What are the dynamic stability properties of an aircraft with wings exactly at the vertical CoG?

Question

Please ignore for a moment the practical issues with building and operating an aircraft like this.

Suppose a fixed-wing aircraft which, when viewed directly along the roll axis, has the wings centered exactly at the vertical center of gravity. Suppose that this aircraft is in stable, level flight in otherwise still air. (Yes, this is a little like the spherical cow in a vacuum from physics.)

If there is a disturbance, such as a localized updraft or downdraft affecting one of the wings more than the other, then broadly speaking and as I understand it so far (feel free to address this if I've got it completely backwards), after exiting the disturbance and with no further control inputs, a high-winged aircraft will tend to return to neutral roll (dynamically stable) and a low-winged aircraft will tend to increase its roll angle further (dynamically unstable), as a result of the differences in localized airflow above and below the respective wings.

However, what would be the behavior in such a situation of our hypothetical mid-winged aircraft that I described above?

My hypothesis is that the aircraft would maintain its roll or bank angle, which (unless compensated for, say by rudder input) in turn will result in a rotation along the yaw axis and a subsequent turn.

Am I right or wrong? Why?

Answer 1

It is a common misconception that the vertical CG position (relative to the wing) plays a major role in roll stability. It doesn't; it's mostly about aerodynamics. (You mention it yourself in "as a result of the differences in localized airflow...")

There are other important factors at play as well: dihedral angle, wing sweep, design of the vertical stabiliser.

Presumably, you are interested in the behaviour of an aircraft with the perfectly neutral roll stability. If so, it needs at least these conditions:

• General symmetry about the horizontal plane, which includes:
  • Zero wing dihedral;
  • Symmetric (e.g. round) fuselage;
  • Wing at the centre of fuselage (vertically);
  • Symmetric vertical stabiliser (e.g. with dorsal fin);
  • Absolute rigidity so that the symmetry remains under load.
• Zero wing sweep (straight wing);
• Lack of any artificial stability augmentation.

(Technically speaking, the geometric symmetry is not necessarily required: we can compensate one effect with another, say, wing sweep with anhedral. But in the spirit of the original question, let's assume the full symmetry).

We should also assume a reasonable stability in pure yaw.

So, if you induce a pure roll disturbance and then remove it,

• the aircraft will start rolling.

Because it has some inertia about the roll axis, it will not simply stop when you remove the disturbance. The only force that will counteract the roll now is roll damping. It naturally occurs because the descending wing has higher angle of attack (and vice versa). The strength of the effect depends on the square of the wingspan, but in theory the spherical cow will keep turning forever, albeit with ever-diminishing speed.

So far, nothing would induce a yaw, and the airflow would still be symmetric.

However, as roll develops, and the aircraft keeps the same pitch balance, the lift will become insufficient to keep it level. The usual thing happens:

• the aircraft enters a spiral dive.

Now, this is not a symmetric situation anymore. There is a side force from gravity; sideslip will develop; the aircraft will yaw due to its yaw stability that we assumed. Things start to get complicated (that is, 'normal'), with many variables involved. Even in the vertical plane, despite postulated symmetry, the shadowing and other cross-flow effects on the top and bottom of the wing will be different (the wing, by definition, cannot be aerodynamically symmetric when it generates lift), and thus some roll effect may be observed.

So it's hard to predict further behaviour in general. At one extreme, with stubby wings and slender body and very strong initial disturbance, the aircraft may barrel roll many times and may even enter a peculiar stable roll motion due to inertia coupling. At the other, the aircraft will just roll a little and will enter a gentle (nearly) coordinated turn with very slight descent.

Answer 2

In order for an aircraft to return to neutral roll the shape of the wings is an important factor as well and a number of other aerodynamic factors will play a role. A positive V-shape for example always helps to return to neutral. Furthermore wings are sometimes also designed to bend under load too which will have the same effect as the V-shape.

But if cows are spherical I would say your hypothesis is right. On high-winged aircraft the leverage of the lower wing when the aircraft is banked is longer than that of the upper wing resulting in a torque in the opposite direction of bank; resulting in a decrease of bank angle. If the wing is placed on the center of gravity the leverages of the forces on both wings would be equal regardless of bank angle and thus the total torque would be zero; the bank angle would stay as it is.