Motor sizing for automated pipetting
A pipetting head indexes in fractions of a second. Acceleration torque runs several times the friction torque, and the gearhead peak rating, not the continuous rating, decides the part.
- Wedge
- Medical and laboratory equipment
- Catalog reviewed
- 2026-08-18
- Configurations screened
- 32 published
A pipetting instrument has two quite different motion problems and they size differently. The plunger axis moves a piston through a stroke set by the aspirate volume, over about a second, against seal friction and a small pressure. The head positioning axes index between well columns in a fraction of a second and then have to be still enough to enter a well. This page is about the second one, because it is the one that is routinely sized wrong. The plunger axis is a low-speed screw drive and follows the same relations as an infusion pump drive.
For an indexing axis the torque is dominated by the inertia term. Reflect the translating mass to the drive shaft as J = m·r² for a belt of pitch radius r, add the pulley and belt inertia, and take the angular acceleration as α = a/r. Then T_peak = (J_load + J_drive)·α + T_friction. The friction term is µ·m·g·r for a horizontal axis. Because acceleration goes as 1/t² for a fixed move distance, halving the index time quadruples the acceleration torque while leaving the friction term untouched. That is why the ordering between the two terms inverts as the move gets short.
The gearhead peak torque rating, and the output speed the index demands.
The continuous torque rating is almost never the binding constraint on a pipetting axis, because the duty is a short move followed by a long aspirate or dispense dwell, and the RMS torque over that period is a small fraction of the peak. What eliminates candidates is that the peak acceleration torque exceeds the gearhead's rated output torque, or that the peak index speed exceeds the configuration's rated output speed. Settling is the third gate: lost motion in the geartrain appears as position error at the moment the head has to enter a well, and no amount of encoder resolution at the motor removes it.
Continuous, peak, starting and abnormal
Sizing on one of them is how a mechanism gets a part that works on the bench and fails in the field.
Failure modes and misleading specifications
What follows is what actually fails, and which published number sends people the wrong way.
| Failure mode | Physical cause | What it looks like in the field |
|---|---|---|
| Output stage tooth breakage after a crash | The axis stalled against an obstruction with no current limit and no compliance in the tool path. | One damaged unit, no wear pattern, and a maintenance record showing a deck collision. Often mistaken for a quality problem with the gearhead. |
| Tip entry errors that appear only at speed | Settling. The axis reaches its commanded position and the structure and geartrain are still ringing when the Z axis starts down. | Errors that disappear when the throughput is reduced, and that track with tip length rather than with the motion axis itself. |
| Position drift over a plate in one direction only | Lost motion taken up in one direction and not the other, combined with an encoder mounted on the motor shaft rather than the load. | Column 1 to column 12 is repeatable, column 12 back to column 1 is offset by a fixed amount roughly equal to the backlash referred to the axis. |
| Belt tooth jump under acceleration | Peak acceleration torque exceeds what the belt tension and wrap can carry, usually because the motor was sized on RMS torque and the peak was never computed. | Sudden whole-tooth position offsets, always during acceleration, always in the same direction. |
| Specification | Why it misleads in this application |
|---|---|
| Continuous output torque | On an indexing axis the RMS torque is often under a fifth of the peak. Sizing on continuous torque selects a part that cannot survive the acceleration it will be asked for. |
| Encoder resolution | Counts at the motor divided by the ratio give a very fine number at the output. It is a display resolution. Repeatability is set by lost motion and structural settling, which are typically one to two orders of magnitude coarser. |
| No-load output speed | The peak index speed is what has to fit, and a trapezoidal profile peaks at 1.5 times the mean speed of the move. Comparing the mean speed against the no-load speed passes candidates that cannot make the move. |
| Rated voltage | The acceleration phase needs current, and current needs voltage headroom above the back-EMF at the peak speed. A configuration rated at the supply voltage has no headroom left to accelerate at. |
Three decisions carry most of the risk, and each one costs something real.
Spur or planetary reduction
Brushed coreless or brushless
Where the feedback goes
Worked example
Size the column index axis of an eight-channel head: a belt-driven carriage that steps one well pitch and settles.
