## 2 Methodology
### Development of Nonlinear and Unsteady Aerodynamic Models
The variables on the right hand side of Equation (20) denote the angle of attack (\(\alpha\)), time rate of angle of attack (\(d\alpha\)/\(dt\), or \(\dot{\alpha}\)), pitch rate (\(q\)), longitudinal reduced frequency (\(k_{1}\)), sideslip angle (\(\beta\)), control deflection angle of elevator (\(\delta_{e}\)), Mach number (\(M\)), roll rate (\(p\)), stabilizer angle (\(\delta_{s}\)), and dynamic pressure (\(\overline{q}\)). These variables are called the influencing variables. The roll rate is included here because it is known that an aircraft encountering hazardous weather tends to develop  
Figure 2: Flowchart for the development of nonlinear and unsteady aerodynamic models. rolling, which may affect longitudinal stability. The variable of dynamic pressure is for estimation of the significance in structural deformation effects. For the lateral-directional aerodynamics [22],  
\[C_{y},C_{l},C_{n}=f\ \Big{(}\alpha,\ \beta,\ \phi,\ p,\ r,k_{2},\ \delta_{a},\ \delta_{r},\ M,\ \dot{\alpha},\dot{\beta}\Big{)} \tag{21}\]  
The coefficients on the left-hand side of Equation (21) represent the coefficients of side force (\(C_{y}\)), rolling moment (\(C_{l}\)), and yawing moment (\(C_{n}\)). The variables on the right-hand side of Equation (21) denote the angle of attack (\(\alpha\)), sideslip angle (\(\beta\)), roll angle (\(\phi\)), roll rate (\(p\)), yaw rate (\(r\)), lateral-directional reduced frequency (\(k_{2}\)), control deflection angle of aileron (\(\delta_{a}\)), control deflection angle of rudder (\(\delta_{r}\)), Mach number (\(M\)), the time rate of angle of attack (\(\dot{\alpha}\)), and the time rate of sideslip angle (\(\dot{\beta}\)). Most representative of the numerical differentiation theory is the difference method. This paper uses the central difference method as follows. \[f^{\prime}(x)=\frac{f(x+\Delta x)-f(x-\Delta x)}{2\Delta x} \tag{22}\]  
Although the numerical differentiation method is used to obtain the derivative of a certain point of the function curve, it can also be applied to the derivative analysis of experimental data. However, the experimental data are scattered and discontinuous points, and interpolation must be used to connect the points into a continuous curve. Fuzzy logic modeling plays the function of interpolation. Fuzzy logic modeling is extremely important in the aerodynamic performance analysis of actual flight data. Two examples of applying the fuzzy logic modeling model to take derivatives are given. The longitudinal stability derivative (\(C_{ma}\)) comes from the model of \(C_{m}\). The aerodynamic derivative of the unsteady aerodynamic model can be calculated using the central interpolation method. The formula of the central difference method is as follows:  
\[C_{ma}=[C_{m}\ (\alpha+\Delta\alpha,\dots\ )-C_{m}\ (\alpha-\Delta\alpha, \dots\ )]/2\Delta\alpha \tag{23}\]  
\(\Delta\alpha=0.1\) degrees, which means that \(\alpha\) changes up and down by 0.1 degrees, while all other variables remain unchanged. The damping derivative (\(C_{lp}\)) is proposed from the model \(C_{l}\). The formula of the central interpolation method is as follows:  
\[C_{lp}=[C_{l}\ (\ \dots\,p+\Delta p,\dots\ )-C_{l}\ (\ \dots\,p-\Delta\ p, \dots\ )]/2\Delta p \tag{24}\]  
where \(\Delta p\) is in unit of deg/s. The calculations of all other aerodynamic derivatives are derived from the same method in this paper.