## 2 Problem Formulation
The constants \(a_{ijkn}\) and \(b_{ijkn}\) are mode-shape dependent and defined as  
\[a_{ijkn}=\frac{1}{2L^{2}}\!\int_{0}^{1}\ \left\{\sum_{i}^{N}\phi_{i}^{ \prime}(\xi)\int_{1}^{\xi}\left[\int_{0}^{\xi}\sum_{j}^{N}\sum_{k}^{N}\phi_{i} ^{\prime}(\xi)\phi_{k}^{\prime}(\xi)d\xi\right]d\xi\right\}^{\prime}\phi_{n}( \xi)\ d\xi \tag{11b}\]  
Investigating Equation (18), the effective natural frequency of the beam is defined as  
\[\hat{\omega}_{1}=\omega_{1}^{2}+\frac{K_{c}^{2}}{C_{p}}\;. \tag{22}\]  
_Inducing an Internal Resonance into the System_  
Nonlinear coupling between two modes whose natural frequencies are connected by an integer is known as internal resonance. Quadratic nonlinearity is necessary to introduce a 2:1 internal resonance. As proposed in [28], quadratic feedback and control signals can produce nonlinearity in active systems. In the meantime, nonlinear electrical components, like a nonlinear capacitor whose capacitance inversely varies inversely with the amplitude of the charge, can also be used to create nonlinearity in passive and hybrid systems. Operational amplifiers and varactor diodes can be used to physically construct this [35]. The nonlinear capacitance and its voltage can be represented as  
\[C_{nl}=\frac{1}{Q} \tag{23}\]  
and  
\[V_{nl}=N\frac{Q}{C_{nl}}=NQ^{2}. \tag{24}\]  
The constant \(N\) acts as a nonlinear gain or proportionality factor that dictates how the voltage scales with the square of the charge. Nonlinear capacitors arise from dielectric materials whose permittivity varies with the electric field. The form given in Equation (24) is not standard for common nonlinear capacitors like varactors [36]. Equation (19) becomes  
\[\ddot{Q}+2\zeta_{e}\omega_{e}\dot{Q}+\omega_{e}^{2}Q+\frac{K_{c}}{LC_{p}}q+ \frac{N}{L}Q^{2}=0\;. \tag{25}\]  
Now, Equations (18) and (25) govern the dynamics of the coupled system; however, the coupling terms in both equations are not of the same form. Therefore, we introduce the following dimensionless parameters to unify the coupling terms:  
\[\tau\;=\;\;\hat{\omega}_{1}\;t,\;\;\widetilde{q}\;=\;q\;,\;\widetilde{Q}\;=\; \sqrt{L}\;Q,\;\widetilde{\Omega}\;=\;\frac{\Omega}{\hat{\omega}_{1}}\;. \tag{26}\]  
As result, the nondimensional governing equations are defined as follows:  
\[\widetilde{\dot{q}}+2\zeta_{1}\widetilde{\dot{q}}+\widetilde{\dot{q}}+ \alpha\;\widetilde{Q}\;+\frac{a}{\hat{\omega}_{1}^{2}}\;\widetilde{q}^{3}\;= \;\widetilde{f}\;cos\bigg{(}\widetilde{\Omega}\tau\bigg{)} \tag{27}\]  
and  
\[\widetilde{\dot{Q}}+2\zeta_{e}\omega_{r}\dot{\widetilde{Q}}+\omega_{r}^{2} \widetilde{\dot{Q}}+\alpha\;\widetilde{q}+\frac{N}{L^{\frac{3}{2}}\hat{\omega }_{1}^{2}}\widetilde{Q}^{2}=0 \tag{28}\]  
where the electromechanical frequency ratio, \(\omega_{r}\), the coupling coefficient, \(\alpha\), the electromechanical coupling factor (EMCF), \(k_{1}\), and the forcing amplitude, \(\widetilde{f}\), are defined as  
\[\ddot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^{N}a_{ijkn}\ q_{i}\ \frac{\partial^{2}}{ \partial t^{2}}\big{(}q_{j}q_{k}\big{)}+\sum_{i,j,k}^{N}b_{ijkn}\ q_{i}q_{j}q_{k }\ =\ f_{n}\cos\Omega t,\ n\ =\ 1,\ \ldots,N. \tag{11a}\]0  
For simplicity, the hat (\(\sim\)) notation will be omitted in the following equations, and the dot refers to differentiation with respect to the nondimensional time \(\tau\). One may note that Equations (27) and (28) govern the coupled electromechanical model of a cantilever beam equipped with a nonlinear shunt circuit using the physical coordinate \(q(t)\) for the beam and \(Q(t)\) for the circuit. Yet, in terms of the physical coordinates, the system of equations is linearly coupled. To introduce the nonlinear coupling that makes the internal resonance possible, we use modal analysis technique to transform the governing equations into a new set of modal coordinates. The nonlinearly coupled modal equations can be derived as  
\[