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9 Roll-Axis Torque Budget

Non-pilots have the idea that pilots keep the airplane in the air by firm intervention and lightning reflexes. In fact, though, pilots are judged more on smoothness than quickness.

This chapter considers the various forces that could impart a rolling moment to the airplane.¹

9.1 Dihedral

Back in section 8.2, we discussed how an uncoordinated relative wind² will affect the yaw axis; now let's see how it will affect the roll axis.

The first thing that people think of in this connection is dihedral. The word comes from the Greek word for ``two planes'' and just means that the two wings are not coplanar, as shown in figure 9.1.

Figure 9.1: Definition of Dihedral

In the presence of dihedral, any uncoordinated (side-to-side) airflow will hit the bottom of one wing and the top of the other wing, as shown in figure 9.2. This means one wing will be forced up and the other forced down. If you work out all the angles between the total relative wind and the wings, you find that indeed the angle of attack is increased on the upwind wing and reduced on the downwind wing. The difference in lift produces a rolling moment. Any process whereby uncoordinated airflow produces a rolling moment it is called a slip-roll coupling; dihedral is a good example of this.

Figure 9.2: Dihedral — Slip Produces Rolling Moment

The rolling moment (the torque about the roll axis) will be proportional to the dihedral angle, and proportional to the amount of slip.

9.2 Other Forms of Slip-Roll Coupling

Dihedral is only one of several reasons why an airplane might have a slip-roll coupling. A high-wing airplane has a certain amount of slip-roll coupling because of interference effects. That is, when the airplane is in a slip, the fuselage interferes with the airflow over the wing. As shown in figure 9.3 and figure 9.4, the stream lines have to bend a little in order to flow around the fuselage. This creates an updraft at the root of the upwind wing, and a downdraft at the root of the downwind wing. This creates a rolling moment that tends to raise the upwind wing.

Figure 9.3: Redirection — High-Wing Airplane in a Slip (1)

Figure 9.4: Redirection — High-Wing Airplane in a Slip (2)

As shown in figure 9.5 and figure 9.6, on a low-wing aircraft the effect is reversed. There is an updraft at the root of the downwind wing, and a downdraft at the root of the upwind wing. This contributes a negative amount of slip-roll coupling.

Figure 9.5: Redirection — Low-Wing Airplane in a Slip (1)

Figure 9.6: Redirection — Low-Wing Airplane in a Slip (2)

A related interference effect is shown in figure 9.7. A fuselage moving sideways through the air is a very non-streamlined object. Downstream of such an object you expect to find a large, messy wake. The air in the wake is less capable of producing lift when it flows over the wing.

Figure 9.7: Turbulence — High-Wing Airplane in a Slip

Figure 9.8: Turbulence — Low-Wing Airplane in a Slip

Because the air that the wing really cares about is coming from ahead and below,³ this type of interference is more pronounced in a high-wing airplane — the fuselage is in a stronger position to disturb the relevant airflow. This is can be seen by comparing figure 9.7 with figure 9.8.

The magnitude of the effect of the wake is very difficult to predict. It will depend not only on the general shape of the fuselage, but also on the details of the surface finish.⁴ It will also depend very nonlinearly on the airspeed and slip angle.

In general, interference effects mean that (to achieve an adequate amount of slip-roll coupling) low-wing airplanes typically need more dihedral than high-wing airplanes. You can check this by looking at typical airplanes at your local airport.

A third effect is illustrated in figure 9.9. If you put a swept-wing airplane into a slip, the more-forward wing produces more lift. That wing (the left wing in the figure) presents effectively more span to the airstream. It is a common mistake to think that the increased span explains the increased lift. The mistake is to overlook the fact that when it presents more span, it necessarily presents less chord. Lift, other things being equal, is proportional to wing area, and it is well known that area is not changed by a rotation. The correct explanation has more to do with the direction of airflow. Air flowing spanwise along an airfoil doesn't produce lift. The key idea is that the chordwise component of the airflow is bigger for the left wing.

Figure 9.9: Swept Wing Airplane in a Slip

A fourth effect is shown in figure 9.10. In practically all aircraft, the rudder sticks up above the roll axis. When the aircraft is in a slip, the rudder produces a substantial force. This force times this lever arm produces a torque about the roll axis.

Anything else that sticks up above the roll axis and produces sideways drag or sideways lift contributes the same way. This includes the wings of a high-wing airplane, although the effect is small since spanwise flow along a wing doesn't create much force — just a little bit of sideways drag. This is another reason why high-wing airplanes can get by with less dihedral (for the same amount of slip-roll coupling).

