Library / Fundamentals / FUN-07
Fundamentals
Shaft speed and angular frequency
Convert revolutions per minute to radians per second.
- Ref
- FUN-07
- Rev
- v1
- Field
- Fundamentals
- Result
- ω [rad/s]
ω in rad/s, f in hertz (revolutions per second), N in revolutions per minute (rpm). The period is T = 1/f.
A shaft turning at N revolutions per minute turns N/60 revolutions each second, and each revolution is 2π radians. Most rotating-machine formulas need ω in radians per second.
Calculator
- Frequency f25 Hz
- Period T40 ms
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.
- To convert quickly, multiply rpm by about 0.1047 to get rad/s.
- The mains-synchronous speed of a 2-pole machine at 50 Hz is 3000 rpm, i.e. 314 rad/s.
- Always convert to rad/s before using ω in power (P = Tω) or vibration formulas.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (rad/s) | Local step | Change in result |
|---|---|---|---|---|---|---|
| N Shaft speed | 60 to 30,000 rpm | 6.283 | 3,142 | 3,135 | +10% | +10 % |
Sweep of Shaft speed
| Shaft speed (rpm) | Angular speed (rad/s) |
|---|---|
| 60 | 6.283 |
| 130.5 | 13.66 |
| 283.7 | 29.71 |
| 617 | 64.61 |
| 1,342 | 140.5 |
| 2,917 | 305.5 |
| 6,344 | 664.4 |
| 13,800 | 1,445 |
| 30,000 | 3,142 |
Sweep one variable
Variables and how each one contributes
Angular speed
What the equation gives you. Everything else feeds it.
Shaft speed
Revolutions per minute. Directly proportional: 1000 rpm is about 104.7 rad/s.
Worked example with the values above
- Revolutions per second: f = 1,500 / 60 = 25 Hz.
- Each revolution is 2π rad, so ω = 2π × 25.
- ω = 157.1 rad/s.
Assumptions and limits
- Steady rotation.
- Whole revolutions counted as 2π radians.
- Speed measured at the shaft concerned (not a geared-down one).
Common mistakes
- Using rpm directly in P = Tω.
- Forgetting the 60 in the denominator.
- Mixing Hz and rad/s: they differ by a factor of 2π.
Statistics and uncertainty
The conversion is exact. The only uncertainty is in the measured speed, and the relative error carries straight through: δω/ω = δN/N.
Where it is used
- Motor and turbine calculations.
- Gearbox and propeller speed matching.
- Vibration analysis of rotating machinery.
Maths you need
- Radians: one revolution is 2π rad
- Frequency: cycles per second in Hz