When pitching, how do planes establish a pitch rate which corresponds to the centripetal acceleration?

Question

Coordinated flight requires that the yaw rate and the centripetal acceleration correspond. In other words, for a given $a_{centripetal} = v^2/r = \omega ^2 r$ this means that the yaw rate must equal $\omega$. We do this by using two independent controls, one for lateral acceleration (linked to the bank angle) and one for yaw rate. When the plane is not coordinated, it is slipping or skidding.

What about in a continuous pitch-wise rotation? In this case, there is only one control, and that's the elevator. In the controls domain this is known as an underactuated system, and it is both theoretically and practically impossible to directly control two system outputs with only one control input.

So if the elevator, by changing the plane's AoA, controls centripetal acceleration, it can't simultaneously control the pitch rate. And the vice-versa is equally problematic, if the elevator controls pitch rate than it cannot directly control centripetal acceleration.

So in a sense, the plane is skidding[*] through the pitch change. So what are the elements which balance the pitch "skid"? The elevator provides a torque, and this torque must 1) ultimately be balanced or else the plane would angularly accelerate and 2) must be counteracted early enough that the plane does not over-rotate.

My hunch is that the wings have such an incredibly strong tendency to align themselves with the airflow that the skidding action is very tiny.

Is this correct?

[*] Skidding, in the sense of a car or ice skater, where the heading vector over-rotates relative to the motion vector. @Koyovis makes the point that for any non-0 AoA, an airplane is always skidding in the vertical direction.

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P.S. For a great example of a full-on vertical skid, which occurs when the plane's pitch rate and centripetal acceleration diverge in an extreme manner, check out the SU-27's cobra maneuver

Answer 1

What about in a continuous pitch-wise rotation? In this case, there is only one control, and that's the elevator. In the controls domain this is known as an underactuated system, and it is both theoretically and practically impossible to stabilize two system outputs with only one control input.

Not if the two outputs are connected. They are not independent of each other!

What connects them is the lift force. All lift that exceeds the weight force is accelerating the airplane into the pitching motion, so that is the centripetal force. Now imagine that this force produces a path that is narrower than what the pitch rate corresponds to: The angle of attack will immediately decline, and so will lift.

Equally, if the pitch rate is higher than the loop produced by the centripetal force, angle of attack will increase, and so will lift.

Again, both are not independent of each other and pitch damping will make sure that both change smoothly through the full trajectory. At least as long as the airplane flies within the limits of linear flight. Once the wing stalls, a different dynamic evolves (where still both are coupled, but now with a hugely varying ratio, depending on angle of attack).

So if the elevator, by changing the plane's AoA, controls centripetal acceleration, it can't simultaneously control the pitch rate.

Yes, it can. AoA determines lift and, therefore, centripetal acceleration. Elevator deflection controls tail lift and, therefore, pitch rate. The elevator does not control AoA directly - it merely trims a pitch rate which moves the AoA to the desired value.

In a first order approximation, all lift is created by the wing and all pitch moments are created by the tail surface. Closer inspection, of course, reveals that the tail also contributes lift and the wing also contributes a pitch moment, but for explaining basic principles let's disregard those niceties. The much larger distance of the tail (or canard) to the center of gravity (CoG) makes its pitch contribution dominant compared to its lift contribution, and vice versa the opposite for the wing.

AoA changes will affect both wing and tail equally, changing both lift and pitch moment. Stability (lift per area) determines whether there is a pitch moment change connected to that lift change.

Pitch changes will also change AoA eventually but add a second component at the tail resulting from pitch damping: Due to its distance from the CoG there is another local AoA and, therefore, lift change on the tail. This lift change is directly proportional to pitch rate and the square of the tail's distance to the CoG, while only the integration of pitch rate over time will change AoA.

Looking at only AoA and pitch rate neglects a third influence: Vertical speed. A change in vertical speed directly changes AoA. Vertical speed changes when there is an imbalance between lift and the sum of weight and centripetal acceleration times mass. The result is a strong damping of any change in AoA which leads to such an imbalance. When you add this effect, you get the coupling between AoA and centripetal acceleration which you seem to have overlooked.

So is the plane skidding through the pitch change?

Depends how you define it. You may view the angle of attack as "skidding", just like a propeller travels less length in one revolution than its blade pitch would suggest. Or you define skidding as a difference between pitch change and change of a tangent to the airplane's trajectory (the first derivative of both over time): Then all skidding is the change in angle of attack due to the change in the direction of gravity (in reference to the airplane) and the change in speed due to the altitude change throughout the trajectory.

Answer 2

So is the plane skidding[*] through the pitch change?

Yes, absolutely. In fact, if we define skidding in pitch as a difference in pitch angle and heading vector angle, the aeroplane is always skidding when Angle-of-Attack ≄ 0.

My hunch is that the wings have such an incredibly strong tendency to align themselves with the airflow that the skidding action is very tiny.

Is this correct?

