## 3 Results and Discussion
### Optimization Results and Optimal Wing Performance
Due to the lower requirement for the lift there, the optimal single wing length is much shorter, and the flapping amplitude reaches the maximum, up to 90 deg. However, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Variables** & **Description** & **Value** \\ \hline \(r_{0}\) & Wing root offset & 72.27 mm \\ \(L\) & Wing length & 179.81 mm \\ \(C_{R}\) & Root chord & 51.95 mm \\ \(f\) & Frequency & 6.54 Hz \\ \(\phi_{m}\) & Flapping Amplitude & 65.88 deg \\ \(\theta_{m,r}\) & Pitching Amplitude at the root & 39.01 deg \\ \(\theta_{m,t}\) & Pitching Amplitude at the wingtip & 75.96 deg \\ \(C_{\theta}\) & Regulating parameter of rotation angles & 3.18 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Optimal parameters obtained for the optimization problem. Figure 7: Comparison for wing models that are simplified as quarter-elliptical planforms, obtained through three different approaches: (**a**) live Calliope hummingbird measurement [54], (**b**) statistical analysis based on the target quality [45], and (**c**) optimization in this study. The scale of subfigure (**a**) is 3.35 times larger than the scale of the other two subfigures. optimal aspect ratio is close to that obtained in this work. The differences in these results also fully illustrate the existence of a strong coupling between these parameters. The performance of the optimized wing was also evaluated using the UBET, and all obtained results were plotted in Figure 8c-f. Wherein, Figure 8c shows the time-varying curve of the total vertical force and the respective force components generated by a single wing. It can be seen that inertial force has a significant negative contribution to that in the vertical direction, which is especially obvious when the wing rotates rapidly due to pitching acceleration or deceleration. In addition, during pronation, negative and downward rotational forces are produced by pushing air upward, away from the wing surface. During the whole stroke cycle, other force components promote the generation of positive vertical forces. Among them, the translational force component has the most significant effect. After weighted averaging this force over one stroke period, the total generated by a single wing is approximately 15.05 g. So, the constraint on the vertical force is satisfied. To further quantify the time-varying power, we plotted the power output of the optimal wing along the flapping axis and the pitching axis in Figures 8d and 8e, respectively. These two power outputs include aerodynamic power consisting of various components caused by translational, rotational, and added-mass loads, and inertia power of the wing model itself (displayed in Figure 8). Additionally, the positive values of these power components were summed to obtain the corresponding instantaneous contribution of aerodynamic power to total power. The associated inertial contribution was obtained by truncating the negative portion of the inertial power curve in Figure 8d. These two quantities were then added to calculate the total mechanical power utilized, to estimate the cycle-averaged mechanical power. All components are displayed in Figure 8f. The time-history curves of vertical and horizontal forces are also plotted in Figure 9, and those can be better understood by combining the distribution of \(\theta_{r}\) along the span. For the power around the flapping axis in a stroke cycle (see Figure 8d), the translational component is more dominant than the inertial power, which is positive when the wing accelerates and negative when it decelerates. Drag (the \(\eta\)-axis force component), always in the opposite direction of the translational velocity, is generated in the forward or backward reciprocating cycle. The translational power is thus positive during the forward and backward strokes to overcome the drag. It peaks in the middle of each half-stroke when the angular speed of the flapping angle is at its maximum.