Understanding Scale Speed
By Bob Boucher

The Technique of building beautifully detailed electric scale models has advanced greatly in the past decade. Unfortunately we have all witnessed the flight of under powered scale models whose performance can best be described as a crash waiting to happen. Armed with the proper knowledge we can avoid making these expensive mistakes. This paper will derive the basic physical scaling laws which if followed will insure good flying behavior. With proper scaling a model can be built that replicates the flying characteristics of the original full size prototype. It will execute all maneuvers in scale dimension. If you want to build your next scale model so that the model will take off in scale runway length and wll perform a scale loop, and fly at scale speed, then read on!

Keith Shaw with his famous Gee Bee
Astro 25 Geared Motor with 14 cells

SCALE WEIGHT

If a construction drawing of a full size airplane was reduced in scale and used to build a scale miniature from the very same materials as the full size airplane, then all surface areas would be reduced by the SQUARE of the scale factor and the thickness of all the materials would be reduced by the scale factor. The net result would be that the Volume of materials used and therefore the Weight of materials would be reduced by the Cube of the dimension. This scaling law is expressed by rule 1.

RULE #1...The scale weight of the model should be equal to the weight of the real airplane times the cube of the scale factor. For example a Piper J-3 Cub weighs 1000 pounds and has a 36 foot wingspan. A six foot model would be 1/6 scale and should weigh 1000/(6x6x6)=4.6 pounds to be at scale weight. A Quarter Scale Cub should weigh 1000/(4x4x4)= 15.6 pounds. The model may very well be built from different materials than the real airplane, still the weight of the finished model should be scaled in conformity with Rule #1.

SCALE WING LOADING

Since the wing area of the model is reduced by the square of the scale factor while the weight of the model is reduced by the cube of the scale factor it follows that the Wing Loading is directly proportional to the scale factor. For example a quarter scale model will have a wing loading of 1/4 that of the real airplane.

Rule #2..The wing loading of a model built to scale weight will be reduced from that of the real airplane by the ratio of the scale factor.(The quarter scale cub will have a wing loading of 20 ounces per square foot instead of the 80 ounce wing loading of the real airplane)

SCALE THRUST

The thrust to weight ratio of an airplane determines how long it takes to accelerate to flying speed during take off, how steeply it can climb,and how fast it can fly. If we want our model to fly like the real airplane we need to make sure that the model has the same thrust to weight ratio. That is the thrust of the model should be reduced by the same ratio as the weight of the model. By applying Rule#1 we get Rule #3.

Rule #3...The thrust needed by the model airplane propeller is equal to the thrust of the real airplane times the scale factor cubed. For example the thrust required for a quarter scale model Cub needs to be 1/64 the thrust of the full size airplane.

SCALE RPM

Assuming that our model in constructed to exact scale and therefore is fitted with an exact scale size propeller, the question is what RPM is required of our scale propeller to give us scale thrust? The thrust of any propeller varies as RPM squared times Prop Diameter raised to the fourth power. This means that as diameter is reduced the RPM must increase.

Rule #4...Scale Thrust requires the propeller RPM to be increased by an amount equal to the square root of the scale factor. For example if the 78 inch prop on the full size Cub spins at 2,000 RPM our quarter scale cub needs to have its 19 inch propeller turning at 4,000 RPM to produce scale thrust.

SCALE TIME

Another way of thinking about this RPM shift is to say that TIME has been made to run faster in our model than it does in real life. The model's clock is ticking at a faster rate than ours. This is an exact literal truth. If a miniature clock were to be made exactly 1/4 scale it would tick exactly twice as fast. This same scaling law causes the heart of a child to beat much quicker than an adult, and the wings of a humming bird to beat much faster than the wings of that of an eagle.

Rule #5...Scale time for the model is faster than real time for the real airplane by an amount equal to square root of scale factor. This relationship between scale time and real time is used by Hollywood camera men to create realistic film footage of train wrecks, explosions, and other disasters. They run their cameras at high speed and play the film back at normal speed. This is known as slow motion and the effect is very realistic.

SCALE SPEED

Suppose our Piper J-3 Cub flies at 60 miles per hour (one mile a minute). We want our quarter scale model to fly a scale mile in a scale minute. Since the DISTANCE is divided by four but the TIME is divided by two, the scale model will fly at one half the speed of the real airplane. Scale speed for quarter scale model of a 60 mph J-3 cub is 30 miles per hour. Like scale time, scale speed varies as the square root of the scale factor.

Rule #6...To fly at scale speed our model needs to fly slower than the real airplane by an amount equal to the Square root of the scale factor.

SCALE POWER

Before we can select the right motor and gear box for our electric scale model we have to know how much torque will be needed to spin our scale propeller at scale RPM so that it will produce the needed scale thrust. The propeller is a transmission device that converts torque to thrust. Propeller thrust is proportional to shaft torque and inversely proportional to propeller diameter. Since the required thrust is proportional to the model weight and therefore to scale factor cubed, the required motor torque if proportional to scale factor raised to the fourth power. Power is torque times RPM. Since RPM varies as the square root of scale factor, the horsepower needed for the model airplane would be reduced from the horsepower of the real airplane by the scale factor raised to the 7/2 power.. For example if the real Piper Cub had a 65 horsepower engine, then our quarter scale model of the cub would require about 1/2 HP.

Rule #7...The horsepower required of the motor in the model airplane is less than the horsepower of the real airplane by the Scale factor raised the 7/2 power.

SCALE PERFORMANCE

In the first paragraph of this paper, I promised that if a model were built to these scale rules then this model would have SCALE PERFORMANCE. What I mean by scale performance is that the model would execute the same maneuvers as the real airplane in scale dimensions. To prove this assertion lets examine a few basic scale maneuvers.

SCALE TAKE OFF DISTANCE

In order to take off from the ground the real airplane must accelerate to flying speed from a standing start. Since the thrust to weight ratio of the model is the same as the real airplane, the acceleration of the model will be the same as the acceleration of the real airplane. Scale flying speed will be reached in scale time and therefore the model will travel scale distance down the runway.

SCALE CLIMB ANGLE

Since the thrust to weight ratio of the model is the same as the thrust to weight ratio of the real airplane, the climb angle for the model airplane will be the same as the climb angle of the real airplane.

SCALE SIZE LOOPS

Both the propeller thrust to weight ratio and the wing lift to weight ratio is the same for the model as for the real airplane. So all maneuvers will be scale size. A quarter scale model will loop in one fourth the diameter as the full size airplane. Since the scale clock in the model ticks faster than in the real airplane things will happen faster in the model world than in the real world. A quarter scale model will perform all its maneuvers in one half the time required by the real airplane.

Various Scale Factors

Example 2. North American P-51D

These very high power military aircraft are very difficult to model at scale speed. We have electric motors of up to two horsepower and gasoline engines of over ten horsepower, but most modelers would find that flying a model with a five pound per square foot or 80 ounce wing loading is like holding a tiger by the tail. Most scale modelers prefer to build fighter planes on the light side at about one half of scale weight and one half of scale power.

Example 3. Piper J-3 Cub

The Astro 40 geared motor will power a one quarter scale 65 hp J-3 Cub in a scale manner. Most modelers would prefer a bit more power and would use the geared 40 on a one fifth scale model. Almost any light plane makes a good scale subject. The scale models have very manageable wing loading and low power requirements.

Example 4. Douglas DC-3