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Copyright © 1996-2001 jsd
8 Yaw-Axis Torque Budget
In aircraft (unlike cars, bikes, or sailboats) you
have separate control over which way it is pointing relative
to which way it is going.
8.1 OverviewThis chapter discusses the motion
of the airplane around the yaw axis.1 The conventional definitions
of the three principal axes are shown in figure 19.9 in section 19.6.1. For a discussion of the terminology and general principles
of forces and moments, you can refer to section 19.7.
(The motion around the other axes is discussed in the following
For a precise definition of terms such as yaw
angle, heading, and slip angle, please
refer to section 19.6.3.
It is important to keep the slip angle to a
minimum, that is, to make sure the aircraft is pointing
the same direction as it is going through the air. This
is important for several reasons:
Maintaining zero slip angle while maneuvering requires
coordinated use of the
ailerons and rudder, so pilots speak of
``zero slip angle'' and ``good coordination'' almost
- Efficiency: If your objective is to turn to the
left, it doesn't make sense to let the maneuver begin with a big
inadvertent yaw to the right. Also, all that slipping
creates lots of unnecessary drag.
- Comfort: Passengers really hate being sloshed
from side to side. In many trainers, the pilots are seated so
close to the pivot point (the center of mass) that they don't
appreciate the effect yawing motion can have at other locations.
- Safety: Whereas if you stall in coordinated flight
the nose will just drop straight ahead, if you manage to stall
in sufficiently uncoordinated flight, you will get a spin (see
chapter 18) or a snap roll (see chapter 11), which
is much harder to recover from.
This section considers, one by one, the various phenomena
that affect the airplane's motion around the yaw axis. There
are surprisingly many such phenomena, including the helical propwash,
yaw-axis inertia, adverse yaw, P-factor, and gyroscopic precession
— plus the stability and damping created by the vertical fin and
8.2 Yaw StabilityAn airplane is not always pointing in the
same direction as it is going. This is a new concept for
most people, since for ordinary objects such as cars, bicycles
and sailboats, the direction they are pointing is (more or less)
always the same as the direction they are going, and certainly
there is no separate control of the two directions.
As an extreme example of the new concept, take a
Frisbee and draw on it the picture of an airplane. When you throw
the Frisbee, the picture of the airplane will turn
around and around and around. The direction it is pointing has
no connection with the direction it is going.
In a sailboat or airplane, you can change the heading
with the rudder. In the airplane (unlike the sailboat) the resulting
sideways forces are too small to be of much use in changing the
direction of motion (but see section 8.10).
To change the direction of flight of a Frisbee or
airplane, the proper procedure is to put it into a bank and lift
it around the turn.
Unlike a Frisbee, an airplane is not free to turn
completely around the yaw axis. If the slip angle (i.e.
the difference between the direction it is pointing and the direction
it is going) gets to be more than a few degrees, performance is
compromised. The vertical fin is designed to keep the slip
angle from getting too large.
Figure 8.1 shows a situation where the airplane's
heading has been disturbed out of its usual alignment with the
airflow. There are lots of ways this could happen, including
a gust of wind, a momentary uncoordinated deflection of the controls,
In this situation, the relative wind is striking
the vertical fin and rudder at an angle. Like any other airfoil,
the fin/rudder produces lift in proportion to its angle of attack,
so it will produce a force (and therefore a torque) that tends
to re-align the airplane with the wind. We say that the airplane
has lots of yaw-axis stability.
The colloquial name for yaw-axis
stability is ``weathervaning
tendency''. That is, the airplane tends to align itself with the
relative wind, just as a weathervane does. Section 8.11 discusses weathervaning during taxi.
8.3 Yaw DampingPure yawing motions are reasonably well damped. The process is
analogous to the process that produces damping of pure vertical
motions and pure rolling motions (see chapter 5). When
the tail is swinging to the right with an appreciable velocity, it
sees a relative wind coming from ahead and to the right. The
resulting angle of attack produces a leftward force that damps the
A leftward force in proportion to a rightward velocity
is exactly what constitutes damping.
8.4 Helical PropwashOne of the very first things that people find out about when they
start learning to fly is that it takes right2 rudder
(sometimes a lot of right rudder) to keep the airplane going straight
at the beginning of the takeoff roll. The physics of the situation is
portrayed3 in figure 8.2.
It would be nice if the propeller would just take the air and throw it
straight backwards, but it doesn't. The propeller airfoil necessarily
has some drag, so it drags the air in the direction of rotation to
some extent. Therefore the slipstream follows
a helical (corkscrew-like) trajectory, rotating as it flows back over
The next thing to notice is that on practically all
aircraft, the vertical fin and rudder stick up, not down, projecting
well above the centerline of the slipstream. That means the helical
propwash will strike the left side of the tail, knocking it to
the right, which makes the nose go to the left, which means you
need right rudder to compensate.
You don't notice the effect of the helical propwash
in cruise, because the aircraft designers have anticipated the
situation. The vertical fin and rudder have been installed at
a slight angle, so they are aligned with the actual airflow, not
with the axis of the aircraft.
In a high-airspeed, low-power situation (such as
a power-off descent) the built-in compensation is more than you
need, so you need to apply explicit left rudder (or dial in left-rudder
trim) to undo the compensation and get the tail lined up with
the actual airflow.
Conversely, in a high-power, low-airspeed situation
(such as initial takeoff roll, or slow flight) the helix is extra-tightly
wound, so you have to apply explicit right rudder.
Helical propwash sometimes contributes to left/right asymmetry in
multi-engine aircraft, as discussed in section 17.2.12.
