth in the first local resonance band gap region is similar in both hierarchies. For the inward configuration, the first attenuation region shifts to the left when \(\beta_{1}\) is less than 0.5, indicating its ability to attenuate vibrations at lower frequencies. The zero-order and first-order hierarchies show equivalent sizes of the first attenuation regions when \(\beta_{1}\) is greater than 0.5. The first-order hierarchy produces a first attenuation zone of similar size to the zero-order hierarchy at certain \(\beta_{1}\) such as \(\beta_{1}=0.85\). This is one of the benefits of the first-order hierarchy, as it retains the same effective frequency attenuation, while also introducing the potential for additional attenuation through the second attenuation region. The second local resonance vibration attenuation zone in the first-order configurations is located within the Bragg scattering attenuation zone of the zero-order hierarchy. Local resonance attenuation is stronger than Bragg attenuation, which is crucial for effective vibration attenuation. The local resonance attenuation regions can be tuned through \(\beta_{1}\) to target specific frequencies and broaden the band of attenuated frequencies compared to the fixed band gap of the zero-order hierarchy.