Motor Sizing for Machine Builders: Torque, Inertia, and the Duty Cycle
Motor Sizing for Machine Builders: Torque, Inertia, and the Duty Cycle
Motor sizing looks like a catalog exercise, and that is exactly why so many machines end up with a motor that runs hot, a drive that trips, or an axis that cannot hold position. The catalog gives the rated torque, but the application gives the real demand. This article covers the practical motor sizing logic for machine builders, from the torque calculation to the inertia match.
The Torque Profile Is the Design
The motor does not need one torque; it needs a torque over time. The sizing starts with the torque profile of the motion cycle, not with the peak number.
Build the profile from the motion segments:
| Segment | Torque (Nm) | Time (s) |
|—|—|—|
| Accelerate | 8.5 | 0.4 |
| Constant speed | 3.2 | 1.2 |
| Decelerate | -6.1 | 0.4 |
| Dwell | 0.5 | 0.8 |
The RMS torque over the cycle is the number that matters for heating. The motor rating is a thermal rating, and a motor that peaks at 150% of rated torque for a fraction of the cycle is fine, as long as the RMS stays under the rated continuous torque.
The RMS torque calculation is the root mean square of the torque segments weighted by their durations. It is a spreadsheet formula, not a thesis, and every motor catalog expects it.
The Torque Components
The total torque on the motor has four parts, and each one is calculated separately:
Acceleration torque: inertia times angular acceleration. This dominates on fast cycles.
Load torque: the steady force from the process, gravity, friction, and cutting.
Friction torque: guideways, seals, bearings, and the screw nut.
Gravity torque: for vertical axes, the load weight times the lever arm.
The designer who lists all four before picking the motor will rarely undersize. The designer who picks the motor from the load torque alone will have a machine that stalls on acceleration.
Inertia Matching: The Ratio That Keeps Servo Systems Stable
Servo systems work best when the load inertia reflected to the motor is within a range of the motor inertia. The industry rule of thumb is a load-to-motor inertia ratio below 10 for standard servos and below 3 for high-performance positioning.
The inertia ratio affects the tuning: a high ratio makes the system harder to stabilize, and the gain must drop, which makes the axis feel soft. The practical fix for a high ratio is a gearbox or a larger motor, not more tuning effort.
The reflected inertia of a ball screw drive is roughly the screw inertia plus the slide mass translated through the screw lead. The designer can calculate it from the screw diameter, the lead, and the moving mass, and compare it to the motor rotor inertia from the catalog.
The Gearbox Decision
A gearbox changes the reflected inertia by the square of the ratio and multiplies the torque at the load. It is the classic solution for the mismatch between a fast, low-torque motor and a slow, high-torque load.
The practical guidance:
• Use a gearbox when the required load speed is much lower than the motor’s efficient speed.
• Use a gearbox when the inertia ratio is too high for stable tuning.
• Use a gearbox with backlash low enough for the positioning accuracy.
• Avoid a gearbox when the motor can be direct-driven with an acceptable inertia ratio.
The backlash of the gearbox adds to the positioning error. For precision axes, a low-backlash planetary gearbox or a direct drive is the choice. For feed axes with moderate accuracy, a standard gearbox is fine.
Duty Cycle and Thermal Reality
The motor datasheet gives the rated torque at a rated duty, usually continuous at a specific ambient temperature. The real machine runs a cycle with peaks and dwells, and the thermal behavior depends on the RMS torque and the cooling.
The practical checks:
• The RMS torque should stay under the continuous rated torque with a safety margin of 20 to 30%.
• The peak torque should stay under the peak rating, which is typically 2 to 3 times the continuous rating.
• The motor’s thermal time constant matters: a short high-torque burst does not heat the motor if the dwell is long enough to cool it.
• The drive must supply the peak current, and the cable and connector must carry it.
A motor that runs warm but not hot in the prototype is a motor that will run hot in the production machine with a longer duty cycle. Test at the worst-case duty, not the average.
Vertical Axes: Gravity Changes Everything
Vertical axes add gravity torque that never goes away. The motor holds the load at rest, accelerates it against gravity, and decelerates it with gravity helping.
The critical numbers for a vertical axis:
• The holding torque at rest: the load torque from gravity, which the motor or the brake must hold continuously.
• The acceleration torque: the load inertia times the acceleration plus the gravity load.
• The deceleration torque: gravity subtracts from the braking demand, but the motor still must control the descent.
The brake is the other half of a vertical axis. The motor brake must hold the load at rest with the power off, and the brake torque rating should exceed the gravity torque with a safety factor of at least 1.5. A vertical axis without a brake is a falling hazard.
The Drive and the Power Supply
The motor and the drive are a pair, and the drive’s current capability and the power supply’s headroom are part of the sizing.
The practical rules:
• Size the drive from the peak current, not the RMS current, because the peak happens during acceleration.
• Check the bus voltage sag during the peak, especially on a shared power supply.
• Use regenerative resistors or a shared bus when the deceleration energy is high.
• Verify the cable length limit for the drive-motor pair; long cables need filters or a different drive.
The drive that trips on overcurrent during acceleration is a drive that was sized from the continuous torque, not the peak. The fix is a bigger drive or a softer acceleration ramp.
Verification: The Prototype Test
The final check on any motor sizing is the prototype test. The calculated numbers get verified with a current probe and a thermocouple.