8.5 P-FactorThe term P-factor is defined to mean ``asymmetric disk
loading''. It is an extremely significant effect for helicopters.
When the helicopter is in forward flight, the blade on one side has a
much higher airspeed than the other. If you tried to fly the blades
at constant angle of attack, the advancing blade would produce quite a bit more lift than the retreating
8.5.1 Blade SpeedFor airplanes, the same effect can occur, although
it is usually small. For the effect to occur at all, you need to
have an angle between the propeller axis and the relative wind.
To be specific, imagine that the aircraft is in a nose-high attitude,
but its direction of motion is horizontal (i.e. the relative wind
is horizontal). Then the downgoing blade
will be going down and a little bit forward, while the upgoing
blade will be going up and a little bit backward.
The downgoing blade will effectively have a slightly higher airspeed.
Since this blade is on the right-hand side of the airplane (once
again assuming a typical American engine) it will tend to torque
the airplane around to the left and you'll need right rudder to
The situation is depicted in figure 8.3.
The airplane is in level flight, with a 10 degree nose-up attitude.
The motion of the blade through the air is shown in magenta.
It consists of the rotational motion (shown
in green) plus the forward motion of the whole airplane (shown in
red). The motion of the downgoing blade is shown with solid lines,
while the motion of the upgoing blade is shown with dotted lines.
You can see that the speed of the downgoing blade is larger than the
speed of the upgoing blade.
This is the main contribution to P-factor: the advancing
blade sees more relative wind, while the retreating blade sees
less relative wind.
8.5.2 Blade AngleThere is a widespread misconception that P-factor arises because the
angle of the right (downgoing) propeller blade is larger
than the angle of the left (upgoing) propeller blade. Many
books erroneously call attention the angle of
the blade relative to the ground. The blade doesn't care about the
ground; the only thing that matters is the angle of attack, i.e. the
angle between the blade and its own motion through the air.
The correct analysis is shown in figure 8.4. As a point
of reference, the left panel shows level pitch attitude in normal
level flight, where no P-factor occurs. Meanwhile, the right
panel shows the airplane in a 10 degree nose-up attitude (still in
level flight). Since we want to discuss angle of attack, I have attached a
``reference line'' pointer to each of the blades, just like the
reference line used in section 2.2. The angle of attack of
the propeller blade is just the angle between the reference line and
the blade's motion through the air.4
When the propeller disk is inclined to the direction
of flight (so that P-factor really is occurring) the downgoing
blade has a slightly greater angle of attack (compared to the
upgoing blade) as shown in figure 8.4. This occurs
because the vector representing the airplane's motion has ``better
leverage'' when it meets the resultant, because the resultant
is shorter and because it is more nearly perpendicular to the
This angle-of-attack effect is of course zero when
propeller axis is aligned with the direction of flight.5
The effect is never very large, because
This angle-of-attack effect is in addition to (and
usually smaller than) the airspeed effect discussed previously.
Both are small compared to the helical propwash effect.
At low speeds, the airplane's forward velocity
(as represented by the horizontal red arrow in figure 8.4
is so small that it can't have much effect on anything.
- At high speeds, the airplane has a low angle
of attack, so the angle between the propeller disk and relative
wind is necessarily small (except for helicopters, tilt-rotors,
- At very high speeds, when you are going fast enough to
over-run the geometric pitch of the propeller (so that the resultant
coincides with the reference line in figure 8.4), you
might think that a small difference in angle of attack would be a
100% effect. I suppose that's true, but in this case the total
thrust is practically zero, and 100% of nothing is nothing.
Remember, we don't care whether the downgoing blade makes a bigger angle to the vertical than does the
upgoing blade. The blade doesn't care which way
is up — all it cares about is where the relative wind is coming
from. Imagine a tailwheel-type airplane stationary in the run-up area
on a windless day. You can incline the propeller disk as much as you
want relative to vertical, but there will be no P-factor unless there
is wind blowing through the propeller disk at an angle.
8.5.3 Initial Takeoff RollThere are quite a lot of myths surrounding P-factor. For some reason,
P-factor gets blamed for the fact that typical aircraft
require right rudder on initial takeoff roll. This is impossible
for several reasons.
The real reason that you need right rudder on initial
takeoff roll is because of the helical propwash, as discussed
in section 8.4. P-factor exists in some
circumstances, but it cannot possibly explain the behavior we observe
during initial takeoff roll.
Nearly everybody these days learns to fly in nose-wheel type
aircraft, which means the propeller disk is vertical during the
initial the takeoff roll. Since there is no angle between the
relative wind and the propeller axis, P-factor obviously cannot occur.
- Now let's suppose, just for sake of argument,
that you are flying a taildragger, in which the propeller disk
is actually non-vertical during the initial takeoff roll. Common
experience is that the most right rudder is required at the very
beginning of the takeoff, before much forward speed has been achieved.
The Flight Training Handbook says this is because P-factor
is worst at low airspeeds. But wait a minute — real P-factor
is proportional to airspeed. In the initial moments of the takeoff
roll, there is no relative wind, so there can't possibly be any
P-factor. Of course, if you are taking off into a headwind, there
could be a little bit of P-factor — but does that mean if you
take off with a slight tailwind there will be a negative amount
of P-factor, requiring left rudder? Don't bet on it.
8.5.4 Observing P-FactorIt is not easy to observe P-factor. It is hard to separate P-factor
from other yaw contributions such as helical propwash
(section 8.4) and twisted lift
An important preliminary experiment is to observe what happens during
the takeoff roll in a multi-engine aircraft. (To be specific, let's
consider the case where the engines rotate clockwise as seen from the
rear.) In some airplanes, the propwash hits the tail, and you must
apply right rudder to compensate, just like in single-engine planes.
