Your question is very confusing. First, over rotation in pitch is not what the term "skid" normally describes. Second, you say the elevator controls AOA, but can't simultaneously control pitch rate - where do you get this notion from?! Of course the elevator controls pitch, if not what does? And finally, "how do planes establish a pitch rate..." the plane doesn't establish a pitch rate, the pilot does - By moving the elevator. It's not clear at all what your confusion actually is...

@MichaelHall have to agree with you here. I actually started with the "iceskater" analogy before realizing the wing really runs the show (and gravity), and (with increasing AoA) pulls harder into the turn. In reality (why fighter pilots move their weight back), you don't want "an incredibly strong tendency to align themselves with the airflow". That is the job of the tail. (Which is why elevators and rudders must be deflected).

Oh, and Mike a plane with pitched wings (like the B-52) need not have the nose and fuse out of the relative wind, only the wing need have a (dragging) AoA.

@MichaelHall, I understand this is a novel usage of the word "skid", however that does not make it incorrect. If a skid is over-rotation of the rotational heading vs the translational heading, which is it in every other field, then I think that applies equally here. If you have a better word for describing the behavior, perfectly exemplified in the Cobra maneuver, I'll happily use it.

And Kenn (if I may use first names) in a loop pitch rate will be that which maintains desired line of flight. You must consider gravity as part of "centripetal force" (negative to positive back to negative), which determines the amount of lifting force required.

@MichaelHall regarding the elevator control, it is a fundamental rule of controls engineering that a single control input cannot directly control two control outputs. This is known as an underactuated system, and it a topic of continuing active research. The elevator controls one part of this system, but something else stabilizes the rest. So either it controls pitch rate and some downstream effect stabilizes centripetal acceleration or the vice versa. I will update my question to make this clearer.

@RobertDiGiovanni, I respond to pretty much every name, including "Hey, stupid!" Re this Q, gravity plays no role here on rotation as it affects all parts of the plane equally. A plane in a hypothetical freely falling atmosphere would have the same issue that the system is underactuated.

Vectored thrust may also be used in the "Cobra". Change in relative wind stops "over rotation", unless one would have aircraft revolving on their yaw axis while turning around a point. That seems more like what the Earth does around the Sun. (LOL, sorry). It may be possible if the plane is directionally unstable.

Kenn, gravity does play a role in whether AoA changes or direction changes. You have a good question. Try to grasp it by considering that role of gravity in that which changes direction. (If the aircraft were buoyant, would this be different?)

Kenn, changing AOA to centripetal acceleration in the third paragraph does nothing to clarify the question, and I think you are overthinking any "fundamental rule" about controlling two outputs. The elevator directly controls pitch. Pitch is defined as rotation about the lateral axis. AOA is a byproduct (or "downstream effect") of pitch and power, so the elevator does control AOA. If centripetal force lies in this same plane, then the elevator would (indirectly) control it as well.

In case it helps, pilots don't think about controlling centripetal force. It isn't really in our vocabulary, and there is no instrument for providing feedback. We use the elevator to control pitch, AOA, and G force. Centripetal force is a downstream effect, or engineering concept, that is simply a byproduct of control inputs made to achieve the desired aircraft attitude and performance. No student pilot ever asked his/her instructor "how do I establish a pitch rate that corresponds to the centripetal acceleration?" Not sure if this perspective helps or not...

And I understand what you mean by "skid" in this context, but slip and skid have very distinct meanings in aviation, specific to the horizontal plane, yaw, and rudder control. I don't have a better term off the top of my head, but over rotation in pitch beyond critical AOA is a very different concept. But even removing a coordinated level turn from the discussion I'm not sure what the base questions is: Loops, Cobras, or mathematical confusion?

@Michael Hall: Finally someone with a clue! For a parallel, if you study the biomechanics of walking, you run in to a lot of complicated equations balancing forces, accelerations, &c. (I'd have to go re-read the text to be more specific.) But somehow kids manage to learn to walk before learning differential equations, or even basic calculus. FTM, insects manage to handle 6 legs, and sometimes wings, without much of a brain at all :-)

@jamesqf, Thanks, and an excellent comparison! The internet tends to foster a lot of book-smart overthinking of basic principles. Correction to my comment above though, I believe a G-meter (for aircraft equiped with one...) would directly measure centripetal force, right?

@RobertDiGiovanni - "Oh, and Mike a plane with pitched wings (like the B-52) need not have the nose and fuse out of the relative wind, only the wing need have a (dragging) AoA." If this is your attempt to explain angle of incidence to me, please stop.