No, it is not. Wings don't have a large tendency to align themselves with the airflow, in fact they (and the fuselage) have a strong tendency to position themselves perpendicular to the airflow, that is why an aeroplane needs a horizontal tail.

So if the elevator, by changing the plane's AoA, controls centripetal acceleration, it can't simultaneously control the pitch rate. And the vice-versa is equally problematic, if the elevator controls pitch rate than it cannot directly control centripetal acceleration.

The controlled variable in loops is not pitch rate but the trajectory: a perfect loop is perfectly round, like in OPs picture. This circle trajectory output is achieved using a combination of inputs: airspeed, elevator deflection, load factor (g-forces). As described in this link:

There are loops and then there are round loops. To make a loop round and, well, loop-like, it must be flown with a linear flow and control feel. After you pull back on the stick to enter the loop, you can’t continue to pull the same amount of G all the way around because the speed is decreasing. If you pull 5 G’s with an entry speed of 160 knots, you won’t be able to pull 5 G’s at the apex with a speed of only 100 knots. The pilot has to relax the stick pressure before the top of the loop; then, as the airplane starts downhill again and the speed is increasing, the pilot needs to increase the back pressure so not to over-speed the airplane and to keep the loop round.

Answer 3

How do aircraft establish a pitch rate which corresponds to the centripetal acceleration.

It is a combination of throttle and elevator, with gravity and aerodynamic forces considered. If your loop is flown at constant speed, AoA would have to be changed as well as throttle because of the changing position of the gravity vector.

Mathematically, we have the answer in: G force = v$^2$/r We can see that airspeed squared and radius are the determining factors.

Starting with the easiest case, straight level flight is where the AoA is enough to cancel gravity (1G, no centripetal force).

Amazingly, it is the banked circle we most commonly fly that may be confusing (banking and yawing), second only to the loop (variable gravity vector), so let's consider the 2 simplest turning models, the pure "skid" turn (0 degrees bank) and the pure "slip" turn (90 degree "knife edge" bank). The mechanisms of skidding and slipping become apparent.

In the case of the horizontal circle, at constant speed with constant throttle, we have that which rotates and that which creates centripetal acceleration (in the direction of the turn).

First, the skidding turn. With 0 degrees of bank, all we have to accelerate sideways is the crude airfoil created by the fuselage being pitched (yawed) into the turn with the rudder (plus the inward component of the thrust vector). The rudder is rotating the aircraft to a certain "pitch" to create a certain amount of sideforce and directional stability holds that pitch as the relative wind changes in the turn. It is the sideways motion created by the airfoil that swings the tail 'round by changing the relative wind.

we can see the relationship between the controlling surface and the stabilizing surfaces is no different in a turn than in straight flight!

On to the pure "slip" knife edge 90 degree bank circle. This time the crude fuselage airfoil (and a lot of power, helps to be light) is holding the aircraft up and the wing is creating lateral acceleration. Little doubt why many aerobatic pilots like this manuever, by adding in a little pitch with the elevator the wing would rapidly pull it sideways, and again, directional stability swings the tail around, maintaining the pitch set by the elevator.

The common banked turn is perhaps intuitively easier to explain, but we must realize the rudder is simply fine tuning (compensating for adverse yaw) what a good tail should already be doing: following the relative wind.

if the elevator controls pitch (AoA) than it cannot directly control centripetal force

And neither does the rudder in a horizontal turn!. "Coordinating" a turn merely means reducing drag as much as possible by aligning the aircraft into the relative wind. The flight path (around the circle) is determined by the thrust and the lateral force vector created by the wing.

Answer 4

The comments on sideslips or skids in the question tend to obscure the fact that no matter what we want to say about yaw rate, turn rate, centripetal force, bank angle, etc, the fundamental cause of a slip or skid is generally that some aerodynamic asymmetry is present which is causing the fuselage to fly in some attitude other than pointing head-on into the airflow.^1,2 Thus the "sideways angle-of-attack" of the fuselage is non-zero. The displacement of the slip-skid ball is the result of this, not the cause. If we fix this -- often with the rudder-- we'll end the slip or skid.³

And there's simply nothing equivalent to a slip or a skid in regards to purely pitchwise maneuvering, unless we want take the position that any non-zero angle-of-attack of the wing is somehow the "vertical equivalent" of a slip or skid. Or unless we simply mean flying with the wing at some angle-of-attack that is not the one that yields the flight path we want. We can fix that by moving the elevator to change the angle-of-attack of the wing -- problem solved -- assuming that the goal lies within the aircraft's attainable performance envelope!

To a first approximation, the pitch rate we'll see at any point in a maneuver is pretty much "baked in" to the trajectory we are flying, which is determined largely by the combined effects of angle-of-attack, airspeed, and bank angle.⁴

Perhaps the simplest answer to the question is as follows: "If the intent is to suggest that we can't use the elevator to independently vary the centripetal acceleration and the pitch rotation rate, that is absolutely correct. Fortunately, we have no need to do this."

