Aerodynamic Center

[Mr Irons]

In simple terms can someone please explain what the Aerodynamic Center of an aerofoil is.

Thanks

[rigpiggy]

a la wiki

The torques (http://en.wikipedia.org/wiki/Torques) or moments acting on an airfoil (http://en.wikipedia.org/wiki/Airfoil) moving through a fluid (http://en.wikipedia.org/wiki/Fluid) can be accounted for by the net lift (http://en.wikipedia.org/wiki/Lift_%28force%29) applied at some point on the foil, and a separate net pitching moment about that point whose magnitude varies with the choice of where the lift is chosen to be applied. The aerodynamic center is the point at which the pitching moment (http://en.wikipedia.org/wiki/Pitching_moment) coefficient for the airfoil does not vary with lift coefficient (http://en.wikipedia.org/wiki/Lift_coefficient) i.e. angle of attack (http://en.wikipedia.org/wiki/Angle_of_attack), so this choice makes analysis simpler [1] (http://en.wikipedia.org/wiki/Aerodynamic_center#cite_note-NASA_GRC_2006-0).

long story short the point where all aerodynamic forces appear to be acting, will move according to configuration, angle of attack, etc.....

[zerozero]

I've always referred to it as the "Center of Pressure". The concept makes more sense when viewed as a cross section.

Center of Pressure (http://www.grc.nasa.gov/WWW/K-12/airplane/cp.html)


http://www.grc.nasa.gov/WWW/K-12/airplane/Images/cp.gif

[HazelNuts39]

NASA: Center of pressure moves with angle of attackAerodynamic center does not. As rigpiggy explains.

regards,
HN39

[henra]

I've always referred to it as the "Center of Pressure". The concept makes more sense when viewed as a cross section.


Hmmm, that could be identical to what is called the aerodynamic cneter but it doesn't have to. In fact usually it isn't.
The pitching moment in the aerodynamic center doesn't have to be Zero.
The trick is the pitching moment remains constant even if Alpha (or Lift being a function of Alpha) changes.

Center of pressure would be the point where the pitching moment is Zero for a given Alpha. If Alpha changes the pitching moment around this point will change, i.e. the center of pressure will move.

[Microburst2002]

Let my try:


When a number of forces are exerted on different points of a body, the net effect of all those forces on the body (both lineal and angular) is the same as the effect of a single force acting on a particular point.

This resultant force is the sum of all the forces, and the moment that this force creates about any point is the sum of the moments of all the forces about that point.

Each air molecule exerts a force on the airfoil. The resultant of all the forces is Lift, and the point at which Lift acts is the center of pressure.

The relative position of the CP with respect to the CG determines the necessary tailplane lift to trim the airplane.

We use the center of pressure for static situations. But it is better to use the aerodynamic center when it comes to non-static situations.

When there is a change in Lift we can think of the Lift after the change as being the sum of the Lift before the change plus the Lift increment. Understand? think of two vectors, one being the "old" Lift and the other the lift increment. The point at which the lift increment is acting is the Aerodynamic Center.

The relative location of the AC with respect to the CG will determine the degree of stability of the airplane.

[Pugilistic Animus]

I must say you are ALL wrong this is an aerodynamic center...I'll refrain from showing where the Cp is located...:ouch:
WALEG Photo Galleries - Angelina Jolie Swimsuit Photoshoot/Angelina Jolie (http://www.waleg.com/photos/displayimage.php?album=1280&pos=0)


:}:}:}

[CliveL]

Yep, they are all wrong.
The original question was related to the aerodynamic centre of an airfoil. This is the point at which the lift due to angle of attack acts. Theoretically, for subsonic airfoils this is at 25% of the chord aft of the leading edge. In practice it is within 1 or 2% of that point up to Mach Numbers where drag rise and shock waves become important. Camber produces a pitching moment even at zero lift so the centre of pressure, which depends on the resultant pitching moment and lift, varies with angle of attack.
The aerodynamic centre of a wing is usually close to 25% of the mean aerodynamic chord.
The aerodynamic centre of the aeroplane depends on tail area and position relative to the wing.
You can find details in almost any standard textbook.

CliveL

[Wind Shear Ahead]

The location of the airfoil where, if an aerodynamic force was to be applied, the resultant pitching moment would be the same for all angles of attack. For low speed airfoils, 1/4 back from LE. For Supersonic airfoils, about 1/2 back from LE

WSA

[henra]

This is the point at which the lift due to angle of attack acts.


