## 3 Results and Discussion
### Effect of Wing Rotation Angle Distribution  
The wing-root and wing-tip pitching amplitudes (\(\theta_{rm,r}\) and \(\theta_{rm,t}\)) were selected to analyze their effects on vertical force and mechanical power in the optimized wing and to investigate how spanwise torsion affects wing performance in a broader design space. In Figure 11a, the vertical force contours have an elliptical shape, with higher forces concentrated in smaller ellipses. The force gradually decreases from inside to outside, beginning with the center of the innermost ellipse. However, the contour line of the mechanical power shows an inclined curve. The power consumption is less when both the root and tip sections of the wing model have larger pitching amplitudes, as shown in Figure 11b. When the wing model has no pitching motion and is at an angle of attack of 90 deg, there is almost no vertical force, but only the force component inside the stroke plane is produced. Therefore, a large amount of power is required to overcome the aerodynamic drag. That corresponds to the case in the lower left corner of the contour plots. Compared to that case, the wing model with an angle of attack of 0 deg also produces almost no vertical force. That is because the wing surface is parallel to the airflow direction. Thus, the contribution of the aerodynamic component to the mechanical power is reduced, while only the inertial contribution is retained. The upper-right corner of the contour plots indicates this case. For the mechanical power, those results imply a gradual increase from the lower left corner to the upper right corner of the contour plots. In addition, for a non-twisting wing where \(\theta_{rm,r}=\theta_{rm,t}=45\) deg, the maximum vertical force is generated while maintaining moderate power consumption. However, a lower power consumption and a comparable force were obtained for a twisted wing with the combination of a smaller \(\theta_{rm,r}\) and a larger \(\theta_{rm,t}\), such as the case where \(\theta_{rm,r}=20\) deg and \(\theta_{rm,t}=60\) deg. By contrast, a higher power was consumed in the case where \(\theta_{rm,r}=60\) deg and \(\theta_{rm,t}=20\) deg, although the same vertical force is generated. This comparison study shows that an energy-efficient wing has the feature of a larger pitching amplitude at its tip but a lower one at its root. The optimal model obtained in the current optimization has just such characteristics. However, because the requirement for vertical force is not the highest here, the pitching amplitudes at these two spanwise positions vary slightly in value. In addition, the wing-root section of a hummingbird wing is usually at a high angle of attack in nature due to the limitation of the shoulder joint on the rotation motion. That forces them to improve efficiency by lowering the angle of attack in the wing-tip section. This point is qualitatively consistent with the current result.  
In a previous comparison study [32], the power loading (\(F_{\zeta}/\overline{P}\)) of the optimal twisted wing was approximately 5.9% higher than that of the optimal flat wing. The twisted wing with the largest power loading has an average angle of attack of 44 deg at the root and 25 deg at the tip. However, in the current work, the optimized wing has an angle of attack of 50 deg, slightly higher than that obtained in Ref. [32]. This difference comes from the distribution of the rotation angle along the wingspan. A linear distribution was specified on the current wing in our study. Despite that, the average geometric angle of attack for all wing cross-sections is approximately 30 deg in that study, approaching the 32 deg found in the currently obtained wing.