Figure: Airplane Reference Axes, from [Hurt, pg. 250]

Eddie Sez:

If you've been blessed by flying only stable aircraft, much of this may be lost on you. If you are currently flying an aircraft that often misbehaves, you need to read this. In either case, knowing what makes an airplane stable helps to deal with the times it isn't. Having grown up in two aircraft that had negative directional stability, the KC-135A and the Boeing 707, I became an expert at dealing with dutch roll. Most modern aircraft have no dutch roll tendency at all, due either to superb design or superb yaw dampers. If your aircraft has a yaw damper, or more than one in some cases, you might want to be familiar with how to get out of a dutch roll when the yaw damper isn't working.

Oh yes, most of this comes from my notes at the USAF Flight Safety Officer Course but I did lean heavily on the references listed at the bottom of the page.


Stability and Control Definitions

An aircraft is said to be in a state of equilibrium when the sum of all forces and all moments comes to zero. There are no accelerations and the aircraft is said to be in steady flight.

Static Stability

[Hurt, pg. 245] The static stability of a system is defined by the initial tendency to return to equilibrium conditions following some disturbance from equilibrium. Only its initial tendency is considered.

Figure: Positive Static Stability, from [Hurt, pg. 244]

Imagine a marble in a bowl which is perfectly rounded, inside and out. The marble tends to gravitate to the bottom and stay there. If the marble is pushed to one side, it will roll back to the center, perhaps overshooting but eventually returning to its original state in the center. The marble is said to have positive static stability.

I've heard that the U-2 has superb longitudinal stability. If you trim the airplane and push the yoke forward or aft, it will easily return itself to the original pitch. It has good static pitch stability.

Figure: Negative Static Stability, from [Hurt, pg. 244]

Now consider the same marble atop the same bowl inverted. The marble sits up there precariously and even the slightest force will cause it to move away from the center, its state of equilibrium. Not only does the marble move, it continues to move. The marble is said to have negative static stability.

The early KC-135As had terrible Dutch roll tendencies. If you had the aircraft in trimmed, level flight and let go of the ailerons, they would induce an oscillating roll on their own. The airplane had negative lateral stability.

Figure: Neutral Static Stability, from [Hurt, pg. 244]

A final case of static stability places our marble on a perfectly flat, frictionless surface in a vacuum. (Ridiculous, I know, but that's how engineering professors think.) If you push the marble in any direction, it will move and continue to move.

Dynamic Stability

While static stability concerns the initial tendencies of an object, dynamic stability concerns the resulting motion over a span of time.

Figure: Non-Oscillatory Dynamic Stability, from [Hurt, pg. 246]

If the initial tendencies tend to decrease, the object is said to have positive dynamic stability. If they increase, it is negative dynamic stability. If they stay the same, of course, that is neutral dynamic stability.

If the dynamic motion is in agreement with the static tendency, you have a non-oscillatory stability.

Figure: Oscillatory Dynamic Stability, from [Hurt, pg. 246]

If the motion oscillates but gets smaller over time, you have positive static and dynamic stability.

If the motion oscillates but gets greater with time, although you had positive static stability this was only the initial tendency. You will have a negative dynamic stability.

If the motion oscillates without end at the same magnitude, your initial static stability as been countered with negative dynamic stability.


Longitudinal Dynamic Stability

You have dynamic stability if the amplitude of any oscillations dampen out with time. These are further classified as "stick fixed" or "stick free." A stick fixed condition is where the controls do not move freely in response to air loads. Most aircraft with hydraulically actuated controls are stick fixed and should exhibit greater stability. A stick free airplane is one that will allow control movement in response to air loads, gusts, or other stimuli.

Figure: Longitudinal Dynamic Stability, 1st Mode from [Hurt, pg. 280]

phu·goid [fyoo-goid] adjective Aerospace.
of or pertaining to long-period oscillation in the longitudinal motion of an aircraft, rocket, or missile.
Origin:
1905–10; irregular < Greek phyg ( ḗ ) flight + -oid

[Dole, pg. 256] The phugoid oscillation is one where the airspeed, pitch, and altitude of the airplane vary widely, but the AOA remains nearly constant. The motion is so slow that the effects of inertia forces and damping forces are very low. The whole phugoid can be thought of as a slow interchange between kinetic and potential energy.

