## 2 Materials and Methods
### Estimation of Force and Power Consumption
Once the morphological and kinematic parameters (\(R_{2}\), \(\overline{c}\), \(\varphi_{m}\), \(f\), and \(\dot{\varphi}\)), the density of air (\(\rho_{air}\)), and its kinematic viscosity (\(\mu\)) are determined, then, for the spanwise blade element with a radial radius of \(r\), these equations will be reduced to a set of non-linear equations with two unknowns: \(V_{i}\) and \(\alpha\). The corresponding numerical solutions can be obtained by using the Newton-Raphson method. After that, the translational force generated by a wing section can be determined according to  
Figure 3:0Figure 3:1  
where \(\mathrm{d}F_{T\eta}\) and \(\mathrm{d}F_{T\zeta}\) are the \(\eta\)- and \(\zeta\)-axis components of the translational force produced by the blade element with a width of \(\mathrm{d}r\), and'sign[\(x\)]' is a function of \(x\), defined as  
\[\mathrm{sign}[x]=\left\{ \begin{array}{cc}1&(x>0)\\ -1&(x<0)\end{array} \right.. \tag{16}\]  
The added mass forces stem from a change in the flapping and pitching speeds. In this model, only the air within an elliptical cylinder that is determined by the chord length \(c(r)\) as the major diameter, the projected chord length in the motion direction \(c(r)\sin\theta\) as the minor diameter, and the thickness \(\mathrm{d}r\) as the height are called the 'added mass'. The mass element \(\mathrm{d}m_{f}\) can be calculated as  
\[\mathrm{d}m_{f}=\rho_{air}\frac{\pi}{4}c(r)^{2}\sin\theta\mathrm{d}r. \tag{17}\]  
A point at the half chord \(c(r)/2\) is selected as a reference point, and it is considered to be the point of action of the resultant force of the added-mass forces. The normal acceleration \(a_{n}\) of this point is expressed as  
\[a_{n}=r\ddot{\varphi}\sin\theta_{r}+\frac{c(r)}{2}(\dot{\varphi}^{2}\cos \theta_{r}\sin\theta_{r}+\ddot{\theta}_{r}), \tag{18}\]  
where \(\ddot{\varphi}\) and \(\ddot{\theta}_{r}\) are the angular accelerations of flapping and rotation motions, respectively. Therefore, the added-mass force perpendicular to the wing can be decomposed into \(\eta\)- and \(\zeta\)-axis components as  
\[\mathrm{d}F_{A\eta}(r,t)=-\mathrm{d}m_{f}a_{n}\sin(\pi-\theta_{r})=-\frac{\pi }{4}\rho_{air}c(r)^{2}a_{n}\sin\theta\sin\theta_{r}\mathrm{d}r \tag{19}\]  
\[\text{and}\;\mathrm{d}F_{A\zeta}(r,t)=-\mathrm{d}m_{f}a_{n}\cos(\pi-\theta_{r })=\frac{\pi}{4}\rho_{air}c(r)^{2}a_{n}\sin\theta\cos\theta_{r}\mathrm{d}r. \tag{20}\]  
The rotational force is related to the pitching motion. The expression for the rotational force acting on a radial cross-section is  
\[\mathrm{d}F_{\mathrm{R}}=\rho V_{T}\mathrm{d}\Gamma_{\mathrm{R}}, \tag{21}\]  
where \(\mathrm{d}\Gamma_{\mathrm{R}}\) is the circulation induced by the rotation motion and is determined by  
\[\mathrm{d}\Gamma_{\mathrm{R}}=C_{\mathrm{Rot}}\dot{\theta}_{r}c(r)^{2} \mathrm{d}r. \tag{22}\]  
In Equation (22), \(C_{\mathrm{Rot}}\) is the rotational force coefficient and can be expressed as  
\[C_{\mathrm{Rot}}=\pi[\frac{0.75-x_{f}}{c(r)}], \tag{23}\]  
where \(x_{f}\) is the distance from the pitching axis to the leading edge. The straight leading edge is the pitching axis in this work, and \(x_{f}\) is thus zero. Eventually, the rotational force can be decomposed into \(\eta\)- and \(\zeta\)-axis components as  
\[\mathrm{d}F_{\mathrm{R}\eta}(r,t)=\mathrm{d}F_{\mathrm{R}}\cos(\frac{\pi}{2}+ \theta_{r})=-\rho_{air}V_{T}C_{\mathrm{Rot}}\dot{\theta}_{r}c(r)^{2}\sin \theta_{r}\mathrm{d}r \tag{24}\]  
\[\text{and}\;\mathrm{d}F_{R\zeta}(r,t)=\mathrm{d}F_{\mathrm{R}}\cos\theta_{r}= \rho_{air}V_{T}C_{\mathrm{Rot}}\dot{\theta}_{r}c(r)^{2}\cos\theta_{r}\mathrm{d}r. \tag{25}\]The inertial force generated by a wing section depends on the chord-wise distributions of the mass and angular acceleration.