In other airplanes, most of the helical propwash misses the vertical
tail in normal flight. This causes no problems and no compensation is
required. See section 17.2.12 for details on this. This
is the perfect way to illustrate that there is no P-factor when the
propeller disk is not inclined.
If you really want to observe nonzero P-factor, you can proceed as
follows: Take a twin-engine (or four-engine)
aircraft with non-counter-rotating
propellers. Attach a slip string. Establish coordinated cruising
flight, with the same amount of power on both sides. Confirm that the
ball and string are centered. Pull up into a nonturning climb at very
low airspeed (i.e. very high angle of attack), maintaining cruise
power. Maintain coordinated flight as indicated by the slip string.
Observe the rolling tendency due to propeller drag. Shift weight
(e.g. fuel) from left to right to get rid of the rolling tendency, so
you can fly straight without deflecting the ailerons,
i.e. without incurring any twisted lift.
You will observe the inclinometer ball will be slightly off-center.
This can be attributed to P-factor. To be explicit:
The effect of P-factor is not very large. You can easily compensate
using a little bit of right rudder and right bank. Indeed, in typical
situations you can just ignore it entirely.
Each engine is producing the same amount of thrust.
- But the result is asymmetric thrust, because the location
of the thrust vectors is shifted (because of asymmetric disk
- We have gone to some trouble to minimize other contributions,
such as twisted lift. Also note that even if there were some helical
propwash hitting the tail, that wouldn't be a good explanation for the
off-center inclinometer. You should be able to compensate for helical
propwash using the rudder alone, without banking.
- To summarize: we have symmetric power settings but asymmetric
thrust. A detailed analysis of the effects of asymmetric thrust can
be found in section 17.2.5.
You can never use rudder deflection as an indication of P-factor,
because any situation that exhibits P-factor will also exhibit a large
amount of helical propwash.
The single-engine version of the previous experiment goes like this:
Put a slip string on each wing, far enough out on the wing that it is
not unduly disturbed by the propwash, yet close enough in that you can
see it. In a high-wing aircraft, you'll have to put the string on the
bottom of the wing. Put strings on both sides in symmetric locations,
so you can tell for sure what string position corresponds to symmetric
airflow. Then confirm that in normal nonturning cruising flight, you
have symmetric airflow (as indicated by the strings) and zero
inclination (as indicated by the inclinometer ball). Finally, set up
a situation in which the largest possible P-factor occurs: flaps
retracted, minimum airspeed, and full power.
Once again, the indication of P-factor in this situation would be to
have the ball be off-center when the strings were centered. I have
tried this experiment, but the P-factor was too small to observe.
Here's another possible experiment. Take your favorite aerobatic
airplane and paint the starboard rudder pedal green and the port
rudder pedal red, just so we can keep straight which is which. Now go
to a safe altitude and set up for inverted slow flight. In this
high-power, low-speed situation, do you need to push the port (red)
pedal or the starboard (green) pedal? If P-factor is more important,
the answer will be port, because that is now the downgoing, advancing
blade. If helical propwash is more important, the answer is
starboard, because the relationship between the propeller, rudder, and
rudder pedals is unchanged by the inversion.
8.6 Gyroscopic PrecessionA spinning object will respond to a torque around
one axis with a motion around another axis. This remarkable and
counterintuitive phenomenon — gyroscopic precession — is
discussed in more detail in section section 19.9.
is often quite noticeable at the point where a taildragger raises
the tail, early in the takeoff roll.6
If the airplane were an ordinary non-spinning object, you could
raise the tail using the flippers alone. The flippers do not
actually dictate the motion of the fuselage; they just
produce a force and a torque around the pitch axis. For
a gyroscope, a torque around the pitch axis produces a motion
around the yaw axis. If you try to raise the tail of a real airplane
using flippers alone, it will yaw to the left because of precession.
To get a gyroscope to actually start moving
around the pitch axis, you need to apply a torque around the yaw
axis. This is what the vertical fin and rudder are for. See
Of course, an airplane has some plain old mass in
addition to its gyroscopic properties. In order to lift this
ordinary mass you need to use the flippers. Therefore, the tail-raising
maneuver requires both flippers and rudder — flippers
to change the pitch of the ordinary mass, and rudder to change
the pitch of the gyroscope.
8.7 Canted EngineOften the engine is mounted in such a way that direction of the thrust
vector is a little to one side of the axis of the airplane. This is
done in order to compensate for various nonidealities such as helical
propwash. It contributes to the yaw-axis torque budget in the obvious
8.8 Rudder Usage During RollsTurning the airplane properly requires coordinated use of ailerons and rudder. Getting it exactly
right is a bit tricky.
Remember that in an airplane, the direction you are
moving is not necessarily the same as the direction you are pointing.
There are several crucial things that happen during
Item 1 is relatively straightforward: you put the airplane into a
bank. The horizontal component of lift will change the direction of
- You use the wings to change the direction your center of mass is
going. I call this the
- You use the rudder to change your heading (i.e.
to overcome yaw-axis inertia, i.e. to
provide yaw-axis acceleration).
- You use the rudder to overcome steady adverse
due to twisted lift, as discussed below.
- You use the rudder to overcome transitory adverse
to differential drag.
