An endless-rack reciprocator uses a continuous oblong rack and a pinion that keeps turning in one direction. The rack slides horizontally while the pinion shaft moves vertically in a fixed slot. As the pinion passes around each curved rack end, its center changes height and the rack reverses. This is the arrangement shown in Brown’s movement 119 and Hiscox’s figure 897.
Endless Rack Reciprocator Interactive Calculator
Choose pinion size, teeth, straight-run pitches and speed. Watch the shaft travel in its fixed vertical slot as the rack reverses, then compare stroke and full-cycle timing.
Equation Used
- Pure rolling of pitch curves.
- Semicircular rack-end radius equals pinion radius.
- Rack translates horizontally; pinion center moves only vertically.
- Whole pitches on each straight rack run.
- No tooth-force, stress, backlash or drive-accommodation calculation.
Semicircular rack-end pitch radius equals pinion pitch radius. Integer tooth counts and common pitch define the geometry. Tooth profiles are explanatory, not production tooling.
The shaft must move as the rack reverses
The blue pinion is carried by the vertical slotted guide. The gold oblong rack is confined to horizontal translation. Along its upper and lower straight portions, the pinion center stays at a constant height and the rack moves at constant speed. Around an end, the center moves vertically and the rack slows to zero, reverses, then accelerates into the opposite straight run.
The animation solves this pitch-line rolling motion. The pinion center remains on one fixed vertical line; the rack does not rotate. A contact marker identifies the instantaneous common pitch point. The close view follows the same contact, rather than showing a separate decorative animation.
The worked geometry uses semicircular rack ends with pitch radius equal to the pinion pitch radius. This is an illustrative geometric assumption, not a dimension supplied by the historical engraving. An even pinion tooth count and whole straight-run pitches keep the closed perimeter consistent with one common pitch.
A mechanical reversing arrangement
The historical sources describe continuous pinion rotation producing reciprocation of a bar carrying an endless rack. The distinguishing feature is its vertically moving pinion shaft, which permits the change from one side of the rack to the other.
This calculator compares stroke, straight-run speed and cycle time for the stated geometry. It does not establish a traverse machine’s load capacity or winding quality. Operating rates and maintenance requirements depend on the manufactured rack, guides, drive and applied loads.
Include the rounded ends in cycle timing
Let D be pinion pitch diameter, r=D/2, z its integer tooth count and m=D/z the module. One tooth pitch is πm. If each straight rack run contains k pitches, its length is L=πmk.
The rack end pitch radius is R=r in this model. The pinion-center path relative to the rack is offset outward by r, so the end-path radius is R+r=2r. That relative path has length P=2L+2Ï€(R+r)=2L+4Ï€r.
At pinion speed N rpm, rolling speed along this relative path is v=Ï€DN/60. Cycle time is P/v and cycles per minute are 60v/P. Rack stroke is L+2(R+r)=L+4r; shaft vertical travel is 2(R+r)=4r.
On the straight portions, horizontal rack speed has magnitude v. On a circular end it varies with the direction of the contact normal and reaches zero at reversal. Consequently, dividing twice the stroke by the straight-run speed underestimates cycle time. Cycle timing includes the curved path at each reversal.
Each full cycle requires 2+2k/z pinion revolutions. At zero rpm, all movement stops and cycle time is reported as stopped. The even tooth count puts a tooth center at each straight/arc join. Tooth flanks have illustrative clearance and simplified roots; force, tooth stress, backlash and guide reactions are not solved.
Example with a 60 mm, 24-tooth pinion
For D=60 mm and z=24, module is 2.5 mm and pitch is 7.854 mm. With k=24 straight-run pitches, L=188.50 mm. The rack stroke is 308.50 mm and the pinion shaft travels 120 mm vertically.
At 60 rpm, straight-run rack speed is 188.50 mm/s. The relative pinion-center path is 753.98 mm long, so a full cycle takes 4 seconds, giving 15 cycles per minute. The pinion turns four times during that cycle.
Using 2×308.50/188.50 would predict only 3.27 seconds because it treats reversal as constant-speed travel. The complete rounded-end geometry accounts for the extra time while the rack slows and reverses.
Geometry assumptions and practical limitations
Rack ends are semicircular with pitch radius equal to the pinion radius. Changing tooth counts adjusts straight-run length through a common module; the displayed total stroke is therefore a result, rather than an independent incompatible dimension.
The full view shows the fixed guides and complete horizontal range. The enlarged view follows the same mesh point so fine details remain visible on long racks. The pinion keeps its direction of rotation throughout the cycle.
The historical mechanism requires an input drive that accommodates vertical shaft motion. That drive and any guide-retaining detail are outside this planar diagram. The contact follows a constrained, no-slip pitch model; it is not a force-closure analysis. Tooth-root and straight-to-curve transitions require generated production profiles. The visual tooth shapes are explanatory, not cutting templates.
Acceleration changes at straight/arc joins in this ideal geometry. A real machine needs appropriate transition design, lubrication, guide strength and dynamic analysis. No load or safe-speed rating is inferred from the slider ranges.
Endless-rack questions
Does the pinion reverse?
No. Its rotation keeps the same direction. The rack reverses as the pinion changes from the upper run to the lower run around an end.
Why does the shaft move vertically?
The fixed slotted guide lets the pinion follow the curved ends while keeping its horizontal coordinate fixed. Holding that shaft at one fixed point would prevent this motion.
Why is stroke an output?
It follows from the chosen pinion size, end radius and straight-run tooth count. Those dimensions must be consistent.
Why is cycle time longer than twice stroke divided by rack speed?
The displayed rack speed applies to the straight runs. Horizontal speed falls to zero at each rounded end and then reverses.
Does the enlarged view use different motion?
No. It is a magnification of the same calculated rack, pinion and contact point.
Can these teeth be manufactured from the picture?
No. The pitch geometry is solved, but the illustrated tooth roots and transitions are simplified and require separate profile generation.
Primary mechanism references
Brown, Five Hundred and Seven Mechanical Movements, movement 119, describes the oblong endless rack, uniform pinion rotation and vertical shaft guide.
Hiscox, Mechanical Movements, Powers, Devices and Appliances, printed page 231, figure 897, shows the same vertically slotted pinion-shaft arrangement. The equal end/pinion radii and integer-pitch sizing used here are explicit geometric assumptions.
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