Reciprocating rectilinear motion means back-and-forth travel along a straight line. This example uses two rotating gear sectors that alternately engage opposite sides of a rack, producing advance and return without reversing the input shaft. Adjust gear module and retained tooth count to explore tooth spacing, pitch radius, ideal engaged travel and the intervals between strokes.
Reciprocating Rectilinear Motion Interactive Calculator
Explore a double-sided rack driven by alternating gear sectors. Module sets physical pitch and radius; retained tooth count also changes engaged travel and the time available for dwell.
Equation Used
- Each front sector shares a shaft with a complete 26-tooth rear gear. The blue 18-tooth input gear drives both rear gears, keeping the sectors synchronized.
- Pitch circles and marks show the rolling relationship; they are not manufactured tooth profiles.
- The model assumes external rack restraint during dwell and does not calculate tooth-entry loads, backlash, interference, contact ratio or permissible speed.
Watch the upper sector move the rack right and the lower sector move it left. Both sector shafts keep rotating during the gaps between engagements. The stationary rack during each gap represents an external holding restraint, not a lock provided by the missing teeth.
How the two sectors reverse the rack
The mechanism has a rear synchronization train and a front rack drive. A blue 18-tooth input gear meshes with two complete 26-tooth rear gears. Each rear gear shares a shaft with a front sector, so both sectors continue turning in the same direction even when their teeth are clear of the rack.
The upper sector engages the top edge of the pink double-sided rack. Its lower contact moves right as the sector turns counter-clockwise. Half a sector revolution later, the lower sector engages the rack’s bottom edge; its upper contact moves left. The contact side produces the reversal.
Parts in the drawing
- Rear gears: synchronize the two sector shafts.
- Front sectors: drive the rack during their retained-tooth intervals.
- Double-sided rack and guide: provide the straight-line output.
- Pitch circles and marks: show rolling radii and spacing without specifying tooth profiles.
When neither sector engages, the drive applies no holding constraint. The animation keeps the rack still during that interval by assuming an external restraint. At 13 retained teeth, the ideal intervals meet and no dwell remains.
Exploring cyclic advance and return
This arrangement demonstrates how continuous shaft rotation can produce a repeated feed-and-return sequence. It is useful for studying cyclic slides and demonstrators where the output should change direction and, with fewer than 13 retained teeth, pause between strokes.
Use the calculator to compare pitch geometry and motion timing. Selecting a mechanism for an actual feeder also requires its load, inertia, holding arrangement and tooth-entry behaviour. Those factors are not determined by module and tooth count alone.
Pitch geometry for the retained sector
For module m, circular pitch is p=πm. A completed gear with z=26 teeth has pitch radius R=mz/2. If k teeth remain, the ideal rolling distance while that sector is engaged is s=kp=kπm, and its angular span is β=360°k/z.
These are pitch-line relationships. The calculator does not predict total machine travel beyond engagement, contact ratio, interference, backlash, load capacity or the transient at sector handoff.
Worked example at the default settings
Enter module m = 4 mm and retained count k = 10. The completed gear count is fixed at z = 26.
- Circular pitch p = π × 4 = 12.566 mm.
- Pitch radius R = 4 × 26/2 = 52 mm.
- