Robot reducers: planetary, strain-wave and cycloidal mechanisms

Explore ratio, mechanical power and reflected inertia

Use the panel above to vary a hypothetical reducer's speed ratio, imposed input speed and torque, assumed motoring efficiency, and load inertia. No local Python is needed. These are arithmetic examples, not measurements, product ratings or a mechanism selector. The original mechanism comparison remains below.

N is the positive magnitude of input/output speed ratio. Input speed n is in revolutions per minute and ωin=2πn/60 rad/s. Under steady positive motoring with constant assumed efficiency η, the model computes nout=n/N, Tout=η N Tin, Pin=Tin ωin, Pout=η Pin and Ploss=(1−η) Pin. It also satisfies Pout=Tout ωout with ωout=ωin/N. All speed and torque values are magnitudes: no shaft direction is inferred from N. Both input speed and torque are prescribed; a real motor may not sustain that pair.

At 3,000 rpm, 0.20 N m, N=50 and η=0.8, output is 60 rpm and 8 N m. Mechanical input is about 62.832 W, output 50.265 W and loss 12.566 W. Doubling N halves output speed and doubles output torque while keeping the assumed input and efficiency fixed. Power does not increase. Changing η changes torque and output/loss power but leaves the kinematic speed ratio unchanged. η=1 is an ideal lossless reference, not a realistic efficiency claim. The chosen control bounds keep speed and torque positive; they do not cover standstill, breakaway friction or backdriving.

The separate ideal rigid-transmission relation is Jload,ref=Jload/N². For the assumed 0.020 kg m² load and N=50, this is 0.000008 kg m² at the input. It follows by equating ½ Jload ωout² and ½ Jload,ref ωin². Efficiency is not inserted in this inertial equivalence. The model omits motor rotor and gearbox inertia. Load inertia changes only this separate result; no acceleration, acceleration torque or transient response is simulated. Zero load inertia is an ideal reference, not proof that a drive has no inertia.

Try half/double ratio, unity ratio, lower efficiency, the ideal lossless case, higher speed, higher torque and zero load inertia. The speed, torque and power bars use separate units and scales, shared across A/B for each quantity. Blue is mechanical input, teal output; the power split adds brown loss. Exact computed values remain in the accessible metrics/table even when a bar is very small. Reflected inertia is shown numerically to avoid hiding the square-law change in a tiny bar. Displayed values are rounded.

Constant efficiency is supplied, not predicted from ratio, speed, load, temperature or mechanism. Loss power is an accounting difference, not a temperature estimate or electrical motor loss. There is no tooth geometry, stage count, backlash, elasticity, torque-speed curve, reverse efficiency, bearing load, thermal limit, life estimate or allowable torque. These bounds are teaching choices. Do not treat a calculated torque as a reducer's load rating or infer backdrivability from N and η.

References reviewed 2026-09-20: Oriental Motor on torque and load inertia through gearing and maxon gearhead selection. The browser uses generic scalar relations, without adopting any manufacturer's product specifications. Continue below for planetary, strain-wave and cycloidal mechanisms; motor FOC discusses how motor torque is controlled.