Hmm, I'm afraid that is not universally correct.
That would be the Center of Pressure.
The Aerodynamic Center is the Point at which the moment is constant, independent of AoA, but not necessarily Null.
Yes, generally that is close to the Quarter Chord Line but it depends on the Airfoil (mainly on the Camber) where it is exactly.

[D120A]

Agree 100% with Henra. Thin aerofoil theory in an ideal fluid shows that the pitching moment coefficient Cm is independent of the lift coefficient CL at the quarter-chord point. For finite thickness wing sections in a real fluid, the point (which will be near to the quarter-chord point) can be found such that Cm about this point is independent of incidence. This point, where d Cm / d CL is zero, is called the aerodynamic centre. Note that this is where the first differential, the rate of change of pitching moment with incidence, is zero. The pitching moment coefficient itself at that point will have a constant value, denoted by the term Cmo.

The aerodynamic centre is useful because when you construct the equations of longitudinal static stability for the whole aircraft, taking moments about that point eliminates any variation in the wing pitching moment with incidence. It makes the equations simpler, although when you see them you'd be hard pushed to believe it.

Acknowledgements to the wonderfully clear 1963 lecture notes from the late F.G. Irving (1925-2005).:ok:

[CliveL]

Hmm, I'm afraid that is not universally correct.
That would be the Center of Pressure.
The Aerodynamic Center is the Point at which the moment is constant, independent of AoA, but not necessarily Null.Sorry guys, but that is not what I said. If you take moments about the point where the lift due to incidence acts then you will get a constant pitching moment as incidence varies and this point is generally close to 25% chord. So the aerodynamic centre is the location where the lift due to incidence acts and is independent of incidence (more or less).

Centre of pressure on the other hand varies with incidence. At zero lift the centre of pressure is at plus/minus infinity if there is any wing camber, because the pitching moment at zero lift is produced by a couple. As you increase incidence (and lift) the centre of pressure will move towards the 25% chord point but it will never get there.

[henra]

So the aerodynamic centre is the location where the lift due to incidence acts and is independent of incidence (more or less).


Hi CliveL,
As D120A already pointed out, there is a moment Cm0 in the aerodynamic center. This in turn implies that it is not exactly the point where the Lift is excerted, otherwise you wouldn't have a moment at this point. Normally the aerodynamic center is somewhat in front of the Center of Lift (i.e. Center of Pressure). The pitching moment of Airfoils usually increases with increasing AoA.

[HazelNuts39]

henra;

As pointed out already in the first response to OP, lift is represented by a force acting in the aerodynamic center AND a moment. The lift force then increases with AoA while the moment is approx. constant. Alternatively, lift can be represented as a force acting in the center of pressure without a moment.

[Microburst2002]

Hi, CliveL

I think I understand what the CP and the AC are, at least for practical purposes. Basically the first is where Lift can be considered to act, the second is where changes in Lift can be considered to act. More or less.

But I have always had difficulties in grasping what the Aerodynamic Moment is. I see that you talk about lift due to AoA or incidence, as it is different than the lift due to camber. That is what I don't understand well.

Symmetrical airfoils have no aerodynamic moment because their camber is null, but I don't understand why camber does creates it and AoA alone does not.

What is the origin of the aerodynamic moment? I thought that the circulation around the airfoil created an equal and opposite "circulation" of the airfoil, as I read in a nice Darroll Stinton book. I mean, it is not a "theoretical" moment of a force about a point, but a "pure" moment, like that of a pair of forces with zero net force but a given resultant moment which "phisically exists".

However I can't explain why the circulation due to camber is different to the circulation due to AoA.

I hope you can enlighten me with this thing that has always puzzled me.

Thanks

[HazelNuts39]

However I can't explain why the circulation due to camber is different to the circulation due to AoA.It isn't.

regards

[CliveL]

As D120A already pointed out, there is a moment Cm0 in the aerodynamic center. This in turn implies that it is not exactly the point where the Lift is excerted, otherwise you wouldn't have a moment at this point. Normally the aerodynamic center is somewhat in front of the Center of Lift (i.e. Center of Pressure). The pitching moment of Airfoils usually increases with increasing AoA.

I don't know what you mean by "in" the aerodynamic centre. Cmo is by definition the pitching moment at ZERO lift and the concept of centre of pressure in conjunction with zero lift is meaningless. As I said, Cmo is produced by a couple, which has no centre of pressure.
You can't talk of airfoil pitching moment without also specifying the point about which the moments are being measured. Usually this is taken as 25% chord for the precise reason that as incidence (AoA) changes the pitching moments about 25% chord don't vary much because the lift acts near there.

