Guide
Gearbox service life calculation: average torque, average speed and rated life
How precision reducer life is rated, how to reduce a duty cycle to average torque and speed, and two worked life calculations for an RV reducer and a harmonic drive.
Key takeaways
- Life Lh = K × (N0 ÷ Nm) × (T0 ÷ Tm)p, with p = 10/3 for roller-bearing designs and 3 for harmonic drives.
- Reduce the duty cycle to one average torque Tm and one average speed Nm.
- Peak torque is a separate check; life only covers wear and fatigue.
How reducer life is rated
Precision reducer makers state a rated life K (hours) at a rated torque T0 and a rated speed N0. The wear-out mode is rolling-contact fatigue in the bearings, so life follows the bearing life relationship of ISO 281: Lh = K × (N0 ÷ Nm) × (T0 ÷ Tm)p. The exponent p is 10/3 for roller bearings (RV and most cycloidal reducers) and 3 for ball bearings (the harmonic drive's wave generator). Makers also rate harmonic drives with input speed and their own K values; always use the figures from the product's datasheet.
Average torque and speed from a duty cycle
For segments with time ti, speed ni and torque Ti:
Dwell time with zero speed counts in Nm (it lowers the average speed) but carries no weight in Tm.
Worked example 1: RV reducer on a positioner
Reducer ratings: T0 = 167 N·m at N0 = 15 rpm, K = 6,000 h, allowable acceleration torque 412 N·m.
| Segment | Time | Average speed | Torque |
|---|---|---|---|
| Accelerate | 0.2 s | 7.5 rpm | 300 N·m |
| Constant speed | 1.0 s | 15 rpm | 100 N·m |
| Decelerate | 0.2 s | 7.5 rpm | 200 N·m |
| Dwell | 1.0 s | 0 | 0 |
- Peak 300 N·m < 412 N·m allowable acceleration torque ✓
- Nm = (0.2 × 7.5 + 1.0 × 15 + 0.2 × 7.5) ÷ 2.4 = 7.5 rpm
- Tm = [(1.5 × 30010/3 + 15 × 10010/3 + 1.5 × 20010/3) ÷ 18]3/10 = 161 N·m (just below rated)
- Lh = 6,000 × (15 ÷ 7.5) × (167 ÷ 161)10/3 ≈ 13,500 h, about 2.8 years at 16 hours a day, 300 days a year.
Notice how the short acceleration peak dominates Tm: cutting the acceleration torque from 300 to 250 N·m lifts life noticeably.
Worked example 2: harmonic drive on a cobot joint
Size 17 unit at 100:1: rated 31 N·m at 2,000 rpm input, K = 10,000 h, repeated peak limit 44 N·m, average torque limit 34 N·m. Exponent p = 3, speeds at the input.
| Segment | Time | Input speed | Output torque |
|---|---|---|---|
| Accelerate | 0.3 s | 1,000 rpm | 40 N·m |
| Constant speed | 1.2 s | 2,000 rpm | 20 N·m |
| Decelerate | 0.3 s | 1,000 rpm | 30 N·m |
| Dwell | 1.2 s | 0 | 0 |
- Peak 40 N·m < 44 N·m repeated peak ✓; Nm = 3,000 ÷ 3 = 1,000 rpm
- Tm = [(300 × 40³ + 2,400 × 20³ + 300 × 30³) ÷ 3,000]1/3 = 24.9 N·m < 34 N·m average limit ✓
- Lh = 10,000 × (2,000 ÷ 1,000) × (31 ÷ 24.9)³ ≈ 38,400 h
Practical tips
- Use S-curve acceleration: lower peaks raise life sharply because of the exponent.
- Counterbalance gravity on vertical joints to cut average torque.
- Keep grease within temperature limits and follow the relubrication interval; life figures assume it.
References
- ISO 281, Rolling bearings — Dynamic load ratings and rating life.
- ISO 6336, Calculation of load capacity of spur and helical gears.
Frequently asked questions
Why is the life exponent 10/3?
Precision reducer life is governed by their rolling bearings. Under ISO 281, roller bearing life varies with load to the power 10/3 and ball bearing life to the power 3. RV and most cycloidal reducers use roller bearings; harmonic drive wave generators use a ball bearing.
What does halving the torque do to life?
With a 10/3 exponent, halving the average torque multiplies life by 2^(10/3), about 10 times. Speed has a linear effect: halving average speed doubles life in hours.
Is rated life the same as warranty life?
No. Rated life is a statistical calculation under stated conditions (usually L10: 90% of units reach it). Warranty is a commercial promise and is usually shorter.
