## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method  
The kinematic simulation of the wing using the multi-plane method enables the modification of wing by adjusting parameters, which is closer to the realistic flapping process than the blade element theory assuming a single rigid plane. After obtaining the wing's deformation during the flapping trajectory, aerodynamic analysis of the wing can be performed by combining it with the quasi-steady blade element theory. In aerodynamic analysis, the conventional blade element theory typically divides a single rigid-plane wing into multiple blade elements and calculates the aerodynamic forces for each blade element, which are then summed. Figure 13a illustrates the schematic diagram of the classical blade element theory for calculating aerodynamic forces on a single rigid plane wing. In the diagram, \(dr\) represents the blade width, \(r\) denotes the distance of the current blade element from the flapping axis, \(R\) represents the spanwise length of the wing, \(c(r)\) represents the chord length of the blade element, \(\hat{x}_{0}\) represents the intersection point between the rotation axis and the flapping axis, and \(x_{mid}\) represents the geometric center of the wing. Neglecting the wing's vertical flapping motion, the flapping plane of the wing coincides with the \(XY\) plane of the wing coordinate system and the body coordinate system with an angle \(\alpha\), which can be called the angle of attack between the wing surface and the flapping plane, as shown in Figure 13b. Figure 12: Shape changes during wing flapping, based on simulation values observed in the positive x-axis direction of the body coordinate system, compared with high-speed photography. Figure 13: Parameters used in blade element analysis: (**a**) definition of blades and (**b**) direction of force. Based on investigation of the quasi-steady aerodynamic model for hovering flapping-wing aircraft, the aerodynamic forces can be categorized into translational aerodynamic forces, rotational aerodynamic forces, aerodynamic forces induced by added mass, inertial forces, and forces resulting from special aerodynamic effects [1; 2; 3]. Among these, the aerodynamic forces caused by clap and fling contribute the most to the forces resulting from special effects. While these have a significant impact on hovering flight with flapping amplitudes greater than 180\({}^{\circ}\)[5; 6], they have a negligible effect on flapping-wing aircraft where the wings do not make contact during the flipping motion, and can be disregarded. Similarly, the effects of delayed stall [7; 8; 9; 10] and wake capture [11; 12; 13] are not often considered in quasi-steady aerodynamic models. The inertial forces are related to the wing mass and flapping angular velocity; however, due to the front-back symmetry of flapping, they mutually cancel out within one cycle and can be omitted from the analysis. Therefore, the translational aerodynamic force and the rotational aerodynamic force are the forces that have the greatest impact on the hovering flapping wing. By combining the defined parameters and previous research, the translational aerodynamic force \(F_{tr}^{{}^{\prime}}\) of a single blade element can be expressed as  
\[F_{tr}^{{}^{\prime}}=F_{Ltr}^{{}^{\prime}}+F_{Dtr}^{{}^{\prime}}, \tag{20}\]  
where \(F_{Ltr}^{{}^{\prime}}\) represents the translational lift force and \(F_{Dtr}^{{}^{\prime}}\) represents the translational drag force, provided by  
\[F_{Ltr}^{{}^{\prime}}=\frac{1}{2}\rho C_{L}(\alpha)c(r)(\dot{\psi}r)^{2}dr, \tag{21}\]  
\[F_{Dtr}^{{}^{\prime}}=\frac{1}{2}\rho C_{D}(\alpha)c(r)(\dot{\psi}r)^{2}dr, \tag{22}\]  
where \(\rho\) represents the air density, \(\dot{\psi}\) is defined as the flapping wing's angular velocity, \(\dot{\psi}r\) is the translational velocity of the blade section at a distance \(r\) from the flapping axis, and \(C_{L}(\alpha)\) and \(C_{D}(\alpha)\) are the lift and drag coefficients, respectively, obtained from experimental data.