Most airfoils have positive camber (mid-thickness line above the line joining LE and TE) and this produces a negative Cmo, so if you measure pitching moments about 25% chord when the airfoil is lifting and that lift acts at the aerodynamic centre (also at 25% chord) then the pitching moment will remain equal to Cmo and the effective centre of pressure will be located at Cmo divided by lift coefficient aft of the measuring point. This means, as has been said, that the CP is behind the aerodynamic centre but that it moves towards the aerodynamic centre as lift increases.

How the pitching moment varies with increasing AoA depends entirely on the datum you assume for pitching moment. If you choose somewhere between the leading edge and 25% chord then the pitching moment will become more negative as AoA is increased. If you choose a point aft of 25% chord the moment will become more positive (nose up).

[henra]

HazelNuts,

As pointed out already in the first response to OP, lift is represented by a force acting in the aerodynamic center AND a moment. The lift force then increases with AoA while the moment is approx. constant.


That is exactly what I was trying to point out.

Hoped that was clear from my post !?

CliveL

You can't talk of airfoil pitching moment without also specifying the point about which the moments are being measured.


The point at which the pitching moment is measured would be the aerodynamic center

Re: Cm0 and Center of Pressure: Yes indeed for Zero Lift it is true, there is basically no Center of Pressure. Was an incorrect over-simplification, as happens often when trying to simplify things- Should not try to simplify too much.


regards,
henra

[CliveL]

The point at which the pitching moment is measured would be the aerodynamic center

Not true I'm afraid. The wind tunnel guys can choose any reference point they like. For convenience they usually use 25% mean aerodynamic chord, but that is not the aerodynamic centre for a complete aircraft or a wing/fuselage combination and may not be the aerodynamic centre for an airfoil or wing if they are testing at moderately high Mach Numbers

Regards

CliveL

[Microburst2002]

The point about which we choose to take moments is not an irrelevant thing.

We choose one point or another for a reason.

We can try to find about which point a given force has zero moment. If we find it, we can consider the force to be acting exactly on that point.

In the case of Lift, if we find the point about the moment of Lift is zero, we have found the CP

M = L * arm
(if we are in the point where Lift acts, the arm is zero, therefore M is zero)

If we find the point about which pitching moments are constant irrespective of lift, we have found the point about which the moment created by changes in lift is zero.

dM = dF * arm

When this arm is zero, the change in moment due to a change in Lift is zero. Therefore we have the point where changes in Lift are acting. That is the AC. I don't remember the mathematical formulae that show that theoretically this point is at the 25% of the chord (subsonic).

As for the aerodynamic moment, which confuses me:

- This moment is like that of a couple of forces, isn't it? the moment is always the same no mater what point you choose, right? It's like a "pure" moment.

- Why is it always zero when the airfoil is symmetrical?

- Does it changes with Lift in cambered airfoils? Why?

[HazelNuts39]

henra;

Sorry if I was confused by: The pitching moment of Airfoils usually increases with increasing AoA.

regards,
HN39

[henra]

HazelNuts,

Ahhh, now that I read it, I see it !
You are right, that was confusing !
What I really wanted to say was actually the Pitching Moment at the Center of Pressure usually increases with AoA. At the Aerodynamic Center it remains constant. Thanks for pointing that out !

[zerozero]

In simple terms can someone please explain what the Aerodynamic Center of an aerofoil is.

Thanks

Holy cow! I wonder if we answered his question.

I certainly learned something.

:}

[Pugilistic Animus]

I'm too lazy now to write more about aerodynamics...but, just to give some assistance wrt visualizing these concepts look up---[not wiki]....ZLA and CGT ...zero lift axis,....CoG track....that should help sort it out for everyone...not involved in {injunearin}:}

Sorry guys, but that is not what I said. If you take moments about the point where the lift due to incidence acts then you will get a constant pitching moment as incidence varies and this point is generally close to 25% chord. So the aerodynamic centre is the location where the lift due to incidence acts and is independent of incidence (more or less).

