I am trying to figure out exactly what causes the pitching moment for the Phugoid mode? We know that alpha remains mostly constant, so what is causing the pitching moment? I thought maybe it is the trim point of the aircraft, but if trim was exactly neutral then how would the phugoid mode still proceed, since alpha is zero and at the design condition there would be zero tail load?

I guess my question then becomes what derivatives are responsible for causing the Phugoid mode or am I missing somehow that is just the dynamic response to the derivatives as a whole?

  • $\begingroup$ "since alpha is zero" -- why? $\endgroup$ – quiet flyer Mar 18 at 11:32

Everything. When we are talking about flight dynamics and stability in particular, it is the balance of forces or moments that we must consider. For simplicity of analysis, we often choose reference points that allow us to isolate some effects and make them independent of some variables, but if you are interested in the "physics" of it, then you should keep in mind that, in general, the forces and moments are everywhere, and they change.

Specifically, you seem to think that the tail load is zero in the trimmed condition. Not true. It could be true only for aircraft with neutral static longitudinal stability, which normally wouldn't display phugoid oscillations.

In the trimmed conditions, both the tail and the wing create moments (with respect to CG), but they balance out to zero. The essence of static stability is that upon the disturbance, the balance is offset such that it leads to restoration of the initial conditions.

In pure phugoid motion, the upset is airspeed but not angle of attack (AoA). Now, an airplane may have a tendency to have a pitch moment directly in response to speed, which we can designate $M^V$, or better as a coefficient $c_m^V$. This can be related to the nature of its thrust and/or moments from other parts. But this is not strictly required, and we can consider the simplest classical system of just a wing and a fixed tailplane. This system will not see a balance change simply due to airspeed. But consider what will happen:

  • Say, airspeed is suddently reduced due to wind shear. This is the only change, and AoA remains truly the same.
  • The moment balance remains the same, becuse lift of the wing and the tailplane changes proportionally. Thus no pitch change yet.
  • Lift will drop and will not sustain level flight anymore.
  • The aircraft will start accelerating down (while still in the same attitude!)
  • This acceleration has an effect of increasing AoA (due to the vertical component of speed). Notice that this angular effect depends on the absolute forward airspeed, which suggests that the effect will be lower at higher speeds.
  • Now the normal short-period static stability (the $c_m^{\alpha}$ or $c_m^{c_L}$ derivative) will cause a nose-down moment (if the system was stable) and will restore the original trimmed AoA.
    • This restoration effect normally happens much quicker than the speed change, especially for larger airplanes. So with a good approximation you could analyse just speed changes and assume AoA = const throughout the process. But physically this is not true, which, I guess, is the source of your confusion.
  • With the nose down, the aircraft will gain airspeed.
  • QED: the aircraft tries to negate the initial airspeed upset, which is the definition of static stability.
  • However, what happens next can be interesting. Due to high inertia, the aircraft will inevitably gain more airspeed than needed, and the process will reverse. Oscillations will ensue, and this is what is usually called "phugoid motion". The dynamics of this motion depends on several factors (absolute speed was already mentioned). Typically, this motion is much less dampened than the short-period AoA stability, and can even be dynamically unstable (i.e. have increasing amplitude). But this is not necessarily a problem: because the motion is slow, pilots can control it.

If you want derivatives, they have already been mentioned. Omitting derivation and with some simplifications, they combine to

$$\sigma_V = c_m^{c_L} - \frac{V}{2c_L} c_m^V$$

This $\sigma_V$ is the measure of static stability by airspeed. It is negative for stable aircraft. As can be seen, aircraft without direct speed-moment response ($c_m^V \approx 0$ - think gliders, for example) will have speed stability fully determined by the short-period AoA stability.