Guys, I'm going to go ahead and delete this question here in a bit, and reopen it from a different tack. I see a lot of misunderstood concepts in the comments above and it can't be addressed in the comments section. This question is fundamentally about the theory and application which goes into aircraft design and control dynamics, not about how we learned to fly our Cessnas. When wanting to model systems, the limitations on what is theoretically possible become relevant, no matter how easy it was to learn how to walk as children.

This is worth addressing, though: "pilots don't think about controlling centripetal force. It isn't really in our vocabulary, and there is no instrument for providing feedback." @MichaelHall you're thinking about this in the wrong way. What's seen as a centripetal acceleration in the world frame is sensed as a centrifugal acceleration in the body frame. Your butt is very good at sensing these loads, and from a zero-G pushover to a snap roll, with a little bit of training it's quite easy to be accurate to +-0.10g. Source: I test this frequently while doing aerobatics.

@Kenn Sebesta: But that's really the point. When you fly, you're not doing equations, you're just going by the seat of your pants, no?

I'm at a loss with this Q. It makes controlled flight seem impossible, yet I know for a fact it is not...

@Kenn, Centripetal (or Centrifugal) forces are only "Felt" by a pilot if they are those forces as measured in a non-accelerated (zero-G free fall) frame of reference. In a 1-G frame of reference, (where we generally do all these calculations/analysis,), they are fictitious forces, only included to balance the force equations because we are doing them in an accelerated frame. Pilots cannot "See" (or "feel" these fictitious forces.

The elevator does NOT control Pitch rate, at least not directly. Pitch rate is determined by the excess (or insufficiency) of the total aerodynamic force on the airframe perpendicular to the Flight Path. if the Lift force is greater than the Mass, (x 32.2 ft/sec), the normal load factor is > 1.0 g and then, if this load factor exceeds the component of god's G which is normal to the flight path, the pitch rate is positive, (nose goes up-relative to wings) if it is less, the normal load factor is < 1.0 g and the pitch rate is negative (nose goes down) –

@Jpe61: It's the "Centipede's Dilemma". The centipede could walk perfectly well, until someone asked how she managed to control all those legs: en.wikipedia.org/wiki/The_Centipede%27s_Dilemma

@CharlesBretana (one more time) rotation in pitch (like when taking off) is caused by deflection of the elevator. This causes a torque that changes AOA until relative wind changes enough (on the entire airframe) to provide sufficient counter torque to zero torque acceleration. So, due to the control surface deflection you wide up "flying sideways" unless another force (such as gravity) holds you in place. That's establishing AoA. Directional stability (the tail) is what rotates the aircraft through the turn. Two different rotations. OK?

So, to clarify (crystalize?), if you set your AOA (in the direction of the turn, by whatever means), the tail will "chase" the plane around the turn attempting to maintain its AoA (because relative wind keeps changing). The centripetal vector (by whatever means) is unbalanced, causing the aircraft to go "around and around".

@jamesqf I am, but the plane isn't. My question is about the aerodynamics of controls, and I think I must have asked it very poorly because the responses are about pilotage.

@Jpe61 I don't understand this comment. The question is about the structural mechanisms which permit a plane to skid through a vertical body-frame acceleration. There is no requirement for two controls here, similar to how a plane doesn't have to have two controls to turn. There are many R/C planes which have only a rudder but no ailerons.

@CharlesBretana Centripetal acceleration is not a fictitious force, and is a misnomer to call centrifugal force fictitious, as it is a very real force in the body-frame. Centrifugal force is what leads to G-lock, which can and has killed people. It is indeed real, even if it is not perceived in the inertial frame.

@CharlesBretana This is incorrect. Angular rates can only be generated by torques, which are forces applied at some distance from the fulcrum. The vertical force can change without a pitching torque, as is evidenced by rotorcraft.

Unlike rotorcraft, the airplane airframe has certain properties which make it point in the direction of flight. The question I have was about the characteristics which make it align itself with its vertical motion.

@RobertDiGiovanni Yes! Exactly this. The elevator generates a force acting on a lever arm. This force is actually in the opposite direction of the pitching arc. Without a stabilizing reaction torque, the plane would increasingly accelerate around its lateral axis. I think the wings are providing that reaction torque, but I thought maybe someone here can confirm.

I'm also interested in the design which goes into the tail, and to understand if the A/C designer cares in the slightest bit about this effect, since it might be automatically solved by any sane set of wings.

@CharlesBretana, when you were an F-4 fighter pilot how did you increase the pitch rate of your Phantom? And did you wear a G-suit? If so, why?

@KennSebesta, "My question is about the aerodynamics of controls, and I think I must have asked it very poorly because the responses are about pilotage." I anxiously await some clarity, and an end to this arcane hand wringing across differing disciplines...

And yes, I caught myself earlier regarding centripetal force being felt as Gs. (despite their oft repeated "fictional" nature...)