What the transmission does
A motor is best at high speed and low torque, while most loads want low speed and high force, and many need straight-line motion. The transmission converts one to the other: it changes the speed-to-torque ratio, turns rotation into linear motion, and supports the load. It also adds friction, backlash, compliance and inertia, so it strongly affects accuracy and the tuning of the servo (Servo Systems and Tuning). The gear and belt geometry is covered in Gears and Belts and Pulleys; this page is about how they behave as part of a motion axis.
Ball screws and lead screws
A screw converts rotation to linear force. The lead is the linear travel per revolution. The force and torque are related by T = F × lead ÷ (2π × η), with η the efficiency. A ball screw uses recirculating balls and reaches 90 percent efficiency, low friction and high accuracy; a lead screw (Acme or trapezoidal thread) slides, is cheaper and quieter, and has a much lower efficiency.
| Property | Ball screw | Lead (Acme) screw |
|---|---|---|
| Efficiency | 85 to 95 percent | 20 to 60 percent, depending on lead and lubrication |
| Back-driving | Yes: a load can turn the screw, so a vertical axis needs a brake | Often self-locking at low leads |
| Accuracy and stiffness | High, can be preloaded to remove backlash | Lower; backlash unless an anti-backlash nut is used |
| Speed limit | Limited by critical (whirl) speed and the ball recirculation | Lower, limited by heat from friction |
| Cost, noise | Higher, some noise from the balls | Lower, quiet |
| Maintenance | Needs grease and protection from chips | Needs lubrication; wear changes backlash |
Worked example: screw torque and a lead-screw efficiency
- Ball screw load torque. For the axis in Motion Control Basics (5 kg, 2.67 m/s² acceleration, friction coefficient 0.01 on the guides): the accelerating force is 13.3 N plus 0.49 N of friction = 13.8 N. The torque at the screw is T = 13.8 × 0.010 ÷ (2π × 0.9) = 0.0244 N·m.
- Check against the inertia method. The load inertia seen through the screw gives the same accelerating torque: Jload × α = 1.27 × 10-5 × 1 676 = 0.0212 N·m, and Facc × lead ÷ 2π = 0.0212 N·m with η = 1: the two methods agree.
- Lead-screw efficiency follows from the thread angle λ and the friction angle φ: η = tan λ ÷ tan(λ + φ). For an 8 mm lead on a 9 mm mean diameter, λ = atan(8 ÷ (π × 9)) = 15.8°; with a friction coefficient of 0.15, φ = 8.5°, so η = 0.63 (63 percent). The screw is not self-locking because λ is larger than φ: it can be back-driven. A fine-lead screw of the same diameter would be self-locking but far less efficient.
- Resolution and speed. A 10 mm lead moves 10 mm per turn: 1 m/s needs 6 000 rpm, which is near the critical speed of a long, slender screw. A shorter lead and a slower motor, or a thicker screw, avoids it.
- Critical speed (the whirl speed of the unsupported length) and buckling load in compression limit long screws: use the manufacturer’s tables and a support bearing at the far end.
- Preload removes backlash but adds friction and heat; the drive torque must overcome it.
- Accuracy grade (C3 to C10 or ISO grades) gives lead error per 300 mm: a rolled C7 screw is cheaper and less accurate than a ground C3.
- Support the ends correctly: fixed-fixed gives the stiffest and fastest, fixed-free the cheapest.
Belts, chains and rack and pinion
| Type | Strengths | Weaknesses | Use |
|---|---|---|---|
| Toothed (synchronous) belt | Light, quiet, no backlash if tensioned, long strokes, cheap | Stretch and elasticity, lower stiffness, limited force | Printers, gantries, fast pick and place |
| Roller chain | High force, rugged, tolerant of dirt | Noise, wear and stretch, lubrication | Conveyors, heavy machines |
| Rack and pinion | Unlimited length, high speed, high stiffness | Backlash (dual-pinion preload), wear | Long gantries, machine tools |
| V-belts and flat belts | Cheap, damp shocks | Slip, so not for position | Fans, pumps, power transmission |
Worked example: a toothed-belt axis
A 5 mm pitch belt on a 20-tooth pulley moves 20 × 5 = 100 mm per revolution (a pulley diameter of 31.8 mm). With the motor on a 2:1 reduction the load moves 50 mm per motor turn, and the reflected inertia of a load mass m is m × (100 mm ÷ 2π ÷ ratio)². Belts stretch under load: a longer span is softer, and a low-stiffness belt limits the loop bandwidth (see Servo Systems and Tuning).