Figure 9.10: Tall Rudder in a Slip

All four effects just mentioned are in the same direction, and can be combined: You can have a high-wing, swept-wing airplane with lots of dihedral and a really high tail — in which case you would probably have more slip-roll coupling than you need.⁵

The propwash contributes a negative amount of slip-roll coupling when the engine is producing power. If you yaw the nose to the right, the uncoordinated component of the wind will blow more of the propwash to the right wing. The extra lift on the right wing will roll you to the left.

Slip-roll coupling is the reason why you can make a relatively normal turn with the rudder (inelegant though it is). If you gently press on the right rudder, you will cause a skid that will eventually produce a bank to the right. Of course the skid itself will also cause a boat turn to the right. If you hold a constant rudder deflection, the boat-turn force will only be proportional to the rudder deflection, whereas the bank (and the associated non-boat turn) will keep getting larger and larger because of the slip-roll coupling.

9.3 Roll-Axis Stability

We are now all set to understand how the airplane responds if, for some reason, one wing goes a bit lower than the other.

The airplane will start to turn. If the turn were perfectly coordinated, the airplane would be happy to keep turning around and around and around. Fortunately, as we recall from our discussion of the long-tail slip effect (section 8.9), ``an inadvertent turn will be a slipping turn''. This tiny amount of slip, acting through the slip-roll coupling, will tend to roll the airplane back to wings-level straight-ahead flight.

This process gives the airplane a slight amount of roll-axis stability.

Airplane designers always make sure the airplane has a certain amount of slip-roll coupling, for exactly this reason.

The roll-axis stability is rather weak, because the two necessary ingredients are individually weak: The slip-roll coupling is usually moderately weak, and the long-tail slip effect is so weak that (except for glider pilots) most pilots never notice it unless it is pointed out.

Common experience indicates that roll-axis stability is indeed rather weak. If you are cruising along in turbulent air and take your hands off the controls for a couple of moments, you do not expect the nose to pitch up or down 30 degrees, and you do not expect it to yaw left or right 30 degrees, but you would not be at all surprised to have a 30 degree bank develop.

Even in the best of conditions, the stability generated by the long-tail slip with slip-roll coupling can only overcome a small amount of uncommanded bank. For larger bank angles, the overbanking tendency (section 9.4) takes over and makes the roll axis unstable.

* Dihedral in the Absence of Slip

Before going on, let's take another look at what happens in a coordinated turn. Sometimes it is argued that when the airplane is in a bank, the lowered wing has a bigger footprint (a bigger projection on the ground) than the raised wing, as shown in on the left in figure 9.11. A similar argument was used back in section 9.2 to explain why swept wings produce a slip-roll coupling. There is one slight difference: the swept-wing effect is real (because it involves the direction of the air) whereas the supposed effect of dihedral in a coordinated bank is completely imaginary. The wing doesn't know or care where the ground is. It cares only where the air is coming from. In a coordinated turn, the air is coming from straight ahead, so dihedral has no effect.

Figure 9.11: Dihedral Has No Effect in the Absence of Slip

Other myths about dihedral involve the angle of the lift vectors of the two wings. The correct answer is the same: in the absence of slip, dihedral has no effect. As long as the air is coming from straight ahead, the lift vectors are symmetrically disposed, as shown in figure 9.12.

Figure 9.12: Symmetric Lift Vectors

In a coordinated turn, the aircraft is happy to continue turning forever; it will definitely not have any tendency to return to wings-level flight. Indeed, it will have the opposite tendency, called the overbanking tendency, which we now discuss.

9.4 Differential Wingtip Speed; Overbanking

Figure 9.13 shows the aircraft in a coordinated turn. The outside wingtip follows a path of length 2 p R (big R) while the inside wingtip has the proverbial ``inside track'' — its path is only 2 p r (little r). Since the outside wingtip travels farther in the same amount of time, it must be moving faster.

Figure 9.13: Overbanking Tendency

The same fact is depicted a second time in the figure — the relative wind is depicted to be stronger on the outside wingtip. Since the lift generated by an airfoil depends on the square of the airspeed, the outside wing would produce more lift (other things being equal). This means that the aircraft in a turn (especially a properly coordinated turn) will tend to bank into the turn more and more. The steeper the turn, the more pronounced this overbanking tendency becomes. The next thing you know, you are in a spiral dive (as discussed in section 6.2).

In order to combat this tendency, you need to deflect the ailerons against the turn.

The strength of this effect depends on the ratio of the wingspan to the radius of turn. If you have stubby wings, high airspeed, and shallow bank angle, you'll never notice the effect. On the other hand, in a glider you might have long wings, low airspeeds and steep turns — in which case you might need nearly full outside aileron just to maintain a steady bank angle.

