# Wing Geometry and Kinematic Parameters Optimization of Flapping Wing Hovering Flight
## 4 Extended Quasi-Steady Aerodynamic Model
### Aerodynamic Forces  
As seen in Figure 1, the blade-element method assumes that the aerodynamic force acting on the wing planform is the summation of force acting on each infinitesimal spanwise strip, and the local pressure drag acting on each strip is the summation of normal pressure drag acting on the chordwise differential elements. So the aerodynamic forces in the wing planform fixed frame, which are produced by translational circulation, rotational circulation, and added mass effect, can be given by  
\[F_{\text{trans,y}}=\frac{1}{2}\rho R_{\text{eff,vari}}{}^{3}C_{\text{aver,vari}} \hat{F}_{\text{trans}}C_{\text{N}}\left(\alpha\right)\omega_{\text{pal}}{}^{2}, \tag{12}\]  
\[F_{\text{rot,y}}=\frac{1}{2}\rho R_{\text{eff,vari}}{}^{2}C_{\text{aver,vari}}{}^ {2}\hat{F}_{\text{rot}}C_{\text{R}}\omega_{x}\omega_{\text{pal}}, \tag{13}\]  
\[F_{\text{add,y}}=\frac{\pi}{4}\rho C_{\text{aver,vari}}{}^{2}R_{\text{eff,vari}}{ }^{2}\hat{F}_{\text{rot}}\left(\dot{\omega}_{z}+\omega_{x}\omega_{y}\right)+ \frac{\pi}{4}\rho C_{\text{aver,vari}}{}^{3}R_{\text{eff,vari}}\hat{F}_{\text{ coeff,add,y}}\dot{\omega}_{x}, \tag{14}\]  
where \(\hat{F}_{\text{trans}}\) and \(\hat{F}_{\text{rot}}\) can be termed non-dimensional translational and rotational aerodynamic force, respectively [25]. And \(\hat{F}_{\text{coeff,add,y}}\) can be analogously termed non-dimensional rotational added-mass aerodynamic coefficient. Their complete expression is detailed in Table 2. \(C_{\text{R}}\) is the theoretical rotational coefficient with expression of \(C_{\text{R}}=\pi(0.75-\hat{x}_{0,\text{vari}})\), and \(\omega_{\text{pal}}\), which is equal to \(\dot{\phi}\), is defined as the angular velocity of the pitch axis line of wing planform. Here, for translational aerodynamic force, its tangential component is neglected due to its small contribution [30; 41; 42]. The normal translational aerodynamic force coefficient (\(C_{\text{N}}\left(\alpha\right)\), \(\alpha\) is angle of attack) is obtained from lift and drag coefficients using trigonometry. Within the community of the flapping wing, the two-dimensional (2D) quasi-static lift and drag coefficients in the local flow for translational circulation mechanism adopted broadly are from simple harmonic fitness relationships to the experimental data for the dynamically scaled fruit fly wing model [34],  
\[\left\{ \begin{array}{l}C_{\text{L}}\left(\alpha\right)=0.225+1.58\text{sin} \left(2.13\alpha-7.2\right)\\ C_{\text{D}}\left(\alpha\right)=1.92-1.55\text{cos}\left(2.04\alpha-9.82\right) \end{array} \right., \tag{15}\]  
These formulas might be appropriately used to evaluate translational component of aerodynamic force for dynamically scaled wing planform with non-dimensional conformal feature. However, the translational aerodynamic lift and drag coefficients for prolonged attachment of the leading edge vortex (LEV) might be influenced by the variation of wing shape. As concluded and remarked by [PERSON] and [PERSON] [43], the prolonged attachment of LEV on insect-scale wings may be due to the attenuating effect of downwash induced by the tip vortex and wake vorticity but not spanwise flow under lower \(Re\) for fruit flies, such as 100 \(\pm\) 250. Thus, it is necessary to try to analytically explore whether the variation of wing shape will indirectly have much influence on lift and power density by probable way of influencing translational aerodynamic coefficients. In order to account for the possible effect of any variation of wing shape on aerodynamic coefficients thus lift and power, an approximate expression to experimental data for translational steady lift coefficient is also utilized to model the lift for flapping wing biomechanics of insect or FWMAV [30; 44], this expression for 3D translational lift coefficient is given by  
\[C_{\text{L}}\left(\alpha\right)=0.5C_{La}\text{sin}\left(2\alpha\right), \tag{16}\]  
where \(C_{La}\) is 3D wing lift curve slope.