The test checklist:
• Measure the actual RMS current over the full cycle and compare to the drive rating.
• Measure the motor temperature at the winding and the housing after the worst-case cycle.
• Verify the axis reaches the required speed and position within the cycle time.
• Check the settling time and the stability at the highest gain.
• Test the brake hold on the vertical axis with the power off.
A motor sizing that passes the calculation but fails the prototype is a motor sizing that missed something in the duty cycle. The prototype test is the place to catch it, not the first production run.
The Motion Profile Comes First
The torque profile is built from the motion profile, and the motion profile is the design’s answer to the cycle time requirement. Before any torque number, the designer should write down the move segments: the distance, the speed, and the time for each.
The motion profile choices:
Trapezoidal: constant acceleration, constant speed, constant deceleration. Simple, hard on the mechanics at the transitions.
S-curve: the acceleration ramps up and down. Smoother, less shock, longer cycle time for the same average speed.
Triangular: no constant-speed phase. The fastest for short moves, but the acceleration peaks are the highest.
The trapezoidal profile is the default for most machines because it is simple and the jerk at the corners is acceptable for moderate speeds. The S-curve is the choice for high-speed, high-precision axes and for heavy loads where the jerk would shake the structure.
The acceleration value comes from the move distance and the time, not from a guess. The formula for a trapezoidal move is standard, and the designer who derives the acceleration from the profile will size the motor from a real number.
The Reflected Load: From the Slide to the Motor
The load at the motor is not the load at the slide; it is the load transformed through the drive mechanism. The designer has to reflect the slide mass, the screw inertia, and the process force back to the motor shaft.
For a ball screw drive:
• The slide mass reflects as inertia through the screw lead: mass times the square of (lead divided by 2 pi).
• The screw inertia is calculated from its diameter, length, and material density.
• The process force reflects as torque: force times (lead divided by 2 pi) divided by the efficiency.
• The efficiency of a ball screw is high, around 90%, but it is not 100%, and the loss shows up as heat.
The reflected numbers are what the motor catalog compares against. The designer who reflects correctly will pick the motor from the right column.
Holding and Braking Torque
Some axes must hold position with the power on, and some must hold with the power off. The holding torque requirement decides the motor’s continuous rating or the brake’s rating.
For horizontal axes:
• The holding torque is usually small, just enough to counter the friction and the drift.
• The servo motor’s continuous torque normally covers it with margin.
For vertical axes:
• The holding torque is the gravity torque, and it is sustained, so it counts against the continuous rating.
• The brake must hold the full gravity load with the power off, with a safety factor.
The common mistake is sizing the brake from the static load and forgetting the dynamic case: the brake may be applied while the axis is still moving, and the stopping energy heats the brake. The brake duty cycle matters for frequent stops.
Cooling and Mounting Conditions
The motor datasheet rating assumes a mounting condition and an ambient temperature. A motor mounted on a warm machine in a warm shop derates from the catalog number.
The practical derating factors:
• Mounting on a heat sink or a cooled plate: full rating.
• Mounting in free air: slight derating.
• Mounting inside an enclosure: significant derating, because the motor heats the enclosure and the enclosure heats the motor.
• Ambient above 40 degrees: derate by roughly 2% per degree above the standard.
The motor thermal protection is a trip, not a design point. A motor that runs at the thermal limit is a motor that will trip on a hot day, and the machine will stop at the worst time.
The Control Loop and the Tuning
The motor sizing is not complete without the control loop. A motor that is sized correctly but tuned poorly will overheat, vibrate, or drift, and the fix is not a bigger motor.
The tuning parameters that matter:
• The loop gains: proportional, integral, and derivative, set from the inertia ratio and the mechanical stiffness.
• The feedforward: compensates the tracking error at constant speed.
• The notch filter: removes a mechanical resonance from the loop.
• The velocity and acceleration limits: protect the mechanics and the motor from the command demands.
The inertia ratio drives the tuning difficulty. A machine with a ratio of 2 tunes easily; a ratio of 20 fights the operator. The designer who fixes the ratio at the sizing stage saves the commissioning engineer weeks.
The Sizing Sheet as the Handover Document
The motor sizing work should end in a sheet that the next engineer can read, and the sheet is the handover document for the build, the commission, and the service.
The sizing sheet contents:
The motion profile: the segments, the speeds, and the times.
The torque profile: the acceleration, the load, the friction, and the gravity components.
The RMS torque and the peak torque, with the motor ratings compared.
The inertia ratio and the gearbox decision.
The drive, the cable, and the power supply selection.
The brake selection for the vertical axes.
The tuning parameters and the expected gains.
The prototype test results and the final verification.
The sizing sheet is the record that answers the question “why this motor” months later, at the commissioning stage or at the first field failure. The sheet that is written at the sizing stage is worth more than the memory of the engineer who left.
Conclusion
Motor sizing is not a catalog pick; it is a torque profile calculation with a thermal check and an inertia match. The RMS torque, the peak torque, the inertia ratio, the gearbox, the vertical axis gravity, the drive, and the prototype test all work together.
Build the torque profile, calculate the components, check the inertia, decide on the gearbox, respect the thermal limits, handle the gravity load, size the drive, and verify on the prototype. The machine will run cool, hold position, and last longer than the motor that was picked from the “looks about right” column.
