Library / Driveline and transmission / DRV-13
Driveline and transmission
Carrier speed of a simple planetary gear
Speed of the carrier in a simple planetary gear set, from the speeds of the ring and the sun and their tooth counts.
- Ref
- DRV-13
- Rev
- v1
- Field
- Driveline and transmission
- Result
- ωc [rad/s]
For a simple planetary set with Zr ring teeth and Zs sun teeth, the speeds of the ring, sun and carrier are tied by (Zr + Zs)ωc = Zrωr + Zsωs. Solving for the carrier speed gives the equation above. Holding one member still, or locking two together, gives each ratio of the set: with the ring held (ωr = 0) the sun-to-carrier ratio is 1 + Zr/Zs; with the sun held, the ring-to-carrier ratio is 1 + Zs/Zr; with the carrier held the sun-to-ring ratio is −Zr/Zs. Any two members locked together give a ratio of 1. Speeds are signed: a negative value turns the opposite way.
Speed of the carrier in a simple planetary gear set, from the speeds of the ring and the sun and their tooth counts. This single relation produces every ratio of the set by holding a different member still.
Calculator
- Sun-to-carrier ratio if the ring is held, 1 + Zr/Zs4
- Ring-to-carrier ratio if the sun is held, 1 + Zs/Zr1.333
- Carrier speed in rpm238.7 rpm
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 the ring held (0 rad/s) and the sun driven at 100 rad/s, the carrier turns at 25 rad/s: a reduction of 4, equal to 1 + 72/24. This is the large reduction of the set.
- Drive the ring at 50 rad/s with the sun held and the carrier turns at 37.5 rad/s: a reduction of only 1.33. Drive both at the same speed and the carrier turns at that speed too: 100 rad/s, ratio 1.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| ωr Ring speed | −300 to 300 rad/s | −200 | 250 | 450 | +12 rad/s | +36 % |
| ωs Sun speed | −300 to 300 rad/s | −75 | 75 | 150 | +12 rad/s | +12 % |
| Zs Sun teeth | 12 to 80 | 14.29 | 52.63 | 38.35 | +1 | +3.09 % |
| Zr Ring teeth | 30 to 150 | 13.79 | 44.44 | 30.65 | +1 | −1.03 % |
Sweep of Sun speed
| Sun speed (rad/s) | Carrier speed (rad/s) |
|---|---|
| −300 | −75 |
| −225 | −56.25 |
| −150 | −37.5 |
| −75 | −18.75 |
| 0 | 0 |
| 75 | 18.75 |
| 150 | 37.5 |
| 225 | 56.25 |
| 300 | 75 |
Sweep one variable
Variables and how each one contributes
Carrier speed
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 carrier speed towards the ring speed, as the ring has more weight in the average. Raising it by 6 (to 78) moves the result by −5.88 %, from 25 to 23.53 rad/s.
Sun teeth
The number of teeth on the sun gear, a whole number from 12 to 80. Raises the carrier speed towards the sun speed, as the sun has more weight in the average. Raising it by 6 (to 30) moves the result by +17.6 %, from 25 to 29.41 rad/s.
Ring speed
The ring angular speed, in rad/s, signed. Zero holds the ring still. Raises the carrier speed by Zr/(Zr + Zs) of the change. Raising it by 10 rad/s (to 10 rad/s) moves the result by +30 %, from 25 to 32.5 rad/s.
Sun speed
The sun angular speed, in rad/s, signed. Zero holds the sun still. Raises the carrier speed by Zs/(Zr + Zs) of the change. Raising it by 10 rad/s (to 110 rad/s) moves the result by +10 %, from 25 to 27.5 rad/s.
Worked example with the values above
- Weighted sum: Zrωr + Zsωs = 72 × 0 + 24 × 100 = 2,400.
- Total teeth: Zr + Zs = 96.
- ωc = 2,400 / 96 = 25 rad/s.
Assumptions and limits
- A simple planetary set with the carrier, sun and ring on one axis.
- No slip and ideal meshing; speeds measured on the same axis with the same sign convention.
Common mistakes
- Dropping the sign of a member that turns in the opposite direction.
- Using the ring and sun tooth counts the wrong way round.
Statistics and uncertainty
Tooth counts are exact, so the kinematic ratios are exact too; the uncertainty comes from the measured speeds. A 1 % error in the sun speed passes straight through to the carrier, scaled by Zs/(Zr + Zs), which is 25 % at the defaults.
Where it is used
- Analysing automatic transmissions, hybrid power-split units and final drives.
- Checking the speeds of the members of an epicyclic gear.
Maths you need
- Angular speed: in rad/s or rpm, with a sign
- Gear tooth counts: ratio equals the ratio of teeth