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CALCULATORS / GEARS AND GEAR TRAINS

Rack-and-pinion travel and force calculator

Ideal rack and pinion, neglecting tooth friction

gears-rack-pinion.napkin / editable
✓ checks pass
RACK AND PINIONTravel 20 mmRack force 100 N · torque 2 N·mPinion r 20 mm · rotation 1 rad

Dashed circles mark pitch geometry. Tooth outlines are schematic, not manufacturing profiles. Labels come from the evaluated .napkin model.

GEARS / TOOTH SCHEMATICSAME SOURCE

Change an input to see the consequences.

force budget100.00 N✓ passes
whole pinionTeeth0.00 ✓ passes
positive pinionTeeth20.00 ✓ passes

BASELINE INPUTS

CALCULATED

rackForce100 N
rackTravel20 mm
module2 mm

Edits change the baseline source.

Calculated from your source · not saved
Baseline · Baseline
force budget100.00 N ✓ passes
whole pinionTeeth0.00 ✓ passes
positive pinionTeeth20.00 ✓ passes
smaller pinion · smaller pinion
force budget166.67 N ✓ passes
whole pinionTeeth0.00 ✓ passes
positive pinionTeeth20.00 ✓ passes

Illustrative first-order example. Checks are editable requirements, not design standards. The broader topic may require additional equations, data, and validation.

Formula and units

This example uses the following equations. Input names and units match the editable source above.

rack force
rackForce = torque / pinionRadius [N]
rack travel
rackTravel = rotation * pinionRadius [mm]
module
module = 2*pinionRadius/pinionTeeth [mm]

Original worked example

These are the baseline values in the downloadable source. Edits above let you test your own values.

Given

pinion radius
20 mm
torque
2 N*m
rotation
1 rad
pinion teeth
20

Calculated

rack force
100 N
rack travel
20 mm
module
2 mm

Assumptions and limits

Ideal rack and pinion, neglecting tooth friction

The requirement checks use the limits written in this example. Passing a check does not establish that the model covers every load, material behavior, or operating condition in a real design.

Download this .napkin example ↓

Take it further with your agent

Napkin keeps the inputs, equations, checks and assumptions together in one study. Your agent sets it up; you change the numbers.

Try with Claude ↗

codex 'Use Napkin (https://napkin.engineering/mcp, or the local Napkin plugin) to build a study: Rack-and-pinion travel and force. Inputs: pinion pitch diameter; RPM; torque; efficiency. Outputs: linear speed; travel per turn; rack force. Model scope: Rigid involute-gear kinematics; qualified stress approximations. Read napkin_capabilities and the current snapshot. Ask me for the values that matter, label assumptions with units, author .napkin source, validate it and apply it with napkin_source using the current revision, then open the study. Add checks for the limits I supply. Keep unsupported analyses explicit.'

Claude needs the Napkin connector; Codex needs the Napkin plugin. Set up your agent ↗

The broader study

The topic brief below describes the broader scope. The worked example above states the particular calculation it implements.

  • Inputs: pinion pitch diameter, RPM, torque, efficiency
  • Outputs: linear speed, travel per turn, rack force
  • Modes: calculate, compare, sweep
  • Drawing: Animated gear assembly

EXECUTABLE WORKED EXAMPLE · FIRST-ORDER MODEL · CHECK THE SCOPE & ASSUMPTIONS

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