## 4 Aerodynamic Analysis Based on the Multi-Plane Simulation Method
Incorporating  
Figure 18: **(a)** The average aerodynamic force reflected in units of gf and **(b)** the average aerodynamic force as reflected by the lift coefficient. these unique aerodynamic effects inherent in the flapping process can significantly narrow the gap between simulated and empirical values. In this context, our study estimates that for flapping frequencies up to 40 Hz, the quasi-steady-state aerodynamic model utilizing the multi-plane approach aligns with the actual measured data in predicting average lift with a 90% degree of similarity. However, with further frequency elevation, incorporating additional aerodynamic forces previously disregarded in simulations may be imperative in order to more precisely mirror the actual findings. The lift-to-power ratio is a parameter of great interest in the design of flapping-wing micro air vehicles. Typically measured in g/W, it evaluates the generated lift efficiency per unit power of the wing. Based on previous research [24; 47; 48] and the aforementioned analysis, it is known that the main sources of lift and drag in flapping flight are the translational components, which are closely related to the power consumed by the wing in generating lift and overcoming aerodynamic drag during translation. By employing the multi-plane approach and quasi-steady aerodynamic model, the variation in the lift-to-power ratio can be further analyzed under different slack angles and wing deformation conditions, providing a valuable reference for wing design. Figure 19 illustrates the aerodynamic effects of wing translation, where the variations in transient lift, transient drag, transient power, and lift-to-power ratio with respect to \(y_{2-w}\) under different slack angles are displayed separately in Figure 19a-d. The transient power in the translational effect can be approximated as the power consumption of the wing, as the wing only moves in the drag direction. Its calculation can be expressed as follows:  
\[P_{Dtr}=\int\limits_{0}^{R}F^{{}^{\prime}}_{Dtr}\dot{\psi}r=\int\limits_{0}^{R }\frac{1}{2}\rho C_{D}(\alpha)c(r)(\dot{\psi}r)^{3}dr \tag{41}\]  
and the lift-to-power ratio \(\eta\) can be simplified as  
\[\eta=\frac{F_{Ltr}}{P_{Dtr}}. \tag{42}\]  
Figure 19: Comparison of data for changing \(y_{2-w}\) at different slack angles: (**a**) transient lift; (**b**) transient drag; (**c**) transient drag power; (**d**) ratio of transient lift to drag power. As mentioned earlier, under different slack angles, the range of \(y_{2-w}\) values in the wing coordinate system varies to ensure a full wing shape. With an increasing slack angle, the translational aerodynamic lift of the wing shows the increasing trend displayed in Figure 19a, while the drag and drag power exhibit a decreasing trend, as shown in Figure 19b,c. Consequently, the lift-to-power ratio increases, indicating an improvement in wing efficiency. However, at the same slack angle, as shown in Figure 19a, the trend of lift variation is related to the absolute value of \(y_{2-w}\). This is because under different slack angles the wing's angle of attack changes differently around 45 deg, where the lift coefficient reaches its maximum value. In contrast, in Figure 19b,c it can be seen that the drag increases as the absolute value of \(y_{2-w}\) decreases, because the drag coefficient increases with increasing angle of attack. The variation in wing shape is closely related to the slack angle. By combining this variation, a simulated schematic diagram of the \(y_{2-w}\) values and lift-to-power ratios under different slack angles can be plotted, as shown in Figure 19d. After reaching a certain range of slack angles, the wing exhibits a full shape, characterized by the maximum lift-to-power ratio.