[Comments or questions]
Copyright © 1996-2001 jsd
10 Equilibrium, Stability, and Damping
Three of the most useless things
in aviation are:
- The airspace above you.
- The fuel not on board.
- The runway not in front of the wheels.
Several parts of this book make use of the concepts
of equilibrium, stability, and damping. This section defines
the concepts a little more precisely and clarifies the relationships
10.1 EquilibriumThe word equilibrium is quite ancient.
The word has the same stem as the name of the constellation ``Libra''
— the scale. The type of scale in question is the two-pan balance
shown in figure 10.1, which has been in use for at least
7000 years. The compound word ``equilibrium'' translates
literally as ``equal balance'' and means just that: everything
in balance, no unbalanced forces.
The wheel is more modern than the balance; it has been in use for
``only'' some 5500 years. It provides some more sophisticated
illustrations of equilibrium and related concepts. If we make the
wheel lopsided by attaching a weight to the rim at one point, the
system will be out of equilibrium unless the weight is exactly at the
top or bottom. If we prepare the system in an unbalanced state and
then let go, it will immediately start rotating.
To reiterate, and as indicated in figure 10.2,
there are three ways to make the wheel be in equilibrium: (1)
position the weight at the bottom, (2) remove the weight entirely,
or (3) position the weight at the top.
10.2 StabilityStability has to do with how the
system responds if we move it a little ways from its equilibrium
position. There are three possibilities:
It usually doesn't make much sense to talk about
stability except for systems that are in equilibrium or nearly
Positive stability means that if the system is displaced
a little ways from its equilibrium position, it will generate a force
tending to push it back towards equilibrium. The wheel with the
weight positioned at the bottom is an example of positive stability.
- Neutral stability (also called zero stability)
means that if the system was in equilibrium and you displace it
slightly, it remains in equilibrium. No force is generated. The
perfectly balanced wheel is an example of this.
- Negative stability means that if the system is displaced
a little ways from its equilibrium position, it will generate a force
that tends to push it farther from equilibrium. The wheel with the
weight at the top is an example of negative stability.
For a multi-dimensional system, we get to ask about the stability of
each axis separately. For example, consider an egg resting on a horizontal table. An ideal
egg has zero stability against motion in one direction: it is free to
roll around its axis of symmetry. On the other hand, it has positive
stability against motion in the end-over-end direction; if you rock
the egg slightly by pushing its nose down, it will tend to return to
its original state.
A system exhibits damping if motion of the system produces a
force that opposes the motion.
A bicycle wheel provides a good demonstration of
a system with very little damping. Assuming the bearings are
good and the wheel is not touching anything, when you spin the
wheel it will keep going for more than a minute. Air friction
produces very small forces that eventually cause the wheel to
A bicycle wheel that is rubbing against something
is much more heavily damped. When it is in motion, rubbing friction
can create large forces that oppose the motion and bring the motion
to a stop.
A dynamical system can exhibit negative amounts of damping, but this
is harder to demonstrate with a simple system. Negative damping tends
to make the motion increase, which means that energy is being added to
the system from somewhere; therefore simple friction can never produce
Nose wheel shimmy of an airplane is a good example of what
happens if a system has a negative amount of damping. If the aircraft
is moving along the ground at high speed, the nosewheel will
eventually hit a pebble or something. The nosewheel is then no longer
aligned with the direction of travel. By the usual ``castering''
principle, this causes a force that tends to return the wheel to its
proper position (that is, the wheel exhibits positive stability).
Unfortunately, many aircraft have too much stability, and too much
inertia in the castering mechanism. The result is that the wheel
tends to overshoot its equilibrium position and continue to the other
side, going out of alignment in the opposite direction by an even
greater amount. The result is an oscillation that quickly grows to
Note the relationship of stability and damping: when
the wheel is being forced back toward alignment, the force is
toward the equilibrium position (positive stability) but is in
the same direction as the motion (negative damping).
To eliminate the shimmy problem, a hydraulic ``shimmy
damper'' is installed on the nose wheel. Figure 10.3
is cutaway drawing showing how a hydraulic damper works. It consists
of an oil-filled cylinder, plus a pushrod attached to a disk inside
the cylinder. When the pushrod moves from side to side, oil is
forced to flow through the small holes in the disk. This creates
a force proportional to the velocity of motion — i.e. damping.
Sometimes the fluid leaks out of the damper, and even more commonly
the linkages connecting the damper to the wheel become worn and loose.
