# Could a plane be constructed to be fly in fixed-stick roll-stable circles?

According to https://aviation.stackexchange.com/a/12511/973 it is not possible for a plane to aerodynamically recover from a roll disturbance. Is that an inherent limitation of how planes could be constructed, or is that merely a design trait that is accepted in exchange for other considerations like fuel economy and ease of control? Would it be possible to construct a plane such that if the controls were eased into a certain setting, the plane would settle into a stable spiral climb until it reached an equilibrium altitude set by the thinning air, and such that mild disturbances in the roll axis would settle out [possibly changing the plane's heading]?

I would expect that maintenance of roll stability during straight and level flight would be complicated by the fact that roll won't immediately affect the flow of air over a plane in the same way as would a change in pitch or yaw. If a plane was flying in deliberate circles, however, I would think that a change in roll would rather quickly change the rate at which the plane would want to rotate about the yaw axis, and that aerodynamic surfaces cause such change to produce torque about the roll axis, thus counteracting the disturbance.

My intuition would suggest that even if roll stability could be achieved, a plane designed for such stability would probably be annoying to fly in normal usage. If, however, it was possible to activate a control which would induce such stability, it would seem such a thing might be helpful in some situations where a pilot might otherwise be in trouble [e.g. unexpected sudden IMC or other conditions causing loss of visual horizon reference]. Would such a thing be aerodynamically impossible, theoretically possible but practically unworkable, or workable but not insufficiently useful to be worthwhile?

• Hydraulicly boosted ailerons helped the P-38 Lightning. – Robert DiGiovanni Jun 20 at 10:17

This will become another of my long posts; after all, lateral stability is more complex than longitudinal stability and involves many more factors. Short answer: It is possible, but not by aerodynamic means, and would incur a drag penalty.

## Starting condition

Assume we have an aircraft with fixed controls, trimmed for straight flight, and make it a glider to remove propulsion effects. Next, let it fly through an asymmetric gust which lifts one wing. For what follows we assume calm air again.

## Initial sideslip

For an outside, earth-fixed observer, a component of lift is pointing sideways and not compensated by weight, so the aircraft will accelerate to that side. From the point of view of the aircraft, lift is still acting in the plane of symmetry, but gravity does not and will cause it to sideslip. This will mainly evoke these reactions:

• Directional stability $$c_{n\beta}$$: The aircraft will yaw into the wind because the vertical tail will create a side force ("weathervane effect").
• Dihedral effect $$c_{l\beta}$$: The additional lift on the windward wing will lift this wing. Also, the additional side force on the vertical tail, which yaws the aircraft, will cause a supporting rolling moment. Roll damping ($$c_{lp}$$) will limit the amount by which the aircraft rights itself up, however.
• Yaw moment due to dihedral and wing sweep: The lift difference due to dihedral and wing sweep will retard the upwind wing, causing a yawing motion into the wind. This effect can be added to the directional stability $$c_{n\beta}$$.

## Eventual (almost) coordinated turn

The aircraft will start to fly a coordinated turn, because all forces try to drive sideslip to almost zero, and on their way to do so start a yawing motion. In contrast to that, the roll angle will not change further once sideslip stops, because the sideslip-induced rolling moments stop as well. However, if a roll angle remains, the aircraft will now begin to turn. This opens up a new bunch of effects, because now we have asymmetric flow: Due to the yawing motion, airspeed varies over wingspan and sideslip angle varies over length:

• Rolling moment due to yawing motion $$c_{lr}$$: The outer wing (which was the leeward wing during sideslip before) now travels faster, creating more lift. This causes a rolling moment which increases roll angle.
• Since the angle of attack would be the same in a coordinated turn over the whole wingspan, drag will behave proportional to lift and increase with the local turn radius. This drag difference will create a small yawing moment which will let the inner wing advance, until the dihedral effect will kick in and create enough of a difference in angle of attack that the drag on the inner and on the outer wing are equal again. Since induced drag is proportional to the square of the lift coefficient, the wing will still create less lift on the inner wing than on the outer wing, but the difference will be reduced. If the airplane climbs or descends during the turn, angle of attack will vary over span and will create an uprighting moment on the descending airplane (and vice versa).
• Yawing moment due to yawing motion $$c_{nr}$$: This is also called yaw damping and creates a yawing moment opposed to the yawing motion. Contributing factors are the vertical tail which sees a sideward flow component due to the yawing motion, and the drag distribution along the wingspan. Aircraft with a long tail have high yaw damping and, in combination with plenty of dihedral, tend to be more roll stable than short-tailed aircraft. Here the long tail forces the wing into a sideslip which uprights the aircraft. This configuration is found on free-flying models which do indeed upright themselves from shallow banks.
• Centrifugal forces: The sideward component of lift is now balanced by the centrifugal force due to the turn. However, the centrifugal forces also will create an uprighting rolling moment which depends on the spanwise distribution of masses.

