Wikipedia's MAC Equation

[Tim Wescott]

Here's a definition and a reasonable discussion: http://en.wikipedia.org/wiki/Chord_%28aircraft%29 .  If you use this for a control line airplane, you oughta toss an r^2 term inside the integral to account for the outside tip going faster than the inside tip. 

Howard:

That equation for MAC must come with an accompanying equation for effective span; if it doesn't, then the equation does not match the statements in the text.

The integral?  It's in there.  Just go until there's no more wing. 

Well, yes, but you end up with a number which, when multiplied by the wing span, does not equal the wing area.  In the nicely easy-to-calculate, yet still not rectangular case of a triangle wing that starts with a chord of c_r and ends with a chord of zero, I get a MAC equal to b * cr^2 / (3 * S).  And, I don't get any position information.

So, if you wanted to use that equation to reduce things down to an equivalent rectangular wing of the same area, as the text says, then you need to use some other span than b.

[Howard Rush]

I'll attempt to answer that without going to any effort to speak of.  MAC for a rectangular wing ought to be the chord.  MAC actually has some aerodynamic significance.  The middle of the wing does more than the part near the tips.  MAC takes that into account.  Look further on the World Wide Web to find the formula for spanwise MAC location, or search here.  I think Serge posted it awhile back.

[Tim Wescott]

The middle of the wing does more than the part near the tips.  MAC takes that into account..

Yet Wikipedia's MAC equation pays no attention to the spanwise direction in their integral, other than as a variable to integrate over.

[Howard Rush]

They were kinda loose about defining their symbols.  The integral is from the airplane center to the right wing tip.  They assume symmetry.  The source cited is George Aldrich's cousin.

[Bob Reeves]

I used this web site when I was designing (if that is what one can call it) the FJ-4 series of Carrier airplanes.

http://www.geistware.com/rcmodeling/cg_calc.htm

Allowed a dumb guy like me to build 3 versions in different sizes that actually worked.

[Jim Thomerson]

To quote one of our past students, "I didn't understand a word you sayed." Does anyone understand MAC well enough to describe it in terms intelligible to someone who is not an aeronautical engineer?

[Tim Wescott]

The middle of the wing does more than the part near the tips.  MAC takes that into account.

That doesn't answer my concern, which is that the comment in the text does not match the calculation, nor does the calculation give you a way to find the chordwise location of the MAC -- only the actual length of the MAC.

For that matter, that integral will give you the same answer for a triangular wing with zero chord at the tip, and a triangular wing with zero chord at the root.  So while the MAC may take the action of the tips vs. the root into account, the Wikipedia article certainly doesn't.

They were kinda loose about defining their symbols.  The integral is from the airplane center to the right wing tip.  They assume symmetry.

I got that part.  My concern was that the text doesn't match with the equation, and that the equation given doesn't tell you anything about the longitudinal location of the MAC -- only it's length.

[Tim Wescott]

To quote one of our past students, "I didn't understand a word you sayed." Does anyone understand MAC well enough to describe it in terms intelligible to someone who is not an aeronautical engineer?

When I understand it, I'll let you know.  Howard's comment about MAC having a spanwise meaning is causing me all sorts of distraction until I understand it.

In the chordwise direction, though, the MAC is -- for the purposes of some calculations, like stability and maneuverability -- the chord of the rectangular wing that would act the same as the wing that you have.  It's a way of lumping all of the effects of each part of the wing into one entity, so that you can treat your three-dimensional airplane as a two-dimensional entity that only moves forward and up or down, and only rotates in pitch.

[Tim Wescott]

I'll attempt to answer that without going to any effort to speak of.  MAC for a rectangular wing ought to be the chord.  MAC actually has some aerodynamic significance.  The middle of the wing does more than the part near the tips.  MAC takes that into account.  Look further on the World Wide Web to find the formula for spanwise MAC location, or search here.  I think Serge posted it awhile back.

Any suggestions for a correct and not-too-complicated discussion of the MAC?  Formula for calculating it that include integral signs are OK.  The internet is preferred, but an aerodynamics book that's at least remotely accessible to someone with a good background in classical mechanics and calculus would do.

I have five books that touch on stability and control in airplanes, but don't go into it deeply enough:

"Circular Airflow" by Ziac doesn't mention MAC as far as I can tell (no index).

