## 3 Numerical Results and Discussion  
In the present study, the accuracy of the established unsteady aerodynamic models with six aerodynamic coefficients by using the fuzzy logic modeling technique is estimated by the sum of squared errors (\(SSE\)) and the square of multiple correlation coefficients(\(R^{2}\)). All the aerodynamic derivatives in the study of dynamic stability characteristics are calculated with these aerodynamic models of aerodynamic coefficients. Figure 3 presents the main aerodynamic coefficients of normal force \(C_{z}\), pitching moment \(C_{m}\), rolling moment \(C_{l}\), and yawing moment \(C_{n}\) predicted by the nonlinear and unsteady aerodynamic models. Figure 3: Main aerodynamic coefficients predicted by the nonlinear and unsteady aerodynamic models. (**a**) Time history of normal force \(C_{z}\); (**b**) Time history of pitching moment \(C_{m}\); (**c**) Time history of rolling moment \(C_{l}\); (**d**) Time history of yawing moment \(C_{n}\). The changes in the normal force coefficient (\(C_{z}\)) are large for both model predicted and input data before the modeling in Figure 3a due to the large variations in normal acceleration (\(a_{z}\)) and angle of attach (\(\alpha\)), referred to in Figure 3a,b. The magnitudes of az and \(\alpha\) (AOA) are obviously large to cause strong variations in the normal force coefficient (\(Cz\)) that started at roughly t = 7650 s.  
The scattering of \(C_{m}\) data in Figure 3b is most likely caused by turbulence-induced buffeting [12; 13] on the structure, particularly on the horizontal tail. The predicted results of others by the final models match well with the flight data. Once the aerodynamic models are set up, one can calculate all the necessary derivatives by a central difference scheme to analyze the stability characteristics. The final main aerodynamic models of aerodynamic coefficients consist of many fuzzy rules for each coefficient, as described in Tables 2 and 3. In Tables 2 and 3, the numbers below each input variable represent the number of membership functions. The total number of fuzzy cells (\(n\)) in each model is the product of each number presented in column 3. The last column shows the final multiple correlation coefficients (\(R^{2}\)). The accuracy of the established aerodynamic model through the fuzzy logic algorithm can be judged by the multiple correlation coefficients (\(R^{2}\)). The twin-jet transport aircraft encountered clear-air turbulence during a commercial flight. As a result, several passengers and cabin crew members sustained injuries, and this event was classified as an aviation accident. To examine the dynamic stability characteristics, it is imperative to understand the flight environment in detail. The corresponding flight data for twin-jet transport with plunging motion is presented in Figure 4. The dataset of the time span t = 7480-7739 s used for the modeling was extracted from the FDR. This twin-jet transport encounters severe clear-air turbulence twice during this time span. In Figure 4a, the variations in normal acceleration (\(a_{z}\)) show the highest \(a_{z}\) of 2.05 g around t = 7483 s and the lowest \(a_{z}\) of 0.08 g around t = 7484 s in the first turbulence encounter; in the second turbulence encounter, the highest \(a_{z}\) was 1.83 g around t = 7682 s and the lowest was 0.06 g around t = 7684 s. Figure 4b shows that the variation in \(\alpha\) (AOA) is approximately in phase with \(a_{z}\) during those two turbulence encounters. The variation ranges of AOA are 4 deg. to \(-\)1.8 deg. in the first turbulence encounter and 5.5 deg. to \(-\)2 deg. in the second. The time history of \(\beta\) is about \(-\)2 deg. with the magnitude of small fluctuation, as indicated in Figure 4c. The aircraft rapidly plunges downward during the turbulence encounter. The largest drop-off height reaches 57.3 m in the time span t = 7484-7486 s. The Mach number (\(M\)) drops from 0.83 to 0.75 in the first turbulence encounter and from 0.80 to 0.70 in the second, as shown in Figure 4d.