DERIVATION of the
GEAR RATIO, SPEED, MOTOR RPM AND DRIVER DIAMETER EQUATIONS


NOTE: These equations apply to all scales, where driver diameters are measured in scale inches.

Lets look at what we know:

INCHES/MILE = 5280 * 12 =63360

As a wheel with a diameter (D) rolls for one revolution along the track it moves:

INCHES/REVOLUTION = ¶ * D

Since we want to work with RPM (Revolutions Per Minute) and driver inches, we convert to:

1 MPH = 63360 / 60 IN/MI = 1056 IN/MIN

Now we find how many revolutions a 1 inch diameter wheel needs to cover the distance. This is also the RPM necessary for 1 MPH.

RPM = 1056 / ¶ / D = 1056 / ¶ / 1 = 336.1352398 = A

Use 336 for an error of 0.04%.

The constant, A = 336 AXLE RPM for 1 MPH per Driver diameter in inches.

If N (axle RPM) =772: S (MPH) = N / A = 772 / 336 = 2 for a 1 inch wheel.

If D = 2 IN. and N = 336 then S = 2 MPH since it moves twice the distance in 1 minute. Thus:

S = D * N / A
or
N = S / D * A

In our locos the axle RPM = Motor RPM (M) / Gear Ratio (G):

N = M / G

Substituting for N, we get:

S = D * M / G / A

Use this to predict speed with a selected driver diameter, motor and gear ratio.

Solving for other parameters:

To select motors when remotoring.

M = S * G * A / D

To select potential ratios when regearing.

G = M * D / A / S

To select driver diameter when scratch building.

D = S * G * A / M


Writing a program makes these much more usable. See -BASIC- example.


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