Item 2a is important because if the airplane didn't
have any vertical tail, banking would cause it to just slip
off in the new direction without changing its heading. It is
much nicer to yaw the plane to align its axis with the new direction
of motion, so you apply the rudder, thereby creating a yaw rate
that matches the CM-turn rate.7
Now we come to item 2b. We must consider adverse
yaw. As discussed in section 8.8.4, during a steady
roll, the aerodynamic forces produced by the two wings are equal
in magnitude, but one force vector is twisted slightly forward
while the other one is twisted slightly rearward. This causes
a yawing moment in exactly the wrong direction: if you are rolling
to the right it tries to make the airplane yaw to the left. To
compensate you must deflect the rudder whenever the ailerons are
Finally, we come to item 2c. Suppose you are flying an airplane where
there is a lot of mass out on the wings. Whenever you are starting or
ending a roll maneuver, you need to accelerate one wing upward and the
other wing downward. As discussed in
section 8.8.3, this briefly requires extra lift on
one wing and reduced lift on the other wing. This unequal lift produces
unequal induced drag. This drag
causes additional adverse yaw.
For any given rate of roll, you need to use lots
more rudder at low airspeeds, for reasons discussed in section 8.8.6.
Procedures for maintaining coordination during turns
are summarized in section 8.8.7; the intervening
sections describe in a little more detail what is the problem
we are trying to solve.
8.8.1 Analysis of a RollTo make the discussion more concrete, let's consider
a roll starting from straight-and-level flight and rolling to
the right. As we can see from figure 8.5, there
are multiple timescales in the problem.
Let's analyze what happens if you move the ailerons
fairly abruptly. Although generally I recommend flying with a
smooth, gentle touch, (1) there will be times when you want to
roll the airplane on short notice, so let's learn how to do it;
and (2) the abrupt case makes it easier to understand what is
- [t0, t1] the time it takes you to move the ailerons;
- [t1, t2] the time it takes for the roll rate
to reach the value commanded by this aileron deflection;
- [t2, t3] the time it takes the yaw rate to reach
the value corresponding to the rate of CM-turn;
- [t3, t4] the time you hold the ailerons deflected.
In some airplanes, such as a Piper Cub, the roll
rate will reach its final very quickly (within a small fraction
of a second), because the airplane has very little inertia about
the roll axis. Practically all the mass
(pilot, passenger, fuel, and engine) is arranged in a straight
line right on top of the roll axis, so they don't contribute much
moment of inertia. In other airplanes, such as a Cessna 310,
the roll rate responds much more slowly, because lots of mass
(engines and tip tanks) is situated far from the roll axis.
Before the roll rate is established
(i.e. during the time [t1, t2]) the plane will experience transitory
adverse yaw due to differential induced drag. The nose will swing
a little toward the outside of the turn. The effect is usually
rather small, since
The rest of the discussion applies no matter how
slowly or abruptly you moved the ailerons.
- these differential drag forces are typically
small (during slow flight) or very, very small (during cruise),
compared to the differential lift forces that cause the roll,
- these forces must act against the yaw-axis inertia,
which is at least as large as the roll-axis inertia.
After the time t2, a steady roll rate exists. Even
though the ailerons are deflected, there is no difference in lift
from one wing to the other, for reasons discussed in section 8.8.4. Since there is no difference in lift, there will be
no difference in induced drag, hence no transitory adverse yaw.
However, one wingtip is diving, so its force vector
is twisted slightly forward. The other wingtip is rising, so
its force vector is twisted slightly rearward. Even though each
force has practically the same magnitude as it would in
non-rolling flight, the twist means there is a slight component
of force in just the right direction to produce a steady adverse
In addition, because the airplane has been rolling
for a while, a bank has developed. This bank causes a CM-turn.
The airplane is changing its direction of motion. In order to
keep it pointing in the same direction it is moving,
you need to use the rudder, as discussed in section 8.8.5.
At time t6, the ailerons are neutralized, but the
rolling motion has not yet stopped. (Again, there is a delay
due to roll-axis inertia.) At this point there are several things
In practical situations, the first item (transitory
adverse yaw) is usually smaller than the other two.
- There is a difference in lift between the two
wings, as needed to damp out the roll. This creates a negative
amount of transitory adverse yaw. This requires a left-rudder
contribution to compensate.
- However, the airplane is still rolling, and a
still-increasing rate of yaw is needed to coordinate with the
still-increasing rate of CM-turn. This requires a right-rudder
- Similarly, because the airplane is still rolling,
the twisted lift requires a right rudder contribution.
Analogous statements would apply if you started from
a left turn and used right aileron and right rudder to roll out
of the turn. Similarly, it is easy to do a similar analysis for
rolling into a left turn and/or rolling out of a right turn.
8.8.2 Designers' TricksImagine an airplane without a vertical fin. It would
behave be more like a Frisbee than a sailboat —
if you gave it a yaw rate, inertia would make it just keep on
yawing until some torque acted to stop it. Even if it wasn't
yawing, there would be no reason to expect the yaw angle (i.e.
heading) to be anywhere close to the desired value.
In a real airplane, of course, the vertical fin and
rudder supply the forces required to keep the yaw angle and yaw
rate under control. An overview of how you use the rudder during
turns can be found in section 8.8.
