## 3 Numerical Results and Discussion
To be stable, \(C_{z\dot{\alpha}}>0\), \(C_{m\dot{\alpha}}<0\), \(C_{l\dot{\beta}}<0\), and \(C_{n\dot{\beta}}>0\). The magnitudes of \(C_{z\dot{\alpha}}\) and \(C_{m\dot{\alpha}}\) have significant variations and \(C_{z\dot{\alpha}}<0\) in the period of t = 7482.5-7491 s in Figure 6b. It should be noted that \(C_{z\dot{\alpha}}\) represents the virtual mass effect and is particularly large in transonic flow to affect the plunging motion [28]. The magnitudes of \(C_{l\dot{\beta}}\) in Figure 7c and \((C_{lp})_{osc}\) in Figure 7a are small; \(C_{lp}\) is not shown because it is small throughout. The values of \(C_{n\dot{\beta}}\) are from positive at t = 7483 s to negative at t = 7484 s; those of \((C_{nr})_{osc}\) are from positive at t = 7483.5 s to negative at t = 7485 s. This implies that the effect of \(\dot{\beta}\)-derivative is to cause the directional stability to become more unstable. Figure 4.0 Time history of main lateral-directional oscillatory derivatives along the flight path. (**a**) Time history of \((C_{4\dot{\alpha}})_{osc}\) and \(C_{4\dot{\beta}}\); (**b**) Time history of \((C_{nr})_{osc}\) and \(C_{nr}\); (**c**) The effects of \(\dot{\beta}\)-derivatives on \(C_{n}\) and \(C_{r}\). In essence, the effects of \(\dot{a}\)-derivative on \((C_{2q})_{osc}\) and \(\dot{\beta}\)-derivative on \((C_{lp})_{osc}\) are small in Figures 6 and 7. The effect of \(\dot{\alpha}\)-derivative on \((C_{mq})_{osc}\) is to improve the stability in pitch; the effect of \(\dot{\beta}\)-derivative is to cause more directional instability. These results indicate that the turbulent crosswind has some adverse effects on directional stability and damping. Although the dynamic derivatives tend to be small for the present configuration, these are helpful to understand the unknown factors of instability characteristics. In the present study, the longitudinal and lateral-directional motion modes are analyzed based on the damping ratio (\(\xi\)) and undamped natural frequency (\(\omega_{n}\)). The roots of the complex conjugate are as follows:  
Figure 4.1  
where \(-\zeta\omega_{n}\) is a real part (i.e., in-phase) and \(\pm i\omega_{n}\sqrt{1-\xi^{2}}\) are imaginary (i.e., out-of-phase) parts. \(\lambda_{r}\) and \(\lambda_{i}\) represent eigenvalues of real and imaginary parts, respectively. If \(\lambda_{r}\) is positive, the system is unstable; if it is negative, the system is stable [13]. Figure 8 presents the eigenvalues of the short-period and phugoid modes. The eigenvalues of the real part in Figure 8a are small negatives, except two portions at t = 7486.2 s and t = 7492.0-7493.0 s are small positives. The short-period mode is typically a damped oscillation in pitch about the body \(y\)-axis in normal flight conditions. The short-period mode is mostly in a stable condition due to the value of \((C_{mq})_{osc}\) being mostly negative to have adequate pitch damping in oscillatory motions in the time span t = 7484-7486 s with largest drop-off height, the highest a\({}_{2}\) being 2.05 g around t = 7483 s and the lowest being \(-1.05\) g around t = 7484 s in Figure 4a, and the highest AOA being about 4 deg. in Figure 4b. The eigenvalues of the real part in Figure 8b are positive and the magnitude varies like a mountain chain with pinnacle. The value of \((C_{2q})_{osc}\) is negative and the magnitudes of \(C_{z\dot{\alpha}}\) also have negative values; \((C_{2q})_{osc}\)is insufficient in oscillatory damping with virtual mass effects during sudden plunging motion to cause the phugoid mode in an unstable condition. Note that part of the conventional phugoid mode has degenerated into the plunging mode. Figure 4.2 Eigenvalues of longitudinal modes of motion. (**a**) The short-period mode; (**b**) The plunging mode. Figure 9 presents the eigenvalues of the Dutch roll, spiral, and roll modes.