A double-rack crank substitute uses a continuously turning pinion to drive alternate sides of a closed, internally toothed rack. The rack shifts across the pinion at each rounded end, reversing its horizontal travel while the pinion keeps rotating in one direction.
In the fixed-shaft arrangement shown here, the pinion axis stays still. The rack moves both horizontally and transversely, and a separate output carriage remains in its straight horizontal guide. Removing that transverse motion would make the illustrated end transitions impossible.
Double-rack Crank Substitute Interactive Calculator
Set the pinion and rack proportions. Watch the shifting rack reverse around a fixed shaft while the output bar stays in a straight guide.
Equation Used
- The fixed pinion turns at constant selected speed.
- Rack end pitch radius is twice pinion pitch radius.
- Each straight section contains an integer number of pitches.
- The rack translates without rotating.
- A transverse cross-slide isolates the horizontal output.
- Loads, friction, production tooth generation and cam stresses are outside this calculation.
Hiscox 891 / Rose 214: a fixed pinion and transversely shifting internal rack. The blue shaft turns continuously in one direction.
A shifting rack around a fixed pinion
Hiscox figure 891 describes Parson’s crank substitute: a double rack meshes alternately with a pinion, while a cam faceplate running in smooth ways guides the rack across at the ends. The end teeth maintain engagement during the change.
Joshua Rose’s Modern Machine-Shop Practice, figure 214, explains the fixed-pinion version more fully. Its rack is mounted to a reciprocating piece in a way that permits a short transverse movement. The pinion can then pass from one row of teeth to the other without moving its shaft.
The animation uses a vertical cross-slide between the shifting rack and horizontal output carriage. It represents the required transverse freedom directly; it is not a dimensional copy of Rose’s linked support. A guide disc coaxial with the pinion follows the closed guide groove on the rack’s front plate.
The blue pinion turns continuously counterclockwise. The gold rack runs left on its upper row, moves around the rounded end, runs right on its lower row, then completes the opposite reversal. The output bar follows only the horizontal component.
Explore the rack geometry
Gear module and pinion tooth count determine pinion size. The straight section is entered as a whole number of tooth pitches so the closed tooth pattern remains consistent. Its millimetre length is displayed alongside the results.
Both end pitch arcs have radius twice the pinion pitch radius in this selected construction. The pinion-center path relative to the rack therefore has end radius equal to one pinion radius. The extra end motion contributes to both total stroke and cycle time.
Pinion speed changes the calculated velocities and cycle rate. At zero rpm, the mechanism stays visible and stationary. The animation is slowed when needed, with the display factor stated beneath the drawing.
Stroke and cycle time including the ends
Let module be m, pinion tooth count z, straight-section pitch count n and shaft speed N in rpm. Pinion pitch radius r=mz/2; circular pitch p=πm; straight length L=np. The selected rack end pitch radius is R=2r, so the relative pinion-center path has radius ρ=R−r=r at either end.
The full horizontal stroke is L+2r. Rack transverse travel is 2r. The closed relative-center path length is C=2L+2πr, including both straight runs and both semicircular reversals.
At constant pinion speed the path speed is v=2πrN/60. Thus complete cycle time is C/v, cycle rate is 60v/C, and pinion turns per output cycle are C/(2πr)=1+2n/z.
On each straight portion horizontal speed is ±v. In a rounded end, horizontal and transverse components change continuously with the path tangent. Horizontal speed passes through zero at maximum stroke while the pinion continues turning.
The rack contains 2n+2z tooth pitches. After one output cycle the pinion has turned through 2n+z of its own tooth pitches, so the tooth pattern returns to the same meshing phase. Drawing profiles include clearance; they are explanatory outlines, not a generated cutter specification.
Example: 24-tooth pinion and 12-pitch straight sections
At module 2 mm, the pinion pitch radius is 24 mm. Each straight section is 12π×2=75.40 mm long. Full horizontal stroke is 123.40 mm and transverse rack travel is 48 mm.
The relative center travels 301.59 mm per complete cycle. At 12 rpm, pitch-line speed is 30.16 mm/s and a complete output cycle takes 10 seconds, giving 6 cycles per minute. The pinion makes two revolutions during that cycle.
A calculation based only on twice the straight length would omit both reversals and overstate cycle rate. Likewise, the full stroke is greater than the straight-section length because the rounded ends contribute additional horizontal travel.
What the model does and does not resolve
The model follows exact pitch-line rolling and a constant shaft direction. The output cross-slide removes the rack’s transverse displacement from the output bar. The guide groove and coaxial disc keep the required path visible.
The straight and curved path sections meet tangentially. Velocity is continuous, but curvature changes at their joins; the ideal model consequently has an acceleration step there. It does not claim shock-free operation.
Tooth widths are relieved for visual clarity. End transitions, cutter geometry, backlash, friction, bearing loads, tooth strength and a production cam profile require further design. The calculator gives geometry and motion, not a manufacturing-ready or load-rated mechanism.
Double-rack crank substitute questions
Does the pinion reverse?
No. Its shaft keeps turning in one direction. Switching between opposite sides of the rack reverses the output.
Why does the rack move up and down?
The pinion shaft is fixed. The rack must shift transversely to bring the opposite row of teeth into engagement. The output carriage receives only horizontal travel.
Why enter a number of tooth pitches?
The closed tooth pattern must remain phase-consistent. Whole straight-section pitch counts retain that relationship; the corresponding physical length is calculated.
Is the full stroke just the straight rack length?
No. For the selected end geometry it is L+2r, including the horizontal reach of both end transitions.
Why is there an enlarged view?
It makes the current tooth engagement and guide relationship visible even when the rack is long. Both diagrams use the same physical pose.
Primary references
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, figure 891, printed page 230: Parson’s double-rack crank substitute, alternate engagement and cam-guided end transitions.
Joshua Rose, Modern Machine-Shop Practice, volume I, figure 214 and accompanying explanation: a fixed-axis pinion, internally toothed shifting rack, guide groove and separate straight-moving output piece. The cross-slide and dimensional ratios are illustrative choices for this model.
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