## 3 Numerical Results and Discussion
\begin{table}
\begin{tabular}{c c  
Figure 4: Time history of main flight variables in severe clear-air turbulence. (**a**) Time history of normal acceleration (\(\mu_{2}\)); (**b**) Time history of angle of attack (AOA); (**c**) Time history of sideslip angle (\(\beta\)); (**d**) Time history of Mach number. Figure 5 presents the twin-jet transport aircraft response to dynamic flight motion. The magnitudes of pitch rate (\(q\)) and roll rate (\(p\)) are shown in Figure 5a; the variation of \(p\) is larger than that of \(q\). The yaw rate is not shown because it is small throughout. This implies that pitch angle (\(\theta\)) does not vary as much as AOA and the variations in roll angle (\(\phi\)) are large, especially during the second turbulence encounter. Figure 5b,c presents the time rate of the angle of attack (\(da/dt\), or \(\dot{\alpha}\)) and the time rate of the sideslip angle (\(\hat{\beta}\)), respectively. Figure 5b shows that variations in \(\dot{\alpha}\) are approximately in phase with \(a_{z}\) and AOA during the two turbulence encounters. Figure 5c describes that the time history of \(\beta\) is about a certain value with the magnitude of small fluctuation. It  
Figure 5: Time history of twin-jet transport aircraft response to dynamic flight motion. (**a**) Time history of the pitch rate (\(q\)) and the roll rate (\(p\)); (**b**) Time history of the variable \(da/dt\); (**c**) Time history of the variable \(d\beta/dt\). implies that the twin-jet transport aircraft has a crosswind encounter, but this crosswind does not have variations in flight directions. In general, the configuration of the static stability assessment is based on enough values of steady damping derivatives to maintain a stable condition. The longitudinal, roll, and directional damping derivatives have to be associated with \(\dot{\alpha}\)- and \(\dot{\beta}\)- derivatives as oscillatory motion in a stability assessment. The longitudinal oscillatory derivatives are defined as [19]:  
\[\left(C_{mq}\right)_{osc}=C_{mq}+C_{m\dot{\alpha}} \tag{38}\]  
\[\left(C_{2\dot{\alpha}}\right)_{osc}=C_{2\dot{\alpha}}+C_{2\dot{\alpha}} \tag{39}\]  
The roll and directional oscillatory derivatives are:  
\[\left(C_{lq}\right)_{osc}=C_{lq}+C_{l\dot{\beta}}\sin\ \alpha \tag{40}\]  
\[\left(C_{mr}\right)_{osc}=C_{mr}-C_{n\dot{\beta}}\cos\alpha \tag{41}\]  
The values of oscillatory derivatives are equivalent to the combinations of steady damping and dynamic derivatives in Equations (38)-(41). The use of oscillatory derivatives instead of steady damping only is more consistent with the actual case of plunging motion. To be stable in oscillatory motion, \((C_{2\dot{\alpha}})_{osc}>0\), \((C_{mq})_{osc}<0\), \((C_{lp})_{osc}<0\), and \((C_{mr})_{osc}<0\). Physically, if it is unstable, the motion could be divergent in oscillatory motions within a short time period. The \((C_{2\dot{\alpha}})_{osc}\) and \((C_{mq})_{osc}\) would enlarge the variations in \(C_{z}\) and data scattering of \(C_{m}\). The \((C_{2\dot{\alpha}})_{osc}\) and \((C_{mq})_{osc}\) are insufficient in oscillatory damping, as shown in Figure 6a,b in the periods of t = 7484-7486 s and t = 7490-7494 s. The effect of \(\dot{\alpha}\)-derivative on \((C_{mq})_{osc}\) is to improve the stability in pitch in the periods of t = 7482.5-7484 s and 7486-7488.5 s. The values in the period of plunging motion have some differences between oscillatory and steady damping derivatives in Figure 6a,c due to the effects of the dynamic derivatives (i.e., \(\dot{\alpha}\)-derivatives), and in Figure 7a,c due to the effects of the dynamic derivatives (i.e., \(\dot{\beta}\)-derivatives). Figure 6: _Cont._  
Figure 6: Time history of main longitudinal oscillatory derivatives along the flight path. (**a**) Time history of (\(C_{2q}\))\({}_{osc}\); (**b**) Time history of (\(C_{mq}\))\({}_{osc}\) and \(C_{mq}\); (**c**) The effects of \(\dot{a}\)-derivatives on \(C_{2}\); (**d**) Time history of the dynamic derivatives of stability. Figure 6c,d and Figure 7c show dynamic derivatives of stability.