Gearboxes
| Gearbox | Efficiency | Backlash | Features |
|---|---|---|---|
| Planetary | 90 to 97 percent per stage | 1 to 15 arc-minutes | Compact, in-line, high torque density: the servo standard |
| Harmonic (strain wave) | 70 to 90 percent | Near zero | Huge ratios (50 to 160) in a thin package; robot joints |
| Cycloidal | 85 to 95 percent | Less than 1 arc-minute | Very rugged, high shock load |
| Worm | 30 to 90 percent | Variable | Right-angle, may be self-locking; low efficiency at high ratios |
| Spur or helical (parallel shaft) | 95 to 98 percent per stage | Moderate | Simple, cheap, parallel shafts |
| Bevel and hypoid | 90 to 97 percent | Moderate | Right-angle drives |
A gearbox of ratio N and efficiency η turns a motor torque T into an output torque T × N × η and divides the speed by N. With a 0.2 N·m motor and a 10:1 planetary of 95 percent efficiency the output torque is 0.2 × 10 × 0.95 = 1.90 N·m at one tenth of the speed. Use the Gear Ratio Calculator for gear trains.
Reflected inertia and matching
A rotating or moving load seems lighter to the motor through the transmission: the reflected inertia is the load inertia divided by the square of the ratio, Jreflected = Jload ÷ N². For a screw, the equivalent inertia of a mass m is m × (lead ÷ 2π)². A gearbox is therefore a very effective way of reducing the inertia ratio (see the target ratios in Servo Systems and Tuning).
Worked example: choosing a gear ratio
- A rotary load of 0.01 kg·m² is driven by a motor with a rotor of 2.00 × 10-5 kg·m². Direct drive has an inertia ratio of 500:1; with a 10:1 gearbox it is 5.0:1.
- The ratio that gives exactly 1:1 (the “matched” condition that maximizes power transfer for a given acceleration) is N = √(Jload ÷ Jmotor) = √500 = 22.4. In practice a ratio of 3:1 to 10:1 is accepted, because very high ratios need very high motor speeds and add the gearbox’s own inertia and friction.
- Check the motor speed at the output speed you need: 120 rpm at the load with a 10:1 gearbox needs 1 200 rpm at the motor, which is comfortable; with 50:1 it would need 6 000 rpm.
Backlash, stiffness and windup
- Backlash is the lost motion when the direction reverses, measured in arc-minutes (gearboxes) or mm (screws, belts). It causes position error on reversal, limit cycles in a servo and noise.
- Stiffness (torsional, N·m per rad, or axial, N per µm) decides how much the load lags the motor under force. A flexible link between motor and load creates a resonance and limits the loop gain.
- Windup is the twist in a shaft or belt under load; it is not backlash but also moves the load relative to the encoder when the encoder is on the motor.
- Reducing backlash: preloaded nuts and gearboxes, dual-pinion racks, zero-backlash couplings, direct drive, or load-side feedback (see Encoders, Resolvers and Feedback).
To translate: 3 arc-minutes of backlash on a gearbox output of radius 100 mm is 3 ÷ 60 × π ÷ 180 × 100 = 0.087 mm of lost motion at the rim, comfortable for a conveyor and not for a precision stage.
