# practical gear theory for modellers

Gearwheels shouldn't have any secret for a modeller.

The standardized way to gear design is by using the modulo principle of gearwheels. You probably know that circles have a relation with a number called Pi. This not different for gearwheels. But for gearwheels someone (who probably did not like Pi either) thought up an elegant method for working with Pi. The Modulo M of a gear is the ratio of the pitch(distance between two teeth) to Pi. So for a M = 1 wheel the pitch is Pi = 3.14 and for M = 0.5 the pitch is 1.57. Now you normally never calculate the pitch of gearwheel so this is no problem, you may forget this. For work in our scale M has nice round values of 0.2, 0.3 or 0.4 with of course an exception for Englishmen. In using gearwheels there are two important dimensions both are diameters. That is the overall diameter, which you need to see if a gearwheel fits in a certain space and the effective working diameter which you need to calculate axle distances between gearwheels. The working diameter is the diameter at half height of the teeth. As the height of the teeth is a constant these two are linked. In fact gearwheels can be regarded as developed from cylinders rolling off against each other with teeth introduced to prevent any slip between the two. The diameter ratio is also the gear ratio. So let:

N = Number of teeth
Do = Overall diameter
Dw = Working diameter
M = Modulo

then
Dw = N * M
and
Do = (N+2) * M

See how elegant this is with nice numbers for M there is no need for a calculator!

The distance Ad between two gearwheels 1 and 2 is:

Ad = ( Dw1 + Dw2 )/2 =( N1 + N2 ) * M / 2

We sometimes have use for the axle distance between worm and wormwheel:

Aww = Doworm /2 + (N-2)/2 * M

That's all there is to gearwheels for using them in practice. Note however that the above formulas for the axle distances give the ideal distance. If your gearwheels don't have a central axle hole or are a bit oval, you will end up with a bind. So when designing a new gear train put in a few hundredths of a mm extra for non ideal dimensions.

Now there is only one thing left and that is the English DP value. As it is of English origin you will not be surprised that there is a connection with another awkward number (obsolete, no SI standard unit) namely 1 DP = 1" = 25.4 mm. This means that 100 DP gear is in practice M=0.25 (25.4/100) and 64 DP gear will match with M=0.40 (25.4/64=.397). Turns out not too bad, isn't it?