Library / Driveline and transmission / DRV-16
Driveline and transmission
Planetary ratio, sun to ring (carrier held)
Speed ratio from the sun gear to the ring gear when the carrier is held still.
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- DRV-16
- Rev
- v1
- Field
- Driveline and transmission
- Result
- ωs/ωr
With the carrier held still, the planet gears simply reverse the direction between the sun and the ring: ωs/ωr = −Zr/Zs. The minus sign means the ring turns the opposite way to the sun, which is how a planetary set provides reverse. It follows from setting ωc = 0 in the carrier-speed relation. The ratio is the tooth ratio of the ring and sun, without the +1.
Speed ratio from the sun gear to the ring gear when the carrier is held still. The minus sign means the ring turns the opposite way, which is how a planetary set provides reverse.
Calculator
- Ring speed when the sun turns at 100 rad/s−33.33 rad/s
- Magnitude of the ratio3
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, the carrier held gives a ratio of −3: the sun turns 3 times for each turn of the ring, in the opposite direction. This is a reverse gear with a reduction of 3.
- The reduction is smaller than the sun-to-carrier ratio, 4, from the same gears: the ratio is Zr/Zs here, one less than 1 + Zr/Zs, because the held carrier takes no part of the motion.
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 | −6 | −0.9 | 5.1 | +1 | −4 % |
| Zr Ring teeth | 30 to 150 | −6.25 | −1.25 | 5 | +1 | +1.39 % |
Sweep of Ring teeth
| Ring teeth | Speed ratio, sun over ring |
|---|---|
| 30 | −1.25 |
| 45 | −1.875 |
| 60 | −2.5 |
| 75 | −3.125 |
| 90 | −3.75 |
| 105 | −4.375 |
| 120 | −5 |
| 135 | −5.625 |
| 150 | −6.25 |
Sweep one variable
Variables and how each one contributes
Speed ratio, sun over ring
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. Makes the ratio more negative: a larger ring turns more slowly than the sun. Raising it by 6 (to 78) moves the result by +8.33 %, from −3 to −3.25.
Sun teeth
The number of teeth on the sun gear, a whole number from 12 to 80. Moves the ratio towards zero: a larger sun speeds the ring up relative to the sun. Raising it by 6 (to 30) moves the result by −20 %, from −3 to −2.4.
Worked example with the values above
- Tooth ratio: Zr/Zs = 72/24 = 3.
- Change the sign for the opposite direction.
- ωs/ωr = −3.
Assumptions and limits
- A simple planetary set with the carrier held fixed.
- The sun is the input and the ring the output.
Common mistakes
- Dropping the minus sign, which loses the direction reversal.
- Adding 1 as in the other arrangements. Holding the carrier means no 1 + term.
Statistics and uncertainty
The ratio is exact for given tooth counts. Tooth numbers are integers, so the realised ratio differs from the ideal only if the design was based on a non-integer target.
Where it is used
- The reverse gear in automatic transmissions.
- The fixed-carrier ratio of a simple planetary set.
Maths you need
- Gear ratio: input speed over output speed
- Planetary gear: sun, planets, carrier and ring