This makes the damper ineffective, whereupon the you get a vivid
demonstration of negative damping. A preflight check of the damper and
linkages is easy and worthwhile.
Also... as discussed in chapter 5, the airplane's
rolling motion and pure vertical motion are normally very heavily
damped, but this damping goes to zero and becomes negative at
10.4 Relationship of Stability and DampingTo reiterate: stability refers to a force that arises
depending on the position of the system; damping refers
to a force that arises depending on the velocity.
In old-fashioned terminology, what we call ``stability'' was sometimes
called ``static stability'', and what we call ``damping'' was
sometimes called ``dynamic stability''. What's worse, occasionally
both terms were shortened to the single word, ``stability'',
which was unnecessarily confusing.
Also, modern usage prefers ``damping'' not
``dampening'' — if you start talking about a ``dampener''
people will think you want to moisten the system.
Stability can be positive, zero, or negative; damping
can also be positive, zero, or negative. A dynamical system can
display any combination of these two properties — nine possibilities
in all, as shown in figure 10.4. In the top row, the
bicycle wheel is dipped in molasses, which provides damping.
In the middle row, there is no damping. In the bottom row, you
can imagine there is some hypothetical ``anti-molasses''
that provides negative damping.
: Stability and Damping —
10.5 Oleo-Pneumatic StrutsA great example of a device that provides a force that depends on
position and a force that depends on velocity is the
oleo-pneumatic strut, which is widely used on landing gear as a
combination spring and shock absorber. It consists of a piston in a
cylinder filled with both oil (``oleo'') and air (``pneuma''), as
shown in figure 10.5. If the piston is moved up into the
cylinder, the air at the top of the cylinder is compressed. (The
hydraulic oil is essentially incompressible.) This ``air spring''
creates a force that depends on the position. As the piston moves,
the oil in the hollow part of the piston is forced to flow through the
holes in the disk, creating a force that depends on the speed of
motion, using the same principle as the damper
It is important that the strut contain the right
amount of air and the right amount of oil. Problems can
arise more easily than you might think.
Suppose that over time, some of the oil leaks out of the strut on your
airplane.1 Your friend, Murgatroyd
Fudpucker, borrows the plane and notices during preflight that one of
the struts is low — that is, not enough of the piston is protruding
from the cylinder. Murgatroyd gets out a bicycle pump and adds air to
the strut. Everything now looks fine. During future preflight
checks, a passive glance the strut will give you the impression that
everything is OK.
Alas, things are not OK. The problem is that oil
has been replaced with air. Since air is a thousand times
more compressible than hydraulic oil, the amount of force it takes
to make the strut ``bottom out'' has been greatly reduced.
If you or Murgatroyd makes even a slightly hard landing, the
piston will smash against the end of the cylinder, metal to metal.
This has roughly the same effect on the airframe as hitting it
with a sledgehammer. Repairs could be very, very expensive.
Therefore, if there is any chance that the airplane
has been mis-serviced since the last time you flew it, you should
check not only the height of the struts, but also their springiness.
To check a main-gear strut, lift up the wing a few inches and
then let it drop. Similarly, to check the nose strut, lift up
the nose (perhaps by pushing down on the tail) a little ways and
then let it drop. If any strut compresses more than it should
(e.g. if it comes anywhere close to bottoming out), do not fly
the airplane until the strut has been properly serviced with air
There is a thin coating of oil on exposed part of
the piston, which collects dust. When the piston is shoved into
the cylinder, the O-ring will scrub the dirt down the piston and
cause it to collect in a ring called the scrub line. Observing
the scrub line can tell you how
close the strut has come to bottoming out recently.
Please do not get the impression from the foregoing
discussion that ``air is bad'' and ``oil is good''.
I discovered an airplane recently where nose strut contained
no air at all, but contained several inches too much oil instead.
Once again, the preflight checklist calls for checking the height
of the strut — which was completely normal. Fortunately, I noticed
that the strut had no springiness whatsoever; trying to compress
a solid column of hydraulic oil is like trying to compress cast
To reiterate: you should make sure that the struts
contain the right amount of air and the right amount of oil.
Servicing a strut isn't very tricky; it just has to be done right.
10.6 OscillationsWhenever a system has positive stability but not enough damping, you
can expect to see oscillations.
10.6.1 Analysis of Dutch RollAs remarked in section 9.3, the airplane has only a
small amount of stability around the roll axis. You may be wondering
why designers don't fix this problem by increasing the slip-roll
coupling. The answer is that they are worried about Dutch roll.
