KEVOS® MAKEOS · Mechanics
Rack and Pinion Axis Sizing
What pinion torque, motor torque and gear ratio does this rack-and-pinion axis need, and are the teeth strong enough?
Sizing of a rack-and-pinion linear axis (gantry, transfer or vertical axis): forces from friction, gravity on an incline, external load and acceleration, pinion pitch diameter, torque and speed, reducer ratio for the motor speed, motor torque including rotor inertia, and the Lewis tooth bending stress with the velocity factor (as the Lewis bending tool) against an allowable.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Load and motion | ||
| Moving mass | kg | Mass of the carriage and load. |
| Guide friction | Friction coefficient of the guides. | |
| External force | N | Process force opposing motion. |
| Axis inclination from horizontal | ° | Angle of the axis from horizontal (90° for a vertical axis; no counterbalance assumed). |
| Maximum speed | m/s | Top speed. |
| Acceleration time | s | Time to reach top speed. |
| Pinion and drive | ||
| Module | mm | Pinion and rack module. |
| Pinion teeth | Teeth on the pinion. | |
| Face width | mm | Face width in mesh. |
| Lewis form factor | Lewis form factor for the pinion tooth count. | |
| Tooth quality | choice | Manufacturing quality, for the Barth velocity factor. |
| Allowable bending stress | MPa | Allowable bending stress for the rack and pinion material. |
| Rack and pinion efficiency | % | Efficiency of the rack and pinion mesh. |
| Reducer ratio | Reducer ratio between motor and pinion. | |
| Reducer efficiency | % | Reducer efficiency. |
| Motor inertia | kg·m² | Motor rotor inertia. |
Method and formulas
- Running force = m g (sin α + μ cos α) + F_ext; accelerating force = running + m a.
- Pinion d = m z; pinion torque = F d ÷ 2 ÷ η_rp; pinion speed = v ÷ π d (as the rack and pinion tool).
- Motor torque = pinion torque ÷ (i η_g) + J_m α_motor; motor speed = pinion speed × i.
- Lewis: σ = K_v F ÷ (b m Y) with the Barth factor for the tooth quality (as the Lewis bending tool).
- Example: 300 kg at 1.5 m/s with a module 3, 20-tooth pinion needs 9.3467 N·m peak at the motor.
Limitations
- Vertical axes need a counterbalance or brake check beyond this; gravity is included only as a running force.
- Lewis is a first check; use ISO 6336 for final rack ratings.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size motor, reducer and pinion for a long-travel axis
- Check rack teeth for a heavier carriage
- Plan a vertical rack drive with counterbalance
Who it is for
- Machine designers
- Automation engineers
- Design engineers
What you get
- Axial forces running and accelerating
- Pinion and motor torque and speed
- Tooth bending stress
Interpreting the result
- A larger pinion lowers tooth load but needs more motor torque or a higher ratio.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2140, 2219. Topic reference; no source tables or recommended settings are reproduced.
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