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Rack-and-Pinion Calculator

The following online calculator computes the basic dimensions and tooth profiles of a meshing rack and pinion based on the pinion's module, number of teeth, pressure angle (usually 20°) and profile shift. The rack's teeth are trapezoidal, while the pinion has an involute tooth profile.

The profiles of the rack and pinion can be fully defined by the number of teeth on the pinion, module, profile shift and pressure angle. A positive profile shift is required for the pinions with 18 teeth or fewer to avoid a collision with the teeth of the rack. If the specified profile shift is insufficient, the calculator displays a warning.

The rack tooth profile is a trapezoid which dimensions are provided by the calculator. The base angles of the trapezoid are the same as the overall pressure angle (usually 20°). The calculator also provides the parameters for the rounded area between the rack teeth, known as fillet.

Basic Dimensions
ItemSymbolValue
Module m
Number of teethz
Pressure Angle (°) α
Profile shiftx
Width, Helix Angle (optional)w, β
Tooth thickness at base (°)ψb

Pinion Tooth Profile
X equation
Y equation
Z equation
Umin, Umax, Ustep          

Rack Dimensions
ItemSymbolValue
Slanted Addendum & Dedendumhα
Tooth Tip LengthL
Pitchp
Fillet Center Shiftaρ
Fillet Angleρ
Mounting Distancea
Pinion's Spiral Angle Per Divisionφp 8 divisions
Rack's Shift Per DivisionSr 4 divisions

Related: How to Model a Rack-and-Pinion in Blender