Chain length and sprocket spacing calculator

Approximate chain pitch count, rounded to an even number.

belts-chain-length.napkin / editable

SCENARIO COMPARISON

Compare the numbers.

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

Scenario	pitchCount	evenPitches	lengthmm
	93.7159	94	1193.8
	109.319	110	1117.6

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.

length budget1193.80 mm✓ passes

BASELINE INPUTS

pitch12.7 mmdriver15 driven45 center400 mm

CALCULATED

pitchCount93.716

evenPitches94

length1193.8 mm

Edits change the baseline source.

THE SOURCE

Edits below update the same model.

# Chain length and sprocket spacing
# Scope: Approximate chain pitch count, rounded to an even number.
# Status: illustrative worked example; engineering review pending.
# Inputs are example values. Checks are editable example requirements, not standards.

pitch = 12.7 mm
driver = 15
driven = 45
center = 400 mm

pitchCount := 2*center/pitch+(driver+driven)/2+(driven-driver)^2*pitch/(4*pi^2*center)
evenPitches := 2*ceil(pitchCount/2)
length := evenPitches*pitch [mm]

check length budget: length <= 1300 mm

scenario alternative: pitch = 10.16 mm

Calculated from your source · not saved

Baseline · Baseline

length budget1193.80 mm ✓ passes

alternative · alternative

length budget1117.60 mm ✓ 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.

pitch count: pitchCount = 2*center/pitch+(driver+driven)/2+(driven-driver)^2*pitch/(4*pi^2*center)
even pitches: evenPitches = 2*ceil(pitchCount/2)
length: length = evenPitches*pitch [mm]

Original worked example

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

Given

pitch: 12.7 mm
driver: 15
driven: 45
center: 400 mm

Calculated

pitch count: 93.7159
even pitches: 94
length: 1193.8 mm

Assumptions and limits

Approximate chain pitch count, rounded to an even number.

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: Chain length and sprocket spacing. Inputs: pitch; tooth counts; center spacing. Outputs: link count; adjusted spacing. Model scope: No-slip kinematics with explicit tension and efficiency assumptions. 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: pitch, tooth counts, center spacing
Outputs: link count, adjusted spacing
Modes: calculate, compare, sweep
Drawing: Animated belt or chain

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