# Question about calculation of planetary gear ratios

• mrhall1923
In summary, the conversation discusses a formula for calculating the gear ratio of a planetary gear, specifically when the planetary gear carrier is fixed and either the ring gear or sun gear is the input/output. The formula is (R + S) * Ty = R * Tr + Ts * S. The conversation also explores a two-stage planetary gearbox design and the possibility of finding a single formula for solving the overall gear ratio. The speaker suggests using kinematic constraint equations or referring to machine design texts for such formulas. They also provide a potential solution for the case of a fixed planetary gear carrier and varying tooth quantities.
mrhall1923
So I've found the following formula for calculating the gear ratio of a planetary gear.

Tr Turns of the ring gear
Ts Turns of the sun gear
Ty Turns of the planetary gear carrier
R Ring gear teeth
S Sun gear teeth
P Planet gear teeth
( R + S ) ×Ty = R × Tr + Ts × S

The example I saw on the same website used a fixed ring gear so Tr=0 and the formula worked great.

My question is about having a fixed planetary gear carrier (Ty) with the ring gear being the input and the sun gear being the output. If Ty=0 in the above formula, then the left side of that equation will always be 0 and I'm at a loss at how to solve for Ts.

I have a feeling that there is a better formula for solving this. Can someone help?

Next,

I'm trying to design a 2 stage planetary gearbox with the following.

First planetary gear:
Planetary gear carrier = input
Ring gear = fixed
Sun gear = output (to second planetary gear)

Second planetary gear:
Planetary gear carrier = fixed
Ring gear = input (from first planetary gear)
Sun gear = output

Can anyone provide a single formula for solving the overall gear ratio? So one turn of the first planetary gear carrier will provide how many turns of the second sun gear? I'm looking for a formula that I can use for different size gears of varying tooth quantities.

Machine design texts frequently provide formulas of the sort you are looking to find. You can also work it out for yourself simply by writing the relevant kinematic constraint equations.

mrhall1923 said:
My question is about having a fixed planetary gear carrier (Ty) with the ring gear being the input and the sun gear being the output.

I think this is correct..

Let Nr, Np, Ns be the number of teeth on the ring, planet and sun gears.

For each turn of the ring the planets rotate (about their fixed centre) Nr/Np times.
For each turn of the planets the sun rotates Np/Ns

So for each turn of the ring the sun turns:

(Nr/Np) * (Np/Ns) = Nr/Ns

Edit: In case it's not obvious, these two set ups are equivalent (except for the direction of rotation).

Last edited:

## 1. How do you calculate the gear ratio for a planetary gear system?

The gear ratio for a planetary gear system can be calculated by dividing the number of teeth on the sun gear by the number of teeth on the planet gear. This can be further multiplied by the number of planet gears in the system to get the overall gear ratio.

## 2. What is the purpose of calculating the planetary gear ratio?

The planetary gear ratio is important for determining the speed and torque output of the system. It also helps in determining the efficiency and overall performance of the gear system.

## 3. How does the gear ratio affect the overall performance of a planetary gear system?

The gear ratio directly affects the speed and torque output of the system. A higher gear ratio results in a slower output speed but higher torque, while a lower gear ratio results in a faster output speed but lower torque.

## 4. Can the gear ratio be changed in a planetary gear system?

Yes, the gear ratio can be changed by altering the number of teeth on the gears or by changing the number of planet gears in the system. However, this may require redesigning and replacing the gears.

## 5. Are there any other factors that can affect the gear ratio in a planetary gear system?

Apart from the number of teeth and planet gears, the gear ratio can also be affected by the pitch diameter of the gears, the angle of the planet gears, and the placement of the gears in the system.

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