## 2 Mathematical Modeling
By substituting the assumed solution form into Equations (1)-(3), we obtain the following results:  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]1  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]2  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]3  
Upon substituting the assumed solution into the equations of motion for the inward configuration, presented in Equations (4)-(7), the system simplifies to Equations (11)-(14). \[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]4  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]5  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]6  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]7  
The determinants given in Equations (A1) and (A2) were expanded using Mathematica to obtain the dispersion equations used in the simulations presented in this study. Then, the roots of the dispersion relation, along with the finite chain transmissibility and power output, solved in MATLAB R2024a. A local resonance band gap is created as a result of mode separation caused by the local resonance effect. In an undamped single degree-of-freedom resonator, the natural frequency of the resonator marks the center of the local resonance band gap and is called the tuning frequency [10]. In this context, an undamped resonator refers to a local resonator without piezoelectric patch, shunt circuit, or mechanical damping element. A local resonance band gap tuning frequency equation can be derived by assuming the outer mass as a fixed boundary [53]. In the outward configuration, motion of the resonators are not coupled; therefore, the undamped tuning frequency, \(\omega_{n}\), of the \(n^{\text{th}}\) local resonator can be treated as a single degree-of-freedom system and expressed as  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]8However, the coupled motion of the resonators in the inward configuration yields a multiple degree-of-freedom system. The multiple degree-of-freedom local resonator undamped tuning frequencies in the inward configuration are the eigenvalues of Equation (16), which is the characteristic polynomial determined from the coefficient matrix of Equations (12) and (13). \[ \begin{vmatrix}-\omega_{n}^{2}m_{1}+k_{1}+k_{2}&-k_{2}&0&\dots&0\\ -k_{2}&\ddots&\ddots&\vdots&\vdots\\ \vdots&\ddots&\ddots&-k_{N-1}&\vdots\\ &0&-k_{N-1}&-\omega_{n}^{2}m_{N-1}+k_{N-1}+k_{N}&-k_{N}\\ 0&\dots&\dots&-k_{N}&-\omega_{n}^{2}m_{N}+k_{N}\end{vmatrix} =0 \tag{16}\]  
The tuning frequency of the \(n^{\text{th}}\) electromechanical resonator, \(\omega_{em}\), shunted with an electrical circuit in the outward configuration is derived from Equations (9) and (10). Equations (17) and (18) are used to characterize the tuning frequency of the electromechanical resonator in the zero-order hierarchy, as discussed in the next section. \[\left(k_{n}-m_{n}\omega_{em}^{2}\right)A_{n}+\theta_{n}B_{n}=0 \tag{17}\]  
\[i\omega_{em}R_{n}\theta_{n}A_{n}+(1-i\omega_{em}R_{n}C_{n})B_{n}=0 \tag{18}\]  
For the analysis, the parameter values listed in Table 1 are utilized [49]. In addition, the dimensionless ratio of the total mass of the electromechanical resonator to the outer mass is defined as \(\eta=\frac{m_{r}}{m_{0}}\), where \(m_{r}\) represents the total mass of the electromechanical resonators within the unit cell. The ratio of the \(n^{\text{th}}\) electromechanical resonator mass to the resonators total mass is defined as \(\beta_{n}=\frac{m_{n}}{m_{r}}\). Based on this generic definition, for the first-order hierarchy, \(\eta=\frac{m_{1}+m_{2}}{m_{0}}\) and \(\beta_{1}=\frac{m_{1}}{m_{1}+m_{2}}\). For the zero-order configuration \(\beta_{1}\) is 1.