For a fixed-shaft metric pinion meshing with a straight rack, one pinion revolution advances the ideal rack by the circumference of the pinion reference circle: s = πd = πmz. The direction reverses with pinion rotation.
Rack Travel per Pinion Revolution Calculator
Enter module, tooth count and RPM to calculate rack travel and speed. The diagram runs at a fixed eight seconds per revolution; its playback speed does not follow RPM.
Equation Used
- The pinion axis is fixed and the rack translates along a straight reference line.
- Metric module and tooth count define reference pitch diameter d = mz.
- Ideal no-slip rolling occurs at the pinion reference circle and rack reference line.
- Backlash, forces, stress, accuracy, vibration, noise, efficiency, wear and ratings are excluded.
The companion figure repeats the same fixed-shaft pinion and straight rack. An orbiting pinion requires a different calculation.
How pinion rotation moves the rack
The pinion shaft is fixed in place while the rack translates. At the ideal reference geometry, the pinion reference circle is tangent to the rack reference line and rolls without slip. The tooth outlines are simplified symbols. The calculation uses the reference-circle relation, rather than calculating the contact between individual tooth profiles.
This is not the orbiting-pinion or epicyclic case sometimes used to explain the coin-rotation paradox. A moving carrier or a pinion orbiting another gear requires an additional carrier-motion term and is outside this calculator.
What this calculator demonstrates
Enter metric module, pinion tooth count and pinion speed to calculate reference pitch diameter, ideal rack travel per pinion revolution, ideal rack speed and tooth-passage frequency.
The results are kinematic reference values. They do not size a gear or predict load capacity, backlash, positioning error, vibration, noise, efficiency, lubrication, wear, life or safe operating speed.
Fixed-shaft metric rack-and-pinion relationships
For module m and pinion tooth count z, reference pitch diameter is d = mz. One revolution gives ideal rack travel s = πd = πmz. At n revolutions per minute, ideal rack speed is v = sn/60 and the tooth-passage frequency is f = nz/60.
These equations assume a fixed pinion axis and ideal no-slip motion at the reference circle and rack reference line.
Worked example
With module 3 mm, 22 teeth and 1,500 rpm, the reference pitch diameter is 66.0 mm. Ideal rack travel is 207.3 mm per pinion revolution and ideal rack speed is 5,183.6 mm/s. The tooth-passage frequency is 550.0 Hz.
The 550 Hz value is only the ideal tooth engagement event rate. It is not a predicted vibration or acoustic frequency spectrum.
Reading the drawing and sizing a drive
Playback shows one pinion revolution in eight seconds regardless of entered RPM. The rack tooth pattern moves with the pinion, but the rack body remains fixed on screen. The green arrow labels a travel distance; its direction does not track the moving tooth pattern.
The displayed radius is limited to keep the pinion visible, so use the numeric diameter for dimensions. The tooth shapes do not calculate involute contact or interference. A real drive also needs a specified pressure angle, tooth profile, face width, materials, mounting stiffness, lubrication, force and duty cycle.
The four results describe ideal motion. They do not establish load capacity, positioning accuracy or an allowable operating speed.
Pinion revolution questions
Why does one revolution move the rack by πmz?
The pinion reference diameter is module times tooth count, mz. Ideal rolling advances the rack by one circumference, πmz, per full pinion turn.
Does RPM change the animation speed?
No. The diagram completes one revolution in eight seconds for readability. RPM changes the calculated rack speed and tooth-passage frequency, while playback stays at its fixed display speed.
Is tooth-passage frequency a noise prediction?
No. It counts tooth passages per second. Vibration and sound also depend on tooth contact, load, accuracy and the surrounding structure.
Does this cover an orbiting pinion?
No. The shaft center is fixed in this calculator. A pinion carried around another gear needs an additional carrier-motion calculation.
Can these results select a rack and pinion?
They establish ideal travel and speed, but do not check tooth strength, backlash, accuracy, wear or mounting loads. Use those requirements and manufacturer ratings to select the components.
Technical references
- KHK, Calculation of Gear Dimensions — metric module and reference-diameter relationships; one gear rotation displaces a rack by one reference-circle circumference.
- KHK, Gear Rack Technical Reference — rack circular pitch p = πm and rack assembly reference geometry.
- SDP/SI, Spur Gear Technical Information — module as pitch diameter per tooth and circular-pitch relationships.
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