## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method
Based on previous research [36; 37; 38], for flapping-wing aircraft at the decimeter scale, the following equations can be used for approximate fitting:  
\[C_{L}(\alpha)=0.255+1.58 sin(2.13\alpha-7.2), \tag{23}\]  
\[C_{D}(\alpha)=1.92-1.55 sin(2.04\alpha-9.28). \tag{24}\]  
In some studies, a small difference between the geometric angle of attack and the actual angle of attack has been considered, referred to as the induced angle \(\beta\), which has a small value. In this paper, consistent with most of the literature, \(\beta\) is assumed to be 0, implying that the geometric angle of attack is equal to the actual angle of attack. The rotational aerodynamic force of a single blade can be expressed as  
\[F_{rot}^{{}^{\prime}}=\rho C_{rot}\kappa(\dot{\psi}r)c^{2}(r)dr, \tag{25}\]  
with the value of the rotational aerodynamic coefficient \(C_{rot}\) being dependent on the position of the axis of rotation, which can be expressed as follows:  
\[C_{rot}=\pi(0.75-x_{0}) \tag{26}\]  
where the parameter \(x_{0}\) represents the dimensionless distance of the axis of rotation, which is dependent on the wing morphology and the position of the axis of rotation. In nature, birds and insects typically have values of \(x_{0}\) ranging from 0.25 to 0.35; however, in the majority of existing flapping-wing prototypes, where the leading edge serves as the axis of rotation, the dimensionless position of the axis of rotation \(x_{0}\) can be set to 0. Thus, the aerodynamic forces resulting from the rotation of a single blade can be expressed as follows:  
\[F_{rot}^{{}^{\prime}}=\frac{3}{4}\pi\rho\dot{\alpha}(\dot{\psi}r)c^{2}(r)dr. \tag{27}\]Similar to the direction of the force indicated in Figure 13b, the direction of the rotational aerodynamic force is perpendicular to the wing surface, and can be decomposed into the rotational lift \(F^{{}^{\prime}}_{Lrot}\) and rotational drag \(F^{{}^{\prime}}_{Drot}\), which can be expressed as follows:  
Figure 13:0  
Figure 13:1  
The source of the added mass aerodynamic force is the reaction force exerted by the surrounding air on the wing during the flapping motion. The added mass aerodynamic force of a single blade can be expressed as follows:  
Figure 13:2  
where \(S_{air}dr\) represents the equivalent volume of air on the blade and \(S_{air}\) represents the equivalent surface area of the modified cylinder. These can be expressed as  
Figure 13:3  
where \(a_{air}\) is the acceleration of the considered part of the air, which can in turn be expressed as  
Figure 13:4  
The aerodynamic forces caused by added mass are similar to the aerodynamic forces resulting from rotation, as they both act perpendicularly to the wing surface. Similarly, they can be decomposed into the lift \(F^{{}^{\prime}}_{Ladd}\) and drag \(F^{{}^{\prime}}_{Dadd}\), which can be expressed as follows:  
Figure 13:5  
Figure 13:6  
By integrating the translational aerodynamic force, rotational aerodynamic force, and added mass-induced aerodynamic force on a single blade, the quasi-steady model under the assumption of a rigid flat plane can be obtained. This model represents the transient aerodynamic forces during wing flapping. The transient lift \(F_{L}\) and transient drag \(F_{D}\) can be expressed as follows:  
Figure 13:7  
Figure 13:8  
From the parameters in the above equations, it can be observed that under the assumption of a rigid flat plane in the quasi-steady model, the aerodynamic forces of wing flapping can be estimated when the morphological parameters of the wing such as the chord length \(c(r)\), angle of attack \(\alpha\), and flapping angular velocity \(\psi\) are known. Under the assumption of a rigid flat plane, the angular velocities, angles of attack, and their derivatives for each blade on the wing are assumed to be the same; however, during the actual flapping process of a wing, the wing's flexibility and nonlinear characteristics may cause variations in wing's deformation and aerodynamic force distribution.