Dynamic Longitudinal Modes Questions and Answers

Answer: C

A vertical gust acts directly on incidence, and it is the short period that governs how that incidence, and therefore the load factor, builds and decays. The peak load arrives within the first fraction of a second, long before the phugoid could respond at all. The phugoid's duration is irrelevant to a peak load, since its incidence excursion is negligible. Gust loads are emphatically not static: the dynamic overshoot is the reason the calculation is done at all. Treating both as equal would mean expecting a ninety-second mode to matter in a one-second event.

Answer: D

A damping ratio of unity is critical damping: the two roots are real, equal and negative, and the aeroplane returns to trim as quickly as it can without overshooting. Below unity the motion oscillates, above it the aeroplane is overdamped and sluggish, which is why handling requirements put an upper bound as well as a lower one on short-period damping. An undamped oscillation is the zero-damping case. A divergence needs negative damping. And the boundary of static stability is set by the pitching moment derivative with incidence, not by a damping ratio.

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.

Answer: B

Both statements are correct. Any elevator input rich enough in frequency content will put energy into both modes, which is why flight-test records so often show a fast oscillation riding on a slow drift: the short period decays within a couple of seconds and leaves the phugoid running for the next minute. The two are then separated in analysis rather than in the air, by fitting each portion of the trace. Choosing the input shape carefully, a sharp doublet against a slow throttle change, is what makes each mode dominant enough to measure cleanly.

Answer: C

Gliders have less phugoid damping because lift to drag ratio is higher for gliders.

The phugoid is the slow exchange of speed and height. Drag is what damps it, and Lanchester's approximation gives the damping ratio as roughly

\[ \zeta \approx \frac{1}{\sqrt{2}\,(L/D)} \]

so the higher the lift to drag ratio, the lower the damping. Gliders are designed for very high L/D, so their phugoid is lightly damped. Being unpowered, light or slow does not appear in this relation; the damping is set by the aerodynamic efficiency.