## 2 Mathematical Modeling  
The schematic diagram in Figure 1 illustrates two types of hierarchical configurations, namely the outward and inward configurations. The outward configuration Figure 1a arranges the electromechanical resonators in parallel; however, the inward configuration Figure 1b arranges them in a nested manner such that the \(n^{\text{th}}\) electromechanical resonator is connected to the \((n-1)\) and \((n+1)^{\text{th}}\) electromechanical resonators [25]. In this setup, a unit cell consists of \(N\) electromechanical local resonators with masses \(m_{1},m_{2},\dots,m_{n},\dots,m_{N}\), arranged within an outer mass \(m_{0}\). Adjacent unit cells are linked by a spring with a stiffness constant, \(k_{0}\), creating a continuous metamaterial chain. The stiffness of the resonator connectors is denoted as \(k_{1}\), \(k_{2}\), \(\ldots\), \(k_{N}\). A resistor-loaded shunt circuit is integrated into each piezoelectric patch. The parameters \(\theta_{n}^{(j)}\), \(C_{n}^{(j)}\), and \(R_{n}^{(j)}\) are the electromechanical coupling coefficient, the capacitance of the piezoelectric materials, and the resistance of the shunt circuit, respectively. The maximum number of local resonators, \(N\), is determined by the hierarchy order, which is defined as \(N-1\). In this work, in this work, the zero-order and first-order configurations, with \(N=1\) and \(N=2\), respectively, will be studied. Accordingly, the equations of motion for the \(j^{\text{th}}\) unit cell in the outward configuration can be expressed as:  
\[m_{0}\ddot{u}_{0}^{(j)}-k_{0}u_{0}^{(j-1)}+2k_{0}u_{0}^{(j)}-k_{0}u_{0}^{(j+1) }+\sum_{n=1}^{N}m_{n}\ddot{u}_{n}^{(j)}=0 \tag{1}\]  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]  
\[R_{n}C_{n}\ddot{v}_{n}^{(j)}+v_{n}^{(j)}+R_{n}\theta_{n}u_{0}^{(j)}-R_{n} \theta_{n}u_{n}^{(j)}=0 \tag{3}\]  
Further, the governing equations of motion for the inward configuration's unit cell are  
\[m_{0}\ddot{u}_{0}^{(j)}-k_{0}u_{0}^{(j-1)}+(2k_{0}+k_{1})u_{0}^{(j)}-k_{0}u_{0} ^{(j+1)}-k_{1}u_{1}^{(j)}-\theta_{1}v_{1}^{(j)}=0 \tag{4}\]  
\[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{n-1}^{(j)}+(k_{n}+k_{n+1})u_{n}^{(j)}-k_{n+1}u _{n+1}^{(j)}+\theta_{n}v_{n}^{(j)}-\theta_{n+1}v_{n+1}^{(j)}=0 \tag{5}\]  
\[m_{N}\ddot{u}_{N}^{(j)}-k_{N}u_{N-1}^{(j)}+k_{N}u_{N}^{(j)}+\theta_{N}v_{N}^{( j)}=0 \tag{6}\]  
\[R_{n}C_{n}\ddot{v}_{n}^{(j)}+v_{n}^{(j)}+R_{n}\theta_{n}\dot{u}_{n-1}^{(j)}-R _{n}\theta_{n}\dot{u}_{n}^{(j)}=0 \tag{7}\]  
Here, \(u_{0}^{(j)}\) and \(u_{n}^{(j)}\) are the displacements of the outer mass and \(n^{\text{th}}\) resonator, respectively, and \(v_{n}^{(j)}\) is the induced voltage across the \(n^{\text{th}}\) shunt circuit, all of which are within the \(j^{th}\) unit cell. A general harmonic solution of the form  
Figure 1: Schematic diagrams of the hierarchy: (**a**) outward and (**b**) inward configurations of electromechanical resonators shunted with resistor loaded circuits. \[u_{0}^{(i)} =A_{0}e^{i(\kappa-\omega t)},\] \[u_{n}^{(i)} =A_{n}e^{i(\kappa-\omega t)},\text{ and}\] \[m_{n}\ddot{u}_{n}^{(j)}-k_{n}u_{0}^{(j)}+k_{n}u_{n}^{(j)}+\theta_{n}v_{n}^{(j) }=0 \tag{2}\]0  
is assumed to establish the frequency-wavenumber relations for the infinite lattice [53; 54]. In this solution, \(t\) is time, \(\omega\) is excitation frequency of the system, and \(\kappa\) is the normalized wavenumber. Substituting the presumed solution into Equations (1)-(7) provides \(2N+1\) simultaneous equations for each configurations.