  • $\begingroup$ Re "Specifically, you seem to think that the tail load is zero in the trimmed condition. Not true. It could be true only for aircraft with neutral static longitudinal stability"-- hasn't it been established elsewhere on ASE that the tail load for an aircraft with positive static stability may be either uplifting, downlifting, or neutral? $\endgroup$ – quiet flyer Mar 18 at 11:31
  • $\begingroup$ From the static stability perspective, yes. That is basically just the airframe weathervaning in the vertical plane due to a gust. The AOA/speed seeking restorative tendency is TRIM and trim is a function of the opposing pitching moments, of ND moment from the wing/CG balanced by tail downforce, reaching equilibrium at trim AOA, whatever it's been set to with the trim control. The airplane pitches when displaced from trim AOA/speed because the trim forces were moved out of balance, and the interaction of the out of balance moments creates pitch movements as it seeks to regain balance. $\endgroup$ – John K Mar 18 at 13:22
  • $\begingroup$ Inertia and tail volume are also contributory, adding variables. A very large tail torque can contribute to AOA overshoot, too small and we get airspeed overshoot. Zeus bullet 7 is a gem. $\endgroup$ – Robert DiGiovanni Mar 18 at 14:55
  • $\begingroup$ @quietflyer, generally, you are right, although for stable aircraft this (zero tail load) will happen only at a certain AoA. Only for neutral stability this can be achieved throughout the whole range of AoA. (It then raises a question why would we need such tail at all. Perhaps for damping, which Robert hints at). I implied this general case. I should have worded more carefully, but I always fear excess of side details when explaining basics: I have a propensity for this and it tends to confuse people... $\endgroup$ – Zeus Mar 18 at 23:40
  • $\begingroup$ With a large speed (lift) change, more tail volume can make the oscillation worse by overcorrecting, too small a tail does not correct fast enough. Tails must be "right sized", but also large enough for directional stability and to assist stall recovery. @quiet flyer is right that a plane can be staticly stable with an uplifting tail (all the way out to delta tail/straight canard), but the torques around CG must make the nose rise when it goes faster than trim and vice versa. $\endgroup$ – Robert DiGiovanni Mar 19 at 1:10

So that all brings us to:

alpha remains mostly constant

The old constant alpha variable speed phugoid caused by pitch inertia.

speed increases, which increases downward lift on the tail,
pitching the nose up.

With a staticly stable aircraft, the $torque$ of the tail (around the center of gravity) reacts more strongly than the $torque$ of the wing to a change in lift caused by a deviation from trim speed. This means, as speed increases, the tail "pitches the nose up". This is an increase in Angle of Attack, resulting in greater lift and a curving flight upwards.

As the gravity vector swings behind the plane (aircraft reference), it works with drag to slow the plane down. This process repeats as the plane "seeks" its trim speed.

There is a very interesting interplay between changing speed (increases lift by its square) and changing AOA of the wing. Phugoid is observed in very "slick" gliders, which have very low drag.

In the event of loss of headwind, lift drops, the plane sinks, AOA of wing increases, wing/tail torque imbalance lowers nose. But the glider rapidly gains speed, resulting in a lift increase (and overshoot). A draggier plane might be more "in balance" between AOA change and speed change.

This is where one might look to better understand longer period phugoid motion. With weak tail downforce trim (such as when weight is moved back), the plane is more susceptible to speed change/lift differential oscillation. Combined with very low drag, this would increase the chances of phugoid motion. The "short period" motion may be other side of the spectrum, caused by excessive trim/weight forward causing a more poorly damped variation in AOA.

  • $\begingroup$ No, the torque (moment) doesn't change directly with the speed (for the simple wing-tail system). It only changes indirectly because of the AoA changes. In phugoid motion, all the lift changes are caused by speed changes (to the first approximation). $\endgroup$ – Zeus Mar 18 at 2:13
  • $\begingroup$ @Zeus The tail "reaction" is caused by changes in relative wind caused by a change in lift due to change in airspeed. Thanks Zeus, absolutely correct that the moments initially are the same when airspeed changes. Interesting it is broken into "short period" and "phugoid". What actually happens may be somewhere in between. $\endgroup$ – Robert DiGiovanni Mar 18 at 2:20
  • $\begingroup$ The difference between short and long period motion just "happens" for real airplanes (esp. larger ones; it is not valid for paper airplanes). It turns out that the angular motion (AoA/pitch variation) is much quicker than linear motion (speed/lift/altitude), and this is handy for analysis: we can focus on them separately. It's not "in between", its both. But as a result, we can consider AoA=const throughout phugoid motion if we are only interested in its dynamics rather than physics. $\endgroup$ – Zeus Mar 18 at 3:13

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