- Index distance
- 9 mm, one well pitch
- Index time
- 150 ms
- Moving mass
- 1.2 kg
- Pulley pitch radius
- 12 mm
- Period
- 500 ms, index plus dwell
| Step | Expression | Result |
|---|---|---|
| Peak carriage velocity | v = 0.009 m ÷ (0.150 s × 2/3) | 0.090 m/s |
| Acceleration | a = 0.090 m/s ÷ 0.050 s | 1.80 m/s² |
| Peak drive speed | n = 0.090 m/s ÷ 0.012 m = 7.5 rad/s | 71.6 rpm |
| Reflected load inertia | J = 1.2 kg × (0.012 m)² | 1.728 × 10⁻⁴ kg·m² |
| Total drive-shaft inertia | J = 1.728 × 10⁻⁴ + 0.300 × 10⁻⁴ | 2.028 × 10⁻⁴ kg·m² |
| Angular acceleration | α = 1.80 m/s² ÷ 0.012 m | 150 rad/s² |
| Acceleration torque | T = 2.028 × 10⁻⁴ kg·m² × 150 rad/s² | 30.4 mN·m |
| Friction torque | T = 0.05 × 1.2 kg × 9.81 m/s² × 0.012 m | 7.06 mN·m |
| Peak torque, and the ratio that matters | 30.4 + 7.06; then 30.4 ÷ 7.06 | 37.5 mN·m peak, acceleration 4.3 times friction |
| RMS torque over the period | √[(37.5² + 7.06² + 23.4²) × 0.05 ÷ 0.50] | 14.1 mN·m |
| Reflected inertia at the motor, 77.66:1 | 2.028 × 10⁻⁴ ÷ 77.66² | 3.36 × 10⁻⁸ kg·m² |
Result
MM-C1329-P00777-120C resolves it. It is a 13 mm coreless brushed configuration, 77.66:1 planetary, 12 V, rated 123.6 rpm at the output against the 71.6 rpm peak demand, 294 mN·m gearhead rating against the 37.5 mN·m peak, and 113.4 mN·m continuous against the 14.1 mN·m RMS. Note which check did the work. MM-C1226-S01002-120A has more than enough speed at 100.8 rpm rated and more than enough continuous torque at 9.3 mN·m against a 14.1 mN·m RMS demand, and it is eliminated because its 24.5 mN·m gearhead rating is below the 37.5 mN·m peak. So is MM-C1219-S00720-030B, and so is MM-C1626-S00311-060B at 29.4 mN·m. Every published spur configuration fails this axis on peak torque while passing on speed and on continuous torque.
What would change it
Index time is the strongest lever, because acceleration torque goes as 1/t². Relaxing 150 ms to 200 ms cuts the acceleration torque by 44 percent and brings the spur candidates back into range. Reducing the pulley radius reduces the reflected inertia with the square of the radius but raises the required speed in proportion, so it trades a torque problem for a speed problem. Reducing the moving mass helps linearly. Rotor inertia is the missing input: if it is a large fraction of the reflected load inertia, the optimum ratio shifts and the motor-side torque rises. Ask for it before committing.
A short qualification plan
Each step names the measurement, not the intention.
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Confirm the profile the controller actually runs
Recorded command and encoder trace for a real index, not the nominal profile. S-curve limits and controller jerk shaping change the peak acceleration.
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Measure the peak torque
Current trace through the acceleration phase on the assembled axis, converted through the torque constant, compared against the calculated peak.
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Measure settling at the tool point
Displacement or vision measurement at the tip, not encoder counts at the motor, for the first 100 ms after the index completes.
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Characterize lost motion
Bidirectional positioning error over the full plate, reported as a reversal offset. Repeat after the run-in the geartrain will see.
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Prove the crash case
Deliberate obstruction at full index speed with the current limit active, then teardown. Confirm the output stage saw no more than the gearhead rating.
Everything on this page connects to something you can calculate, buy or read.
Send the axis for an application review
Send the move distance, move time, moving mass and drive geometry. You get back the peak and RMS torque, the candidates that pass each gate, and the ones that fail and why.