Discussion
Flight test identifies the longitudinal roots as \( -2.4 \pm 3.2j \) and \( -0.008 \pm 0.070j \). The short-period natural frequency and damping ratio are
Answer: C
For a root \( -\sigma \pm j\omega_d \) the modulus is the natural frequency and the real part divided by it is the damping ratio:
\[ \begin{aligned} \omega_n &= \sqrt{2.4^{2}+3.2^{2}} = 4.0 \ \mathrm{rad/s} \\ \zeta &= \frac{2.4}{4.0} = 0.60 \end{aligned} \]
so 4.0 rad/s and 0.60, taking the fast pair as the short period. 3.2 rad/s uses the imaginary part, which is the damped frequency rather than the natural one. 5.6 rad/s adds the parts arithmetically instead of vectorially. The last option pairs the right frequency with the damping ratio computed as \( 2.4/3.2 \), which is \( \tan \) of the root angle, not its cosine.
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