cyclogramFrom module and teeth to motor speed
The pinion rolls along the rack on its pitch circle, whose diameter is the module times the number of teeth. One turn of the pinion moves the carriage π·D; a gearbox in between divides that by its ratio.
D = m · z module × teeth k = π · D / i travel per motor turn n_pinion = v / (π · D) · 60 rpm n_motor = n_pinion · iModule 2 with 20 teeth makes a Ø40 mm pinion. One turn moves 125.7 mm, and the default move's 0.6 m/s turns it at 286 rpm.
Motor on the carriage, or the rack moving
A rack-and-pinion axis is built one of two ways, and they move different masses. With the motor on the carriage, the rack is bolted to the frame and the pinion runs along it: the motor, the gearbox and their bracket ride with the load, so their mass moves too — step 2 asks for it. With the motor on the frame, the pinion stays where it is and pushes the rack, and the rack moves with the load: its own mass is what moves too, estimated from its length and face width unless you type the catalogue figure.
motor on the carriage: m = m_load + m_motor motor on the frame: m = m_load + m_rack m_rack = 7850 · w² · LThe default 20 kg load with 3 kg riding on the pinion makes 23 kg. The same axis with a fixed motor moves a 400 mm rack of 1.26 kg instead: 21.3 kg, and less torque. In exchange, a moving rack must be the stroke plus the pinion's mesh long, and it sweeps that length out past both ends of the axis.
The inertia the motor sees
Everything that moves is a mass on a drum of radius D/2. The pinion adds its own inertia; left blank, the page estimates it as a solid steel disc of the pitch diameter and the face width.
J_mass = m · (D / 2)² / i² J_pinion = m_p · D² / 8 / i² m_p = 7850 · π/4 · D² · w J_load = J_mass + J_pinion + J_coupling + J_gearbox23 kg on a Ø40 mm pinion: J_mass = 23 · 0.02² = 9.20 × 10⁻³ kg·m². A steel pinion 20 mm wide weighs 0.197 kg and adds 3.95 × 10⁻⁵ kg·m². The motor sees 9.24 × 10⁻³ kg·m².
Torque, and the force at the teeth
The torques follow the same rule as a belt's, with the pitch radius as the lever and η the mesh's and the gearbox's efficiency together. The force at the teeth is what the carriage needs — gravity, friction, the external force and the mass times the acceleration — and it is what the rack's catalogue rates for the module and the face width.
T_static = (m·g·sin α + μ·m·g·cos α + F) · D / (2 · η · i) T_accel = T_static + (J_mass + J_pinion) · α / η + J_trans · α / η_t + J_rotor · α F_teeth = m·g·sin α + μ·m·g·cos α + F ± m · aFor the default move with the motor on the carriage, friction costs 0.501 N·m at the motor; accelerating takes 2.35 N·m, cruising 0.501 N·m, braking −1.00 N·m, and the RMS over the cycle is 1.06 N·m. The teeth carry 22.6 N of friction and 105 N at the peak. Type the rack's allowable force in step 2 and step 3 checks it.
The gearbox
Straight on the pinion, the 400 W servo drives 355 times its own rotor inertia — less than a belt drive's, for the smaller diameter, but far over the limit of 10. The reflected inertia falls with the square of a gearbox's ratio:
i = √( (J_load / J_rotor) / 10 ) = √( 355 / 10 ) ≈ 6.0The page suggests about 6.0 : 1. With it the ratio is 9.87 — just under the limit, so the check reads Close — and 6.3 : 1 brings it to 8.95. Through 6 : 1 the motor turns at 1719 rpm and the peak torque drops to 0.440 N·m. With the motor fixed and the rack moving, the lighter 21.3 kg gives a ratio of 329 direct.
Choosing the pinion
The force at the teeth does not depend on the pinion: it is the carriage's force. The pinion's size changes what that force costs at the motor — the torque rises with D, the reflected inertia with D², the speed falls with 1/D. So pick the module for the force, from the rack's catalogue, then the fewest teeth that still run smoothly: below about 17, a standard 20° pinion undercuts its own teeth.
Worked example
| Quantity | By hand | Result |
|---|---|---|
| Pitch diameter | 2 mm · 20 | 40 mm |
| Travel per turn | π · 40 mm | 125.7 mm |
| Pinion and motor speed | 0.6 / 0.1257 · 60 | 286 rpm |
| Moving mass | 20 kg load + 3 kg motor and gearbox | 23 kg |
| Moving mass at the motor | 23 · 0.02² | 9.20 × 10⁻³ kg·m² |
| Pinion | 0.197 · 0.04² / 8 | 3.95 × 10⁻⁵ kg·m² |
| Torque accelerating / cruising / braking | α = 3.6 / 0.02 = 180 rad/s² | 2.35 / 0.501 / −1.00 N·m |
| RMS torque over 1 s | √((2.35² + 0.501² + 1.00²) / 6) | 1.06 N·m |
| Force at the teeth at the peak | 23 · 3.6 + 22.56 | 105 N |
| Inertia ratio, direct | 9.24 × 10⁻³ / 2.6 × 10⁻⁵ | 355 |
Questions
Is the module the same as the pitch?
No. The module is the pitch diameter over the number of teeth, in millimetres; the pitch along the rack, tooth to tooth, is π times the module — 6.28 mm for module 2.
The motor rides on the carriage — does its mass count?
Yes. Choose “On the carriage” in step 2 and type what rides with the pinion — the motor, the gearbox and their bracket. The page adds it to the moving mass, to the friction and, on a climbing axis, to the weight the motor lifts.
Motor on the carriage or on the frame — which is better?
On the carriage, the rack can run the whole length of the machine and several carriages can share it, but the motor's cable travels with it and its mass moves. On the frame, the cable stays put and only the rack moves, but the rack must be the stroke plus the mesh long and sweeps out past both ends — usually the choice for shorter strokes.
How do I find the pinion's inertia?
From the pinion's catalogue, about its own axis. Left blank, the page takes a solid steel disc of the pitch diameter and the face width — close enough for sizing.
What is the allowable force at the teeth?
The largest tangential force the rack and pinion may carry, from the catalogue for the module, the face width, the quality and the hardening. Step 3 compares the force at the peak of the move with it.
Why does a smaller pinion help the motor?
The torque for the same carriage force falls with the diameter and the inertia the motor sees with its square. The motor turns faster in exchange, which a servo usually welcomes.
Sources
- Oriental Motor, Technical Reference, selection calculations — the rack-and-pinion load torque and inertia, the same lever as a belt's pulley.
- ISO 54 — the series of modules for cylindrical gears.
- The involute gear's undercut limit, z_min = 2 / sin²α, about 17 teeth at a 20° pressure angle.
Related guides
- Axis sizing calculatorThe whole guided flow: motion, mechanism, motor, regeneration, and the move as a bar.
- Belt drive motor sizingPulley and motor speed, reflected inertia, torque, belt force — and the gearbox a belt axis needs.
- Inertia ratioThe load's inertia at the motor through a screw, a belt or a gearbox, and the ratio that fixes it.
- Servo motor sizingPeak torque at speed, RMS over the cycle, maximum speed and inertia ratio, each with its margin.
- Move time and motion profilePeak speed, acceleration and jerk from a distance and a move time; trapezoid, triangle and S-curve.
Every term on this page, defined: the axis and motor sizing glossary.