How this calculation works
A rack is the straight-line counterpart of a pinion’s pitch circle. When the ideal pitch circle rolls without slip along a matching rack, one pinion revolution advances one pitch circumference.
Pitch diameter = module × teeth; travel/revolution = π × pitch diameter; rack force = torque/pitch radius × efficiencyA worked example
A module-two, twenty-tooth pinion has a 40 mm pitch diameter and travels 40 pi mm per revolution, about 125.664 mm. At sixty rpm the nominal rack speed is the same number in millimetres per second. Five N m divided by the 0.02 m pitch radius and multiplied by ninety-percent efficiency gives 225 N of modeled rack force.
Good to know
Ideal metric pitch geometry and constant efficiency; no backlash, strength, positioning accuracy, pressure-angle loads or machine approval. Choose components, lubrication and support from appropriate engineering guidance. Travel and force conversions are not substitutes for a complete motion-system design.
Reference: KHK — gear dimensions and rack geometry (opens in a new tab) ↗
What this model answers
A rack is the straight-line counterpart of a pinion’s pitch circle. When the ideal pitch circle rolls without slip along a matching rack, one pinion revolution advances one pitch circumference.
Metric module relates tooth count to pitch diameter. It is not the outside tooth-tip diameter or the circular pitch itself. Circular pitch equals pi times module, so both quantities are shown to distinguish the dimensions.
The Lead screw speed and command pulse calculator describes a different rotary-to-linear mechanism. Screw lead replaces pinion pitch circumference as the distance advanced per turn. For a helical mesh carrying known torque, the Helical gear tangential radial and axial force calculator resolves tangential, separating and axial magnitudes.
Measure the right inputs
Enter the pinion shaft speed and revolutions after any gearbox. A motor speed before a reduction stage would overstate rack speed unless the reduction has already been applied.
Torque is the input at that same pinion shaft. Efficiency reduces the ideal tangential force; it does not change the geometric distance moved per complete revolution. The rack and pinion are assumed to have matching tooth geometry.
Read the result in context
Linear speed follows travel per revolution times revolutions per second. Total travel follows the entered revolution count independently of speed. A zero RPM scenario can still specify a nonzero intended travel count, but it does not imply that travel occurs in finite time.
The force metric is a tangential torque balance. It excludes tooth stress, radial pressure-angle force, acceleration and rail friction variation. A large computed force from a small pitch radius is not evidence that the teeth can sustain it.
Work through the numbers
A module-two, twenty-tooth pinion has a 40 mm pitch diameter and travels 40 pi mm per revolution, about 125.664 mm.
At sixty rpm the nominal rack speed is the same number in millimetres per second. Five N m divided by the 0.02 m pitch radius and multiplied by ninety-percent efficiency gives 225 N of modeled rack force.
Where the model stops
Ideal metric pitch geometry and constant efficiency; no backlash, strength, positioning accuracy, pressure-angle loads or machine approval.
Choose components, lubrication and support from appropriate engineering guidance. Travel and force conversions are not substitutes for a complete motion-system design.
A few questions, answered
Is module the distance between teeth?
Module is pitch diameter divided by tooth count. Circular pitch along the rack equals pi times module.
Does efficiency change travel per turn?
No. In this model it changes transmitted force, while the pitch geometry fixes nominal travel.