Discussion
For a conventional aeroplane, which mode does the natural frequency \( \sqrt{-M_\alpha} \) approximate?
Answer: A
That expression is the one-degree-of-freedom estimate of the short-period frequency, treating the aeroplane as a torsional spring in pitch whose stiffness is the pitching moment derivative with incidence. It shows immediately why the mode slows as the centre of gravity moves aft. It is not the phugoid, whose frequency is \( \sqrt{2}g/u_0 \) and contains no aerodynamic derivative at all. The heave subsidence and the tuck divergence are aperiodic, so neither has a natural frequency in this sense.
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