I would not say gravity is "ficticious" as it acts on the aircraft and must be counteracted by an aerodynamic force (or the ground if standing still). But gravity is different from aerodynamic and thrust forces as it acts on every single molecule in the same way (one may stretch a bit falling into a black hole, but within the confines of a 172, it's similar). So, thrust and drag connot act on every molecule so the parts

attached are either stretched or squished, depending on whether they are in front or behind the thrust. These "fictitious" G forces may be better characterized as "inertial resistance" to acceleration (or decelleration), but the effects are real.

@CharlesBretana That's a really neat point, thanks for the very clear exposition.

I'd say the question got off to a wrong start in the first sentence. It says "Coordinated flight requires that the yaw rate and the centripetal acceleration correspond.". But note that this is not a sufficient condition for coordinated flight (zero slip or skid).

As a thought experiment, say we start in a fully coordinated right turn, and then we suddenly "jump" to a condition where the yaw rate has not changed, nor has the turn rate, nor has the net centripetal force, but the nose is now pointing 5 degrees to the right of the actual direction of the flight at any given instant. The aircraft is skidding. Some of the centripetal force is now provided by the airflow striking the right side of the fuselage.

(Note that the bank angle must have decreased, or the net centripetal force will have increased, which contradicts the conditions specified above.)

Anyway, setting aside the complication of turning /slipping/ skidding dynamics, a much simpler version of your question would be "It is possible to fly a loop by manipulating the elevator only, without changing the position of any other controls. How can a single control, the elevator, control both pitch rate and centripetal force?" The answer, clearly, involves the way that the aircraft's pitch stability dynamics "tie" together angle-of-attack and pitch rate.

Think of the elevator as fundamentally controlling angle-of-attack, not pitch rate. The angle-of-attack (together with airspeed) will determine the magnitude of the lift vector, which will determine the curvature of the flight path, which will determine pitch rate.

Of course that's a bit of an oversimplification because if you have two aircraft of identical shape and size and weight, but one has much more pitch rotational inertia than the other, then the pitch input required to, say, smoothly increase angle-of-attack by 5 degrees (at a precisely specified rate of increase) to begin the entry into a loop will not be exactly identical in each case.

So we can't really say that the elevator position is exactly tied to angle-of-attack in a lockstep fashion, even for a given airspeed. So, it's complicated. I think it would be correct to say that the elevator can't independently control both pitch rate and total lift force, but yet it's still a true statement that the elevator can be used to control pitch rate, and it's also a true statement that the elevator can be used to control total lift force.

(Oops, the third occurrence of the word "right" at the beginning of this block of comments should have been "left". The airflow is striking the left side of the fuselage.)

Back to the slip-skid thing-- I think you are being overly simplistic to say that a coordinated turn involves two different variables (lateral acceleration and yaw rate), and therefore requires two controls. We could identify additional variables as well, but none of them are independent of each other, they are all linked together in a complex way.

These additional variables could include heading, slip/skid angle as measured by a yaw string or weathervane, angle of deflection of inclinometer ball, bank angle, angle-of-attack, and airspeed. And we can note that suddenly yanking the stick back (while turning) to change the g-load does not, to a first approximation, create a change in the position of the inclinometer ball, though it does change the net centripetal force.

And we can note that rather than controlling yaw rate per se, the rudder can be thought of as a more of a "trim tab" that adjusts the aircraft's slip/skid angle through the airflow, as measured by a yaw string or weathervane on the nose. Just as the elevator is less of a pitch rate control and more of a "trim tab" that adjusts the aircraft's angle-of-attack in relation to the airflow. Though again, it all gets a little more complicated when you recognize that rotational inertia plays at

Robert, Let me give you an example. If you are in closed room on a rocket ship, 380K km from the earth, which is accelerating (under thrust) away from the earth at 1.0 G, and tried to analyze the motion of an object in the Frame of reference of that room, you would discover that in order to make the force equation work, you would have to include an additional "Force" equivalent to 1 G.

least a small role in the aircraft's flight dynamics, both in pitch and in yaw. Finally, we can note that some aircraft manage to fly fairly coordinated turns without a rudder at all -- e.g., a hang glider.

Now, in that room, gravity is certainly not 1 G (it's way far away from the earth), so is that 1 G force "real", or is it fictitious? It is fictitious because it is only there to account for the acceleration of the frame of reference you are doing the analysis in.

The key hurdle here is to understand that the acceleration of a frame of reference is no more absolute than it's velocity. Relativity applies to acceleration as well as to Velocity. Just as you understand for Velocity, no frame of reference can be considered to be THE universal baseline frame of reference, from which all others should be compared to see what acceleration they have.

