The historical circular rack has ring-shaped teeth around a straight shaft. A side pinion moves the shaft axially. Hiscox illustrates this arrangement as a variable thrust bearing in figure 886.
Circular Rack Interactive Calculator
Set pinion pitch radius, rack pitch radius, pinion angular range and tooth module. The rack moves straight along its axis; its own diameter changes the envelope, not the travel per pinion turn.
Equation Used
- Rigid pinion and axially guided annular-tooth rack, with no slip at the pitch line.
- Common module; illustrative 20° working flanks and simplified tooth roots.
- Pinion is fixed in position while the rack translates.
- No force, wear, strength, backlash or contact-stress prediction.
This is the historical circular rack: annular teeth around a straight sliding shaft. It is not a curved sector gear. Pitch travel is calculated without slip; tooth illustrations are explanatory and do not specify a manufactured tooth form.
The animation below follows the annular-tooth circular rack shown in Hiscox figure 886.
A circular rack on a straight shaft
In Hiscox figure 886, the rack is a straight cylindrical member with annular teeth. The pinion engages one side and drives axial translation. The word circular describes the teeth surrounding the shaft, not a curved path of travel.
The animation shows the rack guided vertically, the fixed pinion shaft and a marked ring moving with the rack. The pinion turns forward and back through the selected angle. The rack displacement is tied directly to rotation at the pinion pitch radius.
A curved sector gear is a different geometry. Its output turns about a pivot; this circular rack moves along a straight axis.
Axial positioning with annular teeth
Hiscox describes the illustrated circular-rack arrangement as a variable thrust bearing. Ring teeth permit side engagement around the shaft circumference, while the pinion supplies axial positioning.
This calculator explains the pitch kinematics. It does not establish a thrust capacity, bearing arrangement or suitability for a particular machine.
From pinion angle to straight-line motion
At the pitch point, rack displacement equals pinion pitch-circle distance: s = rp θ, with θ in radians. One full pinion turn gives 2πrp of axial rack movement.
For tooth module m, circular pitch is πm and pinion tooth count is z = 2rp/m. The number of tooth pitches passed during a stroke is s/(πm); this is not the number of teeth simultaneously in contact.
Rack pitch radius R determines the shaft pitch diameter 2R. It is independent of pinion radius. Nominal shaft-center to pinion-center distance in the drawing is R + rp.
Default example
A 50 mm pinion pitch radius turning through 90° gives 78.54 mm of rack travel. At module 2, the pinion has 50 teeth and the stroke traverses 12.5 tooth pitches.
A 50 mm rack pitch radius gives a 100 mm shaft pitch diameter. Increasing that radius changes the shaft size and center distance, while the same pinion rotation still gives 78.54 mm travel.
Doubling the pinion radius doubles travel for the same angle. Doubling module halves the equivalent pinion tooth count and pitches traversed, but does not change pitch-line travel.
Geometry limits
The entered pitch radius and module must yield an integer pinion tooth count for a complete standard gear. A fractional value is shown as a pitch-geometry comparison; it does not specify a complete standard pinion.
Low tooth counts can need profile correction. Tooth roots, annular contact geometry and bearing loads require separate design work. The displayed rack length provides room for the chosen motion; it is not a specified stock length.
Circular-rack questions
Is this a curved rack?
No. This page follows the historical annular-tooth circular rack. It has a straight axis and translates along that axis.
Must the rack and pinion radii match?
No. Rack radius sets shaft diameter. Pinion radius sets the distance moved for a given input angle.
Why is a fractional pinion tooth count flagged?
A full pinion needs a whole number of teeth. Adjust module or pitch radius so 2rp/m is an integer. The pitch travel calculation remains useful as a geometric comparison.
Are tooth pitches traversed the same as teeth in contact?
No. Pitches traversed measure how far the rack moves relative to tooth spacing. Contact ratio requires additional tooth geometry.
Does this calculate thrust capacity?
No. It calculates motion and nominal pitch dimensions only.
References & Further Reading
- Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page 230, figure 886, circular rack and pinion.
- KHK Gears, Calculation of Gear Dimensions, module, pitch diameter and tooth count definitions.
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