## 2 Methodology
### Development of Nonlinear and Unsteady Aerodynamic Models  
This section describes the development process of nonlinear and unsteady aerodynamic models, as shown in Figure 2. The flowchart of Figure 2 is based on the data of the nonlinear and unsteady aerodynamic database. The thrust data in aerodynamic database have been separated from aerodynamic coefficients. Model strucg-n, the numerical model structure, is set up through parameter selection, time segment selection, and sampling rate setting, and then goes through the refining process of fuzzy logic modeling to establish the required models. The operational efficiency enhancement can be studied after model defuzzification. \(T_{x}\),\(T_{y}\),\(T_{z}\), and \(T_{m}\) are the axial thrusts along the body \(x\)-, \(y\)-, and \(z\)-coordinates and the pitching moment about the \(y\)-axis, respectively, in Equations (12)-(14) and (16). It is known from the various thrusts that the force and moment acting on the aircraft will be affected by the engines. Both thrust and aerodynamic forces are generated together, and the two effects cannot be accurately separated only from the flight record data during analysis. To accurately estimate the aerodynamic coefficient, it is necessary to obtain an accuratethrust value to separate the thrust effects; then, the accurate aerodynamic coefficients can be obtained [18]. Since the values of thrust for aircraft in flight cannot be directly measured in the current state of the art, they are not recorded by the QAR nor the FDR. The manufacturers of engines agreed that using parameters such as the Mach number, airspeed, flight altitude, temperature, rpm of the pressure compressors, and engine pressure ratios are adequate to estimate the engine thrust [21]. To study aircraft performance with fuel consumption, data from the flight manual for the fuel flow rate (\(\dot{m}_{f}\)) at various altitudes (\(h\)), weights (\(W\)), Mach numbers (\(M\)), calibrated airspeed (CAS), and engine pressure ratios (EPRs) during cruise flight are utilized. For the Pratt and Whitney turbofan engines, thrust (\(T\)) is defined by EPR, and the thrust model is set up as:  
\[T=f\left(h,\ W,\ M,\ \text{CAS},\ \text{EPR},\ \dot{m}_{f}\right) \tag{18}\]  
For GE or CFM turbofan engines, the rpm of the low-pressure compressor (\(N_{1}\)) is used to set the level of thrust, so the thrust model is set up as:  
\[T=f\left(h,\ W,\ M,\ \text{CAS},\ \text{1},\dot{m}_{f}\right) \tag{19}\]  
In the present study, a twin-jet with GE turbofan engines under study is illustrated. The actual thrust in operation is obtained by using the recorded variables in the QAR, particularly the fuel flow rates. Once the thrust models are generated as a function with the flight conditions of climbing, cruise, and descending phases, we can estimate the thrust magnitude by inserting those flight data into the aerodynamic database. When studying the nonlinear and unsteady aerodynamic characteristics of motion, it can be found that the nonlinear and unsteady aerodynamics with hysteresis [6; 22; 23] are affected by many parameters of motion state. The aerodynamic models developed by the fuzzy logic modeling method are suitable for aerodynamic research of coupling motion and can be used to enhance the operational efficiency of transport aircraft. The development process of nonlinear and unsteady aerodynamic models is shown in Figure 2.  
Modeling means to establish the numerical relationship among certain variables of interest. In the fuzzy logic model, more complete necessary influencing flight variables can be included to capture all possible effects on aircraft response to structure deformations. The longitudinal main aerodynamics are assumed to depend on the following ten flight variables [22]:  
\[C_{x},C_{z},C_{m}=\ f\big{(}\alpha,\dot{\alpha},q,k_{1},\beta,\delta_{e},M,P, \delta_{s},\overline{q}\big{)} \tag{20}\]  
The coefficients on the left-hand side of Equation (20) represent the coefficients of axial force (\(C_{x}\)), normal force (\(C_{z}\)), and pitching moment (\(C_{m}\)).