# Analytical Electromechanical Modeling of Nanoscale Flexoelectric Energy Harvesting
## 3 Closed-Form Expressions of Electromechanical Frequency Response Functions  
### Electromechanical Governing Equations in Modal Coordinates  
In the real vibration structures, damping is an important part of the system and also a very complicated problem. Here we take two types of damping mechanisms into account in order to emphasize the flexoelectric effect. These two types of damping are called viscous air (or external) damping and Kelvin-Voigt (or strain-rate) damping [37; 42]. With the damping effects, the electromechanical dynamic Equation (15) becomes  
\[G_{\rm p}\frac{\partial^{4}w}{\partial x_{1}^{4}}-\mu_{3113}B\frac{\partial^{ 2}\psi}{\partial x_{1}^{2}}+c_{\rm s}\frac{\partial^{5}w}{\partial x_{1}^{4} \partial t}+c_{\rm a}\frac{\partial w}{\partial t}+[m+M_{\rm t}\delta(x_{1}-L )]\frac{\partial^{2}w_{\rm b}}{\partial t^{2}}+m\frac{\partial^{2}w}{\partial t ^{2}}=0, \tag{19}\]  
where \(c_{\rm a}\) is the viscous air damping coefficient and \(c_{\rm s}\) is the strain-rate damping coefficient [42]. Viscous air damping is a simple way of modeling the force acting on the beam due to the air particles displaced during vibration, while strain-rate damping accounts for the structural damping due to energy dissipation of the beam material. Both of them satisfy the proportional damping criterion and they are mathematically convenient for the modal analysis [42]. Based on the proportional damping assumption, the vibration response of the flexoelectric cantilever beam can be represented as an absolutely and uniformly convergent series of eigenfunctions as  
\[w(x_{1},t)=\sum_{r=1}^{\infty}\varnothing_{r}(x_{1})\eta_{r}(t), \tag{20}\]  
where \(\varnothing_{r}(x)\) is the mass normalized eigenfunction of the \(r\)th vibrational mode, which satisfies the mechanical and the short circuit electrical conditions, and \(\eta_{r}(t)\) is the modal mechanical response coefficient in the modal coordinates. For the piezoelectric or flexoelectric Euler-Bernoulli cantilever beams, the eigenfunctions \(\varnothing_{r}(x)\) can be obtained from the corresponding undamped free vibration problem [41]  
\[\varnothing_{r}(x_{1})=C_{r}\left[\cos\frac{\beta_{r}}{L}x_{1}-\cos\hbar\frac {\beta_{r}}{L}x_{1}+\zeta_{r}\left(\sin\frac{\beta_{r}}{L}x_{1}-\sin\hbar \frac{\beta_{r}}{L}x_{1}\right)\right], \tag{21}\]  
where \(\zeta_{r}\) is  
\[\zeta_{r}=\frac{\sin\beta_{r}-\sin\hbar\beta_{r}+\beta_{r}\frac{M_{\rm t}}{mL} (\cos\beta_{r}-\sin\hbar\beta_{r})}{\cos\beta_{r}+\cos\hbar\beta_{r}-\beta_{ r}\frac{M_{\rm t}}{mL}(\sin\beta_{r}-\sin\hbar\beta_{r})}, \tag{22}\]  
and \(\beta_{r}\) is the \(r\)th root of the transcendental characteristic equation  
\[1+\cos\beta_{r}\cos\hbar\beta_{r}+\beta_{r}\frac{M_{\rm t}}{mL}(\cos\beta_{r} \sin\hbar\beta_{r}-\sin\beta_{r}\cos\hbar\beta_{r})=0. \tag{23}\]In Equation (21), the constant \(C_{r}\) is the modal amplitude constant, which could be solved by normalizing the eigenfunctions using the following orthogonality condition [33]  
\[\int_{0}^{L}\varnothing_{r}(x_{1})m\varnothing_{s}(x_{1})\mathrm{d}x_{1}+ \varnothing_{r}(L)M_{\mathrm{t}}\varnothing_{s}(L)=\delta_{rs}, \tag{24}\]  
where \(\delta_{rs}\) is Kronecker delta, defined as being equal to unity for \(s=r\) and equal to zero for \(s\
eq r\). It should be noted that the undamped natural frequency of the \(r\)th vibration mode in the short circuit condition is  
\[\omega_{r}=\beta_{r}^{2}\sqrt{\frac{G_{\mathrm{P}}}{mL^{4}}}. \tag{25}\]  
It is important to mention that the electric potential difference between the two surface electrodes due to the flexoelectric effect is a function of time \(t\) and as a result, the spatial derivative of \(\psi\) would vanish in Equation (19). To include the flexoelectric effect in the governing Equation (19), [PERSON] and [PERSON] [41] assumed that \(\psi=\psi(x_{1},t)=V(t)[H(x_{1})-H(x_{1}-L)]\), where \(V(t)\) is the output voltage and \(H(x)\) is the Heaviside function.