## 2 Mathematical Formulation
\begin{array}{l}\int_{t_{1}}^{t_{2}}\int_{0}^{L}\left\{\begin{array}{l} \Big{(}N_{xx,x}-l_{0}\bar{u}-\frac{\partial p}{\partial x}\Big{)}\delta u- \frac{B_{xx}}{h_{p}}\delta V\\ \Bigg{[}-l_{0}\bar{w}-l_{0}\bar{w}_{b}-\frac{m_{s}}{L}\Big{(}x-\frac{l}{2} \Big{)}\Big{(}\ddot{w}+\ddot{w}_{b}\Big{)}\\ +\Big{[}\frac{\partial}{\partial x}\Big{(}N_{xx}\frac{\partial w}{\partial x} \Big{)}+M_{xx,xx}+G_{xx,xx}\Big{]}-\frac{\partial}{\partial x}\Big{(}P\frac{ \partial w}{\partial x}\Big{)}-c_{v}\dot{w}\end{array}\right\}dxdt\\ +2\int_{t_{1}}^{t_{2}}Q(t)\delta V(t)dt=0\end{array} \right\}dxdt\end{array} \tag{17}\]  
where \((x-x_{0})\) is defined as:  
\[ \begin{array}{l}\left\{\begin{array}{l}\Big{(}x-x_{0})=\left\{\begin{array} []{l}0,x\
eq x_{0}\\ \infty,x=x_{0}\end{array}\\ \int_{0}^{L}\Big{(}x-x_{0}\Big{)}dx=1\end{array}\right.\end{array}  \tag{18}\]  
The two piezoelectric layers are connected in series in the circuit. Therefore, the current in the circuit is calculated as \(i(t)=\frac{2V}{K}=\dot{Q}(t)\). Then, the nonlinear electromechanical coupling equations can be obtained as:  
\[N_{xx,x}-P_{,x}=I_{0}\ddot{u} \tag{19}\]  
\[ \begin{array}{l}G_{xx,xx}-\Big{(}P\frac{\partial w}{\partial x}\Big{)}_{,x}+ \frac{\partial}{\partial x}\Big{(}N_{xx}\frac{\partial w}{\partial x}\Big{)}+ M_{xx,xx}\\ =\Big{[}I_{0}+\frac{m_{s}}{L}\Big{(}x-\frac{l}{2}\Big{)}\Big{]}\Big{(}\ddot{w}+ \ddot{w}_{b}\Big{)}+c_{v}\dot{w}\end{array}  \tag{20}\]  
\[\frac{4V}{R}-\int_{0}^{L}\frac{\dot{B}_{xx}}{h_{p}}dx=0 \tag{21}\]  
By substituting Equation (13), into Equations (19)-(21), the nonlinear electromechanical coupling equations are rewritten as:  
\[bA_{11}\Big{(}1-l^{2}\
abla^{2}\Big{)}\frac{\partial}{\partial x}\Bigg{[} \frac{\partial u}{\partial x}+\frac{1}{2}\bigg{(}\frac{\partial w}{\partial x }\bigg{)}^{2}\Bigg{]}=I_{0}\Big{[}1-(e_{0}a)^{2}\
abla^{2}\Big{]}\ddot{u} \tag{22}\]\[-P\Big{[}1-\left(e_{0}a\right)^{2}\
abla^{2}\Big{]}\frac{\partial^{2}w}{ \partial x^{2}}-bB_{11}\big{(}1-l^{2}\
abla^{2}\big{)}\frac{\partial^{4}w}{ \partial x^{4}}\] \[+\frac{\partial}{\partial x}\bigg{\{}\bigg{(}bA_{11}\big{(}1-l^{2} \
abla^{2}\big{)}\bigg{[}\frac{\partial u}{\partial x}+\frac{1}{2}\Big{(} \frac{\partial w}{\partial x}\Big{)}^{2}\bigg{]}+2 be_{311}V\bigg{)}\frac{ \partial w}{\partial x}\bigg{\}} \tag{23}\] \[=\Big{[}I_{0}+\frac{m_{\rm s}}{L}\Big{(}x-\frac{k}{2}\Big{)}\Big{]} \Big{[}1-(e_{0}a)^{2}\
abla^{2}\Big{]}\big{(}\ddot{w}+\ddot{w}_{b}\big{)}+c_{ v}\Big{[}1-(e_{0}a)^{2}\
abla^{2}\Big{]}\dot{w}\] \[\frac{bLa_{33}}{h_{p}}\dot{V}+\frac{2V}{R}-\int_{0}^{L}\bigg{\{} b\Big{[}e_{311}\bigg{(}\frac{\partial\dot{u}}{\partial x}+\frac{\partial w}{ \partial x}\frac{\partial\dot{w}}{\partial x}\bigg{)}-\mu_{3113}\frac{ \partial^{2}w}{\partial x^{2}}\bigg{]}\bigg{\}}dx=0 \tag{24}\]  
For the simply supported boundary, the lateral displacements and the moments at both ends of the beam are 0, i.e.,  
\[w(0)=w(L)=\frac{\partial^{2}w(0)}{\partial x^{2}}=\frac{\partial^{2}w(L)}{ \partial x^{2}}=0 \tag{25}\]  
In the axial direction, the compressive load \(P\) equals to the inner force \(N_{xx}\) at both ends, i.e. :  
\[\Bigg{\{}bA_{11}\Big{(}1-l^{2}\
abla^{2}\Big{)}\Bigg{[}\frac{\partial u}{ \partial x}+\frac{1}{2}\bigg{(}\frac{\partial w}{\partial x}\bigg{)}^{2} \Bigg{]}+2 be_{311}V-P\Bigg{\}}\delta u=0 \tag{26}\]  
at \(x=0\) and \(x=L\). Compared with the lateral vibration, the axial vibration of a slender beam is less important with can be neglected, i.e., \(\ddot{u}=0\).