Figure: Longitudinal Dynamic Stability, 2nd Mode from [Hurt, pg. 280]

[Dole, pg. 256] The short period mode is always heavily damped and, if no pilot inputs are made, the oscillations will dampen out rapidly. The short periods correspond closely with the normal pilot response time, and there is the possibility than any attempt to dampen out these oscillations may actually reinforce them and produce instability. This lead to pilot induced oscillations (PIO), which can destroy an aircraft in a matter of a few seconds. Should such an oscillation develop, usually as a result of sharp gusts, the best procedure is to release the controls and allow the natural damping of the aircraft to take over. A positive control force will also eliminate the PIO. The impulse to steady the aircraft should be avoided in all cases.


Directional Static Stability

Figure: Static Directional Stability, from [Hurt, pg. 286]

[Dole, pg. 262] An airplane is said to have positive directional stability if it is trimmed for non-sideslip flight, is then subjected to a sideslip condition, and reacts by turning into the new relative wind so that the sideslip angle is reduced to zero. On the other hand, if the airplane reacts to the sideslip by turning away from the new relative wind, thus increasing the sideslip angle, the airplane has negative directional stability. If no reaction results from sideslip, the airplane is said to have neutral directional stability.

For an example of how this can go wrong, see Mishaps / KC-135A 59-1443.


Dutch Roll

A dutch roll is a combination of yawing and rolling motions where the rolling motion continuously reverses itself with less noticeable yaw moments unless the rolling is significant. It is better called "oscillatory stability." The aircraft's spiral stability — its tendency to return to wings level when the ailerons are released — and its oscillatory stability are always in conflict. One factor makes the aircraft maneuverable, the other makes it stable.

Roll Tendency

Figure: Effect of Wing Sweep on Airflow Normal to the Leading Edge, from [Davies, pg. 97]

[[Davies, pg. 96-99]

  • A wing produces lift by accelerating the air which passes over the top surface to a higher speed than that which passes under the bottom surface. The greater the difference between these two speeds the higher to difference in pressure, hence the larger the lift vector.
  • Because the local speed of the upper surface air is higher than the free stream speed — a good bit higher where the camber is marked — it is clear that this airstream will become sonic before the effect first become apparent free stream reaches the sonic value. At this speed local shock waves are formed on the wing and compressibility effects become apparent; the drag increases, buffeting is felt and changes in lift and the position of the centre of pressure occur which, at a fixed tail angle, are reflected as changes in pitch. The Mach number at which these compressibility effects first become apparent is the critical Mach number; all other parameters aside, this can be quite low for a straight wing, around 0.7 Mach number.
  • By sweeping a wing significantly, it will be remembered, the velocity vector normal to the leading edge is made less than the chord wise resultant. In [the figure] AC is shorter than AB. As the wing is responsive only to the velocity vector normal to the leading edge, for a given Mach number the effective chord wise velocity is reduced (in effect the wing is persuaded to believe that it is flying slower than it really is). This means that the airspeed can be increased before the effective chord wise component become sonic and thus the critical Mach number is raised.

Figure: Change in Effective Aspect Ratio with Yaw, from [Davies, pg. 98]

  • When a straight winged aeroplane is yawing it also rolls. This is because the inside wing is effectively slowed down and the outside wing is effectively speeded up, thus unbalancing the V terms in the lift formulae for each wing taken separately. The lift is unbalanced and the aeroplane rolls.
  • On a swept wing aeroplane this effect is compounded because the angle of slip effectively alters the sweep of the two wings. The faster (outside) wing becomes less swept and generates increased lift at a constant incidence because of the increase in effective aspect ratio. This is because the projected span of that wing is increased. The slower (inside) wing becomes even more swept and loses lift at a constant incidence for the same basic reason. This further unbalances the lift and the rolling tendency is markedly increased. Reference to [the figure] shows that the outside wing has a much greater effective aspect ratio than the inside wing, and is also travelling faster.
  • This imposes a very marked roll on the aeroplane. Note that during the actual dynamic manoeuvre of yawing both the contributions to roll are present; when however a steady state of sideslip is achieved the contribution of asymmetric V2 disappears and only the change in effective aspect ratio remains. This marked roll with yawing is very significant in terms of flying qualities.

Stable Dutch Roll

Figure: Stable Dutch Roll, from [Davies, pg. 100]

[Davies, pg. 100]

  • Oscillatory stability, that is stable dutch roll, can now be defined as the tendency of an aeroplane when disturbed, either directionally or laterally, to damp out the ensuing yawing/rolling motion and return to steady flight.
  • When an aeroplane is yawed it rolls. The fin and rudder then oppose the yaw, slow it down and stop it, and return the aircraft towards straight flight. If the fin and rudder are big enough, the second yaw and roll are less than the first and each excursion gets progressively smaller until the motion damps right out. If, however, the fin and rudder are too small the second yaw and roll are bigger than the first, each overswing gets progressively larger and the motion becomes divergent, i.e., unstable. Although the initial yaw is the trigger for this misbehaviour it is the rolling motion, in most aeroplanes, which is most noticeable to the pilot; this is why the rolling behavior is used as the main parameter in measuring dutch roll.
  • Like any other form of stability, oscillatory stability can be positive, neutral, or negative, which is to say the dutch roll can be stable, neutral, or unstable.
  • Positive dutch roll is basically safe because the aeroplane, left alone, will either quickly or more slowly, finally bring itself under control.

Neutral Dutch Roll

Figure: Neutral Dutch Roll, from [Davies, pg. 101]

[Davies, pg. 101] Neutral stability is safe (enough) because it won't get worse, but undesirable because, if the amplitude is large, or the frequency is small, the aeroplane is tiring and tedious to fly.

Negative Dutch Roll

Figure: Negative Dutch Roll, from [Davies, pg. 101]

a href="#references">[Davies, pg. 101] Negative stability is potentially dangerous because sooner or later, depending on the rate of divergence, the aeroplane will either get out of hand or demand a constant very high level of skill and attention to maintain control. The unstable case, however, needs to be qualified in this way: if the divergence is very slow then it can be tolerated. To the pilot there is no significant difference between a dutch roll which is only very, very slowly divergent over one which is truly neutral.

Manually Damping Dutch Roll

For most pilots, manually damping a dutch roll is either a theoretical exercise or a simulator demonstration. For many of us, however, it was a daily survival drill. So here is the theory followed by the actual procedure for the Air Force Boeing 707 (EC-135C).

[Davies, pg. 102] The control of a divergent dutch roll is not difficult so long as it is handled properly. Let us assume that your aeroplane develops a diverging dutch roll. The first thing to do is nothing — repeat nothing. Too many pilots have grabbed the aeroplane in a rush, done the wrong thing and made matters a lot worse. Don't worry about a few seconds delay because it won't get much worse in this time. Just watch the rolling motion and get the pattern fixed in your mind. Then, when you are good and ready, give one firm but gentle correction on the aileron control against the upcoming wing. Don't hold it on too long — just in and out — or you will spoil the effect. You have then, in one smooth controlled action, killed the biggest part of the roll. You will be left with a residual wriggle, which you can take out, still on ailerons alone, in your own time.

[Technical Order 1C-135(E)C-1, pg. 6-15] Manual damping of dutch roll is accomplished with the lateral controls only. As a first approximation, use a control wheel input roughly equal to the degree of oscillation. For example, if the bank angle oscillations are +/- 10°, use about 10° of control wheel. Lateral control inputs should not be large and should be centered quickly as the rolling motion stops. If the input is held too long, it may excite another large oscillation in the opposite direction. Several cycles may be required to damp out Dutch roll. The next lateral control input should be delayed until the start of the next bank oscillation and then be applied to stop the rising wing, quickly centering the wheel as the roll stops.


Book Notes

Portions of this page can be found in the book Flight Lessons 1: Basic Flight, Chapter 21.


References

14 CFR 25, Title 14: Aeronautics and Space, Federal Aviation Administration, Department of Transportation

Air Training Command Manual 51-3, Aerodynamics for Pilots, 15 November 1963

Connolly, Thomas F., Dommasch, Daniel 0., and Sheryby, Sydney S., Airplane Aerodynamics, Pitman Publishing Corporation, New York, NY, 1951.

Davies, D. P., Handling the Big Jets, Civil Aviation Authority, Kingsway, London, 1985.

Dole, Charles E., Flight Theory and Aerodynamics, 1981, John Wiley & Sons, Inc, New York, NY, 1981.

FAA-H-8083-15, Instrument Flying Handbook, U.S. Department of Transportation, Flight Standards Service, 2001.

Gulfstream G450 Airplane Flight Manual, Revision 35, April 18, 2013.

Gulfstream G450 Aircraft Operating Manual, Revision 35, April 30, 2013.

Hage, Robert E. and Perkins, Courtland D., Airplane Performance Stability and Control, John Wiley & Sons, Inc., 1949.

Hurt, H. H., Jr., Aerodynamics for Naval Aviators, Skyhorse Publishing, Inc., New York NY, 2012.

Technical Order 1C-135(E)C-1, EC-135C Flight Manual, USAF Series, 15 February 1966.

Technical Order 1T-38A-1, T-38A/B Flight Manual, USAF Series, 1 July 1978.