"The Illustrated Guide to Aerodynamics" by Smith doesn't mention MAC (it's for pilots, I guess they don't need to know).

"Incompressible Aerodynamics" by Thwaites doesn't mention the MAC, or if it does it's by a different name.  I had high hopes for that one.

"Aerodynamics" by John E. Allen has to be the most useless book I own.  It's the book that I would choose if I had to teach one of those classes where you get an A for showing up and warming the room slightly.  It's good for gleaning a bunch of buzzwords and has some pretty pictures, but that's about it.  (Is there an aerospace MBA program someplace?).  It doesn't mention the MAC.

"Theory of Wing Sections" by Abbot and Doenhoff doesn't mention the MAC (I wasn't surprised -- it's wing sections, after all).

So, having a good reference that explains the formal, commonly-accepted term for the MAC would be a help for me.

[Howard Rush]

I just posted definitions of MAC on the other thread.  I assumed Jim wanted the real deal, but I put a link to the graphical method, too.  The calculator Bob posted is even easier.

Tim no doubt wants to know where the MAC notion comes from.  I think it's in the first chapter of Abbott and Costello, although it's not obvious to me just looking it over.  There is a definition on page 27.  Seeing the derivation ought to tell you with how many grains of salt it should be taken.  I looked awhile last night on the Web for a satisfactory explanation and didn't find one, but I reckon I can look some more.

[Tim Wescott]

I just posted definitions of MAC on the other thread.  I assumed Jim wanted the real deal, but I put a link to the graphical method, too.  The calculator Bob posted is even easier.

There's even a graphical method that will work for multiple-taper wings, which means you could sorta-kinda use it for ellipticals.  Dunno if that's the one you pointed to.

Tim no doubt wants to know where the MAC notion comes from.  I think it's in the first chapter of Abbott and Costello, although it's not obvious to me just looking it over.  There is a definition on page 27.  Seeing the derivation ought to tell you with how many grains of salt it should be taken.  I looked awhile last night on the Web for a satisfactory explanation and didn't find one, but I reckon I can look some more.

Abbot and Costello???

[Serge_Krauss]

Edit: I haven't read the last comments made while I typed - sorry!

My club mates are probably not going to appreciate this, but let me give this a try. My collection of basic aero texts is mostly older ones that do not focus on compressibility and transonics/supersonics. Five of my seven texts develop the idea of MAC, one assumes its use, and the last seems to over-simplify it. Most texts fail to answer your question, stopping at the idea of it being the chord length of an "equivalent" rectangular wing or similar idea. Futhermore, I've found that some internet "authorities" have it wrong as far as curved wings are concerned, opting for invalid short cuts. I'm going to try here - 'hope I don't #8/?@&!#& this up.

First, every MAC does have a length and spanwise position, BUT the MAC is not necessarily the same length as the actual wing chord at that position. So some expectations should be discarded.

The MAC definitions are all based on the simplifying assumption that every point on a wing exerts the same amount of lift for a given aoa. So it is an approximation, and we can expect any answer to give a chord actually located outboard of what aerodynamic effects like tip losses would otherwise dictate. By this approximation, the lift at any point along a semispan is simply proportional to the chord at that position. That reduces the position of the MAC to a geometrical determination - no tip losses, camber changes, etc. are considered. For a tapered or elliptical wing then, we expect the MAC to be located closer to the root than the tip, because the larger chords are there. For twisted wings, varying lift coefficients have to be factored in later. For the straight tapered wing, it turns out that the MAC is the same size as the actual chord at its position, but for an elliptical wing, the MAC is really shorter than the actual chord at that position (a lot of internet folks get this wrong by short cutting). So those are our limitations and expectations.

The MAC is not the "average" chord. For its location, we would get a better figure if we found the point where the chord divides the half-span area into two equal parts, but even that is not quite the spanwise location. The MAC position computation actually takes moments about the root so that not just chord length, but chord distance from the root are taken into account. MAC is actually located at this idealized center of lift, even though its length is not necessarily equal to the actual wing chord at that point. Think of it as located at the center of gravity of a wing cut-out of uniform material.

The integral definitions for MAC and its location account for this. The MAC equation just takes an infinite number of thin chord-wise rectangular areas making up the right (+) wing half, multiplies each of these area incriments by its local chord (we're finding a 'weighted average' of the chords), and divide their sum by the entire right-wing area. Think of this as multiplying each chord by it's own fraction of the wing area (da/A) and adding them together.

