Power transmission calculator
Rack and Pinion Calculator
Calculate rack and pinion travel per revolution, linear speed, force from torque, and pinion center distance from module or diametral pitch.
Design a laser-cut gear rack
Set the pitch, rack length, and pinion, then place the mounting holes and choose a bore and plate.
A row of holes evenly spaced along the back of the rack.
Pinion bore, holes, and plate
Rack dimensions
- Linear pitch
- 6.283 mm
- Rack length Ends stop mid-gap so racks butt together on pitch
- 125.664 mm
- Overall rack height Back edge to tooth tips
- 17 mm
- Travel per pinion turn
- 125.664 mm
- Pinion pitch diameter
- 40 mm
- Pinion outside diameter
- 44 mm
- Pinion center to rack pitch line
- 20 mm
- Pinion center to rack back
- 35 mm
- Contact ratio
- 1.769
Mesh preview
The rack has straight flanks at the pressure angle and rounded roots. The pinion's involute teeth are generated by rolling that same rack profile through each tooth space. The dashed circle is the pinion's pitch circle, which rolls along the rack's pitch line.
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125.7 mm × 17.0 mm
⌀ 43.6 mm
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Quick answer
How this rack and pinion calculator works
A rack and pinion converts rotation to travel of π·m·z per revolution and force F = 2T ÷ d, where d = m·z is the pinion pitch diameter. A module 2, 20-tooth pinion moves the rack 125.66 mm per turn, and 5 N·m of torque pushes it with about 250 N. The calculator also gives pinion center position and generates both parts.
How to use the calculator
- 1Enter module or diametral pitch and pressure angle.
- 2Enter the pinion tooth count and, optionally, pinion torque.
- 3Read travel per revolution, linear pitch, pinion center distance, and linear force.
- 4Set the rack length in teeth and generate the rack and pinion for download or ordering.
Key formulas
Rack and pinion formulas
- Travel per revolution
- s = π·m·z = π·d
- Linear speed
- v = π·d·n
- n in revolutions per second gives v per second.
- Linear force
- F = 2T ÷ d
- T is pinion torque; d is in the same length unit.
- Linear pitch
- p = π·m
- Pinion center to pitch line
- h = m·z/2
Worked example
Module 2 pinion with 5 N·m
- Inputs
- m = 2 mm, 20-tooth pinion (d = 40 mm), 5 N·m torque, 60 rpm.
- Result
- Travel 125.66 mm per revolution, linear speed 125.66 mm/s, force ≈ 250 N.
Travel = π × 40 mm. At 60 rpm (1 rev/s) that is 125.66 mm/s. F = 2 × 5 N·m ÷ 0.040 m = 250 N before losses.
Common questions
What to know before using the result
- How do you calculate rack and pinion travel?
- Multiply the pinion’s pitch diameter by π: s = π·m·z. A 20-tooth module 2 pinion travels 125.66 mm per revolution; a 15-tooth module 1 pinion travels 47.12 mm.
- How do you calculate rack and pinion force?
- F = 2T ÷ d. A 40 mm pinion with 5 N·m gives 250 N; a 1.000 in 20 DP pinion with 10 lbf·in gives 20 lbf.
- How do you get linear speed from rpm?
- Linear speed is travel per revolution times revolutions per unit time. 125.66 mm per revolution at 60 rpm is 125.66 mm/s, or about 7.5 m/min.
- What pinion size should I choose?
- Smaller pinions give more force and finer motion per turn; larger ones give more speed. Stay at about 17 teeth or more at 20° unless you use profile shift.
- How do I set the pinion height above the rack?
- Put the pinion center d/2 above the rack pitch line, where the pitch line is one module below the rack tooth tips. For a 20-tooth module 2 pinion that is 20 mm.
Reference
Rack travel per pinion revolution (mm)
Travel = π·m·z. Linear pitch p = π·m is the rack tooth spacing.
| Module | Linear pitch | 12-tooth pinion | 15-tooth | 20-tooth | 30-tooth |
|---|---|---|---|---|---|
| 0.5 | 1.571 | 18.85 | 23.56 | 31.42 | 47.12 |
| 1 | 3.142 | 37.70 | 47.12 | 62.83 | 94.25 |
| 1.25 | 3.927 | 47.12 | 58.90 | 78.54 | 117.81 |
| 1.5 | 4.712 | 56.55 | 70.69 | 94.25 | 141.37 |
| 2 | 6.283 | 75.40 | 94.25 | 125.66 | 188.50 |
| 2.5 | 7.854 | 94.25 | 117.81 | 157.08 | 235.62 |
| 3 | 9.425 | 113.10 | 141.37 | 188.50 | 282.74 |
| 4 | 12.566 | 150.80 | 188.50 | 251.33 | 376.99 |
| 5 | 15.708 | 188.50 | 235.62 | 314.16 | 471.24 |
| 6 | 18.850 | 226.19 | 282.74 | 376.99 | 565.49 |
Formula
p = π·m · L = n·π·m · travel per turn = π·m·z
Linear pitch p is π times the module, so a rack of n teeth is n·π·m long. A pinion of z teeth moves the rack π·m·z per revolution and pushes it with F = 2T ÷ d.
Assumptions and limits
- Rack teeth use straight flanks at the pressure angle, with addendum 1.0m and dedendum 1.25m.
- The pinion center sits d/2 from the rack pitch line with no profile shift.
- Linear force is ideal and excludes friction and mesh losses.
- Backlash is applied by thinning teeth by the entered allowance.
- Laser-cut racks suit low to moderate speed and load; critical drives need engineering review.
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