Aerodynamic Center

[mmrassi]

Hi,

I was reading something about AC and I confused about the difference between AC and CP?!

We know that Lift is assumed to be act on CP but in the picture below Lift is acted on AC, Can someone explain it to me please?



Thanks

[Gabriel]

(I thought I had seen a post by Highkeas just prior to pushing "reply"?)

I'll start saying that the picture you've posted is wrong. You'll see why (if you have the patiente to read my post), but let me assure you that the CP and the AC are two very different things.

Ok, let me first simplify the things a bit by using lift only.
In this condition the Center of Pressure (CP) becomes the Center of Lift (CoL). Because the lift is made up of pressure only, the center of lift is in the same position than the center of pressure (or better, the center of lift is the horizontal coordinate of the center of pressure). Drag, on the other hand, is made both of pressure and share.

The CoL is the point on "something" (that must be treated as a rigid body) where the lift force can be concentrated producing the same effect that the distributed lift around that same "something".

This means that the dynamic response of that something will be the same if you use all the elemental infinitesimal lifts applied around the thing that if you use the sigle net lift applied at the CoL.

That's sort of a definition, however when you starplaying with it you get to a propperty that's exclusive of the CoL and that hence can be used in lieu of its definition. You'll see:

If you inegrate the elemental little lifts around the body, you get a net lift L. That's the lift that you'll apply at the CoL. That's the easy part.

Now, if you integrate all the little pitching moment contributions about any given point "O" made by all the elemental little lifts around the body, you'll also get a net pitching moment. Let's call it Mo (the little "O" showing that we are taking moments about "O")

Now, for the CoL to trully represent "the point where the lift force can be concentrated producing the same effect that the distributed lift", as we defined above, we have no choice but to require that the pitching moment moment about "O" made by the net lift L applied at the CoL, be exactly the same Mo as calculated above for the individual-elemental-distributed little lifts. If we we call "(CoL-O)" the vector that goes from the poit "O" where we are taking monents about to the point CoL of the application of the net lift L, then the pitching moment about "O" made by the net lift L applied on the CoL can be written as:

Mo = (CoL-O) X L

Of course, "O" can be any arbitrary point. And that's how things start to get interestring. If for example we take O in the CG we get:

Mcg = (CoL-CG) X L

And if we take O in the center of lift, we get:

Mcol = (CoL - CoL) X L

Oops.... something looks like zero there, doesn't it?
Yes, the CoL has a very important propperty, and that is that the pitching moment made by the lift about the CoL is zero. And because this is true ONLY for the CoL, we can use it as a definition of CoL:

The CoL is the point where the pitching moment made by the lift is zero. How's that?

Now, bear in mind that I've used words like "something" or "body". That something could be for example a wing, and we would be talking about the center of lift of the wing. Ot it could be the tail, and we would be talking about the center of lift of the tail, or it could be the whole airplane, and we would be talikng about the center of lift of the whole airplane.

And here is where things start to become interesting again.

Let's take a simplified case that is however a very good approximation to the real case (well, most of the times at least). We have an airplane that is flying but not turning and not pitching up or down. Let's asume that the drag and the thrust don't make a pitching moment about the CG (that's the simplification). Of course the airplane's weight can't make a pitching moment about the CG either, because it's applied on the CG. The only force left that could make a pitching moment about the CG is the lift (and I mean the lift of the whole plane, not the wing only). Where can the CoL of the whole plane be in relation to the CG?

Because the plane is steadily not pitching up or down, we need it to be in equilibrium in pitch, and thus pitching moment about the CG MUST be zero. And because the lift is the only force with the ability to make a pitching moment, the pitching moment of the lift about the CG must be zero. But wait! The pitching moment of the lift is zero only about the CoL!

Congratulations, you have just discovered that, whenever the plane is in equilibrium, the CG and the CoL ARE on the same plane.

Not the use of ARE. I didn't say, can be or should be. It just IS that way.

