TorqueMath

Rack & Pinion Calculator

Module and pinion teeth in; travel per turn, linear speed, push force from motor torque, turns for a stroke, rack length and stepper resolution out.

Module · ISO 54
m mm
T
Pressure angle
RPM
N·m
mm
steps
: 1
Pinion turns, rack slides

How it is calculated

Straight-tooth rack with standard involute pinion (no profile shift). m is the module, z the pinion teeth, n the pinion speed, T the torque on the pinion.

Pitch diameter d = m · z Travel per turn s = π · m · z (also the rack length per pinion turn) Circular pitch p = π · m rack tooth spacing Linear speed v = s · n / 60 (mm/s) Rack force F = 2 · T / d (tangential, at the pitch line) Separating force Fr = F · tan α Rack length L ≥ stroke + pinion face contact (≈ 2 · p) + end margins Resolution = s / (steps · gearbox ratio)

Pinions under 17 teeth (20°) are undercut unless profile shifted. For CNC gantries, helical racks run quieter; their travel per turn is π · mₙ · z / cos β.

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