## 3 Experiment Results and Discussion
### The Effect of Different Installation Angles on the Lift Generation and Rotational Speed Generation  
In the above cases and results, the FWR installation angle is \(\theta_{0}=30^{\circ}\). In order to evaluate the effect of the installation angle on the lift, the FWR model, with alternative installation angles, including \(\theta_{0}=15^{\circ}\), \(45^{\circ}\) and \(60^{\circ}\) and have also been tested with the results presented in Figure 18.  
According to the test results shown in Figure 18a, the FWR rotational speed decreases when the installation angle \(\theta_{0}\) increases due to the increased drag force. In general, the larger the flapping frequency and power input, the larger the rotational speed. Furthermore, the rotational speed for CFF case 1 remains almost constant when \(\theta_{0}<30^{\circ}\) before decrease afterwards, while the rotational speed for the other cases of smaller power input is decreased in the whole range of \(\theta_{0}\), and even close to zero when \(\theta_{0}\) reaches \(45^{\circ}\). The maximum average lift force, generated by the FWR, is shown in Figure 18b. It is also observed that the lift force increases with the installation angle until it reaches the maximum at \(\theta_{0}=30^{\circ}\) and then decreases when \(\theta_{0}>30^{\circ}\) for all the cases. The results indicate that there is an optimal installation angle to achieve the largest lift for different power, flapping frequency, and rotational speed. In terms of lift-to-power ratio, the higher value is generally associated with higher flapping frequency or input power, as shown in Figure 18c. However the lift-to-power ratio of the FWR at 3 Hz reduces dramatically when \(\theta_{0}>30^{\circ}\) and becomes smaller than the VFF case (3 Hz & 1 Hz) when \(\theta_{0}=43^{\circ}\)-\(53^{\circ}\), and the 1.3 Hz case, which has much smaller input power. Figure 17: (**a**) Mechanical output curves of motor; (**b**) the relationship between lift and output power. Furthermore, the lift efficiency in terms of Strouhal number (\(S_{t}\) number) and lift efficiency \(P_{f}\) used in previous studies [19; 31; 32] are employed to obtain the insight with a quantified analysis of the results:  
\[St=\frac{f\Phi}{\dot{\psi}_{fund}} \tag{7}\]  
\[P_{f}=\frac{\overline{C_{L}^{1.5}}}{\overline{C_{p}}}. \tag{8}\]  
where,  
\[\overline{C_{L}}=\frac{\overline{L}}{0.5\rho{U_{rf}}^{2}S_{tw}},\ \ \ \overline{C_{p}}=\frac{ \overline{P}}{0.5\rho{U_{rf}}^{3}S_{tw}},\ \ \ \ U_{rf}=2\Phi f \tau_{2} \tag{9}\]  
where \(f\) stands for the flapping frequency (for VFF cases, it is a value equivalent to the CFF with the same output power), \(\Phi\) is the stroke angle amplitude, \(\dot{\psi}_{fund}\) is taken to be rotational angular speed of the wing, \(\overline{P}\) and \(\overline{L}\) is the motor's power and average lift of the FWR, \(S_{tw}\) is the total area of two wings, \(\tau_{2}\) stands for the radius of the second moment of wing area, as defined in [PERSON]'s book [32], \(\rho\) is the air density, \(P_{f}\) stands for the lift efficiency. Since the \(\overline{P}\) is defined as the motor output power instead of mean aerodynamic power in a flapping cycle, the resulting \(P_{f}\) in the current study cases is generally smaller than the case in previous study [19]. For the installation angles \(\theta_{0}\geq 45^{\circ}\) in all cases of different flapping frequencies, it is found that the resulting \(0.1>S_{t}>0.6\) is far beyond the high propulsion efficiency around \(S_{t}=0.3\) of flying animals and the associated lift efficiency \(P_{f}<0.02\) is very low. For the cases with installation angles \(15^{\circ}\)-\(30^{\circ}\), the \(S_{t}\) values fall in the range of 0.25-0.6 with most of the associated \(P_{f}>0.05\) as shown in Figure 19. In particular the \(P_{f}\) reaches a value 0.28 associated with \(S_{t}=0.36\) for the CFF case 1 (3 Hz), which is similar to an optimal result in previous study [19].