# Calculating torque in planetary gearbox

It's my understanding that in a planetary gearbox, force moves through the system via four mechanisms: the ring gear, the planet gears, the planetary carrier, and the sun gear. I'm trying to understand how to calculate torque coming into the system by measuring it at only one of these four points.

In my mental model of the problem, I understand that a force transfer will occur from the ring, through the planets and/or the planetary carrier, and into the sun gear.

My intuition tells me that if the resistance in the planetary carrier is greater than the resistance in the sun gear, the planetary carrier will not rotate and instead force will be transferred by the individual planets gears (rather than the planet carrier). In this case, would measuring the torque experienced by the planetary carrier measure the input torque?

To phrase the question in a more tangible way, in this system:
Ring gear (input) (unknown torque)
Sun gear (output) (has some unknown resistance)
Planetary Carrier (fixed to housing with a static torque sensor between)

Would I be able measure the torque produced from the the Ring Gear via the static torque sensor on the planetary carrier?

• Just put a torque sensor on the input, or output, shaft. What exactly do you want , or need, to measure? May 19, 2022 at 5:52
• hey mike thanks for commenting. This is in a system where it's not practical to measure the torque at the input (ring gear) or output (sun gear). In my understanding, the planetary gearbox is acting as a kind of uneven differential, with the the sun gear and the planetary carrier being the normal 'outputs', but the planetary carrier is fixed to housing so all the torque is being carried through to the sun gear. My question is whether torque measured between the carrier and the housing would always be lineally proportional to input torque. May 19, 2022 at 6:25
• So mount the torque sensor fixed relative to the gearbox case and have it measure shaft torque. No different than a normal setup. May 19, 2022 at 13:59
• is this something different than the gear ratio? May 19, 2022 at 14:18

Generally, torque is not applied to the individual planetary gears, just to the carrier that holds them. Unless you have an individual motor attached to each planet gear (physically possible, but rarely practical) you probably only have 3 locations where you need to know torque: sun, carrier, and ring. If you're measuring torque at the ring gear, then you can calculate the other torques if you know the number of teeth on the sun and ring gears ($$N_s$$, $$N_r$$):
$$\tau_s = \tau_r \frac{N_s}{N_r}$$
$$\tau_c = -\tau_r \frac{N_r + N_s}{N_r}$$