Rack-and-pinion travel and force calculator

Ideal rack and pinion, neglecting tooth friction

gears-rack-pinion.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

This source describes an analytical model. Its quantities and checks are the view.

Scenario	rackForceN	rackTravelmm
	100	20
	166.667	12

All calculated quantities are listed with the inputs. Switch to Source to inspect the equations.

SCENARIOS / CALCULATED VALUESSAME SOURCE

Change an input to see the consequences.

force budget100.00 N✓ passes

BASELINE INPUTS

pinionRadius20 mm1050torque2 N*m0.510rotation1 rad0.16

CALCULATED

rackForce100 N

rackTravel20 mm

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Ideal rack and pinion, neglecting tooth friction
pinionRadius = 20 mm in 10..50
torque = 2 N*m in 0.5..10
rotation = 1 rad in 0.1..6
rackForce := torque / pinionRadius [N]
rackTravel := rotation * pinionRadius [mm]
check force budget: rackForce >= 50 N
scenario smaller pinion: pinionRadius = 12 mm

Calculated from your source · not saved

Baseline · Baseline

force budget100.00 N ✓ passes

smaller pinion · smaller pinion

force budget166.67 N ✓ 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]

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

Calculated

rack force: 100 N
rack travel: 20 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