# Electro-Aero-Mechanical Model of Piezoelectric Direct-Driven Flapping-Wing Actuator
## 2 Electro-Aero-Mechanical Model
### Aerodynamics on Wing
\begin{array}{l}M_{\text{tr},y}=-\int_{0}^{R}(r+x_{r})\ \text{d}F_{\text{tr},\text{ Drag}}\text{sign}(\overline{u_{e}})\\ =-\frac{1}{2}\rho_{\text{Air}}C_{D}(\alpha)A_{w}R\left[I_{w3,1}R^{2}\dot{\phi} ^{2}-I_{w2,2}R\bar{c}\dot{\phi}\psi\cos\psi+\frac{1}{4}I_{w1,3}\overline{c}^{ 2}\left(\dot{\psi}\cos\psi\right)^{2}\right]\text{sign}(\overline{u_{e}}). \end{array}  \tag{25}\]  
The translational coefficients are calculated using the equations below [4; 27]:  
\[\left. \begin{array}{l}C_{L}(\alpha)=\frac{1}{2}\pi\frac{R}{\overline{c}} \left(1+\sqrt{\left(\frac{1}{2}\frac{R}{\overline{c}}\right)^{2}+1}\right)^{- 1}\sin 2\alpha,\\ C_{D}(\alpha)=\frac{C_{D,\text{max}}+C_{D,\text{min}}}{2}-\frac{C_{D,\text{ max}}-C_{D,\text{min}}}{2}\cos 2\alpha,\end{array} \right\} \tag{26}\]  
where we used \(C_{D,\text{max}}=5.0\) and \(C_{D,\text{min}}=0.4\) based on [4]. The force \(\text{d}F_{\text{am}}\) due to the added-mass effect on a blade element is expressed as [5; 6; 7; 8; 9]  
\[\text{d}F_{\text{am}}=-\pi\rho_{\text{Air}}\left(\frac{c(r)}{2}\right)^{2}\dot {u_{n}}\ \text{d}r, \tag{27}\]  
where \(u_{n}\) is the normal component of the wing velocity: \(u_{n}=(r+x_{r})\dot{\phi}\cos\psi-\frac{1}{2}c(r)\dot{\psi}\). \(\dot{u_{n}}\) is obtained by a time-derivative of \(u_{n}\) as \(\dot{u_{n}}=(r+x_{r})\left(\ddot{\phi}\cos\psi-\dot{\phi}\dot{\psi}\sin\psi \right)-\frac{1}{2}c(r)\ddot{\psi}\). The total force \(F_{\text{am}}\) is obtained by integration as  
\[ \begin{array}{l}F_{\text{am}}=\int_{0}^{R}\text{d}F_{\text{am}}\\ =-\frac{\pi}{4}\rho_{\text{Air}}\overline{c}^{2}R^{2}I_{w1,2}\left(\ddot{\phi }\cos\psi-\dot{\phi}\dot{\psi}\sin\psi\right)+\frac{\pi}{8}\rho_{\text{Air}} \overline{c}^{3}R\ I_{w0,3}\ddot{\psi}.\end{array}  \tag{28}\]  
\(F_{\text{am}}\) acts along the z''-axis (normal direction of the wing). The added-mass moment about the pitching x'-axis \(M_{\text{am},x^{\prime}}\) and y''-axis \(M_{\text{am},y^{\prime\prime}}\) are respectively given by  
\[ \begin{array}{l}M_{\text{am},x^{\prime}}=-\int_{0}^{R}\frac{1}{2}\text{c}(r) \ \text{d}F_{\text{am}}\\ =\frac{\pi}{8}\rho_{\text{Air}}\overline{c}^{3}R^{2}I_{w1,3}\left(\ddot{\phi} \cos\psi-\dot{\phi}\dot{\psi}\sin\psi\right)-\frac{\pi}{16}\rho_{\text{Air}} \overline{c}^{4}R\ I_{w0,4}\ddot{\psi}\end{array}  \tag{29}\]  
and  
\[ \begin{array}{l}M_{\text{am},y^{\prime\prime}}=\int_{0}^{R}(r+x_{r})\ \text{d}F_{\text{am}}\\ =-\frac{\pi}{4}\rho_{\text{Air}}\overline{c}^{2}R^{3}I_{w2,2}\left(\ddot{\phi} \cos\psi-\dot{\phi}\dot{\psi}\sin\psi\right)+\frac{\pi}{8}\rho_{\text{Air}} \overline{c}^{3}R^{2}\ I_{w1,3}\ddot{\psi}.\end{array}  \tag{30}\]The moment due to the rotational damping on a blade element is calculated as [4]  
\[\mathrm{d}M_{\mathrm{rot}}=-\frac{1}{2}\rho_{\mathrm{Air}}C_{rd}\dot{\psi}^{2} \int_{-c(r)}^{0}\left|y^{3}\right|\mathrm{d}y\ \mathrm{sign}\ \dot{\psi}, \tag{31}\]  
where \(C_{rd}\) is a constant coefficient. Different \(C_{rd}\) values of 2.0-5.0 were used, as reported in several previous studies as well [4; 28; 29]. In this study, we used \(C_{rd}=3.5\). Upon integrating \(\mathrm{d}M_{\mathrm{rot}}\), the total moment \(M_{\mathrm{rot}}\) can be expressed as  
\[M_{\mathrm{rot}}=-\frac{1}{2}\rho_{\mathrm{Air}}C_{rd}R\ I_{rd}\ \dot{\psi}^{2} \ \mathrm{sign}\ \dot{\psi}, \tag{32}\]  
where \(I_{rd}\) is a constant that depends on the wing geometry as follows:  
\[I_{rd}=\int_{0}^{1}\int_{-c(r)}^{0}\left|y^{3}\right|\mathrm{d}y\mathrm{d}\dot {r}. \tag{33}\]  
\(\tau_{\phi}\) and \(\tau_{\psi}\) are respectively given by  
\[\left. \begin{array}{c}\tau_{\phi}=M_{\mathrm{tr},y}+M_{\mathrm{am},y^{ \prime\prime}}\cos\psi,\\ \tau_{\psi}=M_{\mathrm{tr},x^{\prime}}+M_{\mathrm{am},x^{\prime}}+M_{\mathrm{ rot}}.\end{array} \right\} \tag{34}\]  
The total lift force \(F_{\mathrm{Lift}}\) is given by  
\[F_{\mathrm{Lift}}=F_{\mathrm{tr},\mathrm{Lift}}+F_{\mathrm{am}}\sin\psi. \tag{35}\]