## 4 Study Results and Discussion  
### Experimental Results  
From the experiment, the flapping and pitching angles of the wing measured during one flapping cycle vary in the range of \(-26.1^{\circ}\) to \(25^{\circ}\) and \(-18.2^{\circ}\) to \(43.9^{\circ}\), respectively, as shown in Figure 10. It is noted that there is a small asymmetry in the measured flapping and pitching angles and time delay between the wing up-stroke and down-stroke in a cycle. For example, when the flapping motion was started from \(0^{\circ}\) flapping angle where the wing was at the middle position in down-stroke, the pitching angle was about \(-10^{\circ}\); when the flapping wing is back to \(0^{\circ}\) again in up-stroke, the pitching angle was increased to the maximum \(44^{\circ}\).  
Based on the test data, the wing motion can be modelled by Sine function and Fourier function below. The measured flapping motion is idealized as a symmetric flapping and pitching angle functions with the amplitude varying from \(-25^{\circ}\) to \(25^{\circ}\) and from \(-18^{\circ}\) to \(44^{\circ}\), respectively, as shown in Figure 10. The fitted data matches experimental data very well, and the R-squared values between experimental and fitting values of flapping and pitching angles are 0.9991 and 0.9921, respectively. Such slight fitting deviations are mainly caused by measurement errors of OptiTrack cameras and the fitting model as shown in Equations (20) and (21).  
\[\theta=\mathrm{Hsin}(wt) \tag{20}\]  
\[a=A_{0}+A_{1}\cos(wt)+B_{1}\sin(wt)+A_{2}\cos(2 wt)+B_{2}\sin(2 wt) \tag{21}\]where \(\theta\) and H refer to the flapping angle and its amplitude of the wing. \(w\) is the flapping frequency in radian. \(a\) represents the pitching angle. \(A_{0}\), \(A_{1}\), \(B_{1}\), \(A_{2}\), \(B_{2}\) are the correlation coefficients of the pitching amplitude. The pitching angle amplitude is defined as the difference between the smallest pitching angle to the largest pitching angle.  
Figure 11a shows the measured forces of the model produced by the flapping wing with and without the wing skin in a stable flapping cycle at 2.1 Hz and zero flight velocity. To identify the main source that affects the results, the dynamic behavior of the model can be presented in the frequency domain as shown in Figure 11b by Fourier transformation. Three frequencies of 2.1 Hz, 4.3 Hz, and 6.5 Hz below 10 Hz can be identified from the experimental data. Since the 0 Hz mode can be ignored and the dominating 2.1 Hz is the flapping frequency, the additional modes of 4.3 Hz and 6.5 Hz remain as a concern. It can be speculated that those higher-order modes are from the test rig, which has little influence on the motion of the flapping wing and aerodynamic force.  
According to Equations (16) and (17), the difference between the measured force with and without the wing skin results in the aerodynamic lift as shown in Figure 12a. The corresponding drag force (negative value indicates thrust) can be also obtained and shown in Figure 12b. From the calculation, the measured peak value of lift force is 2.02 N and average lift in a flapping cycle is 0.123 N; the peak value of drag force is \(-0.68\) N and the average drag is \(-0.43\) N. It is noted that the average aerodynamic lift obtained from the experimental result is very low for the size of the wing. This is not only due to the relatively low flapping frequency (2.1 Hz) but also at zero flight velocity. The lift will increase  
Figure 11: Measured force with (**a**) and without skin; (**b**) transformed data in the frequency domain.  
Figure 10: Measured and fitted test data of flapping and pitching angles in one cycle (\(\hat{t}\) refers to the non-dimensional time normalized by flapping cycle).  
significantly with the increase in flapping frequency and flight velocity as shown in the parametric study results in Section 4.3.