Aircraft manufacturers know about how turns are affected by twisted
lift and yaw-axis inertia. They generally try to provide the needed
yaw-axis torque automatically, using various tricks. One trick is to
interconnect the rudder and ailerons with
a spring. That means you automatically get a certain amount of rudder
deflection in proportion to the aileron deflection. They choose the
proportionality factor so that you can more or less fly ``with your
feet on the floor'' at cruise airspeeds. Of course, vastly more
rudder is needed at lower airspeeds; fortunately you can easily
overpower the interconnect spring by pushing on the controls in the
Here's another trick, which you may have noticed on many airplanes:
when one aileron goes down a little, the other one goes up a
lot. (This is called differential aileron
deflection.) The designers
were trying to arrange for the upward-deflected aileron to generate a
lot of parasite drag. If they do it just right,
the drag force is just enough to overcome twisted lift and yaw-axis
inertia during a steady roll. The so-called Frise
aileron uses a similar trick.
It has lip on
the bottom, well ahead of the hinge. The lip sticks down into the
airstream when the main part of the aileron is deflected up. Again,
the purpose of the lip is to generate drag on the wing with the
In addition to overcoming yaw-axis inertia (during
a steady roll), the designers also want to overcome transitory
adverse yaw (when ailerons have been deflected but the roll hasn't
yet started). Fortunately, transitory adverse yaw is rather small,
and by adjusting the amount of differential deflection, and the
amount of the Frise effect, pretty good cancellation can be achieved.
The bad news is that this compensation only works
at one airspeed.
The designers arrange it so you can fly with your feet on the
floor during cruise. This is a mixed blessing, because it can
lull you into complacency. At lower airspeeds, where it is most
important, you still need to use lots of rudder to keep things
coordinated. Don't forget!
8.8.3 Transitory Adverse YawSuppose you wish to roll into a right turn. You will
deflect the ailerons to the right, as shown in figure 8.6.
During the brief time after the ailerons are deflected and before
the steady roll is established, this will increase the lift created
by the left wing, and decrease the lift created by the right wing.
Unfortunately, there is no way to produce lift without producing
drag, so the left wing will be dragged backwards while the right
wing lunges forward. This is the exact opposite of what we wanted;
the airplane yaws to the left even though we wanted it to turn
to the right. Being a good pilot, you have anticipated this,
so you apply right rudder as well as right aileron, to make sure
the nose swings the right way.
Even if you don't get the footwork exactly right,
the nose will eventually swing around and point more-or-less the
right way, because of the airplane's inherent yaw stability (as
discussed in section 8.2).
Once a steady roll rate is established (no acceleration
around the roll axis), the two wings are producing the same amount
of lift, so this type of adverse yaw will no longer exist.9
Now let's consider what happens if you wish to roll
out of the turn. The airplane is banked to the right and already
turning to the right. You will deflect the ailerons to the left.
This will cause extra drag on the right wing, and reduced drag
on the left wing. The airplane will yaw to the right, continuing
and exaggerating the turn that you were trying to stop. Anticipating
this, you apply left rudder along with the left
8.8.4 Steady Adverse Yaw -- Twisted LiftNow let's consider what happens
during a steady turn. As illustrated in figure 8.7,
the airplane as a whole is moving forward, but the left wingtip
is moving forward and up while the right wingtip is moving
forward and down (because of the rolling motion).
Let's see what the local angle of attack is
at the wingtip. We use the trusty formula
In the figure, the right wingtip has a negative angle of climb, since
it is going forward and down. But the deflected aileron gives it a
lower incidence, effectively twisting that section of airfoil
nose-down. By the same token, the left wingtip has a positive angle
of climb (due to the rolling motion) and an increased incidence (due
to the aileron).
angle of attack + angle of climb
= pitch + incidence
In a steady roll, the incidences just cancel the
climb angles, so that the left wing and the right wing end up
flying at the same angle of attack. If they didn't cancel, you
wouldn't have a steady roll.
The cancellation means there is no torque around
the roll axis, but the yaw axis is a different story. As you
can see in the figure, the force vector for the downgoing wing
is twisted forward, while the force vector for the upgoing wing
is twisted rearward. This pair of fore-and-aft force components
creates a torque around the yaw axis. You need to deflect the
rudder to compensate.
Some people try to argue that these force-components should be called
``drag'' forces since they are directed fore and aft, in the same
direction as the overall relative wind. However,
it is much better to think of them as components of the local lift,
since the twisted lift remains perpendicular to the local
relative wind. The strongest argument is this: a drag force should
dissipate energy in proportion to force times airspeed, but it is
clear that the twisted lift forces do not dissipate
8.8.5 Yaw-Axis Inertia(In this section, we will assume that you are flying
at such a low airspeed that the designers' tricks discussed in
section 8.8.2 are not sufficient to produce automatically
the airplane is in a bank, it will make a CM-turn. A pure
CM-turn, however, is not what you want. A pure CM-turn means
that even though the airplane is moving in a new direction,
the heading hasn't changed. The airplane has a nonzero
slip angle. The uncoordinated airflow acting on the tail
will eventually set up a yawing motion that matches the CM-turn
rate, converting it from a pure CM-turn to a more-or-less11 coordinated
turn. If the yaw axis damping is weak, as it usually is, the
nose will slosh back and forth several times as it tries to catch
up with the CM-turn.
At any particular CM-turn rate, once the yaw rate
is established, no further yaw-axis torque is required. Like a toy top,
once the airplane starts rotating around the yaw axis it will
be happy to continue rotating.12 The only time you need
a yaw-axis torque is when the yaw rate is changing.
So, we see that during a steady roll,
Conclusion: the rudder should be deflected when the
ailerons are deflected.
The ailerons are deflected.
- The bank angle is increasing and, correspondingly,
the rate of CM-turn is increasing.
- To match the CM-turn rate, the yaw rate must
- To increase the yaw rate, the rudder should be
8.8.6 Amount of Rudder RequiredAs we have seen, there are three reasons you need to apply the rudder
during roll maneuvers: yaw-axis inertia, differential induced drag,
and twisted lift.
