What I need to know about calculating Center of Gravity

By: Dick Tonan

Recently I found a technique for determining the most aft CG (center of gravity) location for a model airplane that will provide an “adequate” stability margin. I thought I would put this information together for a wider audience than those that just read Replica.

Before I go any further, a disclaimer! The word adequate is pretty subjective. An r/c plane with an adequate margin of stability for the r/c pilot with several hundred hours on the sticks is going to be a pile of balsa at the bottom of a smoking hole for the r/c pilot that soloed a week ago. Therefore temper the results with your experience! The fewer hours of stick time, the further forward the CG should be from the MOST AFT location determined by this method.

I did not come up with these formulas, therefore I cannot claim credit for them. What caught my interest was that this method was the only one that I have seen that takes into account the aircraft aft of the wing leading edge. All other methods treat tail feathers as if they didn’t exist. The following formula determines the “CG” location expressed in percent of the Mean Aerodynamic Chord as measured from the M.A.C.’s leading edge:

CG = [0.17 + 0.3 x ((TMA) x (SH) x HTE))] x 100
CG = [0.17 + 0.3 x ((MAC) x (SW

Definitions:

• CG: Aft most center of gravity with adequate stability margin.
• T.M.A.: The Mean Aerodynamic Chord
• S.H.: The area of the horizonal stabilizer.
• S.W.: Wing area including the portion of the wing under or over the fuselage.

H.T.E.: Horizontal Tail Efficiency. Value ranges from 0.5 for a flat tail located in its normal position in the down wash of the wing to 0.9 for a “T-Tail” at the top of the vertical stabilizer. Before we start banging numbers into the calculator, we need to determine 2 critical factors in the formula. They are the M.A.C. and the T.M.A. We will start with the M.A.C. because we need it to determine the T.M.A.

There are two ways I know to find the M.A.C. The first is mathematical, the second is graphic. The graphic method has the advantage of not only determining the M.A.C., but also indicates its location on the wing. If you have a straight rectangular wing, the M.A.C. is the actual wing chord.

Let’s go over the mathematical method first. To use the following formula, you need to know only two things: the wing’s chord at the root (Cr) and the tip (Ct). Plug these numbers into the formula below.

M.A.C.= 2/3 [Cr + (Ct -((Cr x Ct)
M.A.C.= 2/3 [Cr + (Ct -(((Cr + CT)))]

Using the example of a wing with a 15″ root chord (Cr) and a tip chord (Ct) of 10″, the formula cranks out a M.A.C. of 12.7″. This is very close to being the average chord of the wing.

Now. let’s take a look at a graphic method. If you have either full size or scale drawing of your wing, the hard part is already done as the first step is to make a scale drawing of “your” wing. Figure 1 is a scale computer drawing of the wing in the above example. On your drawing, draw two lines; the first extends forward (perpendicular to the wing span) from the leading edge at the tip and it’s length is equal to the length of the root chord (Cr) on your drawing. Next, draw a line extending back (again perpendicular to the wingspan) from the trailing edge at the root that is as long as the tip chord (Cr) on your drawing.

Actually, you can reverse the direction of these lines and the results will be the same but one must go forward and the other back. From the forward end of the line at the tip, draw a line that connects to the aft end of the line at the root. This line will cut diagonally across the wing.

The next step is to draw a line that connects the midpoint of the root chord (Cr) to the midpoint of the tip chord (Ct). The location of the M.A.C. is at the intersection of the lines drawn in these two steps. The length of the M.A.C. is the chord of the wing at that point. Measure your scale or full size drawing.

Now, that we know what the M.A.C. is, we can determine the T.M.A.

To do this, measure the distance from a point at the root that is 25% of the M.A.C. to the midpoint of the horizontal stab root chord. You can do this either on the model itself, the full size plans or your scale drawing.

At this point, we now know the M.A.C. (12.7″) and the T.M.A. (33″). Now, we need to know the areas of the wing and the horizontal stabilizer. For non-rectangular surfaces, the easiest way to calculate the area is to divide the wing into rectangles and triangles. The area of a triangle is equal to 1/2 the length times the width. For example, the wing area is 875 sq.” (SW) and the horizontal stab area (SH) is 165 sq.”.

HTE (Horizontal Tail Efficiency)

For the horizontal tail efficiency, let’s pick a number…we’ll assume that the tail is in the normal location, so we’ll use 0.5 in our formula. Here’s what our formula will look like.

Formula :

CG = [ 0.17 + (0.30 x (( TMA ) x ( SH ) x HTE ))] x 100
MAC         SW

Formula with example values :

CG = [ 0.17 + (0.30 x (( 33 ) x ( 165 ) x 0.5 ))] x 100
12 875

Formula results :

CG = 24.35% of the M.A.C. or 3.09″

It is important to remember that this is measured from the leading edge at the M.A.C., not at the wing root…or event worse, at the wing tip.

For those incline to utilize “T” tail designs, let’s look at what happens when you change to that configuration HTE will now be 0.9 instead of 0.5. All the variables appear below. We have changed the desired location of the CG by almost 1″ just by making changes to the tail feathers. Remember, this formula was championed because it includes consideration for the tail feathers.