## 4 Study Results and Discussion
### Parametric Study
#### 4.3.2 Effect of Flight Velocity  
Based on the above results at zero flight velocity, the study is extended to the influence of flight velocity on the aerodynamic forces to evaluate the short landing behaviors of the aircraft. To validate the UAM used for the flapping wing of large pitching angle at low flight velocity, the lift and drag forces are calculated by CFD and UAM for a case of the pitching angle at 0\({}^{\circ}\)-60\({}^{\circ}\) at 4 m/s. As shown in Figure 20, the variation of the forces by the two methods is in very good agreement. The difference of average lift and drag forces between the UAM and CFD is only 2.3% and 2.1%, respectively. The results indicate that the UAM can also be utilized to calculate the aerodynamic forces of the flapping wing at low flight velocity.  
Figure 21 shows the average lift and drag forces of the flapping wing in full range of pitching amplitude 15\({}^{\circ}\)-90\({}^{\circ}\) at flight velocity 0-5 m/s during landing. The flight velocity in Figure 21 is also translated to the advance ratio (\(J\)), which is the ratio of the flight speed to the speed of the wing tip. The advance ratio can be expressed by:  
\[J=\frac{V}{2\Phi Fb} \tag{22}\]  
where \(\Phi\) is the amplitude of the flapping angle, \(F\) is the flapping frequency and \(b\) is the wing semi-span.  
The average lift and drag forces increase with the velocity above zero for any pitching amplitude. As shown in Figure 21a, the maximum average lift occurs within the range of pitching amplitude 60\({}^{\circ}\)-75\({}^{\circ}\). In this range, the higher the velocity, the smaller the pitching amplitude required for maximum lift force. As shown in Figure 21b, the average drag force increases with the pitching amplitude for any flight velocity except zero. At zero velocity, however, the average drag is negative and reduced (thrust increases) until the pitching amplitude is increased to 45\({}^{\circ}\), and then slowly increases (thrust reduces) to a small positive value at 90\({}^{\circ}\).  
Based on the above study results, it can be revealed that the pitching angle of the flapping wing for maximum lift in the above cases is in the range between 60\({}^{\circ}\) and 70\({}^{\circ}\) at the flight velocity range from 1 m/s to 5 m/s. For a pitching angle of the wing pre-determined by the flapping mechanism, the variation of the aircraft pitching angle can be altered by the changing the tail-plane.  
The pitching angle is only one of the factors that affect the landing of the flapping wing aircraft. The flapping frequency and velocity of the incoming flow also play important roles in landing. Table 5 shows average lift force and drag force produced by the wing of different flapping frequencies with pitching angle 0\({}^{\circ}\)-60\({}^{\circ}\) at zero flight velocity. The average lift varies with flapping frequency in an almost linear relationship, but with the square of flight velocity. If an optimal kinematics of motion including all the above factors is obtained at different flight velocities and attitudes during landing, an optimal landing strategy may be achieved. However, this optimization process is beyond the scope of the present study.  
Figure 21: Average lift (**a**) and drag (**b**) of different amplitudes under different flows.  
Figure 20: The lift force (**a**) and drag force (**b**) of the flapping wing with pitching angle 0\({}^{\circ}\)-60\({}^{\circ}\) at 4m/s by UAM and CFD.