The amount of rudder deflection you need depends
on the shape of your airplane, and also depends on airspeed.
* Twisted LiftExample 1: Consider an airplane with long wings and
with most of the mass concentrated near the middle of the airplane.
A typical glider is an excellent example, but almost any ordinary-shaped
airplane will do. In this case there will be very little roll-axis
inertia, and accordingly very little transitory adverse yaw.
There will also be rather little yaw-axis inertia. Therefore
in such a plane, the dominant effect will be steady adverse yaw
due to twisted lift.
Example 2: Suppose you are flying along in any airplane on a
sunny summer day. You encounter a situation where your right wing is
in an updraft, while your left wing is in a downdraft. You
deflect the ailerons in order to maintain zero bank, zero roll rate,
and constant heading. This combination of non-horizontal relative
wind and deflected ailerons creates twisted lift, the same as shown in
figure 8.7 (except that the roll rate is zero in
this case). Therefore this is a perfect example of steady adverse
yaw, and you must deflect the rudder to compensate. (This rudder
requirement could not be explained by differential drag or yaw axis
inertia. This is pure twisted lift.)
The yawing moment due to twisted lift is essentially
independent of airspeed. It just depends on the deflection-angle
of the ailerons. Meanwhile, though, the force produced by the
rudder is proportional to airspeed squared. Therefore you need
lots more rudder deflection (per unit aileron deflection) when
the airspeed is low.
* Differential Induced DragExample 3: Consider an aircraft where there is a
lot of mass located far away from the roll axis. A twin with
heavy engines mounted way out on the wings, plus tip-tanks full
of fuel, is a good example. Such a plane will have lots of roll-axis
inertia, and therefore lots of transitory adverse yaw. You will
still have to worry about yaw-axis inertia and twisted lift, but
in addition to those effects you will need to apply extra
rudder deflection when ailerons are first deflected,
before the steady roll develops.
The amount of rudder required depends dramatically
on airspeed. In addition to the rudder-force issue discussed
above, the amount of transitory yawing moment itself increases
when the airspeed decreases. The key to understanding this is
to realize that whereas the coefficient of lift is more or less
proportional to the angle of attack (for moderate angles of attack),
the coefficient of induced drag is more or less proportional to
the square of the angle of attack.
The left side of figure 8.8 shows
the same situation as in figure 8.6, along with
the coefficient of drag curve. On this curve I have indicated
the different angles of attack for the two wingtips, and the correspondingly
different amounts of drag. We see that the coefficient of drag
curve is relatively flat on the bottom, so at relatively small
angles of attack (high airspeeds), a difference in angle of attack
doesn't cause too much difference in drag.
: Slow Flight Means
More Transitory Adverse Yaw
In contrast, the right side of figure 8.8
shows the same aircraft in slow flight. Both wings are
operating at a higher angle of attack. Because the coefficient
of drag curve is steeper in this regime, the same difference in
angle of attack (i.e. the same aileron deflection) creates more
difference in drag (i.e. more transitory adverse yaw).
* Yaw-Axis InertiaExample 4: Consider a long, thin, single-engine biplane
carrying lots of cargo. Since it has a rather short wingspan,
there will be rather little twisted lift, i.e. rather little steady
adverse yaw. Similarly, since all the mass is close to the roll
axis, there will be very little roll-axis inertia, i.e. very little
transitory adverse yaw. There will, however, be lots of yaw-axis
Example 5: Let's return to the case where your right wing is in an
updraft, while your left wing is in a downdraft. This
time, however, you don't deflect the ailerons; you just accept the
resulting roll rate. During the steady roll, you will need to deflect
the rudder to supply the yaw-axis momentum to match the
ever-increasing CM-turn rate. (This rudder requirement could not be
explained by twisted lift or differential drag. This is pure yaw-axis
inertia. Also note that no designers' tricks could maintain
coordination in this situation, since the ailerons are not deflected.)
Once again, the amount of rudder required increases
markedly at low airspeeds. There are three main contributions;
the first two essentially cancel each other:
The roll rate depends on the deflection-angle
of the ailerons, times airspeed. That means at low airspeeds
the roll rate is less, which reduces the amount of rudder required.
- The amount of turn that results from a given
bank angle increases at low airspeeds, as discussed in
section 16.6. This increases the amount of rudder required.
- Finally, as always, rudder effectiveness
depends on airspeed squared, increasing the amount of rudder deflection
required at low airspeeds.
8.8.7 Summary: Coordinated Turning ProceduresA proper turn consist of two ingredients: a CM-turn
and a heading change. In an idealized ``basic'' airplane,
you would use the ailerons to bank the airplane and lift the CM
around the corner, and you would use the rudder to change the
heading and combat adverse yaw. In a typical modern airplane
at cruise airspeeds, deflecting the ailerons alone creates
a fair approximation of the proper torques around both axes.
In all airplanes at low airspeeds, proper rudder usage
is vitally important.
