# How can a planetary gearset "work" with gears whose pitch diameters are not tangent?

Somewhat new to the gear-world here, so I am trying to understand something. Most of the literature an examples of planetary gearsets that I've seen so far have described a relationship where the pitch circle of the planet gear(s) is tangent to both the pitch circle of the sun gear and the ring gear simultaneously. That is, the pitch circle of the planet gear exhibits perfect rolling relative to the sun gear pitch circle and the ring gear pitch circle.

Most examples also follow this relationship:

R = 2P + S

Where R, P and S represent the number of teeth of the ring, planet, and sun gears, respectively.

However, the company where I work has a device with a planetary set that has been in production for several years with zero issues, and the specs are as follows:

R = 66
P = 26
S = 12
number of planets = 3
m = 1 (for all gears)


First, this does not follow the axiom of R = 2P + S, because the ring gear by this equation would have 64 teeth. There is another rule I've found that states that to allow equidistant spacing of the planets in the arrangement, R + S / number of planets must be an integer with no remainder. That axiom, in this case, is met, because 66 + 12 / 3 = 26.

But I'm still confused as to why the pitch circle alignment is "allowed" to be "off". Because each gear module is m = 1 in this case, the number of teeth is equal to the pitch circle diameter. So if you draw the pitch circle alignment with S = 12 mm, P = 26 mm, and R = 66 mm, there is no way for the planet circle to be tangent to both the sun and the ring at the same time. But I thought this was a "requirement" of the planetary system, no?

I also noticed that the carrier in this device as a slightly longer arm than you'd expect. Assuming the gear pitch circles were tangent, given S = 12 and P = 26, you'd expect the arm of the carrier to be 19 mm long (radius of the sun + radius of the planet). But the actual carrier part has arms that are 19.85 mm long. So in essence, it's ever-so-slightly "pushing" the planet gears out toward the ring gear.

I am assuming this plays a role in how the device is able to work without completely tangent pitch circles. But I've not been able to find any literature describing a planetary (or any epicyclic arrangement) gearset where the pitch circles can be non-tangent, and/or what other considerations must be taken into account when designing this way. Even AGMA 6123-C16 does not detail this.

Does anyone have experience with this sort of gearset?

• Can you see this in photos of the gearbox? Commented Sep 21, 2022 at 14:12
• Not really. I mean with the naked eye, the gears all look fine. But the actual engineering drawings explicitly state the pitch diameters of the gears as 12 mm (sun), 26 mm (planet), and 66 mm (ring), which mathematically goes against the typical convention of tangent pitch diameters in an epicyclic setup. It obviously works, because the part has been in production for years, but I'm trying to find/learn the mathematical and mechanical justification for why/how this is "allowed," because it goes against most common literature on gears. Commented Sep 21, 2022 at 14:49
• More slop for the tolerance stack
– Abel
Commented Sep 24, 2022 at 10:33