Discussion
Using the same data as the standard short-period approximation, with \( M_q = -2.0 \ \mathrm{s^{-1}} \), \( M_{\dot\alpha} = -0.4 \ \mathrm{s^{-1}} \), \( Z_\alpha/u_0 = -1.5 \ \mathrm{s^{-1}} \) and \( \omega_n = 3.32 \ \mathrm{rad/s} \), the damping ratio is closest to
Answer: A
The damping term collects all three negative derivatives:
\[ \begin{aligned} 2\zeta\omega_n &= -(M_q + M_{\dot\alpha} + Z_\alpha/u_0) \\ &= 3.9 \ \mathrm{s^{-1}} \\ \zeta &= \frac{3.9}{2 \times 3.32} = 0.59 \end{aligned} \]
So 0.59, comfortably inside the Level 1 band for a fighter. 0.29 forgets to halve, or rather divides by \( 2\omega_n \) twice over. 1.18 omits the factor of two altogether and would imply an overdamped mode. 0.39 keeps only \( M_q \) and \( Z_\alpha/u_0 \) and drops the downwash-lag term, which is the most commonly neglected of the three and is worth about a tenth of the damping here.
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