To find the MAC's distance from the root, we just multiply each of the infinite number of local chord-wise rectangular areas by its distance from the wing center (root) and again divide by the total area - same as multiplying the spanwise distances by their fractions of the entire area and adding the results. This just sums up the lift moments about the root and divides them by the entire lift (lift is proportional to the area in the simplified assumption) to leave only a span-wise distance.

So I think these pass through the centroid, although that often confuses this old-er mind. Without showing pictures and math, this is probably pretty obstruse, and I hope I have not misused the language. A couple years ago, I worked all this out for tapered and elliptical wings of varying sweep and dimensions. I checked my work many ways, and the derivations and sample answers are correct. If anyone wants to examine this 7-pg. document, typed out using the Microsoft equation application, I'll be happy to e-mail it to you; my scanner doesn't like that document. I've gotta stop now!

SK

[Serge_Krauss]

Everyone seems to "like" the graphical method - if they've never used it. The accuracy necessary to make the drawing so that intersections are accurate enough can be daunting and doubtful. Errors in drawing are magnified in the answer. I have all equations necessary and more, but the Palos Verde calculators and the one Bob posted are what I use for my own things. They're easy.

SK

[Howard Rush]

I was going to say what I use, then realized that I don't.  Not for stunt, anyhow.  I've flown the same wing design for 12 years, so I don't rescale it or compare it to others.  In comparing stuff like CG and leadout position with my mates who fly the same design, we use easier-to-measure dimensions.  Oh, I did measure the MAC to get the Reynolds number for XFoil.  I did the graphical method with CAD.  I guess I could have used the Y calculation with airspeed correction to calculate tip weight (it worked for combat planes), but it's easier just to put some lead in the box.

[Trostle]

To quote one of our past students, "I didn't understand a word you sayed." Does anyone understand MAC well enough to describe it in terms intelligible to someone who is not an aeronautical engineer?

OK, Jim has asked several times about the MAC for a wing.  I am going to date myself, but for the basic understanding of the MAC, its calculation and its location longitudinally, I will quote some relevant information from Perkens and Hage, Airplane Performance Stability and Control, which was essentially the bible for such information when I was in school a few years ago.  The seventh printing of my book was in 1958.  Perkins was the Chairman, Aeronautical Engineering Department at Princeton University.  Hage was a Senior Engineer at Boeing.

For any wing surface, it "can be represented by a mean aerodynamic chord, the forces and moments on which represent all the forces and moments operating on the surface."  An over simplification of this is that "generally" though not precisely, there is as much wing area inboard of this MAC as there is outboard.  For a constant chord wing, the MAC is the chord of the wing.  If the wing is tapered, the MAC will be inboard and larger than the average chord.  As the wing taper increase, the MAC becomes longer than the average chord and will be further inboard than the average chord.  For the taper we find in most stunt ships, the MAC generally is not much different than the average chord.

Now for the calculation of the MAC.  This applies to a straight tapered wing with any amount of sweep.  (The MAC is independent of span.)

a = root chord,  b = tip chord,

MAC = 2/3(((a + b - (a*b/(a + b)))

Another question has been regarding the longitudinal location, m, of that MAC which is the distance from the root chord leading edge to the leading edge of the MAC.  Here, we will use span as a function where s = span.


m = s((a + 2b)/3(a + b))

Calculation of the MAC for curved wing planforms (i.e. elliptical) is beyond me at this time and the scope of this thread.

For CL stunt model design, the above equations should be adequate and not be concerned with the appearance of an outboard wing being slightly longer or the outboard flap at the tip being slightly larger than the inboard or the fact that the outboard wing travels faster than the inboard.  Only for high performance full size aircraft with more exotic planforms and performance regimes does the calculation of the MAC need to become more refined regarding effective wing areas and planforms.

Keith

[Howard Rush]

...when I was in school a few years ago

Few my foot.

It's a great book. I finally found a copy at an estate sale.  I look at it a lot, hoping it will sink in.  I have found stuff in there that Bill Netzeband used, so it was probably a source for a lot of his articles.

