## 2 Problem Formulation  
We consider the lateral vibration of a cantilever beam attached to a piezoelectric shunt circuit, as shown in Figure 1. The shunt circuit parameters, which are typically resistance, capacitance, and inductance, are selected such that the resulting coupled response minimizes the lateral vibration of the beam as a primary system. Namely, the shunt circuit includes a nonlinear capacitance that is tuned to induce a 2:1 internal resonance between the mechanical and the electrical natural frequencies. An experimental setup for the linear vibration of a cantilever beam coupled with a piezoelectric shunt was presented by [PERSON] et al. [33]. The proposed nonlinear piezoelectric shunt circuit is used to mitigate the lateral vibration of a cantilever beam that is harmonically excited at its fundamental resonant frequency. The equation governing the nonlinear lateral vibration of an Euler-Bernoulli cantilever beam, and the associated boundary conditions are given using [34]  
\[m\ddot{v}+EIv^{\prime\prime\prime}+\frac{m}{2}\bigg{\{}\ \ v^{\prime}\int_{L}^{x}\frac{ \partial^{2}}{\partial t^{2}}\left[\int_{0}^{x}\ (v^{\prime})^{2}\ dx\right]dx\bigg{\}}^{\prime}+EI\Big{[}v^{\prime}\big{(}v^{ \prime}v^{\prime\prime}\big{)}^{\prime}\Big{]}^{\prime}=Fcos(\Omega t), \tag{1}\]  
Figure 1: Schematic of a cantilever beam coupled with a shunted circuit that has a nonlinear capacitor. \[v=0,\ v^{\prime}=0\ at\ x=0, \tag{2}\]  
and  
\[v^{\prime\prime}=0,\ v^{\prime\prime\prime}=0\ at\ x=L, \tag{3}\]  
\(v(x,t)\) is the lateral vibration; \(E\), \(I\), \(m\), and \(L\) are the modulus of elasticity, the second moment of area, the mass per unit length, and the beam's length respectively; \(F\) is the amplitude of the applied harmonic force, and \(\Omega\) is its frequency; \(x\) is the spatial coordinate, and \(t\) is time. The dot and prime denote derivatives with respect to time and length, respectively. Solving the linear free vibration problem yields the following mode shapes and natural frequencies:  
\[\phi_{i}(\xi)\ =\ sink_{i}\xi-sinh_{i}\xi-\sigma_{i}(cosk_{i}\xi-coshk_{i}\xi) \tag{4}\]  
and  
\[\omega_{i}=\sqrt{\frac{EI}{mL^{4}}}k_{i}^{2}\, \tag{5}\]  
where \(\xi\) is the nondimensional spatial coordinate such that \(\xi\ =\ x/L\), and the constants \(k\) and \(\sigma\) are defined as  
\[cosk_{i}coshk_{i}+1=0 \tag{6}\]  
and  
\[\sigma_{i}\ =\ \frac{sinhk_{i}+sink_{i}}{cosh\ k_{i}+cos\ k_{i}}. \tag{7}\]  
The orthogonality conditions are defined as  
\[\int_{0}^{1}\phi_{i}(\xi)\phi_{j}(\xi)\ d\xi= \begin{cases}1,\ i=j\\ 0,\ i\
eq j\end{cases}  \tag{8}\]  
and  
\[\int_{0}^{1}\phi_{i}(\xi)\ \phi_{i}^{\prime\prime\prime}(\xi)\ d\xi= \begin{cases}k_{i }^{4},\ i=j\\ 0,\ i\
eq j\end{cases} . \tag{9}\]  
Let the displacement \(u(\xi,t)\) be discretized as follows:  
\[v(\xi,t)\ =\ \sum_{i=\ 1}^{N}\varphi_{i}(\xi)\ q_{i}(t), \tag{10}\]  
where \(\phi_{i}\) is the linear vibration mode shapes, \(q_{i}\) is unknown generalized coordinates, and \(N\) is the number of modes retained in the discretization. Substituting Equation (10) into Equation (1), multiplying the resulting equation by the mode shape \(\phi_{n}\), integrating over the domain and noting the orthogonality conditions, the discretized equations are obtained as follows:  
\[\ddot{q}_{n}+\omega_{n}^{2}q_{n}+\sum_{i,j,k}^{N}a_{ijkn}\ q_{i}\ \frac{\partial^{2}}{ \partial t^{2}}\big{(}q_{j}q_{k}\big{)}+\sum_{i,j,k}^{N}b_{ijkn}\ q_{i}q_{j}q_{k }\ =\ f_{n}\cos\Omega t,\ n\ =\ 1,\ \ldots,N. \tag{11a}\]  
Equation (11) is written in a compact index form that can be expanded into N differential equations if N modes are used in the discretization. It includes the inertia nonlinearity in the first term and the geometric nonlinearity due to the relatively large deformation in the second term where both are of cubic type.