Discussion
An aeroplane's short period has \( \omega_n = 4.0 \ \mathrm{rad/s} \) and \( \zeta = 0.5 \). Loading it further aft lowers the frequency to \( 2.5 \ \mathrm{rad/s} \) while the real part of the root is unchanged. The new damping ratio is
Answer: A
The real part is \( \sigma = \zeta\omega_n = 0.5 \times 4.0 = 2.0 \ \mathrm{s^{-1}} \), and it is given as unchanged, so
\[ \begin{aligned} \zeta_{2} &= \frac{\sigma}{\omega_{n2}} \\ &= \frac{2.0}{2.5} = 0.80 \end{aligned} \]
The damping ratio therefore rises to 0.80 as the frequency falls. This is the standard trade with aft loading, and it explains an apparent paradox in flight test: an aeroplane can feel better damped and worse controlled at the same time. 0.31 scales the wrong way. 0.50 assumes damping ratio is a fixed property, which it is not. 1.25 is the frequency ratio rather than the damping ratio.
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