# Research on the Flight Performance of Biomimetic Moth Based on Flapping Function Control
## 3 Calculation and Simulation  
### Lift Calculation of Rigid Wing Flapping Motion  
Before making the experimental prototype, it is necessary to establish a lift calculation model to explore the key parameters in the flapping function and analyze their effect on lift generation. The simplified airfoil model is shown in Figure 2a, and the simplified kinematic model of flapping motion is shown in Figure 2b. The flapping-wing aircraft and its flapping motion are simplified. The body of the flapping-wing aircraft is simplified to a line segment coincident with the \(y\)-axis. The length of the line segment is meaningless, and the volume is not considered. The maximum wingspan \(l_{w}\) of the rigid wing is designed to be about 400 mm, and the maximum chord length is about 250 mm; the leading edge is approximately a straight line coincident with the \(x\)-axis. The trailing edge \(c_{r}\) is a function of \(r\), approximately represented by a quadratic function (1), with the unit in m.  
\[c_{r}=c_{r}(r)=-4.5r^{2}+1.485r+0.126 \tag{1}\]  
The flapping motion is simplified to a rigid body motion in which the leading edge of the wing oscillates back and forth around the \(y\)-axis in the plane \(xOy\). Therefore, the flapping function is the function of the angle between the wing's leading edge and the  
Figure 2: Simplified model of a rigid wing and its flapping motion: (**a**) simplified rigid wing model; (**b**) simplified flapping motion model. plane \(xOy\) changing over time, which is the function of the rotation angle \(\phi\) of the wing around the \(y\)-axis at any time with respect to time \(t\). To further simplify the calculation model, the original segmented flapping function is approximated as a sine function, and the expression of the flapping function within a period is given by Equation (2):  
\[\phi=\phi(t)=\phi_{max}sin(2\pi ft+\psi)+\phi_{0} \tag{2}\]  
where \(\phi_{max}\) is the flapping amplitude with the unit in rad, \(f\) is the flapping frequency with the unit in Hz, and \(\phi_{0}\) is the mid-position angle with the unit in rad. From the flapping function Equation (2), it can be seen that the flapping range of the wing is from \(-\phi_{max}+\phi_{0}\) to \(\phi_{max}+\phi_{0}\); at the middle position of the flapping motion, the angle between the wing and the plane \(xOy\) is \(\phi_{0}\), which is called the mid-position angle in this paper. [PERSON] and others [26] derived the lift and drag calculation formula for 3D translational flapping motion (3):  
\[\left\{ \begin{array}{ll}L=\frac{1}{2}C_{L}\rho AU^{2}\\ D=\frac{1}{2}C_{D}\rho AU^{2}\end{array} \right. \tag{3}\]  
where \(L\) is the lift with the unit in N, \(D\) is the drag with the unit in N, \(C_{L}\) is the lift coefficient without units, \(C_{D}\) is the drag coefficient without units, \(\rho\) is the air density with the unit in kg/m\({}^{3}\), \(A\) is the area of the wing with the unit in m\({}^{2}\), and \(U\) is the flapping speed with the unit in m/s. According to the research of [PERSON], the lift coefficient and drag coefficient are functions of the angle of attack, and the empirical formula (4) is obtained:  
\[\left\{ \begin{array}{ll}C_{L}=0.225+1.58 sin(2.13\alpha-7.20)\\ C_{D}=1.92-1.55 cos(2.04\alpha-9.82)\end{array} \right. \tag{4}\]  
where \(\alpha\) is the angle of attack with the unit in \({}^{\circ}\). The angle of attack in the flapping motion usually changes periodically within a flapping period. To simplify the calculation, it is assumed that the angle of attack is constant at 15\({}^{\circ}\) during the downward stroke and 10\({}^{\circ}\) during the upward stroke. Then, according to the empirical formula (4), the lift coefficient is approximately as follows  
\[\left\{ \begin{array}{ll}C_{Ld}=0.9\\ C_{Lu}=0.6\end{array} \right.