Note that the outer wing has a bigger turn radius $$R$$, but the same angular velocity $$\omega$$ as the rest of the airplane. This causes it to experience more centrifugal force $$m\cdot \omega^2\cdot R$$ than the inner wing (symbolized by the length of the parallel arrows). I used a rather steep attitude to get the point across, but the same holds true at shallower bank angles. This is the uprighting inertial moment.

Depending on the relative size of these effects, the aircraft will either right itself up, stay at this roll angle or dive ever deeper into the turn. Agile aircraft with low roll inertia will have too little uprighting moment and are likely to spiral dive. More stable configurations with big vertical tails and ample of dihedral will either continue to turn gently or right themselves up.

## Effects at non-zero roll rates

Now this was all done under the assumption that the roll angle changes very slowly. A too small vertical tail and/or too much dihedral will excite the Dutch roll motion, and now the roll angle oscillates, adding more roll-induced effects. To keep this answer reasonably short, I will not list them here.

Sweptback configurations optimized for fast flight (= with small vertical tails close to the wing) are typical examples where the ratio $$\dfrac{c_{l\beta}}{c_{n\beta}}$$ is too high and the Dutch roll eigenmode has too little damping. Free flying model aircraft, on the other hand, cannot afford to fall into a spiral dive, and by giving them big vertical tails and long lever arms, which allows them to have substantial dihedral, and a high radius of roll inertia, they can be made to upright themselves.

• As a point of clarification with regard to your last paragraph, are you saying that aircraft can be designed with a significant "basin of stability" (meaning the apparent tendency of such models to settle into stable flight is genuine), but such designs are too inefficient and unmaneuverable to be acceptable in full-sized aircraft? Could a pilot control system which had no visual, gyroscopic, or other reference for "up" recognize the oscillation pattern of a dutch roll and act to damp it and restore stability, or would apparent g forces remain too close to constant? – supercat Feb 8 '15 at 18:59
• @supercat: Historically, airplane designers have preferred the unstable, less draggy way and left it to the pilots to stabilize the plane. This made a gyro reference necessary when visual cues were gone - gs alone would not help. I wouldn't go as far as saying that more stable designs were unacceptable - they were not attractive enough to become the standard. Many successful aircraft had a weakly damped dutch roll and tended to "wag their tails" during flight, but the axis of rotation was close to the cockpit and the amplitude small, so the pilots never bothered. – Peter Kämpf Feb 9 '15 at 5:01
• I just remembered another thought: would it be possible for someone to fly a plane using just compass, airspeed, and altimeter, and contact with radar operators (who could notify the pilot of minimum safe altitude for terrain and ensure the airspace in front of the plane is clear)? I would think that if a pilot tried to alternate climbing and descending while watching the compass, the ride might be uncomfortable, but any significant roll angle should make itself apparent in the compass heading. – supercat Dec 14 '15 at 16:56
• @supercat: This needs a lot of concentration - the compass needle is aligned to the magnetic field lines, wich have an inclination versus the horizon on all latitudes except the equator. Thus, when the aircraft starts rolling and circling, the compass will not show the proper heading, but lead or lag, depending on heading and circling direction. The compass is uniquely poor for short-term heading measurement, but with some fancy calculation this could be remedied. – Peter Kämpf Dec 14 '15 at 19:12
• @supercat I've seen video of a Cessna 120 being flown in this manner, with the turn rate indicator covered up. A steady descent was flown through over a thousand feet of overcast. A key point is that in the northern hemisphere, a heading very close to magnetic south works best because the compass error exaggerates any bank and this gives an advance warning of what is going on. A northerly magnetic heading is worthless. The pilot was making control inputs primarily with the rudder, using the planes slip-roll coupling to effect changes in bank; this system tends to avoid overcontrolling. – quiet flyer Jun 17 at 13:31

This is not an answer to your question and I am guessing there are some terminology issues here (for example, what does 'aerodynamically recover' mean) but it is certainly possible for some aircraft in some configurations to have stable roll dynamics.

Different aircraft have different dynamics, so partly the answer is it depends. Furthermore, the dynamics depend on the operating configuration.

Aircraft lateral dynamics have some roll-yaw coupling but it is typically very slow.

For small perturbations in calm conditions the dynamics can be stable (as in the linearisation eigenvalues have negative real parts) but this can be illusory as the basin of attraction can be very small. This means that typical disturbances can pop you into an 'unstable regime' (I am using the term loosely here).

The lateral behaviour of the aircraft around an operating point can be typically characterised by three modes. One mode is what governs the aircraft's primary response and is typically stable and behaves reasonably. Another mode is called the spiral mode; it is possible that this is unstable, and in either case is typically very slow. (Model aircraft typically have a large dihedral to stabilise this mode.) The final eigenvector is a Dutch roll mode which is stable, but is slow and very lightly damped.

As a further illustration, an aircraft in stable flight can (typically, in reasonable conditions, etc, etc.) be piloted by elevator & rudder alone (the linearised dynamics are completely controllable from the elevator & rudder deflection inputs). This would not be possible if these inputs had no effect on roll (because there would be no way to deal with unstable rolling).

As a bad analogy, consider riding your bicycle without hands. Typically it can be done, but it doesn't take much (bump on road, sneeze,...) to destabilise.