Ok, but what if the CG and the CoL are NOT in the same place? Then there is a net pitching moment and the plane is not in equilibrium in pitch. The plane will either pitch up or down (and at an accelerated rate, that is). That simple.

The first time I nearly understood this a lot of questions raised in my head:

How does the pilot ENSURES that the CG and the CoL is in the same place? What happens as fuel burns and the location of the CG changes? Or if a passegenr walks down the aisle to the lavatory? If the above is true and the CG is actually on the CoL and that measn that the plane doesn't pitch up or down, how does the pilot do to initiate a pitch up or pitch down maneuver?

The answer to all these questions is the same: The pilot MOVES the CoL to either match the CG AT ALL TIMES while he wants to keep the plane in equilibrium, or moves the CoL aft or ahead of the CG to pitch down or up respectivelly.

Say again???
The pilot moves the CoL??? Yes.
How? Later.
Wasn't that a fixed point, a sort of geometric propperty of a body??? No.
Take the simple case of an airfoil. Both the lift and the pitching moment are function of the AoA (well, there is one point about which the pitching moment is not a function of the AoA, more on that later). So if you cahnge the AoA of an airfoil, then the pitching moment about what was previously the CoL is no longer zer, and hence that point is not longer the CoL, and hence the CoL is smoewhere else: it moved.
In the more complex casde of a whole airplane, the movement of the CoL is even easier to understand: Say that you have an airplane steadily flying straight and level. In that condition, we now know that the whole airplane's CoL matches the CG. Now the pilot pulls up. What happens? The plaen pitches up. That measn a non-zero pitching moment and that means that the CoL is somewhere else but on the CG. It moved. How? Let's see it frame by frame. Now it's time to use (and abuse) your picture.

If you look at the pitcure you've posted, it reads Neutral point: "position of the CG where tail moment and wing moment are equal".

I can't end counting how many errors are in that sentence alone.

To begin with, we'll have to assume that it means that "where the tail moment and wing moment are of equal dimmension and opposite direction", which would better said "where the tail and wing pitching moments cancel each other", or sum zero. In a simplified "wing and tail" model (i.e. where only the wing and tail make forces and moments), that would be further clearer if we just said "where the pitching moment due to lift is zero".

But wait! Wasn't that the definition (or exclusive propperty) of CoL?
It was, it is and it will for ever be. Tell whoever made that picture that that is NOT THE NEUTRAL POINT, that's the whole airplane's CENTER OF LIFT (the horizontal coordinate of the CP, remember?).

Finally, by now, what does it mean "the position of the CG where..."?
Grab a marker and mark the CoL in that picture (that is matching the CG). Now move the CG somewhere else. Does that changes the pitching moment about the CG? Yes. Does that changes the pitching moment about the dot marked with the marker? Let's see. The AoA has not changed. The wing and tail are still in the same position relative to that dot too. Hence the wing and tail lifts and pitching moments about that dot did not change. It was zero and still is, so this point is still the Center of Lift even when the CG is somewhere else. The Center of Lift doesn't care about where the CG is (although the pilot, even without knowing so, does care where one is relative to the other).

Returning on how a pilot controls the location of the CoL.
In wour picture, the plane is happily flying straigh and level. The CoL (incorrectly depicted as the neutral point) matches the CG. The nose-up pitching moment made by the wing is ballanced by the nose-down pitching moment made by tha tail (always about the CG, that by now it's matching the CoL). Next, the pilot pulls up on the stick, what makes the tail tilt in the trailing-edge-up direction. In the picture you posted, that translates in a reduced AoA on the tail, and hence a reduced lift on the tail, and hence a reduced nose-down pitching moment about the CG. But the nose-up pitching moment made by the wing has not changed. This brakes the ballance and now we have a net nose-up pitching moment, and hence the plane will start tp pitch up (which, not just by chance, is what the pilot probably had in mind when he pulled up).