It is interesting to combine this with what we learned about long-tail slip effect (section 8.9) — in the slow, steeply banked turn in the glider, you would be holding nearly full inside rudder (to prevent the long-tail slip) and nearly full outside aileron (to counteract the overbanking tendency). If you are not expecting this, it will appear very strange. You are holding completely crossed controls, yet the turn is perfectly coordinated. You can confirm this by observing that the slip string is perfectly centered.

You don't want to have to figure this out while sitting in the glider, trying to make a steep turn. Sometimes it pays to read the book before you go flying.

9.5 Rolling Moment due to Propeller Drag

The engine makes a contribution to the roll axis torque budget.

As we remarked earlier, the propeller does not throw the air straight back; there is some rotational drag on the propeller blades. According to Newton's law of action and reaction, you can see that if the prop throws the air down on the right, it tends to make the airplane roll to the left.

To put it more crudely: take a model airplane (where the propeller rotates to the right) and hold it by the propeller. If you start the engine, the airplane will rotate to the left.

As shown in figure 9.14, some of the rotating air hits the top of the right wing and the bottom of the left wing.⁶ This tends to reduce the amount of roll — but it can never reduce it to zero or cause a roll to the right. Similarly, any air intercepted and ``straightened out'' by the tail reduces the rolling moment somewhat. Using Newton's law again, we see that if any air escapes while still rotating down to the right, the airplane will roll to the left.

Figure 9.14: Rolling Moment Produced by Propeller

The only way to restore equilibrium is to take a corresponding amount of air and throw it down on the left. Airplane designers have long since learned about this propeller drag rolling moment, and they take steps to compensate for it. For instance, they set the left wing at a slightly higher angle of incidence than the right wing. This is called, unsurprisingly, asymmetric incidence. It is especially useful to apply this trick to the part of the wing that flies in the propwash, so that the effect increases as engine power increases. On a Piper Cherokee, the roll-axis trim is rather easy to adjust — in the flap extension mechanism for each flap there is a turnbuckle that allows the flap to be raised or lowered until the roll-axis trim is just right.

If the roll-axis trim is just right in cruise, it will be nowhere near right during a soft-field takeoff. In that case, the propeller drag will be worse because of the high power, and the fancy rigging of the airfoils will be less effective because of the low airspeed. The result: you will have to deflect the yoke to the right, using the ailerons to counter the prop drag rolling moment.

9.6 Engine Inertia

Newton's second law asserts that force equals mass times acceleration. There is a rotational version of this law, asserting that the rotational force (i.e. torque) equals the rotational inertia times the rotational acceleration. That means whenever the engine RPMs are increasing or decreasing, a torque is produced.

There is also a rotational version of Newton's third law, asserting that if you impart a clockwise rotational momentum to one thing, you must impart a counter-clockwise rotational momentum to something else.

Consider an airplane which has the engine aligned in the usual way, but where the propeller-drag effects (discussed in section 9.5 are negligible. The easiest way to arrange this is to have a single engine driving two counter-rotating propellers. The Wright brothers used this trick in their first airplane.

While (and only while) the engine speed is changing, the airplane will tend to roll. It will roll to the left if the engine is accelerating, and it will roll to the right if the engine is decelerating.

In steady flight in this airplane, the the engine's rotational inertia has no effect. The fact that the engine / dual propeller system is producing power does not imply that it is producing any net torque.

To clarify the distinction, compare the two situations shown in figure 9.15 and figure 9.16. We have an ordinary single-engine airplane. We have removed the propeller and bolted a huge brake drum onto the propeller shaft. In figure 9.15, the brake shoes are attached to the floor of the hangar. When we run the engine, the brake will produce a huge torque that will make the airplane want to roll to its left. This is completely analogous to the propeller drag effect discussed in section 9.5. There is an instrument called a prony brake that measures the torque-producing capability of an engine in precisely this way.

Figure 9.15: Prony Brake Attached to the Floor

Figure 9.16: Prony Brake Attached to the Airplane Itself

In figure 9.16, the brake shoes are attached not to the floor, but to the airplane itself. Even if the engine is producing torque (straining against its engine mounts) all the torques flow in a closed circuit and cancel. The airplane as a whole exhibits no rolling tendency.

Newton's law is quite explicit about this: if you want to give the airplane some left-rolling momentum, you have to give something else some right-rolling momentum. This ``something else'' could be the air (as in figure 9.14) or perhaps the hangar floor (as in figure 9.15).

The angular motion of the internal engine parts can only affect the rolling moment if you change their rotational speed.