Centre of pressure on the other hand varies with incidence. At zero lift the centre of pressure is at plus/minus infinity if there is any wing camber, because the pitching moment at zero lift is produced by a couple. As you increase incidence (and lift) the centre of pressure will move towards the 25% chord point but it will never get there.
CliveL,
Yes, the exact 3D rendering of the gross airfoil characteristics are indeed the summation of section charteristics along dy....:ok:


Folks, Pay a little more attention to this man...;)

[CliveL]

As for the aerodynamic moment, which confuses me:

- This moment is like that of a couple of forces, isn't it? the moment is always the same no mater what point you choose, right? It's like a "pure" moment.

- Why is it always zero when the airfoil is symmetrical?

- Does it changes with Lift in cambered airfoils? Why?

I think there may be some misunderstanding here. Let me put it a simply as I canhttp://images.ibsrv.net/ibsrv/res/src:www.pprune.org/get/images/smilies/smile.gif

Aerodynamic (pitching) moment is made up of two bits:

The pitching moment at zero lift = Cmo
plus
The pitching moment due to the lift acting at the aerodynamic centre which is normally close to 25% chord, but which may be distant from the point to which moments are referred = CL*(moment arm between a.c and moment reference point)

Cmo is a constant. It does not change as AoA changes and it does not change with pitching moment reference point. It depends on the shape of the camber line, both in height and location of the 'peak'. For a symmetric airfoil (obviously) the camber is zero and Cmo is also zero.

The lift bit is with you always, whatever the camber, so symmetric airfoils are no different to cambered airfoils. But if you choose to measure pitching moments about 25% chord (as is often the case) then the pitching moment due to lift will be zero because there isn't any moment arm. Camber simply shifts the datum AoA at which you get zero lift.

So in the special case of a symmetric airfoil measured about 25% chord the aerodynamic moment will always be zero.

Hope that helps!

CliveL

[Lightning Mate]

I have constructed this graph for instructional purposes.

It illustrates Cm for a positively cambered aerofoil, and may help.

http://i636.photobucket.com/albums/uu82/Lightning_29/aerodcentregraph.jpg

[CliveL]

Thanks Lightning Mate - that says it all.

CliveL

[Microburst2002]

Still puzzled about it.

Is the aerodynamic moment a "theoretical" thing?
¿Can we use any point of reference we want?

Let's assume an airplane with the CG exactly in the CP, in trim, steady level flight. How much tail lift does it have? How much is the ANU moment it has to balance? Does it depend on what point we choose for the moments? That would make no sense...

When I think of a moment, I think of a force and an arm. Moments will vary depending on the point selected, of course. When talking about the aerodynamic center, Where is the force?

[cwatters]

Yes nice one LM.

Note that at the aerodynamic center the pitching moment does not vary but doesn't have to be zero. Graph shows it to be slightly negative which is reasonable for a normally cambered wing section.

Wikipedia has same answer ...
Aerodynamic center - Wikipedia, the free encyclopedia (http://en.wikipedia.org/wiki/Aerodynamic_center)

I guess the next question is do all sections have an aerodynamic center? Are there sections for which the PM is never constant no matter where it's suspended?

[cwatters]

> When talking about the aerodynamic center, Where is the force?

A pitching moment is a torque. The interesting thing about a torque is that it's not a unique force acting at unique distance.

For example if I tell you the torque is 12 units you have no way of knowing if that's created by a 12 unit force at a 1 unit distance or by a 1 unit force at a 12 unit distance or any other combination (2 x 6 or 3 x 4 etc). It's all the same.

Where does the pitching moment come from? Perhaps think of it like this.. On a wing section each part of the chord contributes a bit to the pitching moment. A little bit of lift generated by a patch near the trailing edge on the top surface might contribute a pitch down, a little bit near the leading edge a pitch up etc. To work out the overall pitching moment the lot has in effect to be summed or integrated.

Most conventional sections have a negative (nose down). Consider what you need for a non swept tailless aircraft - you need a positive pitching moment. To turn a conventional section into one suitable for a tailless aircraft the trailing edge it typically "bent up". That modifies the overall pitching moment for the wing and provides stability (unfortunately it also reduces lift).

[John Farley]

Brilliant and timely diagram.

Well done.

[Lightning Mate]

Thank you John.

I have to teach all this stuff. If anyone wants other diagrams I have hundreds. :)

[Lightning Mate]

I'll post you another picture soon. :)

[Lightning Mate]

Here you go -cambered aerofoil zero lift pitching moment:

http://i636.photobucket.com/albums/uu82/Lightning_29/0LiftPitchMoment1.jpg

[CliveL]

Still puzzled about it. Is the aerodynamic moment a "theoretical" thing?
¿Can we use any point of reference we want?