Evidently, since the pitching moment about the CG is not longer zero, the CoL and the CG don't match any longer, and not because the CG moved because it didn't. So where did the CoL go?

We have to look for a point about which the total pitching moment is zero. Since the wing's nose-up pitching moment about the CG got stronger than the tail's nose-down pitching moment about the CG, we have to look for a point that has a shorter arm for the wing and a longer arm for the tail. Yes, the CoL has moved ahead of the CG.

You see how the pilot controls the location of the CoL to control the airplane in pitch? WITH THE ELVATOR!!!

When applied to the whole airplane, the CoL is all about equilibrium and control, but has nothing to do with strability (the ability to return by itself to the equilibrium when disturbed from it).

To finish the CoL chapter, remember that the CoL can be applied to anything, for example to the wing, or for the tail, or for the whole airplane (as we did). Let's give just a short look at the CoL applied to the wing (or an airfoil).

The propperty still applies: The CoL of a wing is the point about which the pitching moment of the wing is zero. Tyically, for reasons that we are going to mention in the next chapter, we represent the action of the wing with a straigh arrow indicating the lift at one point and a turning arrow around the same point indicating the pitching moment.

In that case, evidently said point or NOT the CoL, or there would be no reason for the turning arrow.

As if there were not enough errors, in the picture you posted the lift is applied on the Aerodynamic Center (AC), but there is no turning arrow. Since the AC doesn't match the CoL, and the lift is shown as applied on the AC, and there is no turning arrow indicating the pitching moment made by the wing or tail about its own ACs, something IS wrong: Either the lift is applied in the wing and tail own CoLs (not the AC), or the pitching moment of the wing and tail about their own ACs is missing.

[Gabriel]

AC and Neutral Point time.

Say that you have an airfoil making lift in the upwards direccion.

Take a point of reference "A" way ahead of the airfoil.
If you want to represent the effect of the lift on that point, you'll have to put the lift there plus a nose-down pitching moment.

In other words, you'll have to put a vertical arrow starting on A and pointing up and marked with L, plus a counterclockwise arrow around A marked with M.

Now take a point of reference "B" way aft of the same airfoil.
It's evident that while the same lift still goes up, now the pitching moment about B is nose-up.
Now you still have the same vertical arrow starting on B and pointing up to represent the lift, but a clockwise arrow around B represnting the pitching moment.

Evidently, somewhere between A and B there will be one point about which, while you'll still need the vertical arrow to represent the lift, the pitching moment will be zero. If it takes you more than two second to say what that point is, I'll regret having wasted my time with the previous post. I hope by now you have already correctly said "It's the CoL".

Ok, now let's be a bit tricky. Let's increase the AoA of that airfoil.
Now the lift increases, and hence the vertical arrows both on A and B become longer. The new lift is the previous lift plus an increment.
The same happens with the pitching moments. The pitching moment about A is the previous nose-down pitching moment about A plus a nose-down increment, and the pitching moment about B is the previous nose-up pitching moment plus a nose-up increment (remember that we are talking about the same system, it's not two pitching moments acting at once, it's how the pitching moment changes whether you take it about A or about B).

Now it's time for the abstraction, let the brain sweat.
Forget about the full lift and the full pitching moment. Think in the increments only.

Both A and B receive the same increment in lift. It's evident that whatever point between A and B would receive the same increment.

But A recives a nose-down increment in the pitching moment, while B receives a nose-up increment in the pitching moment. It's evident that the increment in pitching moment will be zero about some point between A and B. What is this point?

Does it sound too similar to the CoL?
Yes, it does. Unfortunately so, because that's why (together with bad teachers and bad texts an pictures) so may people confuse the terms.

This is NOT the CoL. This is, ahem... the Aerodynamic Center (AC).

Let's make a pause here to clarify things a bit. What is the difference between the CoL and the AC?