[John Sunderland]

I always thought Teds MAC equation from his published works on the Imitation article (I think?) was simple enough for practical stunt design, and it works for stunt purposes. For me, actually drawing out the top view, just the outline with the tip shape included, helped put things in perspective for the wing I intended. For example, with a longer inboard wing, this calculation gives you a slightly different location inboard than out, nothing drastic. Simply adding root to the tip and tip to the root and drawing the intersection from point to point from the half span approximates the MAC . Factor in a slightly larger/wider outboard flap/tip to compensate for the asymmetrical wing from there.   

[Trostle]

I always thought Teds MAC equation from his published works on the Imitation article (I think?) was simple enough for practical stunt design, and it works for stunt purposes. For me, actually drawing out the top view, just the outline with the tip shape included, helped put things in perspective for the wing I intended. For example, with a longer inboard wing, this calculation gives you a slightly different location inboard than out, nothing drastic. Simply adding root to the tip and tip to the root and drawing the intersection from point to point from the half span approximates the MAC . Factor in a slightly larger/wider outboard flap/tip to compensate for the asymmetrical wing from there.   

That geometric solution for the MAC and its location gives the same results as the equations quoted above.  The Perkins and Hage book shows that with those equations.

Keith

[Howard Rush]

on page 92.  I had it marked.

[Jim Thomerson]

Thanks Keith. I'll be able to go to sleep tonight after reading your explaination. 

[Serge_Krauss]

...Calculation of the MAC for curved wing planforms (i.e. elliptical) is beyond me at this time and the scope of this thread...

That's a really nice, concise explanation, Keith. Based on that, I do not believe that the ellipse or others are in any way, "beyond" you. Actually, (counterintuitively!) they are easier to derive and immensely easier to use than the straight tapered wing MAC stuff. Here is the top of p.6 of a my collection of derivations of equations (plus alternative ones) for MAC's and their positions for tapered, elliptical, and parabolic wings, including the distances of the a.c.'s behind the root leading edges (x') for all sweeps and chordwise tip positions. I'd also included some wing transformations and comparisons, before I went on to other things. The elliptical MAC location is often wrong on the internet. My symbols are different but defined below, including using "b" for the half-span, rather than the entire span. The third equation gives the distance behind the root leading edge of the a.c. of an "elliptical" wing with any sweep and chordwise tip position (you'd probably have to see the diagram).

SK

[Trostle]

That's a really nice, concise explanation, Keith. Based on that, I do not believe that the ellipse or others are in any way, "beyond" you. Actually, (counterintuitively!) they are easier to derive and immensely easier to use than the straight tapered wing MAC stuff. Here is the top of p.6 of a my collection of derivations of equations (plus alternative ones) for MAC's and their positions for tapered, elliptical, and parabolic wings, including the distances of the a.c.'s behind the root leading edges (x') for all sweeps and chordwise tip positions.
SK

Well yeah, if you already have the equations for the various planforms, then it is easy.  What I was referring to was generating calculations for "non-standard" planforms, form-fitting curves to equations and then working the differential equations to come up with the MAC and its location calculations.  That is way beyond the scope of what we need to do here.  It has been a long time since I tried to do something like that, as Howard gratiously explained to me, it has been more than the few years for me that I had recently claimed.

Keith

[Howard Rush]

We don't do any real calculations now that we have these machines.  Below is how I did the integration to check Serge's elliptical wing y_MAC.  As he said, the answer running around on the Internet is wrong.

[Howard Rush]

It has been a long time since I tried to do something like that, as Howard gratiously explained to me, it has been more than the few years for me that I had recently claimed.

Oh, I didn't mean that you don't calculate things anymore, just that you went to school before I did, and I'm a very old man.

[phil c]

Few my foot.

It's a great book. I finally found a copy at an estate sale.  I look at it a lot, hoping it will sink in.  I have found stuff in there that Bill Netzeband used, so it was probably a source for a lot of his articles.

I found all the equations Wild Bill used in a book titled Aerodynamics for Naval Aviators, probably published about 1950.  It was in the public library about the same time he published his articles.

As for MAC, technically you have to integrate the area function across the span.  The graphical method, using narrow chordwise strips is a simplified integration.  Keith's equations are what you get from the integration for a straight tapered wing.  Doing it for any ellipse gives you Serge's 42.6% of the half span.

CL planes have a problem in that the circular flight actually puts the MAC further out from the center due to the higher airspeed of the outboard tip vs. the inboard tip.  Wild Bill's estimator for this works well enough for our purposes.