BIZOMAT MAKEOS · Gears
Rack and Pinion Speed, Force and Travel
What linear speed, force and travel does this rack and pinion give?
Pitch diameter, travel per revolution, linear speed and thrust force of a rack and pinion from module (or diametral pitch), pinion teeth, speed and torque, with the motor speed for a required linear speed.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Module | mm | Module of the rack and pinion (25.4 ÷ diametral pitch for inch gears). |
| Pinion teeth | teeth | Teeth on the pinion. |
| Pinion speed | rpm | Pinion speed. |
| Pinion torque | N·m | Torque at the pinion. |
| Mesh efficiency | % | Mesh efficiency. |
| Required linear speed (0 = none) | m/min | Linear speed you need, to find the pinion speed. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Module
- What it is
- The module of the rack and pinion teeth: the pitch diameter divided by the number of teeth, which sets the tooth size. Both parts must have the same module.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 50 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 2 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Travel per revolution rises in proportion to the module; thrust for a given torque falls in proportion.
- Get it right
- For inch gears, module = 25.4 ÷ diametral pitch.
What this means: Pinion teeth
- What it is
- The number of teeth on the pinion that drives along the rack.
- Units and format
- Enter a number in teeth: a whole number of teeth. The form accepts 6 teeth to 500 teeth. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 teeth as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Travel per revolution rises in proportion; thrust for a given torque falls in proportion.
What this means: Pinion speed
- What it is
- The rotational speed of the pinion shaft, after any gearbox between the motor and the pinion.
- Units and format
- Enter a number in rpm: revolutions per minute. The form accepts 0 rpm to 100000 rpm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 300 rpm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Linear speed rises in direct proportion to pinion speed.
- Get it right
- Use the pinion speed, not the motor speed, if there is a reducer.
What this means: Pinion torque
- What it is
- The torque delivered to the pinion shaft, after any gearbox, which the teeth turn into a straight-line push on the rack.
- Units and format
- Enter a number in N·m: newton-metres. The form accepts 0 N·m to 10000000 N·m. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 20 N·m as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Thrust rises in direct proportion to torque.
- Get it right
- Use the torque at the pinion, not at the motor, if there is a reducer.
What this means: Mesh efficiency
- What it is
- The share of the pinion’s input power that reaches the rack after friction losses in the tooth mesh, from the gear maker’s data.
- Units and format
- Enter a number in %: percent: the number is read as a percentage, so type 8 for 8 %. The form accepts 1% to 100%. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 97% as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Thrust rises in direct proportion to efficiency.
- Get it right
- This covers the mesh only; guide friction and gearbox losses are separate.
What this means: Required linear speed (0 = none)
- What it is
- The rack travel speed you need, used to work out the pinion speed that gives it. Zero skips this result.
- Units and format
- Enter a number in m/min: metres per minute. The form accepts 0 m/min to 10000 m/min. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 30 m/min as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- The pinion speed needed rises in direct proportion.
- Get it right
- Enter metres per minute; a value in mm/s must be converted (1 m/min ≈ 16.7 mm/s).
Method and formulas
- Pitch diameter d = m z; travel per revolution = π m z.
- Linear speed = travel per revolution × rpm.
- Thrust = 2T ÷ d × efficiency.
- Example: module 2, 20 teeth at 300 rpm moves 37.699 m/min with 970 N thrust.
Limitations
- Mesh efficiency and torque only; guide friction and acceleration forces add to the drive load.
- Check rack tooth strength with the gear tooth tools.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Size a rack drive for a gantry or slide
- Find motor speed for a traverse speed
- Find the pinion torque for a thrust
Who it is for
- Automation engineers
- Machine designers
What you get
- Pitch diameter and travel per revolution
- Linear speed
- Thrust force
- Pinion speed for the required linear speed
Interpreting the result
- A smaller pinion gives more thrust per unit torque but higher speed for the same travel speed.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 2131–2160. Topic reference; no source tables or recommended settings are reproduced.
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