Engine rotational inertia should not be confused with propeller drag. In a direct-drive propeller installation, the propeller-drag torque does act on the fuselage via the engine mounts, but that is a coincidence, not a law of physics. In a gear-drive installation, most of the propeller-drag torque acts on the fuselage via the gearbox.

Finally, consider the case where the engine rotates one way and the propeller rotates the other way (which is easy to arrange using a gearbox). In a steady slow-flight situation, I guarantee you will need to deflect ailerons to compensate for propeller drag; engine inertia per se will have no effect on pilot technique.

9.7 Climbing and Descending Turns

In a level turn both wingtips are moving horizontally. In a climbing turn, both wingtips will be climbing, but they will not make equal angles to the horizon. This is because the climb angle depends on the ratio of the vertical speed to the forward speed. As a result of the different climb angles, we get different angles of attack for the two wingtips. The geometry of the situation is shown in figure 18.6 (in the chapter on spins). Another way to think about this is to recognize that it involves rotating around a non-vertical axis, as discussed in section 19.6.4.

Let's do an example. Consider an airplane with a 35-foot wingspan making a standard-rate turn (3 degrees per second) at 100 K_TAS while climbing or descending at 500 fpm. We can calculate the resulting angle of attack at the wingtips, as shown in table 9.1.

Airspeed (K_TAS)

Vertical speed (fpm)

Angle of climb
Angle of attack

Climbing turn

inside wingtip
99.46 500 2.844^° 4.485^°

outside wingtip
100.54 500 2.814^° 4.515^°

difference 2.2% 0.7%

Descending turn

inside wingtip
99.46 -500 -2.844^° 4.515^°

outside wingtip
100.54 -500 -2.814^° 4.485^°

difference 2.2% -0.7%

Table 9.1: Climbing and Descending Turns

We see that the change in angle of attack typically has less effect than the change in airspeed. In a climbing turn, the angle effect contributes to the overbanking tendency, while in an ordinary descending turn, it somewhat reduces it.

In a spin (which has a higher vertical speed, lower airspeed, and vastly higher rate of turn) the angle effect is extremely significant, as discussed in section 18.6.1. Far from reducing the rolling moment, increasing the angle of attack on the inside wing (which is stalled) only makes the situation worse.

9.8 Roll-Axis Torque Budget — Summary

There are several effects that can give rise to a rolling moment. The most important ones are:

A slip tends to cause a rolling moment, for several reasons, including: dihedral produces slip-roll coupling; the fuselage shadowing one wing (especially on a high-wing airplane) produces slip-roll coupling; swept wings produce slip-roll coupling, and a tall rudder that sticks up above the roll axis produces slip-roll coupling.
This slip-roll coupling combines with the long-tail slip effect (discussed in section 8.9) to give the airplane a small amount of stability around the roll axis.
At medium or large bank angles the overbanking tendency makes the roll axis unstable; the bank angle will tend to get larger and larger. This produces a spiral dive.
There exists some medium-small bank angle where the two just-mentioned effects cancel. (That is, the overbanking tendency cancels the stability due to slip-roll coupling plus long-tail slip.) At this bank angle, the airplane will happily continue turning, at constant bank angle, without any help from the pilot.
The airplane tends to roll left in high-power / low-airspeed situations, because of propeller drag.
If you suddenly change the speed of rotation of the engine, the rest of the airplane will be subjected a brief rolling impulse. (Similarly, if you change the direction of the axis of rotation, gyroscopic precession will cause yawing and/or pitching moments.) If the speed and direction of rotational motion is unchanging, engine torque will have no noticeable⁷ effects.

Some of these ideas will be revisited when we discuss Dutch roll in section 10.6.1.

1: For a discussion of the terminology and general principles of forces and moments, you can refer to section 19.7.
2: i.e. an airflow pattern that is flowing left-to-right or right-to-left over the fuselage.
3: See the discussion of upwash in section 3.1, including figure 3.2.
4: See the discussion of dimples on golf balls in section 18.3 and in reference 17.
5: Excessive slip-roll coupling will cause the airplane to suffer from Dutch roll, as discussed in section 10.6.1.
6: The figure greatly exaggerates how tightly the flow pattern is wound.
7: There will be all sorts of ``closed-circuit'' torques, but they will not affect the handling of the airplane.

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		Airspeed (K_TAS)	Vertical speed (fpm)	Angle of climb	Angle of attack
Climbing turn	inside wingtip	99.46	500	2.844^°	4.485^°
	outside wingtip	100.54	500	2.814^°	4.515^°
	difference	2.2%			0.7%
Descending turn	inside wingtip	99.46	-500	-2.844^°	4.515^°
	outside wingtip	100.54	-500	-2.814^°	4.485^°
	difference	2.2%			-0.7%