Dutch roll is a messy combination of rolling, slipping,
and yawing. As we shall see, this combined motion is less damped
than the pure rolling, slipping, or yawing motions would be.
A moderate amount of Dutch roll never killed anybody,
but it does tend to provoke nausea, especially in passengers.
The Dutch-roll oscillations typically have such a
short period (a couple of seconds) that it is a challenge for
the pilot to overcome them by working the controls. A spiral
dive, on the other hand, develops much more slowly. Therefore
if it comes down to a compromise between roll-axis stability and
Dutch-roll damping, designers generally increase the damping at
the expense of the stability.
To understand where Dutch roll comes from, and how
to fight it, gives us an opportunity to combine and apply most
of the things we have learned about equilibrium, stability, and
The rolling and yawing motions associated with Dutch
roll are shown in figure 10.6; we will discuss the slipping
component in a moment.
The wingtip yaws forward, then rolls up, then yaws
backward, then rolls downward, then repeats. The opposite wingtip
does the same thing, 180 degrees out of phase. Imagine pedaling
a bicycle backwards.
To analyze the damping of the Dutch roll system,
we must remember that energy is force times distance; by the same
token power (energy flow) is force times velocity. The component
of the force in the direction of the velocity is the only thing
that matters; the component in the perpendicular direction doesn't
We begin by using figure 10.7 to analyze
the forces that affect the rolling motion. The velocity and position
of the wingtip is shown in red; net changes in the lift vector
are shown in blue.
At point A in the figure, the wing is going upward.
That means it has less angle of attack than normal (and in particular,
less angle of attack than the opposite wingtip). The reduced
lift corresponds to a net force opposite to the velocity, and
therefore energy is being removed from the system. At point C,
a similar analysis applies. The wingtip is descending, creating
more angle of attack and more lift than normal. This corresponds
to a net force which is once again opposite to the velocity, removing
energy from the system. This is the same roll damping mechanism
as discussed in section 5.4.
At point B, the wingtip has less velocity than normal,
and less lift, while at point D the wingtip has more velocity
and produces more lift. There is no effect on the damping, because
the forces are perpendicular to the velocity.
We continue by using figure 10.8 to analyze
the forces that affect the yawing motion. At point B in the figure,
the vertical fin/rudder is wagging to the right. This changes
the rudder angle of attack, opposing the motion. This is the
same yaw damping mechanism discussed in section 8.3.
Also, at this point, the port wingtip has less drag than the
other, because it is moving backwards. Both of these effects
take energy out of the system, providing damping. The same processes
produce damping at point D also.
At point A, there is a little less induced drag on the port wingtip
because it is flying at reduced angle of attack. This has no effect
on the damping, because the force is perpendicular to the velocity.
Also at point A, there is a yawing force because
the airplane's heading is not aligned with its direction of travel;
the tail is too far to the left. This provides yaw-axis stability
but does nothing for the yaw damping, because the force is perpendicular
to the velocity.
The analysis of point C is analogous to point A.
If the yawing and rolling motions were the whole story, Dutch roll
would be no problem. According to the analysis so far, there is lots
of positive damping. The Dutch roll would quickly die out.
Unfortunately, nature is not so kind, as we discover
when we take the sideways motion of the aircraft into account.
Refer to figure 10.9.
At point B in the figure, the left wingtip is at
the highest point in the cycle. The airplane is banked to the
right. The wings' lift vector is inclined to the right, so there
is a rightward component of lift. In fact, during the whole half-cycle
from point A to point C there is at least some rightward force.
Since the airplane has lots of inertia and not much damping2 with respect to pure sideways motion, the rightward
velocity just increases and increases during the whole half-cycle.
The maximum rightward velocity is achieved near point C.
During the next half-cycle (from C via D to A) the airplane is
banked to the left. The leftward force reduces the
previously-acquired rightward velocity to zero, and then builds up a
leftward velocity. The sideways velocity is zero at point D, and the
maximum leftward velocity is achieved near point A.
Note that like any other lightly-damped oscillator (such as a
pendulum, for instance a playground swing set) the maximum rightward
force occurs when the plane is at is maximum leftward
The final ingredient is the slip-roll coupling.3 A certain
amount of slip-roll coupling is highly desirable because it is
a necessary part of the process that produces roll-axis stability
The bad news is that the slip-roll coupling contributes
a negative amount of damping to the Dutch roll mode. The rightward
velocity is maximum at point C, producing a leftward-rolling moment.
