Library / Driveline and transmission / DRV-15
Driveline and transmission
Planetary ratio, ring to carrier (sun held)
Speed reduction from the ring gear to the carrier when the sun gear is held still.
- Ref
- DRV-15
- Rev
- v1
- Field
- Driveline and transmission
- Result
- ωr/ωc
With the sun gear held still and the ring gear as the input, the carrier is the output: ωr/ωc = 1 + Zs/Zr. It follows from setting ωs = 0 in the carrier-speed relation. It is a small reduction, only slightly above 1, because the ring is large and the sun small. Input and output turn in the same direction. It is the second ratio available from the same planetary set when a different member is held.
Speed reduction from the ring gear to the carrier when the sun gear is held still. It is a small reduction, only slightly above one, because the ring is large.
Calculator
- Carrier speed when the ring turns at 100 rad/s75 rad/s
- Sun-to-carrier ratio with the ring held, for comparison4
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.
- With 72 ring teeth and 24 sun teeth, holding the sun gives a ring-to-carrier ratio of 1.333, only 33.3 % above direct drive. Holding the ring instead gives 4 from the same set.
- This is why a single planetary set offers a few different ratios: the same gears give a large reduction, a small one, a direct drive and a reverse, depending on which member is held.
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 | 1.167 | 2.111 | 0.9444 | +1 | +1.04 % |
| Zr Ring teeth | 30 to 150 | 1.16 | 1.8 | 0.64 | +1 | −0.342 % |
Sweep of Sun teeth
| Sun teeth | Speed ratio, ring over carrier |
|---|---|
| 12 | 1.167 |
| 20.5 | 1.285 |
| 29 | 1.403 |
| 37.5 | 1.521 |
| 46 | 1.639 |
| 54.5 | 1.757 |
| 63 | 1.875 |
| 71.5 | 1.993 |
| 80 | 2.111 |
Sweep one variable
Variables and how each one contributes
Speed ratio, ring 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. Lowers the ratio towards 1: a larger ring makes it closer to a direct drive. Raising it by 6 (to 78) moves the result by −1.92 %, from 1.333 to 1.308.
Sun teeth
The number of teeth on the sun gear, a whole number from 12 to 80. Raises the ratio: a larger sun makes more reduction. Raising it by 6 (to 30) moves the result by +6.25 %, from 1.333 to 1.417.
Worked example with the values above
- Tooth ratio: Zs/Zr = 24/72 = 0.3333.
- Add 1.
- ωr/ωc = 1.333.
Assumptions and limits
- A simple planetary set with the sun held fixed.
- The ring is the input and the carrier the output.
Common mistakes
- Using the same formula as the sun-to-carrier ratio. The roles of ring and sun are swapped.
- Expecting a reduction as large as in the other arrangement.
Statistics and uncertainty
The ratio is exact for given tooth counts. Because the ratio is close to 1, small changes in tooth numbers have little effect, so this arrangement tolerates machining variation well.
Where it is used
- Understanding the second ratio available from the same planetary set.
- Checking the intermediate gears of an automatic transmission.
Maths you need
- Gear ratio: input speed over output speed
- Planetary gear: sun, planets, carrier and ring