Gear Undercut Minimum Teeth and Backlash

Nmin = 2k/sin^2(phi) reproduces the familiar minimums exactly - 32 teeth at 14.5 degrees, 18 at 20, 12 at 25 - which is why the industry moved to higher pressure angles for small pinions; a 14-tooth 20-degree pinion is undercut and weakened right where bending stress peaks. And backlash from opening the center distance is 2 dC tan(phi), not one-for-one: 0.010 in of center distance makes 0.0073 in of backlash.

Undercut minimum Nmin = 2k / sin^2(pressure angle), k = addendum coefficient (1.0 full depth, 0.8 stub); backlash from center-distance change B = 2 x dC x tan(pressure angle).

Involute gear geometry as compiled in Machinery's Handbook and the AGMA standard proportions, by name - closed-form, no table reproduced.

Both relations are elementary involute geometry and are published across free gear-design references.

Estimate. AHJ and licensed professional govern.

Field names used by the API: pressure_angle_deg, teeth, addendum_coefficient, center_distance_change_in, diametral_pitch, min_teeth_exact, min_teeth, shortfall, backlash_in, circular_pitch_in, backlash_pct_of_pitch

• Undercut limit Nmin = 2k/sin^2(phi); reproduces 32 / 18 / 12 teeth at 14.5 / 20 / 25 degreesinvolute generating geometry
• Backlash relation B = 2 dC tan(phi); the factor 2 tan(phi) comes from separation along the line of actioninvolute mesh geometry
• No profile shift standard proportions assumed; profile shift is the usual fix for an undercut pinion and changes both outputsstated scope limit