## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method
When calculating aerodynamic forces using the quasi-steady aerodynamic model, the derivative of the angle of attack is essential; however, when computing the angle of attack and its derivative, discontinuities in the derivative may occur at inflection points due to the linear fitting method used to define points on the plane, as shown in \[F_{tr}^{{}^{\prime}}=F_{Ltr}^{{}^{\prime}}+F_{Dtr}^{{}^{\prime}}, \tag{20}\]3 This can lead to singular values in the aerodynamic force calculations. To address this issue, it is necessary to apply a smoothing technique to the fitted value function in order to ensure that the derivative of the angle of attack does not exhibit extreme values. The optimized values of \(y_{2-w}\) compared with linear fitting and changed based on the flapping angle are shown in Figure 16, which illustrates the cyclic behavior within one flapping period. Inspired by the sinusoidal modulation of the flapping trajectory in wingbeat modulation techniques [53], we employed trigonometric functions in the simulation process instead of the original piecewise linear fitting method to ensure the continuity of the derivative and second derivative at the inflection points. Through trigonometric function modulation, the original fitting Equation (17) and fitting Equation (18) are transformed into the following. \[\left\{ \begin{aligned} y_{2-w}&=\frac{y_{t}+y_{c}}{2}- \frac{y_{t}-y_{c}}{2}\text{sin}\left(\pi\frac{\psi-\frac{\psi_{t}-y_{max}}{ \psi_{c}+\psi_{max}}}{\psi_{c}+\psi_{max}}\right),-\psi_{max}\leq\psi\leq\psi_ {c}\\ y_{2-w}&=\frac{-y_{t}+y_{c}}{2}+\frac{-y_{t}-y_{c}}{2} \text{sin}\left(\pi\frac{\psi-\frac{\psi_{t}+y_{max}}{\psi_{max}-\psi_{c}}}{ \psi_{max}-\psi_{c}}\right),\psi_{c}\leq\psi\leq\psi_{max}\end{aligned} \right. \tag{39}\]  
After applying trigonometric function smoothing to the \(y_{2-w}\) trajectory, extreme values in the derivative and second derivative no longer occur. By utilizing the flapping trajectory simulation method described earlier in conjunction with quasi-steady aerodynamic analysis using the multi-plane approach, it is possible to simulate the transient aerodynamic forces within one flapping period. As shown in Figure 17, transient aerodynamic analysis was conducted for a single wing using the simulation parameters provided in Table 1.  
Figure 16: The value of \(y_{2-w}\) after smoothing throughout a complete cycle. Figure 15: Schematic diagram of the air that plays a role in the added mass effect. As shown in Figure 17, the transient aerodynamic simulation results for one wing within a single period were analyzed using the multi-plane approach and quasi-steady aerodynamic model. The results include the lift and drag components of the aerodynamic forces in Figure 17a,c as well as the contributions of each plane to the lift and drag in Figure 17b,d. Based on the data in the figures, the following conclusions can be drawn. Regarding the aerodynamic lift, the translation component provides the main contribution while the contribution of the rotation component is relatively small, as shown in Figure 17a. At the wing reversal point, which is the half-period position, the rotation component contributes partially to the lift. Different planes exhibit varying contributions to the lift, with plane1 having the largest contribution and plane2 and plane3 having smaller contributions, as shown in Figure 17b. The results are similar to those of the previous section, except for the \(\theta_{rad1}\) and \(\theta_{rad2}\) cases.