No, it is very real

Yes, but it is usual to use 25% mean aerodynamic chord in the wind tunnel and aircraft CG for anything else

Let's assume an airplane with the CG exactly in (i.e. at) the CP, in trim, steady level flight. How much tail lift does it have? How much is the ANU moment it has to balance? Does it depend on what point we choose for the moments? That would make no sense...

Tail lift = zero (at zero thrust)

Dunno - what is ANU moment?

When I think of a moment, I think of a force and an arm. Moments will vary depending on the point selected, of course. When talking about the aerodynamic center, Where is the force?

Lift force effectively acts at the centre of pressure as always

CliveL

[Lightning Mate]

Yes, but it is usual to use 25% mean aerodynamic chord in the wind tunnel and aircraft CG for anything else

Oh dear oh dear.....!!!!

It seems that further explanation is required. There is no way I can give you what amounts to three hours of tuition on this subject on a thread such as this.

Even John Farley will be unable to do this.

However, I will try if the moderators will accept a very big post.

....(you never know, even PPop might learn something - and that should get a reponse!)...... :)

If you are out there John................................

[Lightning Mate]

Can we use any point of reference we want?

Sorry CliveL, we can.

We may use any point aft of the wing trailing edge and any point forward of the wing leading edge.

I use the trailing edge and the leading edge simply because my students grasp that more easily.


Regards to all, LM.

[Lightning Mate]

Tail lift = zero (at zero thrust)

Sorry - I have total understanding failure.........

The tailplane force has absolutely nothing to do with this argument. :uhoh:

[CliveL]

Sorry CliveL, we can.

We may use any point aft of the wing trailing edge and any point forward of the wing leading edge.

I don't think you read the question and my answer:

Question: So we can use any reference point we like right?

Answer: Yes

Pray tell me how that is in any way in conflict with anything you have written about being able to take moments anywhere from in front of the LE to behind the TE

What I wrote is consistent with an earlier post. This time I merely added that in practice aerodynamicists generally use 25% mac as a reference in the wind tunnel and aircraft CG in trim or stability work. Would you disagree with that?

CliveL

[CliveL]

Sorry - I have total understanding failure.........

The tailplane force has absolutely nothing to do with this argument.I agree, tail loads to trim have nothing to do with the original question as to the nature of aerodynamic centre of an airfoil but I was responding directly to a subsequent question posed, and if you read that it may eliminate your understanding failure.

CliveL

[Lightning Mate]

Clive,

Thank you for the PM - I have replied. :)

[Microburst2002]

I must have somthing wrong in my brain but I still don't get it.

Is it correct if I say that the aerodynamic moment is the moment of the Lift force about a point?

[Lightning Mate]

Yes, but.....

In simple terms we use a point termed the centre of pressure, which is the point on the wing chordline where the total lift may be considered to act.

In reality this point cannot exist, but we may make empirical calcultaions based upon its' existence and its' position with respect to percentage of the wing chord.

I am able to supply a diagram showing just how the pitching moments are derived and how they change with changing angle of attack/CL.

Edit to say prolly best done via PM.

[keith williams]

There are many different ways in which we could accurately describe the lift forces acting on the aerofoil.

METHOD 1.
One way is to add together all of the individual lift forces to produce a single lift force. This single lift forces acts at the centre of pressure.

Because it acts at the centre of pressure it does not exert any pitching moments about the centre of pressure.

So we have lift expressed as a single force, (but no pitching moment) acting at the centre of pressure.

But if we were to measure the moments about any other point on the aerofoil we would find that the lift exerts a moments that is the product of the lift force multiplied by its distance from the centre of pressure.

This means that a second method of expressing the effect of lift is a force plus a pitching moment, both acting at a point that is not the centre of pressure.

METHOD 2
As LM’s diagram shows, if we measure the moments about the trailing edge we will find that increasing angle of attack produces an increasing pitch up moment.

If we take moments about a point slightly ahead of the trailing edge we will find that the rate of increase of pitch up moments is less than it was at the trailing edge.

If we gradually move forward from the trailing edge the rate of increase of pitch up moments will gradually decrease.


If we take moments about the leading edge we will find that increasing angle of attack increases the nose down moment.

If we take moments about a point slightly behind the leading edge we will find that the rate of increase of pitch down moments is less than it was at the leading edge.

If we gradually move aft from the leading edge the rate of increase of pitch down moments will gradually decrease.


