# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 2 Electro-Aero-Mechanical Model
### Aerodynamics on Wing  
As stated above, we model the aerodynamic force as the sum of the translational force, force due to the added-mass effect, and rotational damping. The translational forces acting on a blade element of the wing (as shown in Figure 3) are given by [9]  
\[\left. \begin{array}{c}\mathrm{d}F_{\mathrm{tr,Drag}}=-\frac{1}{2}\rho_{ \mathrm{Air}}C_{D}(\alpha)c(r)u_{e}^{2}\ \mathrm{sign}(u_{e})\ \mathrm{d}r,\\ \mathrm{d}F_{\mathrm{tr,Lift}}=\frac{1}{2}\rho_{\mathrm{Air}}C_{L}(\alpha)c(r )u_{e}^{2}\ \mathrm{sign}(u_{e})\ \mathrm{d}r,\end{array} \right\} \tag{19}\]  
where \(\mathrm{d}F_{\mathrm{tr,Drag}}\) and \(\mathrm{d}F_{\mathrm{tr,Lift}}\) are the drag force (along z'-axis) and lift force (along y-axis), respectively. \(\rho_{\mathrm{Air}}\) is the density of air. \(C_{D}(\alpha)\) and \(C_{L}(\alpha)\) are coefficients of the translational forces for \(\mathrm{d}F_{\mathrm{tr,Drag}}\) and \(\mathrm{d}F_{\mathrm{tr,Lift}}\), respectively. \(\alpha\) is the angle of attack of the wing (\(\alpha=90^{\circ}-\psi\)). \(c(r)\) is the chord length of the wing. \(u_{e}\) is the velocity at the center of a blade element in the z'-direction, and it is expressed as \(u_{e}=(r+x_{r})\dot{\phi}-\frac{1}{2}c(r)\dot{\psi}\cos\psi\). The force component normal to the wing plane \(\mathrm{d}F_{\mathrm{tr,N}}\) is also given by [9]  
\[\left. \begin{array}{c}\mathrm{d}F_{\mathrm{tr,N}}=-\frac{1}{2}\rho_{ \mathrm{Air}}C_{N}(\alpha)c(r)u_{e}^{2}\ \mathrm{sign}(u_{e})\ \mathrm{d}r,\\ C_{N}(\alpha)=C_{L}(\alpha)\cos\alpha+C_{D}(\alpha)\sin\alpha.\end{array} \right\} \tag{20}\]  
By integrating \(\mathrm{d}F_{\mathrm{tr,Lift}}\) with respect to \(r\), the total translational lift force \(F_{\mathrm{tr,Lift}}\) can be calculated as  
\[ \begin{array}{c}F_{\mathrm{tr,Lift}}=\int_{0}^{R}\mathrm{d}F_{\mathrm{tr, Lift}}\\ =\frac{1}{2}\rho_{\mathrm{Air}}C_{L}(\alpha)A_{w}\bigg{[}I_{w2,1}R^{2}\dot{ \phi}^{2}-I_{w1,2}R\dot{\phi}\psi\cos\psi+\frac{1}{4}I_{w0,3}\bar{c}^{2}\Big{(} \dot{\psi}\cos\psi\Big{)}^{2}\bigg{]}\mathrm{sign}(\overline{u_{e}}),\end{array}  \tag{21}\]  
where \(\overline{c}\) is the mean chord length (\(\overline{c}=A_{w}/R\), \(A_{w}\) is the area of the wing). \(I_{wi,j}\) is a nondimensional geometrical parameter of the wing, and it is defined as  
\[I_{wi,j}=\int_{0}^{1}\hat{c}(\hat{r})^{j}\ (\hat{r}+\hat{x_{r}})^{i}\mathrm{d}\hat{r}, \tag{22}\]  
Figure 3: Coordinates and dimensional parameters of wing. where \(\hat{r}=r/R\), \(\hat{x}_{r}=x_{r}/R\), and \(\hat{c}=c/\overline{c}\). In the integration in Equation (21), we approximated \(\text{sign}(u_{e})\) as being uniform for the entire wing; we replaced \(\text{sign}(u_{e})\) with \(\text{sign}(\overline{u_{e}})\), where \(\overline{u_{e}}\) is the value of \(u_{e}\) at \(r=R/2\). This approximation can become invalid when the wing spins around itself; however, we concluded that it is acceptable because such motion is rare in the intended flapping behavior. The translational moment about the pitching (x'-axis) \(M_{\text{tr},x^{\prime}}\) is also derived by integration with respect to \(r\) as  
\[ \begin{array}{l}M_{\text{tr},x^{\prime}}=\int_{0}^{R}\text{c}(r)d_{\text{cp} }(\alpha)\ \text{d}F_{\text{tr},\text{N}}\\ =-\frac{1}{2}\rho_{\text{Air}}C_{N}(\alpha)A_{w}\overline{c}\ d_{cp}(\alpha) \left[I_{w2,2}R^{2}\dot{\phi}^{2}-I_{w1,3}R\bar{c}\dot{\phi}\dot{\psi}\cos\psi +\frac{1}{4}I_{w0,4}\overline{c}^{2}\left(\dot{\psi}\cos\psi\right)^{2}\right] \text{sign}(\overline{u_{e}}),\end{array}  \tag{23}\]  
where \(d_{cp}\) is the normalized center position of the air pressure. [PERSON] et al. reported that it can be estimated as [8]  
\[d_{\text{cp}}(\alpha)=\frac{0.82}{\pi}|\alpha|+0.05. \tag{24}\]  
The translational moment about the stroke (y-axis) \(M_{\text{tr},y}\) is obtained in a similar way:  
\[