The basic rule is simple:
The amount of rudder will depend inversely
on the airspeed.
if you are rolling to the right, you must apply
- if you are rolling to the left, you must apply
Another version of the rule substitutes the word
``aileron'' for ``roll'':
In a steady roll, the two versions are more or less
equivalent; at the beginning and end of a roll (when the roll
rate does not match the aileron deflection) the truth lies somewhere
in between. Split the difference.
right aileron requires right rudder;
- left aileron requires left rudder;
The foregoing rule allows you to anticipate the need
for rudder deflection. As discussed in section 11.5,
you have many ways of knowing when you've got it right:
By the way: If you think about it for a moment, you can see that in
inverted flight (negative
angle of attack) you will have a negative amount of adverse yaw
— if you deflect the yoke to the left you will need to push on the
right rudder pedal, and vice versa — just the opposite of what you
would do in noninverted flight. When you are actually in the plane,
hanging upside down, this is not as confusing as it seems on paper. A
little thought and a little practice will make it fairly self-evident
which wing you should lower to make a CM-turn and which rudder pedal
you should push to change the heading.
The acid test involves looking out the window. You should
perceive that the rate of heading change is proportional to the
amount of bank.
- You can also look to the side and perceive that
the wings flap straight up and down, not slicing fore and aft
as you roll.
- You can see that the inclinometer ball remains
- Yet more information comes from the seat of your
As mentioned at the beginning of this chapter, there
are lots of reasons why you should use the rudder properly during
turns. Alas, the learning process is complicated by the fact
that in many cases the airplane will ``cover up'' small
mistakes for you. In particular, whenever the airplane is in
a slip, the vertical fin will automatically try to return
the plane to zero slip angle. This is the yaw-axis stability
discussed in section 8.2. The plane will (under
most conditions) eventually establish an approximately correct
rate of heading change anyway. The goal of correct rudder usage
is to establish the correct yaw-axis motion without a slip
developing even temporarily.
The dependence of adverse yaw on airspeed can lead to trouble. Pilots
spend almost all of their time buzzing around at cruise airspeeds,
where ignoring the rudder is OK or nearly so. Sometimes this leads to
complacency. The problem arises on approach and/or departure, where
airspeeds are much lower. Proper coordination becomes more
challenging, exactly at the place where it is most important (since
the margins for error are also smaller). If you
mishandle the ailerons at low speed and low altitude, you could well
cause a spin or a snap roll, with no chance for recovery.
Section 11.5 describes a few
useful tricks for perceiving exactly how much rudder is needed
to achieve perfect coordination.
8.9 Long-Tail SlipNow let's see what happens while the airplane is
in an established turn. In particular, let's consider an airplane
with a fairly long fuselage, flying in a fairly tight turn. As
shown in figure 8.9, there is no way that the
airflow can be lined up with the front part of the fuselage and
the back part of the fuselage at the same time. The fuselage
is straight, and the path through the air is curved. You can't
have a straight line be tangent to a circle at two different points.
You have to choose.
: Airplane in a Tight
Turn — Rudder Neutral
If left to its own devices, the airplane will choose
to have the vertical fin and rudder lined up with the airflow.
The fin/rudder combination is, after all, an airfoil. Airfoils
are good at producing tremendous forces if the wind hits them
at an angle of attack. Besides, the tail is way back there where
it has a lot of leverage around the yaw axis.
Because of the air hitting the sides of the fuselage,
and other effects, the fin/rudder might not completely determine
the slip angle, but it will be the main determining factor. For
sure, the airflow at the front of the fuselage — and over the
wing — will have a significant slip component.
This will occur whenever the airplane is in a turn (unless you
explicitly deflect the rudder to compensate). I call this the
long-tail slip effect. This
slip sounds like a bad thing, but in fact it can be put to good use;
without it there would be no roll-axis stability, for reasons
discussed in section 9.3. Remember: an inadvertent turn
will be a slipping turn.
You can see from the geometry of the situation that
the amount of long-tail slip is proportional to the length of
the airplane and inversely proportional to the turning radius.
The latter depends on the square of the airspeed, as well
as the bank angle.
In a stubby, fast aircraft like a V-tailed Bonanza
in a 15 degree bank at 165 knots, the long-tail slip effect will
be small fraction of a degree — hardly noticeable. On the other
hand, in a long, slow glider, maneuvering to stay in a thermal using a 45 degree bank
at 50 knots, the effect will be fifty or a hundred times greater!
You will need several degrees of rudder deflection. You may
need to push the rudder pedal all the way to the floor just to
keep the air flowing straight over the wings. (Even if you decide
to accept a little slip over the wings in order to reduce the
crossflow over the fuselage and stabilizer, you will still want
inside rudder, and lots of it.)
: Airplane in a Tight
Turn — Rudder Deflected
I emphasize that even though you are holding inside
rudder (bottom rudder) during the turn, this is definitely not
a skidding turn (unless you get carried away and use
too much inside rudder). This rudder usage is completely
unrelated to the uncoordinated ``boat turn'' discussed
in section 8.10.
We would like the airflow to be aligned perpendicular
to the wings and parallel to the fuselage everywhere, but in a
tight turn this is not possible. We have to compromise and ``split
the difference''. The lowest-drag arrangement is to have
at the nose a slight crossflow from inside the turn, and at the
tail a slight crossflow from outside the turn.
The best way to check the alignment is with a slip
string13 — a piece of yarn exposed to the
airflow where the pilot can see it. If (as is usually the case)
you don't have a slip string, you can refer to the inclinometer
8.10 Boat TurnMy friend Larry has a sailboat. It doesn't have
ailerons. You steer it with the rudder.15
This changes the direction the boat is pointing. As shown in
figure 8.11, this causes the water to flow crosswise
past the hull, creating a sideways force that eventually changes
the direction the boat is going.
All the same words can
be applied to an airplane. Keeping the wings level, you press
the right rudder pedal. This causes the airplane to yaw to starboard.
