# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 2 Electromechanical System and Mathematical Formation
\begin{split}&\int_{0}^{T}\mathrm{d}t\bigg{\{}Bh\int_{0}^{L} \rho\bigg{(}\ddot{w}+\ddot{w}_{\mathrm{b}}\bigg{)}\delta w\mathrm{d}x_{1}+M_{ 4}\ddot{w}_{\mathrm{b}}\delta(x_{1}-L)\delta w\bigg{\}}+\int_{0}^{T}\mathrm{d} t\int_{0}^{L}\bigg{\{}\bigg{[}G_{\mathrm{P}}\frac{\partial^{2}w}{ \partial x_{1}^{2}}-\\ &\mu_{3113}B\frac{\partial^{2}\psi}{\partial x_{1}^{2}}\bigg{]} \delta w-\bigg{[}\mu_{3113}B\frac{\partial^{2}w}{\partial x_{1}^{2}}+\kappa_{ 33}B\frac{\psi}{h}\bigg{]}\delta\psi\bigg{\}}\mathrm{d}x_{1}\mathrm{d}t+\int_{ 0}^{T}\bigg{\{}G_{\mathrm{P}}\frac{\partial^{2}w}{\partial x_{1}^{2}}-\\ &\mu_{3113}B\psi\big{\}}\delta\big{(}\frac{\partial w}{\partial x _{1}}\bigg{)}\bigg{|}x_{1}=L\mathrm{d}t-\int_{0}^{T}\bigg{\{}G_{\mathrm{P}} \frac{\partial^{2}w}{\partial x_{1}^{2}}-M_{4}\frac{\partial^{2}w}{\partial t ^{2}}\bigg{\}}\delta w\bigg{|}x_{1}=L\mathrm{d}t-\int_{0}^{T}\mathrm{d}t\int_ {0}^{L}B\alpha\delta\psi\mathrm{d}x_{1}\mathrm{d}t=\\ & 0,\end{split}  \tag{13}\]  
where \(\delta(x_{1})\) is the Dirac delta function. In deducing Equation (13), the following result has been used  
\[\delta\int_{0}^{T}\frac{1}{2}M_{4}\big{|}\dot{w}^{m}\big{|}^{2}\mathrm{d}t \bigg{|}x_{1}=-\int_{0}^{T}M_{4}\ddot{w}_{b}\delta(x_{1}-L)\delta w\mathrm{d}t -\int_{0}^{T}M_{4}\ddot{w}\bigg{|}x_{1}=L\delta w\mathrm{d}t. \tag{14}\]  
Due to the arbitrariness of \(\delta w\) in Equation (13), the electromechanically-coupled dynamic governing equation of the flexoelectric beam can be obtained as  
\[G_{\mathrm{P}}\frac{\partial^{4}w}{\partial x_{1}^{4}}-\mu_{3113}B\frac{ \partial^{2}\psi}{\partial x_{1}^{2}}+[m+M_{4}\delta(x_{1}-L)]\frac{\partial^ {2}w_{b}}{\partial t^{2}}+m\frac{\partial^{2}w}{\partial t^{2}}=0, \tag{15}\]  
where \(m=\rho Bh\) is the mass of per unit length of beam. Similarly, due to the arbitrariness of \(\delta\psi\) in Equation (13), we obtain  
\[\int_{0}^{L}(\mu_{3113}\frac{\mathrm{d}^{2}w}{\mathrm{d}x_{1}^{2}}+\kappa_{33} \frac{\psi}{h}+\alpha)\mathrm{d}x_{1}=0. \tag{16}\]  
When the external loading resistor \(R\) is connected into the surface electrodes, the electric current \(\frac{\psi}{R}\) must be equal to the time rate of change of the average output positive charges, i.e., \(-\frac{1}{h}\int_{V}\dot{\omega}\mathrm{d}V\). Using Equation (16), from [PERSON]'s law [37; 41; 42], the electromechanically-coupled electrical circuit equation with flexoelectric effect can be written as \[\kappa_{33}BL\frac{\dot{\psi}}{h}+\frac{\psi}{R}=-\mu_{3113}B\left.\frac{\partial^{2 }w}{\partial x_{1}\partial t}\right|_{x_{1}=L}, \tag{17}\]  
where \(\psi\) is assumed to be independent of \(x_{1}\). In addition, the boundary conditions are obtained from Equation (13) as  
\[\left\{ \begin{array}{c}G_{\rm p}\left.\frac{\partial^{2}w}{\partial x_{1}^{2} }\right|_{x_{1}=L}=\mu_{3113}B\psi\\ G_{\rm p}\left.\frac{\partial^{2}w}{\partial x_{1}^{3}}\right|_{x_{1}=L}=M_{ \rm t}\left.\frac{\partial^{2}w}{\partial t^{2}}\right|_{x_{1}=L}.\end{array} \right.. \tag{18}\]  
Apparently, the flexoelectric effect induces an effective bending moment at the end of the beam, whereas the inertia of the tip mass induces a shear force at the end.