Couplings and guides
| Coupling | Properties | Use |
|---|---|---|
| Rigid | Stiff, no allowance for misalignment | Only with very accurate alignment |
| Bellows | Zero backlash, torsionally stiff, tolerates small misalignment | Precision servo axes |
| Disc / diaphragm | Stiff, zero backlash, higher torque | Servo and larger drives |
| Jaw (spider) | Damps vibration, small backlash, cheap | General-purpose, steppers |
| Oldham, flexible beam | Handles more misalignment, less stiff | Light-duty drives |
Linear guides support the load and carry the side and moment forces so that the screw sees only axial force. Profiled rail guides have rolling elements and low friction (coefficients of 0.002 to 0.01), allow high loads and are the standard; round-rail and plain bearings are simpler and cheaper but with more friction. Misalignment of the screw and guide causes binding, noise and early wear.
Choosing a transmission
| If the axis needs… | Use | Because |
|---|---|---|
| Accurate, stiff, short to medium linear stroke | Ball screw | Efficient, preloaded, precise |
| Low-cost linear motion at modest speed | Lead screw, or a stepper with a lead screw | Cheap, self-locking in some designs |
| Long, fast linear stroke | Toothed belt or rack and pinion | Length and speed are unlimited |
| Matching a high-inertia rotary load | Planetary gearbox | Reduces the inertia ratio by N² |
| A joint with no backlash | Harmonic or cycloidal gearbox, or direct drive | Zero or near-zero backlash |
| A vertical load that must not fall | A self-locking element plus a brake | See Braking, Safe Stopping and Troubleshooting |
Common faults
| Symptom | Likely cause | Check |
|---|---|---|
| Position error on reversal | Backlash in the nut, belt, coupling or gearbox | Measure with a dial indicator; adjust the preload; tighten the coupling |
| Vibration or squeal at certain speeds | Critical speed, resonance, a loose coupling | Check the speed against the screw limit; tighten; use a notch filter |
| Axis stalls or is rough | Misalignment, binding in guides, lack of lubrication | Free the screw from the motor and turn by hand |
| Overheating of the screw or nut | Too much preload, no lubrication, too high a speed | Lubricate, reduce the preload and the duty |
| Noisy gearbox | Worn gears, low oil, over-torque | Check the oil and the backlash |
| Belt jumps teeth | Low tension, overload, a small pulley | Tension, accelerate more gently, a wider belt |
| Slow loss of accuracy | Wear, thermal growth, chips in the screw | Seals and wipers, compensate for temperature, recalibrate |
Related
- Motion Control and Actuators: the overview of this guide.
- Sizing a Motion Axis: using these values to size a motor.
- Encoders, Resolvers and Feedback and Servo Systems and Tuning: feedback placement and tuning around the mechanics.
- Gears, Belts and Pulleys, and Fits and Tolerances for mounting.
- Motor Torque and Inertia Matching Calculator and Gear Ratio Calculator.
FAQ
Ball screw or belt?
Use a ball screw for accuracy, stiffness and force on short to medium strokes. Use a belt for long strokes and speed where a little stretch is acceptable. For a vertical load a ball screw needs a brake, and a belt needs one too.
What inertia ratio should I aim for?
Under about 10:1 for general machines and under 5:1 for dynamic ones. A gearbox is the main tool, since it divides the reflected inertia by N².
Does a gearbox always help?
It reduces inertia ratio and raises torque, but adds backlash, friction, compliance and cost, and limits the output speed. For very high accuracy consider direct drive.
How do I find backlash?
Move the axis in one direction, set a dial indicator to zero, reverse by a small amount and note the distance before the indicator moves. Do the same at the motor shaft to separate the gearbox from the screw.
References
- THK, NSK, Hiwin and Bosch Rexroth, Ball screw and linear guide technical catalogs (efficiency, critical speed, accuracy grades).
- Neugart, Apex and Harmonic Drive, Gearbox selection guides (backlash, efficiency, torsional stiffness).
- Slocum, A. H., Precision Machine Design, SME; Shigley, J. E. and Mischke, C. R., Mechanical Engineering Design.
Last reviewed: October 2026. Efficiency and backlash figures are typical; use the manufacturer’s data for the real component.