# Design of Wing Root Rotation Mechanism for Dragonfly-Inspired Micro Air Vehicle
## 6 Experiment and Analysis
### Analysis of Aerodynamics  
The lift and drag forces acting on the object are given by (16) and (17), respectively, in the quasi-steady model. \[F_{L}=\frac{1}{2}\rho C_{L}SU^{2} \tag{16}\]  
\[F_{D}=\frac{1}{2}\rho C_{D}SU^{2} \tag{17}\]  
where \(F_{L}\) and \(F_{D}\) are the lift force and drag force, respectively, \(\rho\) is the density of air, \(S\) is the area of the object, \(U\) is the velocity of the object, and \(C_{L}\) and \(C_{D}\) are the lift and drag coefficients, respectively. Lift force and drag force were defined with respect to wing, and the wing generates a drag force in the opposite direction of the moving direction and a lift force in the vertical direction of the moving direction. In the equation, the lift and drag are proportional to the velocity squared of the object because all parameters except the velocity are constant. \(F_{x^{\prime}}\) and \(F_{z^{\prime}}\), which were measured in the experiment, have the relationship indicated in (18) and (19) with \(F_{L}\) and \(F_{D}\) as shown Figure 1b. Therefore, \(F_{x^{\prime}}\) and \(F_{z^{\prime}}\) should increase with the flapping frequency squared of the object. \[F_{x^{\prime}}=F_{L} \tag{18}\]  
\[F_{z^{\prime}}=F_{D}\sin\left(\theta_{s}\right) \tag{19}\]  
Figure 19a shows the relationship between each \(F_{x^{\prime}}\), \(F_{z^{\prime}}\) and flapping frequency of the MAV for wing root angles of \(0^{\circ}\) and \(30^{\circ}\) during flapping. The red dotted line (\(F_{t}\)) denotes the estimated value of \(F_{x^{\prime}}\) and \(F_{z^{\prime}}\) of the wing over the flapping frequency square through the characteristics of (16) and (17). The blue straight line is the measured lift force, and the black straight line is \(F_{z^{\prime}}\). Figure 19b shows the difference between the estimated and measured values. The difference between the measured force and estimated value is not over 0.006 N regardless of the flapping frequency, as shown in the figure. Therefore, \(F_{x^{\prime}}\) and \(F_{z^{\prime}}\) in the proposed MAV wing are proportional to the flapping frequency squared, and it is possible to estimate the force generated by increasing the flapping frequency. In the left graph in Figure 19a, \(F_{x^{\prime}}\) at a \(0^{\circ}\) wing root angle increases in the positive direction with flapping frequency, while \(F_{z}\) increases in the negative direction or appears near 0. On the other hand, \(F_{x^{\prime}}\) at a wing root angle of \(30^{\circ}\), shown in the right graph, is lower than that of the wing root angle of \(0^{\circ}\) but has a similar tendency, and \(F_{z^{\prime}}\) increases with increasing flapping frequency. A comparison of the two results shows that the magnitude of \(F_{x^{\prime}}\) and \(F_{z^{\prime}}\) that occurs in the wing changes when the wing root angle changes. Figure 18: The measured angles of attack. The top graph in Figure 20 shows \(F_{x^{\prime}}\) at a \(0^{\circ}\) wing root angle when the MAV flaps at a frequency of 15 Hz, while the bottom graph shows \(F_{z^{\prime}}\) at a \(30^{\circ}\) wing root angle. The top and bottom image in Figure 20 show the angle of attack when the wing is horizontal to the ground. The top two images indicate the angle of attack at a \(0^{\circ}\) wing root angle, while the bottom two images indicate the angle of attack at a \(30^{\circ}\) wing root angle. The force generated is an aerodynamic force, which is \(F_{z^{\prime}}\) of the MAV wing. The aerodynamic force at a \(0^{\circ}\) wing root angle does not show a significant difference when up-down strokes are compared. In contrast, the peak value after vibration due to the inertial force in the downstroke at a \(30^{\circ}\) the wing root angle means that the wing receives a large average \(F_{z^{\prime}}\). The decreasing tendency after the vibration due to the inertial force in the upstroke means that the wing receives a small average \(F_{z^{\prime}}\).