I’ve asked Webmaster Aza to put this article on the site because, due to my error in rewriting it for Model Airplane News (May 2001 issue), the formula for P published there was incorrect. By a very unfortunate coincidence, the one example with which I tested the incorrect formula came out with the right result, so that my test didn’t catch the error. See the Human Factors note that I’ve added at the end of this article for a detailed analysis. Also, due to my thoughtless typing, "mean aerodynamic center" got turned into "mean aerodynamic chord". The article, with the right formula, originally appeared in 1991 in the San Francisco Vultures’ Newsletter. The Vultures is San Francisco’s AMA-affiliated model airplane club, flying since 1939. We specialize in RC sailplanes and electrics, with a bit of free-flight.
I originally derived these equations for my own use because all the methods published in the model airplane magazines (and in most books) were graphical, and thus more tedious to use than the few seconds of arithmetic on a calculator that this method requires.
-
- While a more exact, and somewhat longer, formula involves the stabilizer, you can find a safe, that is, a little nose-heavy, initial balance point from just three easily-measured dimensions of your wing. When measuring, treat the wing as if it goes through the fuselage with straight leading and trailing edges. Ailerons are just part of the wing for these measurements. The span of the wing is irrelevant but we do assume that the right and left halves of the wing have the same planform (look the same when viewed from above).
R is the root chord
T is the tip chord
S is the sweep, measured at the leading edge.
The initial balance point is a distance D measured back from the leading edge of the wing at the centerline of the aircraft. This is a safe place to balance a monoplane RC model for a first test flight of an original design. Of course, if the model is a kit or plans-based airplane, use the balance point recommended by the designer.
Here is the equation for D, using an asterisk to indicate multiplication:
D = ((R*R + R*T + T*T)/(6*(R+T))) + (S*(R + 2*T)/(3*(R+T)))
I find it easier to do the calculation in three steps:
Q = (R*R + R*T + T*T)/(6*(R+T))
P = (R+2*T)/(3*(R+T))
D = Q + S*P
For wings with no leading-edge sweep, S = 0. This means that you don’t have to calculate P because it gets multiplied by zero and doesn’t matter. So Q is your answer. For all flying wings, multiply D by 0.9.
Working out where the initial balance point is becomes a mere matter of plugging in the numbers that you get measuring the wing into the formulas.
Here are some worked-out examples:
Rectangular Wing:
R = 6, T = 6, S = 0
Q = (36 + 36 + 36)/(6 * (6 + 6)) = 108/72 = 1.5
If R, T, and S are in inches, then this wing would be balanced 1.5 inches behind the leading edge. If R, T, and S are in centimeters, then this wing would be balanced 1.5 centimeters behind the leading edge. The formula will work with any units of length, so long as the units are the same for the three measurements.
You don’t have to calculate P or D in this example because there is no sweep. As an example of adjusting D for a flying wing with the dimensions given above, you’d multiply D by 0.9 to get a new distance of 1.5 * 0.9 = 1.35 units, or, in the antique English system, 1 and 11/32 inches.
Triangular Wing
R = 10, T = 0, S = 10
Q = (100 + 0 + 0)/(6 * (10 + 0)) = 100/60 = 1.67 (to three significant digits, which is enough for this job)
P = (10 + 2*0)/(3* 10 + 0) = 10/30 = 0.333
D = 1.67 + 10 * 0.33 = 1.67 + 3.33 = 5
This wing gets balanced 5 units behind the leading edge, measured, as always, where the leading edges would meet along the centerline of the fuselage, even if this geometrical point is inside the fuselage.
Swept Wing
R = 12, T = 9, S = 6
Q = (144 + 108 + 81)/(6*(12 + 9)) = 333/(6*21) = 333/126 = 2.64
P = (12 + 2*9)/(3* (21)) = 30/63 = 0.476
D = 2.64 + 6 * 0.476 = 2.64 + 2.86 = 5.5
Forward Swept Wing
R = 4.5, T = 3, S = -1.25
Q = (4.5*4.5 + 4.5*3 + 3*3)/(6 * (4.5 + 3)) =
(20.25 + 13.5 + 9)/(6 * 7.5) = 42.75/45 = 0.95
P = (4.5 + 2*3)/(3*(4.5 + 3)) = 10.5/3*7.5 = 10.5/22.5 = 0.467
D = 0.95 + (-1.25 * 0.467) = 0.95 - .0.584 = 0.37
For a forward-swept wing, D can come out negative, which means that you have to balance the model at that distance ahead of the leading edge. For example, if D = -2, then you’d have to balance the model 2 units in front of where the leading edges would meet at the center of the fuselage.
Note for aficionados:
D is the distance back from the leading edge at the centerline of the wing to the mean aerodynamic center. P is the fraction of the distance from the root toward the tip that you must travel along the mean aerodynamic centerline to find the mean aerodynamic center of each wing panel. For biplanes with no stagger and identical wings, D can be found as for flying wings, that is by multiplying D from the equations by 0.9.
4*Q, by the way, is the mean aerodynamic chord of the wing. If you prefer these equations in standard mathematical notation, they are:
-
-
- Human Factors Footnote:
Considering my profession, it is amusing to note that I was a victim of bad interface design -- along with incredibly bad luck and my own carelessness in this error. Here’s how it happened: I was editing the article, and was copy-and-pasting the formulae from my earlier piece in the San Francisco Vultures’ newsletter. I copied the formula for P into, but I got a phone call or ran into some other interruption before I could paste it into the new article. After the interruption, I started to work on a different essay, and at some point I copied, into the same invisible copy buffer, a different formula for a different variable P that happened to be a function of variables with the same names as those used in the balance article.
As pointed out in my book, "The Humane Interface", the use of a copy buffer is not only violates the principle of visibility, but is also a mode. I had unknowingly changed system state and was not in the mode I thought I was. Thus, when I returned to working on the balance article, I remembered only that I had copied the formula for P, but forgot which. Normally, this just costs you the copy you made, because the newly copied material doesn’t fit the situation and you just have to go and recopy what you really want (unless you did a cut-and-paste, in which case you are at risk for losing it altogether). In this case, the paste looked OK because, after all, the formula had the right variables, and the last time I had worked with it was a decade earlier, so it wasn’t all that familiar.
Nonetheless, it didn’t look quite right, and if I had thought about it, I would have seen that it obviously didn’t make sense in the case that R = T, but I was in a hurry, didn’t bother thinking things through, but in what is normally a wise move I did what I always do: I tested the equations.
I looked at the old article, took a wing’s measurements from it, and ran the numbers through the equations. The old article said that the result, D, was 5.5. By unbelievably bad luck, the example I chose happened to give a result of 5.52. I do not measure balance points to hundredths of an inch, and I usually round my results to a suitable precision. So the answer agreed with what I expected.
Had the word processor been designed in accord with the principles in my book, this embarrassing situation would never have occurred.
Acknowledgements:
Thanks to the many who kindly wrote to tell me of the error in the MAN article, and especially to Larry Renger, who not only wrote about that article, but who found and fixed an error in the note for aficionados: I had Q where it should have been D. Also thanks to the patient folks at MAN who didn’t get too angry at my goof and to Aza Raskin for putting the various corrections up on the web site.