As shown in figure 8.12, air will then hit the
fuselage on the port side, creating a sideways force16
that will gradually shove the airplane around in a right-hand
turn. (There will also be a lot of drag, but that is not our
concern at the moment.) The force of the wind on the rudder (needed
to yaw the plane) is smaller than, and in the opposite direction
to, the resulting force of the wind on the fuselage.
In powered flight, the horizontal component of thrust will make
an additional contribution to the boat turn.
To reiterate: the airplane will turn to the right
if you hold the right rudder pedal down — even if the wings are
not banked. Of course, turning the airplane properly (using the
wings) is ten times more effective and more efficient than a boat
8.11 Weathervaning During TaxiWhen the airplane is on the ground, it feels the
force of the ground and the force of the wind.
Since the tail is far, far behind the wheels, a crosswind
will create a torque around the yaw axis. It will tend to blow
the tail downwind, forcing the nose to turn upwind, just like
Now, suppose you are moving (as opposed to parked).
The weathervaning tendency causes the nose to turn into the wind.
The wheels are still on the ground, making lots of friction,
so the airplane will roll in the direction determined by the wheels,
i.e. the direction it is heading. Therefore the airplane will
travel toward the upwind side of the runway. This may
seem ironic or even paradoxical, but it's true — the crosswind
causes the airplane to move upwind.17 You have to deflect the rudder to downwind to compensate.
8.12 Asymmetric ThrustIn a multi-engine airplane, if the engine on one side has failed, or
for any reason is developing less thrust than its counterpart on the
other side, this will produce a torque (possibly a very large torque)
around the yaw axis. This is discussed in section 17.2.4.
8.13 Yaw-Axis Torque Budget — SummaryWe have finally come to the end of this section,
having covered the most important causes and effects of torques
and motions around the yaw axis. There are quite a number of
Some of these ideas will be revisited when we discuss
``Dutch roll'' in section 10.6.1.
The helical propwash effect is important, especially
in high-power / low-airspeed situations.
- Gyroscopic precession means that deflecting the
flippers will cause a yawing motion (and deflecting the rudder
will cause a pitching motion).
- Adverse yaw means that deflecting the ailerons
will cause a yawing moment.
- The long-tail slip effect means that an inadvertent
turn will be a slipping turn. This effect is very significant
in gliders. It is much less noticeable in typical powered aircraft,
but it has important implications for roll stability, as discussed
in section 9.3.
- P-factor exists in principle but is usually insignificant.
- Actual motion around the yaw axis will create
a yawing moment that tends to damp the motion.
- Yawing the plane changes the direction it is
pointing which does not automatically change the direction
it is going; the ``boat turn'' effect exists but
is feeble and inefficient.
- The pilot can deflect the rudder to oppose the
unwanted yawing effects and create the desired ones.
Perceiving coordination and maintaining coordinated
flight is important. Further discussion of this topic appears
in chapter 11, along with a discussion of how and why to
perform intentional slips.
a discussion of the terminology of forces and moments, you can
refer to section 19.7.
- All the examples
in this section assume a typical
American engine that rotates clockwise as seen from behind.
- The figure exaggerates the
curvature of the streamlines.
can also think of the blade's angle of attack as the angle between
the reference and the blade's relative wind. Relative wind and
direction of motion are the same concept, just reversed 180 degrees.
Be careful though, because there are various different relative
winds, including the instantaneous wind relative to the moving blade
and the average wind relative to the overall airplane.
- Interestingly, it goes to zero again when the axis
is perpendicular to the direction of flight, as in a helicopter.
if you pay attention you can notice it in many other situations
acceleration (which may be a somewhat unfamiliar subject) is discussed
in more detail in section 8.8.1.
- This analysis ignores the
overbanking tendency and various other small effects.
- Although in a steady turn you may need some rudder
deflection because of the long-tail slip effect, as discussed
in section 8.9, and in a steady roll you will
need some rudder deflection because of twisted lift and roll-axis
inertia, as discussed in the following sections.
- The real drag vector gets twisted,
too, but the consequences are too small to worry about.
- It won't be exactly coordinated because of the long-tail
slip effect, as discussed in section 8.9.
- But you will generally need some rudder deflection
to compensate for the long-tail slip effect.
- This is more
commonly called a yaw string, but in fact it measures the slip
angle, not the yaw angle. This is discussed in more detail in
- However, the inclinometer ball
and the slip string actually measure quite different things.
The difference is important if you have asymmetric thrust, as
discussed in section 17.2.4.
lovers' note: there are some ocean liners that do use roll-control
devices rather like ailerons, although they are primarily for
passengers' comfort, not for steering. Also, to be sure, there
are some boats that can be steered by banking them. On my sailboard,
for instance, you have to bank it the wrong way (i.e. to the outside
of the turn) by shifting your weight. On some light racing yachts
you can steer them pretty well just by shifting the weight of
the crew around. Many speedboats bank into the turns. But we're
getting off the subject. The point is that Larry's boat (like
lots of others) leans to leeward whether you are turning left,
turning right, or going straight. The reason it doesn't tilt
any more than it does is because there are a couple thousand pounds
of lead in the keel. You can't bank it by shifting your weight,
and it wouldn't turn much if you did. You steer it with the rudder.
- Technically, this force is classified as a lift
force, since it is perpendicular to the relative wind — even though
it is produced by the fuselage (not the wings), and even though
it is horizontal. See the official definitions in chapter 4.
those rare cases where there is inadequate friction on the wheels
(such as a seaplane, or an airplane taxiing on a slick icy surface)
it is quite possible for the wind to blow the airplane downwind.
This of course has nothing to do with torque; it's just a plain
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