## 3 Results and Discussion
### Sensitivity Analysis of Design Parameters  
To figure out how total vertical force and mechanical power change with the design parameters at the optimal point, we evaluated wing performance at a range of design points by perturbing each design parameter while keeping the others constant and equal to their optimized values. It is necessary to understand the sensitivity of design parameters in the early stages of flapping wing design. The lower and upper limits of the design variable are limited to about 0.8 and 1.2 times its optimal value, respectively. Those values of the design parameters are obtained at equal intervals between their boundary values, forming a series of design points. For each design parameter, the results obtained at these points are plotted in Figure 10. All geometric parameters, and those parameters related to the flapping motion, consistently and monotonically increase vertical force and mechanical power. However, the parameters related to the rotation motion reduce them synchronously, except for \(C_{\theta}\). These two quantities first decrease and then increase slightly with increasing \(C_{\theta}\). When it is improved to a certain extent, the curve in power becomes almost flat. Both of them, by contrast, are minimally affected by \(C_{\theta}\). The vertical force and the mechanical power have the feature of simultaneous increase and decrease, and thus the optimal values for these parameters are loosely restricted by the constraint about the vertical force. The opposite effects between these design parameters indicate that the rotation motion cannot be ignored for the optimal design of energy-efficient wings. Figure 10: _Cont._  
Figure 9: Time history curves of (**a**) vertical and (**b**) horizontal forces obtained in the optimized wing. For geometric parameters, the single wing length has the most significant influence. Secondly, the significance of wing-root offset and root chord is almost coincident. With a 20% increase relative to the optimum, these three design variables increase the vertical force by approximately 47%, 19%, and 21%, respectively. The corresponding power consumption also increases by about 55%, 23%, and 25%, respectively. Compared with the wing with no offset between the root and the flapping axis, that with a wing-root offset has an additional  
Figure 10: Sensitivity analyses of wing parameters at the optimal design point. The vertical force and mechanical power as the function of some given design parameters are illustrated by the blue solid line and red long dash line, respectively. The subfigures are plotted with (**a**) wing-root offset, (**b**) wing length, (**c**) root chord, (**d**) flapping frequency, (**e**) flapping amplitude, (**f**) wing-root pitching amplitude, (**g**) wing-tip pitching amplitude and (**h**) the controlling parameter for pitching motion as independent variables, respectively. The black dashed vertical line denotes the location of the associated optimal parameter, while black dashed horizontal lines indicate the corresponding values of vertical force and mechanical power. velocity. It is added to the translational speed that varies linearly along the chord; this effect also leads to similar results with the same trend. Although the increase in the root chord and wing length results in the changes in the wing aspect ratio, the wing area, and the second moment radius, the force and power changes caused by the wing length effect are nonlinear. Those are similar to the changes arising from the parameters related to the flapping motion, such as \(f\) and \(\phi_{m}\). When these two parameters are increased by 20% based on their optimal values, the associated force and power are raised by approximately 43% and 73%, and 44% and 61%, respectively. The mechanical power is most affected by the flapping frequency, followed by the amplitude of the flapping angle. However, it should be noted that wing length has the most significant effect on the vertical force among all the parameters of concern.