Dynamic Longitudinal Modes Questions and Answers

Answer: A

The quartic normally factors into two oscillations: the phugoid, with a period of tens of seconds and a damping ratio of a few hundredths, and the short period, over in a second or two and damped between about 0.3 and 0.8. Their frequencies differ by a factor of thirty to a hundred, which is exactly why each can be studied with a two-degree-of-freedom approximation that ignores the other. Four real roots describe an aeroplane with no oscillatory longitudinal behaviour at all, which a conventional configuration does not have, and two positive ones would mean divergence. A repeated real pair is the critically damped special case, not the general one. Purely imaginary roots would mean undamped oscillation, so the aeroplane would ring forever after a gust.

Answer: B

The phugoid is a slow trade of height against speed at almost constant angle of attack. The aeroplane noses down, accelerates, generates more lift than weight, climbs, decelerates and noses down again, all while incidence barely moves. That is what makes it a long-period mode: nothing aerodynamically stiff opposes it, only the slow drain of energy through drag. The second option describes the short period, where incidence carries the motion. Pitch attitude and angle of attack are not in antiphase here; attitude and flight path rise and fall together, which is precisely why incidence hardly changes. A steady fall in incidence is not an oscillation at all.

Answer: C

The short period is a fast rotation of the aeroplane about its own lateral axis: angle of attack and pitch rate carry the motion, and forward speed has no time to change. The aeroplane is heavy and the mode lasts a second or two, so inertia in the surge direction holds the speed fixed. That assumption is what lets the mode be modelled with only the incidence and pitch-rate equations. A rapid speed change at fixed incidence is not a natural mode of a conventional aeroplane. The slow energy exchange is the phugoid, at the other end of the frequency scale. A non-oscillatory pitch departure is what remains once static stability has been lost, not the short period itself.

Answer: D

The first statement is the defining property of the mode. The second is nearly but not exactly true, and the difference is the whole reason the phugoid decays. Kinetic and potential energy trade back and forth almost losslessly, but drag removes energy on every cycle, and that slow leak is the damping. If total energy really were constant the motion would be undamped and the aeroplane would oscillate for ever. Lanchester's original idealisation did assume constant energy, which is why it predicts the period well and says nothing about the damping.

Answer: A

Typical numbers make the contrast plain: a transport's phugoid runs at about \( 0.07 \ \mathrm{rad/s} \) with \( \zeta \approx 0.05 \), while its short period sits near \( 3 \ \mathrm{rad/s} \) with \( \zeta \approx 0.5 \). So the phugoid is slower and far less damped on both counts. Calling it heavily damped inverts the one property pilots notice, which is that the oscillation persists for minutes. It cannot be the higher frequency of the two, since its period is measured in tens of seconds. And the modes are not two phases of one motion; they are separate roots of the same characteristic equation.

Answer: B

Both statements are true and the reason explains the observation. Releasing the stick leaves incidence trimmed and lets the aeroplane oscillate along its flight path, and the only thing removing energy is drag, which changes little over a small speed excursion. The result is a slow, weakly damped wandering of speed and height that can take several minutes to settle. The estimate \( \zeta_{ph} \approx C_D / (\sqrt{2}\, C_L) \) says the same thing algebraically: damping is inversely proportional to lift-to-drag ratio, so an efficient aeroplane has a worse phugoid.

Answer: C

Angle of attack carries the short period, together with the pitch rate that produces it. The mode is over in a second or two, so the aeroplane's mass prevents any appreciable change of forward speed, and there is no time to gain or lose useful height either. Speed and altitude are the phugoid's variables, not the short period's, which is exactly the separation the two approximations exploit. Thrust is a control input rather than a state, and it is normally held fixed while the mode is measured; an engine could not respond that fast in any case.

Answer: D

Dutch roll is a lateral-directional mode: a coupled yaw and roll oscillation that comes out of the lateral characteristic equation, alongside the spiral mode and roll subsidence. It has nothing to do with the longitudinal quartic. The other three all do. The phugoid and the short period are the two oscillatory roots, and aperiodic tuck divergence is what appears when the last coefficient of the longitudinal quartic changes sign at high subsonic Mach number. Longitudinal and lateral motions decouple for small perturbations about symmetric flight, which is why they are solved as two separate problems.

Answer: A

The standard description is a constant-incidence oscillation. Because the aeroplane is trimmed and the elevator is fixed, incidence is held close to its trim value while the flight path angle, speed and height all swing. Constant attitude is wrong: pitch attitude changes through the cycle, and it is the near-equality of attitude change and flight path change that keeps incidence still. Thrust is normally constant during the manoeuvre but that is a test condition, not what names the mode. Energy rate is not constant either; it is the small negative average of that rate which damps the motion.

Answer: B

The approximation keeps the two fast equations and freezes the slow one, so statements 1 and 2 are the whole of it: incidence and pitch rate are retained, the speed perturbation is set to zero. Statement 3 is not a requirement at all, and gets the logic backwards. The approximation works because the two modes are widely separated in frequency, whatever the phugoid's stability. It is just as valid for an aeroplane whose phugoid is slightly divergent, since the phugoid barely moves during the second or two that the short period takes to decay.