Here's another simple answer: "Apply forward or aft pressure to control stick or yoke as needed to manage the angle-of-attack as needed to obtain the desired G-load or the desired pitch rate. All the other relevant variables will automatically 'come along for the ride'."

Of all the things a pilot might care to concern himself or herself with during aerobatic maneuvering, having "the wrong pitch rate for the centripetal acceleration", or vice versa, really should not be one of them, since we generally can't change one of these parameters without also changing the other. Of more interest would be g-load versus airspeed, or pitch rate versus airspeed-- and I've actually heard arguments about which of these a pilot should be most concerned with in certain situations such as looping-- (e.g. a hang glider pilot flying loops without the benefit of a g-meter) -- but it's a somewhat academic argument.

Footnotes --

1. Note that this is possible even if the yaw rotation rate is "correctly" matched to the rate of curvature of the flight path in the yaw dimension, so that the sideslip angle (as indicated by a yaw string) is constant rather than increasing.

2. Admittedly it's possible to also imagine a situation where a slip or skid is caused not by an aerodynamic asymmetry, but rather by a sudden change in the rate of curvature in the flight path in the yaw dimension, without an adequate yaw torque being applied to change the aircraft's yaw rotation rate to match. The most familiar example of this would be the pronounced slipping tendency we often see while going "over the top" of a wingover-like maneuver at a steep bank angle, if we choose not to step on the bottom rudder pedal to yaw the nose downward. At the very top of the wingover, as the aircraft flies in a semi-ballistic trajectory with the wing partially "unloaded", the rate of earthward curvature of the flight path (due to gravity) reaches a maximum value due to the very low airspeed and thus the resulting low radius of curvature of the flight path. The low airspeed also minimizes the amount of "weathervane" yaw torque that can be generated by the vertical fin. Unless the aircraft has zero rotational inertia in the yaw axis, this "peak" in the rate of change of the direction of the flight path in the yaw dimension will tend to drive some noticeable amount of sideslip as the nose "lags" behind the actual trajectory of the aircraft. The yaw string, if present, will tend to blow toward the high wingtip as the aircraft "floats over the top" of the maneuver, and the resulting sideways airflow will then eventually generate the yaw torque that is needed to yaw the nose earthward at a high enough rate to center the yaw string again, or possibly even to drive a brief "overshoot" into a slight skid. Similarly, when the pitch rotation rate is changing, pitch rotational inertia may sometimes play a role in placing the wing at an angle-of-attack slightly different from the one we'd otherwise "expect" to correlate to the position of the elevator control at any given instant if the aircraft had zero pitch rotational inertia -- see footnote 4 for more. This will affect the way an aircraft "feels" -- the way it responds to control inputs -- during aerobatic maneuvering.

3. Note that in general, during turning flight, sharply moving the control stick aft to temporarily "load up" the wing with "extra" g's, or sharply moving the control stick forward to temporarily "unload" the wing to a lower-than-normal g-loading in relation to the bank angle, will immediately curve the flight path up or down as well as decrease or increase the turn radius and rate, but will not cause a significant, immediate change in the position of the slip-skid ball and/or yaw string. Evidently these control inputs do not create a significant, immediate "demand" for a change in yaw rotation rate -- i.e. an immediate, significant change in the rate of curvature of the flight path in the yaw dimension --so yaw rotational inertia does not tend to drive a significant sideslip. (Even though, if the new position of the control stick is maintained indefinitely, the aircraft must eventually settle into equilibrium at a different airspeed, and thus a slightly different turn rate and yaw rotation rate, than it had originally, so that a reduction in angle-of-attack might be expected to promote a slight temporary skid, and an increase in angle-of-attack might be expected to promote a slight temporary slip. These dynamics don't seem to be observable in actual practice.) Compare this to the "wingover" case in footnote 2, where the full three-dimensional evolution of the flight path can eventually drive a rather pronounced sideslip soon after the wing is "unloaded" while climbing, due to the eventual "spike" in the curvature in the flight path in the yaw dimension, and likewise the simultaneous "spike" in the yaw rotation rate that would be required to keep the sideslip angle exactly at zero.

4. To elaborate further, to a first approximation the elevator can be thought of an angle-of-attack control, just as the rudder can be thought of as a sideslip angle control. However there are several significant complicating factors. In the case of the elevator, these factors include pitch rotational inertia (only significant when the pitch rotation rate is changing rather than constant), and the effect of aerodynamic damping of pitch rotation, which also can be viewed as a "curvature" of the free-stream relative wind whenever the flight path is curved. In general, due to the "curving relative wind" or "pitch damping" effect, the elevator must be raised somewhat higher-- and more "pull" force or less "push" force must be applied to the control stick or yoke -- to achieve the same angle-of-attack whenever the flight path is curving in the pitch dimension (in the nose-up direction, as in a normal turn or in an inside loop), compared to when the flight path is linear. This is one of the reasons that we often perceive a need to "hold the nose up" in a turn, even in gliders where our goal is to maintain the most efficient angle-of-attack, rather than to maintain a constant altitude.