"Could a plane be constructed to be fly in fixed-stick roll-stable circles?"

Absolutely. Consider a radio-controlled sailplane with lots of dihedral or polyhedral, such as the famous "Gentle Lady". With the rudder set at a fixed deflection off-center, the aircraft will tend to stay at a given bank angle, and will tend to return to that bank angle after a disturbance.

The root of what is going on here is that there are aerodynamic effects that tend to cause sideslip in circling flight, especially in slow-flying aircraft, and the sideslip interacts with dihedral to tend to return the aircraft to wings-level after a disturbance. That's for the rudder-centered case. With the rudder deflected, the aircraft tends to return to a fixed bank angle.

These aerodynamic effects have to do with the "curving" nature of the relative wind in a turn. Since the aircraft is rotating as well as translating, different parts of the aircraft are moving through the airmass in different directions at any given instant. Even if the vertical fin were perfectly streamlined to the flow at any given instant, more forward parts of the aircraft-- including the wing-- would be experiencing some sideslip.

This is highlighted in this section of John S Denker's excellent "See How it Flies" website-- https://www.av8n.com/how/htm/yaw.html#sec-long-tail-slip

Some of these dynamics are discussed-- perhaps with a few simplifying assumptions -- in these articles by Blaine Beron-Rawdon in the magazine--"Model Aviation"-- a 2-part series of articles called "Spiral Stability and the Bowl Effect" (September and October 1990) and a series of 4 articles entitled "Dihedral, a 4-part series" (August through November of 1988).

"Spiral Stability and the Bowl Effect" series--

"Dihedral" series --

There are also transient effects -- yaw rotational inertia is one-- that will tend to cause additional sideslip immediately after an increase in bank angle. These effects are generally less important than the aerodynamic effects described above, if we are interested in whether or not an aircraft will eventually roll back to wings-level (or to a given "trimmed" bank angle.)

You are right that roll stability is useful for cloud flying. I've been able to maintain control of a sailplane somewhat like the "Gentle Lady", but made of EPP foam, while circling in cloud for quite some time with no visual contact. I had telemetry of altitude and climb rate, which helped assure me that the plane had not accidentally fallen into a spiral dive -- due to aeroelastic effects this IS possible even with the basic built-in stability of such an aircraft. In such an event, my "escape route" was to enter an intentional spin or, an inverted turn with the stick full forward and full to one side. Not recommended for full-scale aircraft.

Note that both the Gentle Lady and the similar EPP glider have low moments of inertia in the roll axis.

"My intuition would suggest that even if roll stability could be achieved, a plane designed for such stability would probably be annoying to fly in normal usage"

Yes, generally speaking that is true. Roll control response to aileron inputs will be sluggish if even slightly less than fully "coordinated" w/ rudder, crosswind gusts will cause unwanted rolling, and there may be a tendency toward "Dutch Roll" oscillations in some parts of the flight envelope.

• If there hadn't been an answer that cited model aircraft, I'd have had to write one. I'd go further -- free flight models have been built to fly in circles with fixed surfaces since at least the 1930s. Pretty well has to be "fixed stick" -- further, they used opposed wing warp and rudder (as well as thrust line) to climb turning one way and glide turning the other. – Zeiss Ikon Jun 17 at 17:02
• Out of curiosity, what would be the practicality of having some control surfaces that could be engaged to put an airplane into such a configuration if e.g. there was an unexpected loss of visibility, but would not interfere with normal maneuvering otherwise? – supercat Jun 17 at 19:32
• I think you'd basically talking about the ability to make the wings pop up to an abnormally large dihedral angle-- probably not very practical. – quiet flyer Jun 17 at 19:42

There can't be any such mechanism that works purely by aerodynamic means.

The reason for this is that, at least on a short timescale, the air doesn't know which way is up.

The airflow can feel deviations in pitch and yaw, because they result in the air coming at the aircraft from a different direction (relative to its structure). This gives an opportunity for the shape of the aircraft to be designed such that the altered airflow produces a righting moment.

However, if you imagine a deviation in pure roll, the air will still be coming at the aircraft from the same direction as before, namely from straight ahead. All of the air flows will be the same as in level flight, just with everything turned to match the roll angle of the aircraft.

There is the difference that if the wings are not level, the direction of lift is no longer opposite to the direction of gravity. In the not-completely-short timescale, this will cause the aircraft to accelerate with respect to the air. Eventually that will lead to flying at through the air at an angle (that is, changing the angle of attack and/or sideslip), which the aerodynamics can react to. I think this must be what Peter Kämpf meant when he said that there can be an "inertial mechanism" for roll stability.

One way is to take a plane with a high dihedral wing and low set CG and simply rudder it.

Important design aspect is to put the dihedral in the wing, rather than to try for uprighting effect from a tall tail for roll stability. A large unsymmetric tail away from the center of gravity will promote yaw in a slip, which is destabilizing to the initial roll, creating a spiral.

You do not need a thresher shark tail to have a stable aircraft.

Why low set CG? Here is your roll damper. No need to be grossly pendulous, just a little low.