## 5 Experiments and Analysis  
To observe the shape variations of wings during flapping, we captured the actual flapping wings using high-speed photography. The schematic diagram of the testing environment is shown in Figure 21. Synchronized high-speed video captures of the actual wings were taken from different angles. The black dots labeled as point1-point7 in Figure 22  
Figure 20: Ratio of transient lift to drag power for changing \(y_{2-w}\) at different frequencies. represent marker positions. The purpose of these markers was to calculate the spatial positions using three-dimensional video processing software after the shoot. The markers were placed on three carbon fiber rods (rod1-rod3) and the root point of the wing, dividing the wing into three sections. These markers were intended to facilitate our analysis of the motion of the carbon fiber rods during data processing in order to determine the fitted plane shape of the motion. The parameters of the captured footage are listed in Table 2.  
The structure used to drive the wings is based on the previously designed string-based mechanism [4], which has the advantage of producing wing flapping angles similar to a standard sine wave and lower flapping angular velocities at the ends of the flapping stroke. The processed results of the data captured at 28.7 Hz are shown in Figure 23. Figure 23a reflects the variations in the flapping angles of the three carbon fiber rods observed from the positive z-axis direction, while Figure 23b shows the differences between them. The changes in the angle of attack of the fitted flapping plane using points 0,2,4, 0,4,6, and 0,6,7 are presented in Figure 24. Due to the discontinuity in the interpolated points, spikes may be present in the waveform; therefore, Gaussian averaging was applied to the data to smooth out any spikes. The displayed images are the processed ones. \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 \\ Slack angle \(\theta_{s}\) & 13.5\({}^{\circ}\) \\ Installation angle of rod1 \(\theta_{rod1}\) & 20\({}^{\circ}\) \\ Installation angle of rod1 \(\theta_{rod2}\) & 50\({}^{\circ}\) \\ Shooting frame rate & 5000 Hz \\ Voltage of motor & 4.6 v \\ Flapping frequency & 28.7 Hz \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of experimental parameters. Figure 21: Schematic diagram of experimental environment. Figure 22: (**a**) Schematic diagram of the position of marked points on wing and (**b**) positions of the carbon fiber rods and the planes that they separate. From Figure 23, it can be observed that Rod1, which is the leading edge rod, consistently exhibits a phase lead with respect to rod2 and rod3. The phases of rod2 and rod3 are close to each other, indicating that, when observed from the positive z-axis direction, plane2 and plane3 can be considered to lie in the same plane most of the time. Additionally, plane3 exhibits greater bending during wing flipping, indicating more pronounced undulations compared to plane2. Through analysis of the time-varying angle of attack for the three planes in Figure 24, it can be observed that plane1 maintains a relatively stable angle of attack during translational motion, approximately around 25\({}^{\circ}\). However, it experiences significant instantaneous reversals, accompanied by a certain amount of rebound, at the ends of the flapping stroke. The angles of attack for plane2 and plane3 remain consistently large, resulting in relatively low aerodynamic lift. Plane2 only exhibits significant changes in the angle of attack during brief moments of flipping when approaching the point of maximum lift coefficient, with lower a flapping angular velocity that limits the generation of significant lift.