Peak torque
-RMS torque
-Motor speed
-Inertia ratio
-All results
| Reflected load inertia at motor | - |
| Total inertia at motor shaft | - |
| Motor angular acceleration | - |
| Torque during acceleration | - |
| Torque at constant speed | - |
| Torque during deceleration | - |
| Cycle time | - |
| Peak mechanical power | - |
Method and formulas
Standard rigid-body drive sizing at the motor shaft, so every figure can be checked by hand.
| Quantity | Formula |
|---|---|
| Motor speed | nm = nL i, ωm = 2π nm / 60 |
| Total inertia at motor | Jt = Jm + JL / i2 |
| Constant-speed torque | Tc = TL / (i η) |
| Acceleration torque | T1 = Jt ωm / ta + Tc |
| Deceleration torque | T3 = −Jt ωm / td + TL η / i |
| RMS torque | Trms = √((T12 ta + Tc2 tc + T32 td) / tcycle) |
| Peak power | P = T1 ωm |
| Inertia ratio | (JL / i2) / Jm |
US inputs are converted to metric internally: 1 lb·in² = 2.9264×10-4 kg·m², 1 lbf·in = 0.112985 N·m, 1 hp = 0.7457 kW. Speed is a linear ramp from zero to full speed and back, with a single constant-speed phase.
Worked example: 100 rpm load behind a 10:1 gearbox
With JL = 0.02 kg·m², TL = 1.5 N·m, η = 0.9, Jm = 0.00015 kg·m², ta = td = 0.2 s, tc = 1.0 s and a 0.6 s dwell: nm = 1000 rpm so ωm = 104.72 rad/s, Jt = 0.00015 + 0.02 / 100 = 0.00035 kg·m². Tc = 1.5 / 9 = 0.1667 N·m, T1 = 0.00035 × 104.72 / 0.2 + 0.1667 = 0.3499 N·m, T3 = -0.0483 N·m. The cycle is 2.0 s, so Trms = 0.1624 N·m, peak power is 36.6 W and the inertia ratio is 1.33 : 1.
Questions
How do you size a servo or motor for a move profile?
Reflect the load inertia and torque to the motor shaft through the gear ratio, add the motor rotor inertia, and compute the torque in each phase of the cycle: acceleration, constant speed and deceleration. The motor must supply the peak torque for the short acceleration and the RMS torque over the whole cycle without overheating.
What is reflected inertia?
A load inertia J_L behind a gearbox of ratio i (motor turns per load turn) appears at the motor as J_L / i squared. Gear ratio therefore has a very strong effect on how hard the motor must work to accelerate the load.
What inertia ratio should I aim for?
Common servo guidance is a reflected load to rotor inertia ratio of about 1:1 to 5:1 for good control, with higher ratios possible on low-precision or well-damped axes. Follow your motor and drive manufacturer's recommendation.
Why is RMS torque used?
Motor heating follows the square of current, and current is roughly proportional to torque, so heating over the cycle follows the time-weighted mean square of torque. The continuous (stall) torque rating of the motor must exceed the RMS torque, with margin, at the operating speed.
How does the calculator treat friction and gearbox efficiency?
The constant load torque T_L is reflected as T_L / (i eta) while driving the load and T_L eta / i while decelerating, because friction helps the motor slow the load. Efficiency is taken as constant; real gearboxes lose more efficiency at low speed and cold temperature.
What does this calculator not cover?
Motor torque-speed curves and voltage limits, thermal time constants, drive current limits, belt or screw compliance, vertical-axis gravity holding, and regeneration. It is a first-pass sizing check: confirm with the manufacturer's sizing tool.
Engineering aid for guidance. It covers rigid-body torque and heating only, not motor curves, voltage limits or compliance. Verify critical designs with the motor manufacturer's sizing tool and applicable standards.