# Effect of Wing Membrane Material on the Aerodynamic Performance of Flexible Flapping Wing
## 4 Results and Discussion
### Deformation Results
#### 4.2.2 Effect of Wing's Flexibility  
In this section, considering the similar lift performance of PET-38 and PE-40, only the deformation of PET-38 and TPU-30 is presented. The rotation angles of PET-38 and TPU-30 are presented in Figure 12. It can be found that at almost all flapping frequencies, more flexible TPU-30 shows a larger rotation angle than PET-38. Furthermore, the amplitude of angle fluctuation seems smaller for the TPU-30, suggesting that flexibility plays a dissipative role in the angle fluctuation. As referred to in Figure 8, TPU-30, with slighter angle fluctuation, obtained higher power loading, suggesting that eliminating the fluctuation is beneficial for improving efficiency, which is consistent with the research of [PERSON] et al. (2018). Figure 11: Cycle-averaged angle of PET-38 at different flapping frequencies. Figure 10: Wing deformation of PET-38 at 14 Hz, 18 Hz, and 22 Hz. (**a**) The rotation angles at 0.25\(R\); (**b**) The rotation angles at 0.5\(R\); (**c**) The rotation angles at 0.75\(R\). Apparently, even if the wing membrane materials are different, the fluctuation trends of PET-38 and TPU-30 are still similar. According to [PERSON] et al. [21], the vein position was able to affect the angle fluctuation and the aerodynamic efficiency. As the identical vein configuration was adopted for the PET-38 and TPU-30, it can be concluded that the effect of membrane material on the angle fluctuation trend is less obvious than that of wing vein. Differences in wing deformation with different flexibility can also be reflected by the cycle-averaged angle shown in Figure 13. Obviously, the cycle-averaged rotation angle of TPU-30 with lower elastic modulus membrane is significantly larger than that of PET-38. The increase in rotation angle along the spanwise direction also explains the twisting deformation of TPU-30, which is more prominent at a higher frequency. It can be seen from Figure 13b that PET-38 has a larger cycle-averaged angle of attack, which is close to 65 degrees, whereas the angle of attack of TPU-30 ranges from 20 degrees to 60 degrees. As presented in the study by [PERSON] et al. [1], the optimal translation component of lift can be obtained at an angle of attack of 45 degrees, based on the quasi-steady aerodynamic model, as seen in Figure 14. Under this circumstance, the lift of TPU-30 should be larger than that of PET-38 at a certain flapping frequency, since the angle of attack of TPU-30 is closer to 45 degrees, as seen in Figure 13. However, by looking at Figure 7a, PET-38 with a larger angle of attack obtained a higher lift. This contradiction  
Figure 12: Deformation comparison of different wing sections at: (**a**) 0.25\(R\) of 14 Hz; (**b**) 0.5\(R\) of 14 Hz; (**c**) 0.75\(R\) of 14 Hz; (**d**) 0.25\(R\) of 18 Hz; (**e**) 0.5\(R\) of 18 Hz; (**f**) 0.75\(R\) of 18 Hz; (**g**) 0.25\(R\) of 22 Hz; (**h**) 0.5\(R\) of 22 Hz; (**i**) 0.75\(R\) of 22 Hz. implies that there is another noticeable lift component that contributes to the resultant force apart from the translation lift. According to the quasi-steady aerodynamic model developed by [PERSON] et al. [22] and [PERSON] et al. [30], instantaneous forces can be represented as the sum of translational force, rotational force, and added mass force. Rotational force and added mass force are closely related to the flapping kinematics, especially angular velocity and angular acceleration of the angle of attack at a given spanwise blade element. Since the flapping kinematic is identical for the same flapping frequency, the difference in lift force for the two wings might be attributed to the discrepancy in the angle of attack caused by different flexibility. In the research of [PERSON] et al. [31], the lift is primarily generated by the outboard wing during its hovering flight.