Motor Torque & Inertia Matching Calculator

How it works

A motor sees two loads: the torque to overcome friction or gravity, and the torque to speed up its own rotor and the load. A gearbox divides the load’s inertia by the square of the ratio, and the load’s torque by the ratio (and by efficiency).

T₁ = T_ref + T_acc is the torque while speeding up, T₂ = T_ref at constant speed, and T₃ = T_ref − T_acc while slowing down. Inertias are in kg·m² inside the formulas (1 kg·cm² = 0.0001 kg·m²). The best gear ratio for transferring power to an inertial load is N = √(J_L ÷ J_M), which makes the reflected load inertia equal to the motor’s.

Worked example

A load of 0.40 kg·cm² and a 0.05 kg·cm² servo rotor connected through a 5:1 gearbox (90% efficient). The load must run at 300 rpm with a 0.5 N·m friction torque, accelerating in 0.2 s, running 1.0 s, decelerating in 0.2 s and resting 0.5 s. The reflected inertia is 0.40 ÷ 25 = 0.016 kg·cm², so the inertia ratio is 0.32 : 1, which is a very good match. The motor turns 1 500 rpm, so α = 785.4 rad/s² and T_acc = (0.05 + 0.016) × 10⁻⁴ × 785.4 = 0.0052 N·m. The load torque at the motor is 0.5 ÷ (5 × 0.9) = 0.111 N·m, so the peak torque is 0.116 N·m and the RMS torque is about 0.095 N·m. Any motor with a continuous torque above that, and a peak above 0.12 N·m, will do the job.

Things to know

Inertia ratio guidelines. A ratio near 1:1 is best, up to about 3:1 for fast, precise moves, and up to about 10:1 for general positioning on a well-tuned servo. Beyond that, the motor has little control over the load and the system can oscillate. Always check your drive maker’s limit.

A gearbox is the cheap fix. Because the reflected inertia falls with the square of the ratio, even a 3:1 or 5:1 reducer can bring a hard-to-control load into a good range. The motor must then turn faster, so check its maximum speed.

Include everything on the shaft. Couplings, pulleys, lead screws and the moving mass all add inertia. For a mass m moved by a lead screw of lead p, the inertia at the screw is m × (p ÷ 2π)². For a solid cylinder, J = ½ m r².

Look at RMS, not just the peak. A motor overheats from the average heating of the whole cycle. The peak torque only has to be inside the motor’s peak rating, but the RMS torque has to be inside the continuous rating.

Leave margin. Choose a motor with at least 20–30% spare torque, and allow for gearbox backlash, friction that varies with temperature, and supply voltage dropping under load.

Gravity loads. For a vertical axis, the load torque holds even when stopped, and the motor needs a brake if power is lost. Add the holding torque to the dwell phase.

Stepper motors follow different rules: their torque falls with speed and the usable ratio depends on the speed curve. Use the maker’s curves for steppers.

Common questions

What is an inertia ratio, and why does it matter?

It is the load’s moment of inertia, reflected to the motor shaft, divided by the motor rotor’s inertia. If the load is much heavier than the rotor, the motor cannot accelerate it quickly or hold it steady against disturbances, and the control loop becomes hard to tune.

How do I find the load inertia?

Add up the inertia of every rotating part, plus moving masses converted to the motor shaft: a disc is ½ m r², a point mass at radius r is m r², and a mass on a lead screw is m × (lead ÷ 2π)². CAD programs also report it.

Does this work for direct drive?

Yes. Enter a gear ratio of 1 and the efficiency as 100%.

Why does gearbox efficiency only affect the load torque?

The inertia terms are used for the speeding-up torque, which is a smaller, short-lived effect. The efficiency applies mainly to the torque needed to drive the load. The tool applies it in every phase for a simple, conservative result.

Sources

• Manufacturer sizing guides for servo motors (inertia matching and RMS torque), for example from Kollmorgen, Yaskawa and Siemens.
• Shigley’s Mechanical Engineering Design, moment of inertia and gear train relations.

Last reviewed: November 2, 2026