Gear Span & Chord Measurement

Scope: external involute spur gears, with optional profile shift x (0 for a standard gear). Use it to set or inspect tooth thickness with a gear-tooth or span micrometer, or a chordal gear-tooth caliper.

Span measurement uses W = m cos α [π(k−0.5) + z·inv(α)] + 2x·m sin α, where inv(α)=tan α−α with α in radians and x is the profile shift. Recommended span count is the nearest integer to z·αₓ/180 + 0.5, with αₓ = acos(d cos α ÷ (d + 2x·m)), which puts the contact on the d + 2x·m circle (the pitch circle when x = 0). Chordal values use ψ = π/(2z) + 2x tan α / z: chordal thickness d sin ψ, chordal addendum m(1 + x) + (d/2)(1 − cos ψ).

The measuring faces touch the teeth at the contact diameter √(db² + W²). A span is accepted only when that diameter is below the outside diameter and no lower than the base or root diameter, whichever is larger; otherwise the faces would rest on the tooth tips or miss the tooth flank.

Backlash is made by thinning the teeth at the pitch circle. Normal backlash (feeler gauge) is converted to circular with jt = jn ÷ cos α. Shared evenly, each gear is thinned by Δs = jt/2; all on this gear, Δs = jt. The span becomes W − Δs cos α, and the chordal values are recomputed from the thinner tooth with ψ = (s − Δs)/d. In a range, the minimum backlash gives the largest dimensions. This assumes the gears run at the standard centre distance.

References: FreeCAD Gear Workbench documentation and the BOSL2 gear library.