Motor Acceleration Time Formulas

Motor acceleration time formulas: load inertia referred to the motor shaft, accelerating torque, run-up time and stored kinetic energy, with a 55 kW example.

The formulas

J is in kg·m², N in rpm, torque in N·m and i is the speed ratio of a gear or belt between the motor and the load. The torques are averages over the speed range, taken from the motor and load speed-torque curves. Use a segment-by-segment sum for a curve that changes a lot. The formula gives the time to reach the final speed with a constant net torque, and the motor current stays high during this time, which sets the heating in the motor.

Worked example: 55 kW at 1,480 rpm through a 5:1 drive

The motor inertia is 1.2 kg·m², the load inertia is 220 kg·m² on the slow shaft, the average motor torque is 150% of rated and the average load torque is 50% of rated.

1. Rated torque = 9549.3 × 55 / 1480 = 354.9 N·m and ω = 2π × 1480 / 60 = 155 rad/s
2. Jref = 220 / 5² = 8.8 kg·m², so J = 1.2 + 8.8 = 10 kg·m²
3. Ta = (1.5 − 0.5) × 354.9 = 354.9 N·m
4. t = 10 × 155 / 354.9 = 4.367 s
5. The kinetic energy at full speed is ½ × 10 × 155² = 120.1 kJ

Run-up time by total inertia (same torques)

Frequently asked questions

Why is the load inertia divided by the ratio squared?

Kinetic energy is the same on both shafts. A load turning slowly with inertia J has the same energy as a smaller inertia J / i² turning i times faster on the motor shaft.

Why does the acceleration time matter?

The motor draws a high starting current for the whole run-up, so a long time heats the windings. Compare the time with the locked-rotor withstand time on the data sheet.

Where do I find the load inertia?

In the data sheet of the fan, pump or machine, often as WK² or GD² in kg·m². Convert with J = GD² / 4.

What if the torque changes a lot with speed?

Split the speed range into steps, find the net torque in each, and add the times. A fan load has a torque that rises with the square of speed.

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