So, I think you may be assuming that the earth frame, (or any other frame which is not accelerating with respect to the earth frame), is the absolute acceleration from which all other frames should be measured, simply because it is not accelerating with respect to everything else in your experience living in that frame. This is wrong. The earth frame IS accelerating, at 32 ft/sec2.

The question includes the statement "So in a sense, the plane is skidding through the pitch change. So what are the elements which balance the skid?" It might help to realize that a skid can be viewed as the centripetal force being "too large for the bank angle", but a skid also can be viewed as a mis-alignment between the aircraft's heading and the velocity vector. Use the rudder as needed to eliminate this mis-alignment, and you've eliminated any skid.

So you don't have to independently worry about centering the yaw string and centering the slip-skid ball. The two things are linked together.

Likewise, in your example of the pitch-up-pitch-down maneuver, if you use the elevator as needed to command the angle-of-attack you need to get the flight path you want, that's like using the rudder as needed to get the sideslip angle you want (normally zero). There's no other thing to worry about. There's no need to explain how the maneuver can be "balanced". The concept of "balance" really has no meaning here at all.

You seem to be interested in asking something like "what balances the torque from the elevator"? (For example, you say "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."-- this is not true at all; otherwise planes wouldn't need tails.

Of course, if the pitch rate is changing, then the torque from the elevator is not fully "balanced"-- some of it is used to overcome pitch rotational inertia and drive the change in the pitch rate.

Other things that balance the torque from the elevator-- the planes longitudinal stability dynamics, just as is the case in wings-level flight, where the elevator often will be placed somewhat up or down from the neutral position. Also, when the flight path is curving, so pitch rotation rate is non-zero, we have an aerodynamic effect called "pitch damping" which can be thought of as the aircraft's aerodynamic resistance to the pitch rotation about the cg. The horizontal tail is key

The horizontal tail is a key contributor to "pitch damping"-- another way to say this is to note that for any given angle-of-attack of the wing, the angle-of-attack of the horizontal tail is slightly more positive or slightly less negative when the flight path is curving in pitch (in the looping direction, the nose-up direction) than when the flight path is linear. The instantaneous, local "relative wind" actually can be viewed as

Another way to look at the "pitch damping" phenomena-- the instantaneous, local "relative wind" actually can be viewed as curving to follow the path of the turn or loop. (Strictly speaking, this statement is exactly true only in the case when the aircraft's pitch rate and yaw rate are correctly matched to the flight path, so that the angle-of-attack and sideslip angle at any given point on the aircraft is constant, not changing.)

( If we somehow kept the flight path exactly linear while allowing the aircraft to yaw or pitch, we'd again see the "yaw damping" or "pitch damping" phenomena-- a difference in slip angle or angle-of-attack between the nose and the tail of the aircraft, induced by the rotation-- which again could be described as a "curvature" in the local, instantaneous relative wind. The nose and tail are "feeling" different directions in the local airflow.)

( If we somehow kept the flight path exactly linear while allowing the aircraft to yaw or pitch, meaning that the slip angle or angle-of-attack could not remain constant, we'd again see the "yaw damping" or "pitch damping" phenomena-- a difference in slip angle or angle-of-attack between the nose and the tail of the aircraft, induced by the rotation-- which again could be described as a "curvature" in the local, instantaneous relative wind.)

(And in this case it wouldn't be correct to say that the relative wind is "curving" to conform to the curvature in the flight path itself, because the flight path is linear.)

Anyway, the pitch damping phenomena is a real thing and helps to explain why in curving flight, the elevator is not placed in exactly the same place, to command a given angle-of-attack, as in linear flight. At the top of a loop, sometimes the pitch rate is so high, and the damping effect is consequently so strong, that the stick can be full aft without commanding the stall angle-of-attack-- that's what this question (answer) was about-- aviation.stackexchange.com/a/55876/34686

Anyway I think the long and short of the answer to what you are asking is : The elevator does control centripetal acceleration, and the elevator does control pitch rate. But the elevator does not control centripetal acceleration independently of pitch rate, nor does it control pitch rate independently of centripetal acceleration. Fortunately, it doesn't need to.

Or more expansively: The elevator affects the balance of pitch torques, so the elevator does control centripetal acceleration, and the elevator does control pitch rate, and the elevator does control angle-of-attack. But the elevator does not control centripetal acceleration independently of pitch rate or angle-of-attack, nor does it control pitch rate independently of centr accel or aoa, nor does it control a-o-a independently of centr accel or pitch rate. Fortunately, it doesn't need to.

I think the problem with introducing slipping/skidding into the question is that it (incorrectly) implies that there's something we have to worry about in turning flight, some sort of "imbalance" in forces, that is not fully controlled simply be using the rudder as needed to set the "sideways angle-of-attack" of the fuselage to the desired value (normally zero). But there is no such something. Setting the fuselage to the desired "sideways angle-of-attack" takes care of everything.

