## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Data** \\ \hline Length of wing \(L_{w}\) & 65 mm \\ Height of wing \(H_{w}\) & 26 mm \\ Installation angle of rod1 \(\theta_{rad1}\) & 23\({}^{\circ}\) \\ Installation angle of rod1 \(\theta_{rad2}\) & 50\({}^{\circ}\) \\ Flapping amplitude & 160\({}^{\circ}\) \\ Flapping frequency & 35 Hz \\ \(y_{t}\) & 15 mm \\ \(y_{c}\) & \(-22\) mm \\ \(\psi_{max}\) & 80\({}^{\circ}\) \\ \(\psi_{c}\) & \(-50^{\circ}\) \\ Flapping frequency & 35 Hz \\ Reynolds number & \(2.2321\times 10^{4}\) \\ Strouhal number & 0.0716 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Values of parameters  
Figure 17: Transient values within a normalized flapping cycle: (**a**) translational aerodynamic lift and rotational aerodynamic lift; (**b**) aerodynamic lift of three planes; (**c**) translational aerodynamic drag and rotational aerodynamic drag; (**d**) aerodynamic drag of three planes. shown in Figure 17b. Therefore, the translational force of plane1 plays a crucial role in lift generation. In terms of aerodynamic drag, the translation component again dominates. As shown in Figure 17c, during wing reversal, the rotation component provides drag in the opposite direction to the motion due to the wing's deformation, thereby reducing the overall drag. It can be seen in Figure 17d that in terms of the drag distribution plane1 and plane2 account for the majority, while the contribution of plane3 is minimal. This aligns with our initial expectation that plane3 would be closer to the axis of rotation and have a lower translational velocity, resulting in negligible aerodynamic effects. This phenomenon is evident when observing the wing shapes of most insects in nature, where the wings typically narrow towards the wing root. This design reduces inertial forces while minimizing the impact of aerodynamic effects due to a missing area of wing. Figure 18 presents a comparison between the simulated and experimentally measured lift at different frequencies of mean lift, expressed in units of gf and lift coefficient. The lift coefficient can be computed as follows:  
\[C_{L}=\frac{2F_{L}}{\rho\overline{u}^{2}S} \tag{40}\]  
where \(F_{L}\) is the average lift expressed in Newtons, \(\rho\) represents air density, \(\overline{u}\) represents average the flapping speed of the wing tips, and \(S\) represents the equivalent area of the wings. The wing parameters used in the simulation are consistent with those in Table 1, with the unmarked parameter being the wing's slack angle, which has a value of 13.5\({}^{\circ}\). By comparing the results in Figure 18, it can be observed that there is minimal difference between the simulated data and the experimental results within the selected frequency range. Meanwhile, through comparison with the single-plane blade element method, it can be seen that the multi-plane method is relatively accurate in simulating lift. Additionally, more precise simulations can be achieved by adjusting the wing's deformation parameters at different time points. Therefore, the quasi-steady aerodynamic model analyzed using the multi-plane approach exhibits high accuracy. With increasing flapping frequency, a growing divergence is observed between the experimental data and both the multi-plane and single rigid plane simulation data regarding the average lift. This discrepancy likely arises from the omission of clap and fling in the simulation models, as delineated earlier. Previous studies have indicated that the interaction at the wingtips can initiate this mechanism, potentially contributing up to 30% of the total lift. As the actual flapping frequency escalates, the inertia of the wings augments the flapping amplitude, thereby enhancing the lift generated by clap and fling. However, quasi-steady-state models neglect this lift source, leading to an expanding deviation between simulation results and experimental observations. This amplifies the limitations of quasi-steady-state models in capturing complex aerodynamic behaviors.