# Effect of Wing Membrane Material on the Aerodynamic Performance of Flexible Flapping Wing
## 4 Results and Discussion  
### Lift and Power  
Figure 7 presents the measured results for these three wings at various flapping frequencies. The force data was presented in the form of mean value with the corresponding standard deviations. It should be noted that due to the limitation of DC power supply equipment, the displayed current is always the same and at constant value during the test under a given flapping frequency, and does not fluctuate in any way. Thus, the standard deviation of the power input was not presented in Figure 7b. It can be found that the three lift curves present a similar trend. As the flapping frequency increases, the mean lift increases monotonically. Meanwhile, at a higher flapping frequency, the lift grows faster, implying a nonlinear relationship between the lift and flapping frequency. By referring to the research of [PERSON] et al. [17] and [PERSON] et al. [27], the mean lift might be proportional to the second power of the flapping frequency. For all flapping frequencies, the lift of TPU-30 is the smallest among the three wings. When the flapping frequency is lower than 18 Hz, the lift of PET-38 and PE-40 are basically the same. While at a higher flapping frequency, PET-38 starts to present an advantage in lift generation. Moreover, the higher the frequency, the more obvious the advantages. According to [PERSON] et al. [27], who analyzed the power consumption of the flapping wing system, the power input can be mainly decomposed into aerodynamic power, mechanism power, inertia power, etc. In this study, the power of each component has not been separated. Although comparing the aerodynamic power for these wings is better for evaluating the aerodynamic performance, the power consumption of the flapping system still needs more attention. The power input, which means the power required to drive the wing's movement, can be regarded as the power consumption for the flapping system. As shown in Figure 7b, similar trends in the three power input curves can also be observed, which increase with the increase of flapping frequency. At a higher flapping frequency, the power input grows rapidly, implying a greater power required at a higher frequency. More precisely, the relationship of power input against flapping frequency also seems nonlinear. This phenomenon can also be found from the research by [PERSON] et al. [17], who conclude the power consumption scaled with approximately a third of the power of the frequency. Similar to the result of lift, the input power of PET-38 is the largest of all flapping frequencies, while that of TPU-30 is the smallest. The higher the frequency, the greater the difference in power consumption of the three wings. At 22 Hz, the power consumed by the PET-38 is almost twice as much as that consumed by the TPU-30. Power loading, defined as the lift generated by per unit power (Lift/\(P_{in}\)), can be considered as the efficiency of power input conversion into lift generation, and is shown  
Figure 7: Measured force and power input for the three wings at various flapping frequencies. (**a**) Mean lift force; (**b**) Power input. in Figure 8. The calculation of the standard deviation for the power loading follows the error propagation criterion [28]. Of note, different from the result of the lift and power input, the power loading curves of the three wings present an inconsistent trend. With the increase of the flapping frequency, the power loading of PET-38 decreases monotonously, and the maximum value appears at 10 Hz. The power loading of PE-40 increases first to the peak value at 16 Hz, after which it drops gradually. The power loading of TPU-30 gradually increases and then decreases, and the maximum value is obtained at 20 Hz. In other words, each wing has an optimum flapping frequency to obtain maximum efficiency, which is consistent with the research by [PERSON] and [PERSON] [12].