And likewise, in looping maneuvers etc, setting the angle-of-attack as needed to get the desired pitch rate, or the desired radius of curvature of the flight path, whichever we care to specify, takes care of everything. There's no mysterious "imbalance" to worry about.

The problem is not under-constrained-- there's no need for an additional control surface. Unless we want to do wild maneuvers like the "Cobra" where we are making the pitch attitude vary without causing the normally expected change in the flight path etc-- then we may need thrust vectoring!

I think there's an answer to be constructed out of those last four chat posts or so--

Still obsessing about this -- in Peter Kampfs answer, he said "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." I don't find this to be a useful concept. Sure, starting from a steady-state condition, the immediate effect of a change in elevator position is a net pitch torque, and thus a change (acceleration) in pitch rotation rate.

Just imagine that you had a weathervane, free to pivot on post, with fins on the end, and a little rudder behind the fins whose position you could change with a servo. As you drove down a straight road with this thing on your car, you change the position of the rudder from centered to right-deflected.

In the very short run, you'll create a yaw torque and a yaw rotational acceleration, but on a time scale of any more than say half a second or so, the position of the rudder is controlling the weathervane's "slip angle" (the angle between the direction the arrowhead is pointing and the direction the wind is actually coming from), not the weathervane's yaw rotational acceleration rate, and not the weathervane's yaw rate.

We can think of an elevator in the same way. Sure, there's a relationship between angle-of-attack and pitch rate, but the elevator is controlling the pitch rate only indirectly. The cause-and-effect chain is like this--

The cause-and-effect chain is like this-- change in elevator position > net pitch torque > pitch rotational acceleration > change in angle-of-attack of wing > wing stabilizes at new angle-of-attack > change in magnitude of lift vector > change in rate of curvature of flight path > change in pitch rate.

Again, it's further complicated by several things-- the pitch damping effect which causes the relationship between elevator position and wing angle-of-attack to vary somewhat as the pitchwise curvature of the flight path varies. And effect of pitch rotational inertia, meaning that we need a slightly different elevator position to briskly transition to a new angle-of-attack than to maintain that angle-of-attack.

Still, on any time scale longer than half a second or so, it seems that the most useful concept is that the elevator is primarily governing the angle-of-attack of the wing--

Naturally, in the "cause-and-effect chain" above, "change in rate of curvature of flight path" could mean the introduction of a downward or upward curvature when the flight path had been linear before. And, naturally, we could also add additional elements at the end of the chain, such as

And, naturally, we could also add additional elements at the end of the chain, such as > change in airspeed due to change in direction of flight path > change in magnitude of lift vector > further change in direction of flight path > further change in airspeed . All together this helps explain, why, say, an abupt aftwards pull of the stick leads initially to a steep zoom climb, followed by, perhaps, a round-out to a gentler steady-state climb at a reduced airspeed--

followed by, perhaps, a round-out to a gentler steady-state climb at a reduced airspeed-- you can't explain all that just by saying that the elevator commands a pitch rate--

t

Back to Ken's question - the comments on sideslips or skids in the question fail to bring out 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 that some aerodynamic asymmetry is present which is causing the fuselage to fly in some attitude other than pointing head-on into the airflow. 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. Fix this, and we end the slip. And there's simply nothing equivalent in regards to purely pitchwise maneuvering. Unless you simply mean flying at some angle-of-attack that is not the one that yields the flight path you want. Fix that by moving the elevator- problem solved.

The previous two comments are quite brilliant but oh my god my head hurts with the concept of skid in this questions. As I understand, Ken is using the word skid to describe a rotating plane's motion along z-axis of the plane's frame of reference. So vertical, not lateral as we would (or at least I would) assume based on "standard" terminology of flying.

@Jpe61 I think he is using the idea of "vertical skid" to describe a mismatch between the rate of pitch rotation, and the rate at which the flight path is actually curving in the pitch axis. Thus forcing a change in angle-of-attack. My comments and answer argue against the usefulness of this analogy--

re "this means that the yaw rate must equal $\omega$." -- no, it's less, because of the bank angle. (Imagine a near-vertical bank-- almost no yaw rotation.) But again, a slip or skid can be present even if the yaw rotation rate is correctly matched to the rate of curvature of the flight path in the yaw axis-- see my answer and my earlier comments-

I think I'm getting an aneuryms from all this slipping, skidding and sliding 🤣 I really like this question, but it seem there is a mismatch with what Ken is asking, and what we are explaining. For me the control and flight mechanics are dead clear, there is no mystery or conflict.

If there is a theory that contradicts the flight of a plane, it's not correct. This reminds me of the theory of lift. There are several theories, none of which are CCC, but they are close enough, or at least they are not wrong...