Just how do the underbanking and overbanking tendencies work?

Question

I understand that most GA airplanes and gliders exhibit an underbanking tendency at low bank angles and an overbanking tendency at high bank angles.

According to the Airplane Flying Handbook, chapter 3, page 3-11, if the aircraft is in steady, coordinated flight at a bank angle less than about 20 degrees, then

This shallow bank is such that the inherent lateral stability of the airplane slowly levels the wings unless aileron pressure in the desired direction of bank is held by the pilot to maintain the bank angle.

On the other hand, if the aircraft is in steady, coordinated flight at a bank angle greater than about 45 degrees, then

The airplane continues in the direction of the bank even with neutral flight controls unless the pilot provides opposite flight control aileron pressure to prevent the airplane from overbanking.

What causes these underbanking and overbanking tendencies, exactly? How do they work?

I know that when an aircraft is in a coordinated turn, the situation as experienced by the aircraft is almost identical to straight flight. As far as I know, the only significant differences are the following:

1. The outside wing will see a greater airspeed than the inside wing, which will mean:
   1. The outside wing will produce more lift than the inside wing, producing a rolling moment to the inside.
   2. The outside wing will also produce more drag than the inside wing, producing a yawing moment to the outside.
   3. On the other hand, the outside wing will also see a shallower angle of attack than the inside wing, which will partially negate both of the above effects. I'm guessing that the lift effect will be negated more strongly than the drag effect.
2. The nose will see relative wind from the inside of the turn (which will have no significant effect); and the tail will see relative wind from the outside of the turn, which will produce a yawing moment to the outside.
3. Since the rudder is deflected towards the inside of the turn, the vertical stabilizer will produce a rolling moment to the outside.

So my questions are:

Is my understanding of the rolling and yawing forces in turning flight correct? Are there any significant effects I've missed? And how large is each of these effects—which ones are the big ones, and which ones are insignificant? And most of all, how come there's an underbanking tendency in shallow turns, but an overbanking tendency in steep turns?

Answer 1

I think your reasoning is correct. However, I would say that the change in lift due to effective AOA on each wing from yaw rate is an order of magnitude smaller than that from airspeed. At high bank angle, the wing loading is higher, resulting in more inward turn and decreased spiral stability. While the yaw-out moment from the drag is also increased due to relative airspeed difference on each wing, the effect is smaller since drag is smaller than lift.

Let's look at it more mathematically. Static spiral stability is dictated by:

$$C_{l_\beta}C_{n_r}-C_{l_r}C_{n_\beta}>0$$

where $C_{l_\beta}$ is roll moment due to sideslip, must be negative for a Part 23 aircraft; $C_{n_\beta}$ is yaw moment due to sideslip, must be positive for Part 23; $C_{n_r}$ is yaw-out moment from yaw rate, where, as you reasoned, should be negative due to increased drag on the up wing and reduced drag on the down wing (due to airspeed difference from yaw rate); $C_{l_r}$ is the roll-in moment from yaw rate, where, as you reasoned, should be positive due to increased lift on the up wing and decreased lift on the down wing (due to airspeed difference).

At high bank angle, there is increased load factor (at 45deg, it's 1.4G). Therefore, there is increased lift and induced drag on each wing. The moment arm to induce the moment is about the same for each. However, if you look at the first order effect from the speed increase, it's:

$$\Delta F=\frac{1}{2}\rho (V+\Delta V)^2SC_x-\frac{1}{2}\rho V^2SC_x\approx\rho VSC_x\Delta V$$

So all things being equal, the larger the coefficient, the more pronounced the effect. And since lift is larger than drag, it wins out. (I'm still hand-waving here a bit since induced drag is squared of lift, so it's more nuanced).

With this in mind, you can see that at higher bank, $C_{l_r}$ increases more readily than $C_{n_r}$, therefore leading to reduced spiral stability.

Answer 2

Underbanking tendency at a shallow bank angle will vary with design. High wing dihedralled trainers need aileron to hold a shallow bank angle. A symmetrical, aerobatic plane with little dihedral, not nearly as much.

Overbanking at high bank angle is a result of spiral instability, which is dependent on vertical empennage area (really balance of total fore and aft area relative to CG) and can be exacerbated by excessive static stability created by setting CG too far forward with a lot of nose up trim. As plane rolls and slips, side force on the tail turns it into a spiral and speeds up the outside wing, creating a greater lift imbalance. The elevator trim contributes to the spiral, tightening the turn.

Designers try to balance spiral instability with roll correcting tendency (sweep, dihedral, tall vertical fin, better fore and aft area balance).