Kevmaniac2000,

You are absolutely correct that the torque and speed ratios should be inverses of each other. If they weren’t, a gear train could be used to violate conservation of energy. (I’ll let you have the fun of deriving the proof of that.)

The difficulty you’re having seems to come from the definition of “diameter”. In practice, a gear has more than one “diameter”, and which one you use in calculations depends on what you’re trying to calculate.

The diameters you reported appear to be the outer diameters, that is, the diameters of circles that pass through the most distant points of the teeth. (I measured the outer diameters as 14.7 and 65.3 mm for those two gear sizes.) The outer diameter is important for calculating clearances, but it’s not the correct diameter for calculating either velocity or torque ratios.

For calculating those ratios, you need to consider the diameters at which the teeth of the two gears contact each other. If the gear train is working properly, that’s the “pitch diameter”.

For an excellent tutorial on gears, I refer you to:

http://www.cs.cmu.edu/~rapidproto/mechanisms/chpt7.html#HDR115

For a convenient table of gear formulae, see:

http://www.engineersedge.com/gear_formula.htm

The calculated pitch diameters of the 12- and 60-tooth gears, in mm, are 12.7 and 63.5, respectively.

The ratio of those two is, **not** coincidentally, 1:5. It’s not a coincidence, because pitch diameter is calculated from diametral pitch (24 for Vex gears) and number of teeth.

If you use half the pitch diameter as the lever length in your torque calculations (instead of half the outer diameter, which is what you appear to have done), you should get a torque ratio that is the inverse of the speed ratio.

Another approach is to consider the distance between gear centers. In the Vex system, the holes are on 12.7 mm spacing. That’s consistent with the sizes of most of the gears (The Vexplorer has a 48-tooth gear.), which have pitch diameters that are integral multiples of that. (BTW, if you do all the calculations in inches, the numbers are much “prettier”.) Once you fix the distance between centers, you can calculate the lengths of the lever arms by apportioning the space between centers based on the relative number of teeth. For the 12:60 example, divide the distance between centers (3 holes = 38.1 mm) by the total number of teeth (12 + 60 = 72) to get the “lever arm/tooth” of 0.529167. If you multiply that value by the number of teeth for each gear, you get 6.35 and 31.75, respectively. These are halves of the pitch diameter. That confirms our previous results.

If I can be of further help, please let me know.

Eric