The Center of Lift (CoL) is the point about which the net pitching moment due to lift is zero. Is the pitching moment zero about this point now? Then this is the CoL now. Now if the AoA changes, the CoL can (and mostly does) move.

On the other hand, the Aerodynamic Center doesn't care about what the pitching moment is, but about how it changes with the AoA. Never mind if in this instant the pitching moment about a given point is zero or not, if the AoA cahnges and the pitching moment doesn't, then this is the Aerodynamic Center.

So now we are ready for a definition: The Aerodynamic Center is the point about which the pitching moment due to lift does NOT change when the AoA does.

Note that in the example above, we just used an unamed airfoil changing from an unamed AoA to another unamed AoA (other that the second one was greater), and two generic A and B points, one "way ahead" and the other "way behind" the airfoil. And if you want me to get complicated, I could even think of two airfoils (one representing the wing and one the tail) as one single airfoil "with a fancy shape".

All that just to say, the example above was a very generic one. The resulting definition applies just to "something", which can be an airfoil, a wing, a wing-tail combination, or a whole airplane.

The AC is very interesting for a number of reasons.

When analyzing your picture, we had seen that if you put the airfoil's lift on the airfoil's CoL, you don't need to put an additional pitching moment about that point. You can take the airfoil's pitching moment as zero. While this looks good, it's not, because of the "mooving" nature of the CoL. You change the AoA and the CoL is not where it was, so now you have to either move the lift (that's complicated) or leave the lift where it is and put a pitching moment around that "now-not-the-CoL" point, and the magnitude of the pitching moment varies with the AoA (including being zero when the AoA happens to be such that the CoL is there again).

If we take any other point, the same happens. Not only the lift but also the pitching moment changes whenever the AoA changes.

Except when you take moments about the AC. In that case, while the pitching moment is generally not zero, at least it doesn't change with AoA. In that case, you can represent the effect of the lift whith a straight arrow that changes in size with the AoA, and a turning arrow around the AC that doesn't change with the AoA. That's why the airfoil's pitching moment about the AC is usually taken as the airfoil's pitching moment.

I really think that whoever made the pitcure you posted, he did mean AC in the airfoils where he put the lift arrows, but he wrongly ommited the pitching moment turning arrows around those AC points. That's the usual way to do it.

It is in the whole airplane, however, where the AC becomes most interesting. And this will lead us to understand why, for the whole airplane, the AC is also called the Neutral Point.

Let's go back to our plane in equilibrium flying sraight and level, as depicted in your picture. Say for a moment that the author actaully meant to put the CG in the the Neutral Point (that is, again, in the aerodynamic center of the whole airplane). I don't really think for a second that he mant this, but let's pretend it.

Then we would be in a case that I have mentioned is very rare: The AC and the CoL are in the same place, and the CG too. And the pitching moment about this common point is zero.

What happens if the plane is disturbed from that equilibrium and the AoA changes?

This is the question of stability. If, when disturbed from the equilibrium, the plane tends to return to the previous state of equilibrium, the plane is said to be statistically stable (or to have positive static stability). If it tends to diverge further away from the previous state of equilibrium, it is said to be statically unstable (or to have negative static stability). If it tends to do nothing, that is, to stay in the new disturbed position, it is said to be staticlly indifferent (or to have neutral static stability).

So, we were saying, the plane changes it's AoA. Since we are taking moments about the AC, the pitching moment doesn't change when the AoA changes. The pitching moment was zero before, and will keep being zero. The CoL doesn't move (as it almost always does). We still have the Col, the AC and the CG in the same spot. The lift will have changed, but since we are taking the lift acting on this point too, it makes no pitching moment about the CG either.

In short, the AoA was disturbed from its previous position of equilibrium and there is no pitching moment that will either tend to bring it back to that previous state or take it farther away. The AoA will happily reamin in the new value, or in wahtever value we set it. It's the case of a statically indifferent plane, one with neutral static stability. And we never want a plane like this, that's why I'm sure this is not what the author of the picture intended.