If we repeat this process, gradually moving forward from the trailing edge and aft from the leading edge we will find a point at which the pitching moments do not change with changes in angle of attack. This point is the Aerodynamic Centre of the aerofoil.

So we can express the effects of lift as a single force plus a constant pitching moment acting at the Aerodynamic Centre.

[Microburst2002]

From what I read on the books, I understand that the Lift of an airfoil can be thought of as the sum of the upper Lift and lower Lift.

In symmetrical airfoils, both upper and lower Lift always act at the same point of the chord and this point, the CP, is always the same regardles of angle of attack, and happens to be at the AC.

In cambered airfoils, these lifts act at different points, so that we have a pair of parallel forces, the resultant of which acts at the CP. The CP location changes with AoA.

The existence of this pair of parallel forces reveals an effect that has an aerodynamic origin. Which exists per se. It is not the moment of any force about any arbitrarily chosen point, but a very phisical twisting torque that can be measured in the wind tunnel and expressed in Newtons-meters or whatever unit.

It is this twisting torque that I am interested in. I don't accept that we consider Lift as acting at the AC because that greatly simplifies calculations. Lift acts at the CP. That's where it acts.

I believed that this twisting moment (pitch down in typical airfoils) was decreasing with AoA, which of course meant a negative effect on airplane's stability. However, when taking the AC as the origin of moments and deriving in the stability equation, this effect is "magically" eliminated, and I can't understand why they do that. I just don't buy it.

I accept empirical evidence, but I don't like when mathematical tricks are used. Then I get lost not because of my mathematical ability (I was very good at that years ago) but because they (not you, the books) are cheating.

I know well what the CP is and what the AC is. But I think the aerodynamic pitching moment is unduly simplified by everybody.

[Microburst2002]

Thank you Keith

Now I start to understand better.

So we are substituting
Lift-acting-at-the-CP by
Lift-acting-at-the-AC-plus-a-pitching-moment (which is convenient because it is constant with AoA).

The net effect would be the same as they are equivalent concepts.
I have always accepted that demonstration of the existence of a point with such a property, graphically expressed in the sketch by LM.

hmmmmm
But Is this pitching moment the same as the aerodynamic moment? Or is the sum of the AM plus the moment of Lift about the AC?

To find the required tailplane lift to trim the airplane in steady flight, we need to know the value of the Lift, the location of the CP and the value of the aerodynamic moment (which should be independent of any point we consider, if we are talking about a couple of forces).
Or, alternatively, we can consider Lift as acting on the AC without forgetting to add the pitching moment with respect to AC.

Still don't understand how the effect of the AM "magically dissapears" in the stability equation.

[CliveL]

Microburst,

Can I suggest you 'Google' AA241 which will put you into a Stanford University site that gives course notes for Aircraft Design.
You will find the answers to all your questions there (and much more besides)

CliveL

[D120A]

Constants such as Cmo which are independent of incidence disappear when the equation is differentiated with respect to alpha. It's not a trick or anything, it's just that some engineering equations couldn't be solved at all unless you set them up in an elegant way. Choosing a subtle point about which to take moments, which has the effect of putting a number of nasty terms to zero later on, is part of the art.

[Microburst2002]

I know, but many times the physical meaning of something is lost while making it easier to be calculated mathematically.
This is what happens in this case, to me. I can't see through the equations.

You know what I mean? There are many who understand the equations of Einstein's theory of Relativity, but not so many who actually understand that theory.

CliveL, thank you. Probably Stanford notes are too much for a humble pilot like me. But I will take a look at it anyway


cheers

[CliveL]

Nah! there are lots of diagrams and even a nice little Java applet that lets you change AoA and see what that does to pressure distributions, lift and pitching moment.

CliveL

[Lightning Mate]

Constants such as Cmo which are independent of incidence disappear when the equation is differentiated with respect to alpha. It's not a trick or anything, it's just that some engineering equations couldn't be solved at all unless you set them up in an elegant way. Choosing a subtle point about which to take moments, which has the effect of putting a number of nasty terms to zero later on, is part of the art.

...kinnel...wossat mean mate....................?

I shall now leave the thread because it's becoming silly, but before I go - ZERO LIFT!

http://i636.photobucket.com/albums/uu82/Lightning_29/56_Loop_Down_600-2.jpg

[Microburst2002]

Thank you, proffessor LM

for wasting your valuable time with us

nice pic

[polarbearjim]

Thanks to Clive and LM for helping me understand!