The force is in the same direction as the roll velocity, so it
adds energy to the Dutch roll.
Analogously, the leftward velocity is maximal at point A, producing a
rightward-rolling moment. This, too, is in the same direction as the
roll velocity, contributing negative damping.
So slip-roll coupling presents designers with a dilemma: it increases
roll-axis stability, but decreases (Dutch) roll damping.
The simplest way a designer can resolve this dilemma is to notice that
roll-axis stability depends on both slip-roll coupling and the
long-tail slip effect. Therefore if you have a problem with Dutch
roll, decrease the slip-roll coupling and increase the long-tail slip
effect, for instance by making the tail boom
longer and reducing the rudder area. As a rule of thumb, you
can tell just by looking at a short-coupled airplane that it will have
a problem with underdamped Dutch roll.
The other (all too common) design choice is to sacrifice
stability. Most airplanes wind up with very, very little roll-axis
stability. Consequently spiral dives are a constant threat.
Note: the Dutch roll motion has some similarities
and some differences when compared to the uncoordinated wing rocking
exercise discussed in section 16.7. The
latter is sometimes referred to as the Dutch roll exercise. The
two patterns of motion share the concept of slipping to one side
and then the other. The main difference is that the exercise
calls for using the rudder to prevent any heading change, whereas
true (natural) Dutch roll involves considerable yaw along with
the banking and slipping.
10.6.2 How to Fight OscillationsSince this book is intended for pilots, not designers,
we should discuss how the pilot should use the controls in order
to oppose obnoxious oscillations.
First, bit of simple advice: in an airplane that is susceptible to
Dutch roll, be extra careful to avoid
uncoordinated usage of ailerons and
rudder since that would unnecessarily put energy into the Dutch
Once Dutch roll gets started (due to turbulence, or klutzy
control-usage, or whatever), it may be hard to stop. In some
airplanes you may be able to improve the situation as follows: If the
rudder pedals are moving because of the sideways force that the Dutch
roll puts on the rudder, then you should rest your feet firmly on the
pedals to prevent them from moving. This will increase the stability
and (more importantly) the damping of the yaw axis.
If that doesn't suffice, you can try to fight the
oscillations by direct intervention. This requires some skill
and lots of attention.
You should not think about correcting the
position of the wing. If you deflect the ailerons to the
right at point D, the wings will return to level (point A) sooner,
but you will be applying a force in the same general direction
as the velocity, increasing the velocity and the energy of the
Dutch roll mode.
As we have seen, the airplane has plenty of stability
and not enough damping, so what we need is a force that depends
on the velocity, not the position. Therefore the ailerons need
to be deflected to the left when the left wing has its maximum
upward velocity, near point A. You should apply the deflection
before point A and remove it after point A. Similarly, you should
apply right aileron (smoothly) a little before point C and neutralize
them (gradually) after point C.
A similar analysis applies to rudder usage. Don't
try to correct the position. Instead, you need to apply right
rudder at the point where the nose is swinging to the left with
the maximum velocity (point B); by the same token you need to
apply left rudder when the nose is swinging to the right with
the maximum velocity (point D).
The same logic applies to phugoid oscillation (section 6.1.12), and to pilot-induced
pitch oscillation associated with a botched landing.4 When the
nose is high, you should not push on the yoke to correct the nose-position;
you should anticipate that the position will very soon over-correct
all by itself. If the nose is high and dropping (or about to
drop), you need a judicious pull on the yoke to prevent the pitch
attitude from overshooting.
You should tell yourself that you are fighting the
velocity, not trying to correct the position. This is because
you need to increase the damping, not the stability.
- On a retractable-gear airplane,
you can lose all the oil, even the oil inside the hollow piston, more
easily than on a fixed-gear airplane.
- In a system with lots of damping and not much inertia,
like a spoon in molasses, the velocity tends to be proportional
to the applied force. In the other extreme (lots of inertia,
little damping) we can apply Newton's second law without worrying
about frictional forces — therefore the acceleration
to the force and the velocity accumulates as long as the force
- That is, a slip produces a rolling moment
— by means of e.g. dihedral, sweepback, tall rudder, and/or shadow
effects, as discussed in section 9.2.
- Actually it is much easier to fight pitch oscillations
by direct intervention, since they happen more slowly.
[Comments or questions]
Copyright © 1996-2001 jsd