Now, let's go back to our original equilibrium. The CG matches the CoL (it's a requirement for having zero pitching moment about the CG, and hence for the equilibrium, remember?). But the whole airplane's AC is ahead of the CG by a distance D.

We can think of the plane as having its lift concentrated in the CoL=CG and no pitching moment about that point, or we can make the equivalent reasoning of placing the lift at the AC with a pitching moment about that point. Let's see.

For the plane to be in equilibrium, we need that the pitching moment about the CG is zero. About the CG, the weight makes no moment, so we are only left with the aerodynamic moments (we'll take those due to the lift only).

The lift, placed on the AC a distance D ahead of the CG, makes a nose-up pitching moment about the CG of L*D. Since the pitching moment about the CG is zero (condition for equilibrium), we conclude that, other than the lift, there has to be a ballancing nose-down pitching moment of the same magnitude: L*D. In a more graphical way, on AC we draw a straight vertical arrow representing the lift and a counterclockwise rotating arrow representing the pitching moment. The two together make for a zero pitching moment about the CG.

Now, what happens is the AoA changes? Say that it increases. The pitching moment about the AC won't change by definition of AC. It reamins the same nose-down value as before. On the other hand, the pitching moment that the lift placed on the AC makes about the CG is still a nose-up L*D, only that L has increased ebcause of the increased AoA. The result is a net nose-up pitching moment about the CG. Let's see: the AoA increases and the plane responds with a nose-up pitching moment, that will only tend to icnrease the AoA even further. If that sounds like something unstable, it is.

Finally, what if the CG is now ahead of the AC?
Now we can draw on the AC the vertical arrow representing the lift. That lift will make a nose-down pitching moment about the CG. So there must be again a pitching moment of value L*D, but this time nose-up (a clockwise turning arrow around the AC). When the angle of attack increases, the lift increases and the nose-down pitching moment it makes about the CG increases too (again, it's L*D but with a greater L). Onthe other hand, again, the pitching moment about the AC remains constant by defintion of AC. That's a net nose-down pitching moment. Let's see: The AoA increases and the plane responds with a nose-down pitching moment that will tend to restore it back to its original value. If that sounds like something stable, it is.

So to close this chapter:

If the CG is ahead of the AC of thewhole airplane, the plane is statically stable (or has positive static stability).
If the CG is behind the AC of the whole airplane, the plane is statically unstable (or has negative static stability).
If the CG is on the AC of the whole airplane, the plane is statically indiferent (or has NEUTRAL static stability). Hence the name of netral point. The AC of the whole airplane is the poitn where, if you put the CG on it,the plane has neutral stbility. Move the CG forward and it becomes stable. Move it back and it becomes unstable.

The AC of the whole airplane, or Neutral Point, cannot be controlled by the pilot (as he can do with the CoL of the whole airplane by moving the emevator). What the pilot can and must do, is ensure that the CG is ahead enoguh of said neutral point for the plane to have sufficient static margin (i.e. that it has sufficient positive static stability), a thing that is achieved during the famous "weight and ballance", making sure that the CG is within the airplane's approved envelope.

In the previous chapted we had said that the CoL of the whole airplane is all about equilibrium and control, and nothing about stability.

Well, in the same way the we can say that the AC of the whole airplane (Neutral Point) is all about stability, and nothing about equilibrium and control.

Except that equilibrium, control and stability are somehow related, but that's a whole different chapter that goes light years beyond the original question (as if these two posts hadn't gone far enough)

[mmrassi]

thanks for the reply,

I'm busy with reading and analyzing your reply.

[Gabriel]

I know. I miss a blackboard and a piece of chalk. It's so much easier than plain text.

[3WE]

In short, "If there is a stall warning, nose over immediately and aggressively."

[brianw999]

In short, "If there is a stall warning, nose over immediately and aggressively."

Oh bugger......don't start that again !!