STEM Quiz #1

Here’s a quiz that’s mostly geometrical, but you may need to do a bit of mechanical/engineering research to get to the answers.

The Vex Versaplanetary gearbox offers single stages ranging from 3:1 reduction down to 10:1 reduction. These gearboxes are driven at the sun gear, have a fixed annular gear, and drive output on the planetary carrier. Assuming that there are at least three planet gears of the same size as each other, what are the theoretical highest and lowest possible reductions possible for a single stage gearbox of this type? For purposes of this calculation, you may allow the teeth to be infinitesimally small, so that this becomes a “roller box” rather than a gear box.

As with Ether’s quizzes, I’m not looking for full mathematical rigor, but show enough of your work that I can tell where your numbers are coming from.

The lower limit as Sun Gear diameter approaches Ring Gear diameter should be 2:1, as the ratio for a planetary gearbox as described is R/S+1.

The upper limit requires some geometry constructions of 3 equally spaced tangent inscribed circles. My literal back of the envelope derivation got messy, but the result came out to 1/(1-6/(3+2*sqrt(3)))+1 or 14.928203235:1

I’ll try to post a coherent derivation when I get it worked out nicely on something more legible.

I should point out that these are the limits of the ratio, rather than achievable ratios under the stated conditions, as you would have gears or rollers moving in opposite directions in direct contact.

Side note: I think it’s beautiful that the ratios offered are all clean integer ratios*, not messy repeating decimals. Planetary gear boxes of this variety have single stage ratios of Ring Gear Teeth/Sun Gear Teeth +1. The ring gears on Versaplanetaries are 72 teeth, which has factors 2x2x2x3x3. Using those factors in different combinations results in sun gears of 36, 24, 18, 12, 9, and 8 resulting in 3:1, 4:1, 5:1, 7:1, 9:1, and 10:1 respectively. 6:1 and 8:1 ratios would have required use of prime factors 5 and 7 respectively, which would increase the ring gear tooth count by that many, which is impractical.

*I understand by definition gear ratios are integer ratios, but when reduced to the mechanical advantage factor, only evenly divisible ratios give integer results.

Correct on the limit of large sun gear is 2:1. On the lowest gear ratio, you’re quite close but not quite right.

Forgot the plus 1 in my calculator when transcribing the upper limit. Should be 14.9282032303:1

The one ratio which always bothers me is the 9:1. The sun gear has 9 teeth. Since the ring gear pitch diameter is the sun diameter plus twice the planetary diameter, working backwards from 72 teeth on the annular gear leaves 31 1/2 teeth on each of the planetary gears. I know it’s possible to play some tricks with pitch diameter vs tooth counts, but I hope to be able to work around needing a 9:1 VP stage.

Gear ratios are always rational, that is ratios of integers, but not necessarily integers. It is possible to get close to the 6:1 and 8:1 integer ratios with a single stage 72t annular gear: with 10 teeth on the sun and 31 on the planets, a VP-compatible stage could be made with a gear ratio of 8.2:1; 14 on the sun and 29 on the planets would result in 6.14:1.

If you wanted to make a single “exact” 6:1 or 8:1 ratio stage compatible with VP, the first step would be to make a (most likely) 70t annular gear. Then your sun gears would be 10 and 14 teeth. If you were to do this, your “nearly identical to the standard” annular gears would probably be quite unpopular unless you found some clever way (e.g. color or a milled exterior similar to a U.S. quarter) to make them look quite unlike the 72t annular gear. I concur that this is not likely to be commercially viable, as the existing VP ratios provide a maximum gap of 40% (5:1 to 7:1) for a single stage. Above 9:1 (allowing multiple stages), the maximum gap is 25% (12:1 to 15:1 and 16:1 to 20:1) below 20:1 and 20% above that (50:1 to 60:1). There are few applications that cannot be designed around a 20% gap in capabilities.

That’s the correct numeric value (with way more significant figures than I calculated), but still waiting on a proof.

Here’s my MS Paint proof. I did the ratio in terms of radii of the gears.

Great point about the 9:1 requiring half-tooth planets. I probably will avoid those for anything that has significant torque needs to avoid shearing teeth.

Well done! Rep points awarded. (Remember, they’re just dots!)

They aren’t actually half tooth planets, but the teeth of these gears presumably had to be adjusted somewhat to make this work. While I haven’t examined a 9:1 VP, my guess is that it has 32 slightly shortened teeth.


I was going to assume 31 teeth and 3/4 tooth engagement. might be worth examining for future reference. either way it’s not ideal.

You realize you can have a planetary setup with only two planet gears, right? My old Milwaukee impact which bit the dust (motor side, not gearbox) uses this setup because it only requires 1 stage to achieve such a high ratio. Usually this is uncommon for packaging at the least, and I’m sure they’re prone to jamming if not done properly (the setup for this one was quite good).
Meaning you have infinite possible reduction…