[Comments or questions]
Copyright © 1996-2001 jsd
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 DihedralBack 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.
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.
: 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 CouplingDihedral 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
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.
: Redirection — High-Wing Airplane in
a Slip (1)
: 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.
: Redirection — Low-Wing
Airplane in a Slip (1)
: 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.
: Turbulence — High-Wing
Airplane in a Slip
: 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
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.
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).
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 StabilityWe 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
Airplane designers always make sure the airplane has a certain amount
of slip-roll coupling, for exactly this
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 SlipBefore 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.
: 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.
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; OverbankingFigure 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.
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 DragThe engine makes a contribution to the roll axis
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
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
9.6 Engine InertiaNewton'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
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.
: 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
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 TurnsIn 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 KTAS while
climbing or descending at 500 fpm. We can calculate the resulting
angle of attack at the wingtips, as shown in
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.
Vertical speed (fpm)
Angle of climb
Angle of attack
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 — SummaryThere are several effects that can give rise to a
rolling moment. The most important ones are:
Some of these ideas will be revisited when we discuss
Dutch roll in section 10.6.1.
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
- 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 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.
a discussion of the terminology and general principles of forces
and moments, you can refer to section 19.7.
- i.e. an airflow pattern that is flowing left-to-right
or right-to-left over the fuselage.
- See the discussion of
upwash in section 3.1, including figure 3.2.
- See the discussion of
dimples on golf balls in section 18.3
and in reference 17.
- Excessive slip-roll coupling will cause the airplane
to suffer from Dutch roll, as discussed in section 10.6.1.
The figure greatly exaggerates how tightly the flow pattern is wound.
- There will
be all sorts of ``closed-circuit'' torques, but they will not affect
the handling of the airplane.
[Comments or questions]
Copyright © 1996-2001 jsd