Library / Driveline and transmission / DRV-14
Driveline and transmission
Planetary ratio, sun to carrier (ring held)
Speed reduction from the sun gear to the carrier when the ring gear is held still.
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- DRV-14
- Rev
- v1
- Field
- Driveline and transmission
- Result
- ωs/ωc
With the ring gear held still and the sun gear as the input, the carrier is the output and turns slowly: ωs/ωc = 1 + Zr/Zs. It follows from setting ωr = 0 in the carrier-speed relation. The ring must have more teeth than the sun, so the ratio is always above 2. This is the largest reduction a simple planetary set offers and is the usual low gear. Input and output turn in the same direction.
Speed reduction from the sun gear to the carrier when the ring gear is held still. This is the largest reduction a simple planetary set offers and is the usual low gear.
Calculator
- Carrier speed when the sun turns at 100 rad/s25 rad/s
- Output torque multiplication (ideal), the same as the ratio4
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 72-tooth ring and 24-tooth sun give a reduction of 4 from sun to carrier with the ring held. The torque at the carrier is about 4 times the sun torque, less losses.
- The sun-to-carrier ratio is always above 2 when the ring has more teeth than the sun. To get 5, the ring needs 4 times the teeth of the sun; to get 3, only twice.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing () | Local step | Change in result |
|---|---|---|---|---|---|---|
| Zs Sun teeth | 12 to 80 | 1.9 | 7 | 5.1 | +1 | −3 % |
| Zr Ring teeth | 30 to 150 | 2.25 | 7.25 | 5 | +1 | +1.04 % |
Sweep of Ring teeth
| Ring teeth | Speed ratio, sun over carrier |
|---|---|
| 30 | 2.25 |
| 45 | 2.875 |
| 60 | 3.5 |
| 75 | 4.125 |
| 90 | 4.75 |
| 105 | 5.375 |
| 120 | 6 |
| 135 | 6.625 |
| 150 | 7.25 |
Sweep one variable
Variables and how each one contributes
Speed ratio, sun over carrier
What the equation gives you. Everything else feeds it.
Ring teeth
The number of teeth on the ring (annulus) gear, a whole number from 30 to 150. Raises the reduction: a larger ring against the same sun means more reduction. Raising it by 6 (to 78) moves the result by +6.25 %, from 4 to 4.25.
Sun teeth
The number of teeth on the sun gear, a whole number from 12 to 80. Lowers the reduction: a larger sun against the same ring means less reduction. Raising it by 6 (to 30) moves the result by −15 %, from 4 to 3.4.
Worked example with the values above
- Tooth ratio: Zr/Zs = 72/24 = 3.
- Add 1.
- ωs/ωc = 4.
Assumptions and limits
- A simple planetary set with the ring held fixed.
- The sun is the input and the carrier the output.
Common mistakes
- Using the tooth counts the wrong way round, which gives the ring-to-carrier ratio.
- Forgetting the +1: the ratio is not simply Zr/Zs.
Statistics and uncertainty
The ratio is exact for given tooth counts. Real efficiencies are 95 to 99 % per planetary stage, so the output torque is slightly below ratio times input torque.
Where it is used
- Choosing the tooth counts for a reduction stage or a hub reduction.
- Checking the low gear of an automatic transmission.
Maths you need
- Gear ratio: input speed over output speed
- Planetary gear: sun, planets, carrier and ring