# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 6 Optimization Results and Analysis  
### WGP Optimization Results  
Firstly, the sole optimization of WGP with 3D translational aerodynamic coefficients is executed to minimize the hovering specific power output. Here, the WKP are unchanged and completely equal to the data from approximately steady hovering fruit fly (Table 3, as illustrated in Figure 7a). The optimal results are listed in Tables 3 and 4. Then the pitch power output and flapping power output of a single wing planform separately along the pitch axis of wing planform and flapping axis of the right wing root frame are estimated again in terms of (41) with the optimal WGP and known WKP (as illustrated in Figure 7b,c). Both the pitch power output and flapping power output include aerodynamic power and inertial power, respectively. The aerodynamic power consists of four components, which are from translational circulation effect, rotational circulation effect, rotational damping, and added-mass effect. In addition, the positive mechanical pitch and flapping power output utilized to estimate time-averaged power density in optimization model (41) and (43) are also shown (Figure 7b,c). Here, it is not difficult find that pitch power consumption mainly distributes around the pitch reversal while flapping power consumption distributes during the large range of the midstroke and peaks near to the midstroke. For pitch aerodynamic power (\(P_{x,\text{total}}\)) during the forward stroke (Figure 7b), the added-mass power (\(P_{x,\text{add}}\)) dominates other three terms, the subdominant term belongs to rotational circulation power (\(P_{x,\text{rot}}\)), and the third one is rotational damping power (\(P_{x,\text{rd}}\)), while the translational circulation power (\(P_{x,\text{trans}}\)) has the minimum contribution to pitch power consumption. Their synergistic effect spans a rapidly varying pitch interval composed of three pitch reversal segments, and two positive peaks of pitch aerodynamic power distribute at previous instant of the first and second pitch reversal points, where the pitch velocity or slope of wing pitch angle is maximum (Figure 7a,b). The pitch inertial power (\(P_{x,\text{inert}}\)) is reverse to the pitch aerodynamic power; their mutual subtraction results in a positive peak of total pitch mechanical power output (\(P_{x,\text{total,posi}}\)) around the middle of the first and the second pitch reversal points. For the backward stroke, a similar trend of pitch power distribution can be observed, although with some differences in amplitude. Namely, the amplitudes are larger than those for forward stroke. This is due to a faster stroke being completed in the backward stroke (Figure 7a,b). For flapping aerodynamic power (\(P_{Z,\text{total}}\)) during the forward stroke (Figure 7c), the translational circulation power (\(P_{Z,\text{trans}}\)) dominates the other three terms, the subdominant term belongs to added-mass power (\(P_{Z,\text{add}}\)), and the third one is rotational circulation power (\(P_{Z,\text{rot}}\)), while the rotational damping power (\(P_{Z,\text{rot}}\)) has a minimum contribution to flapping power. The flapping inertial power (\(P_{Z,\text{inert}}\)) is positive during translation acceleration and negative during translation deceleration. Their synergistic effect results in two positive peaks of total flapping mechanical power output (\(P_{Z,\text{total,posi}}\)) occurring around the middle of the forward stroke, which is different from the situation for the backward stroke. In the backward stroke, there is only one positive peak because a faster backward stroke leads to a larger translational aerodynamic power output than inertial power. Eventually, the positive total flapping mechanical power output is nearly consumed during the whole stroke except at the interval of stroke reversal where a minor negative \(P_{Z,\text{add}}\) might be stored in elastic joint or dissipated.