This spreadsheet will give the cross section of a motor that will give a long, slow climb, a long cruise and a long, slow descent. It will give the maximum duration for the weight of the rubber. It is assumed that the flight starts with the motor wound to 100% of maximum capacity and the prop will turn the whole time with no turns remaining on landing, no freewheeling.

This is done as an iterative calculation. We start with the empty weight of the airplane and calculate how much motor will be required to fly it. Then we notice that when we put that motor on the airplane it will weigh more and require more motor, so we do the calculation again with the increased motor weight to get the next heavier motor.

We start the iteration with the empty weight of the airplane, calculate the required cross section, width, weight of indicated motor, then add that motor to the empty airplane weight and repeat the calculation sequence until the numbers converge to the desired precision. For example, when the width changes by less than 0.01″ or the weight changes by less than 0.01 gram, you are done, you chose.

Wa = weight of airplane without rubber, grams. (I know, grams are a measure of mass, not weight, sure. Multiply everything by 980. Then divide it by 980 when you get done.)

W = total weight of airplane and rubber, grams. This changes with each step of the iteration.

Ql/W = level flight torque to weight ratio as determined by flight test. Torque in gram centimeters, this will have units of centimeters.

Ql = W x (Ql/W) is level flight torque for total weight of W. The Ws are not necessarily the same number, because one is from the flight test and the other is modified by adding rubber weight through the following iterative calculation, and the motor weights will not be the same, as you will see in the following example. This is sort of like the X variable in FORTRAN, it is a label for a place in memory that can contain a series of different numerical values. (I guess I could have defined the ratio Ql/W as R or something, to avoid the ambiguity.)

L = length of motor in inches. This is selected by the designer, generally in a ratio to the available distance between the hooks. 2X is a good number, about as long as can be managed. 3X will probably be impossible to manage and have the motor still turn the prop. Up to you.

Kq = dimensionless torque coefficient for level flight. Different for each batch of rubber, I will suggest 23,800 for the example. Good Super Sport. The selection of this point on the torque curve is very involved and I will not go into it for this example calculation. It is somewhere near the middle of the torque curve. Average torque might be a good approximation, although that is not how I derived it. (Mathematically it turns out to be the same thing.) This requires torque testing a sample of the rubber.

S = rubber cross section, square inches. Product of number of strands, thickness and width. Generally assume two strands and modify as necessary, i.e. two strands of 1 ½” might end as 24 strands of 1/8″ or 12 strands of 1/4″ in the airplane, your choice. Thickness assumed to be 0.042″.

w = width of strands in inches.

Wr = weight of rubber, grams.

For this example I will use the hypothetical Wilbur which I just did and can still find the scrap of paper I wrote it on.

Wa = 55 gm (Joshua says it should be more.)

Ql/W = 1.0, clean design with high aspect ratio. (Maybe I am being optimistic, with that fat fuselage.) AMA Cub has a value of 1.24, short flat wings, Dandiflyer has a value of 0.78, high aspect ratio cambered wing, stick fuselage. The prop also comes into play here, but that is another topic.

L = 70″, twice the motor space distance scaled from the plan.

The equations used are:

1. Ql = (Ql/W) x W, (Ql/W) is a constant ratio measured in level flight tests of the airplane. The W in the denominator is the total weight of the airplane in flight testing, not the same as the weight W used in this iteration, which changes as we increase the size of the motor.

2. S = (Ql/23,800)^2/3, ^ is exponentiation, the value 23,000 depends on the batch of rubber.

3. w = S/0.084, width of 2 strands with 0.042″ thickness, in inches.

4. Wr = w x 8/12 x 70 x 2 = 93.38 x w using the rule that one foot of 1/8″ weighs one gram, adjust as desired.

5. W = Wa + Wr add weight of rubber to weight of airplane to get new total weight and repeat as necessary.

Here is the iteration table for the assumed Wilbur calculation. The iteration is robust; if you make a little input mistake, or round off, it will self correct.

W	Ql	S	w	Wr
55	55	0.1748	0.2081	19.42
74.42	74.42	0.02138	0.2546	23.75
78.75	78.75	0.2221	0.2644	24.68
79.68	79.68	0.2238	0.2664	24.86
79.87	79.87	0.02241	0.2668	24.9
79.9	79.9	0.02242	0.2669	24.91

You can see that it converges to the precision criteria on the fifth iteration. This gives us a two strand motor of 0.2669″ width or a four strand motor of 0.1335″ width or an eight strand motor of 0.06673″. No matter how you cut it, it will be a little bigger than a standard width. Or you could make a six strand motor of 0.08897″ width.

Note that this made a lot of unconfirmed assumptions about this particular airplane and rubber and should be regarded as being entirely hypothetical. Verify the inputs before actually using this method to make motors. This assumes that you wind the motor to 1/4 turn short of breaking. Actually, you don’t lose much time by shorting the count a couple percent.

You may download the spreadsheet at the head of this article. The cells highlighted in yellow are for inputs.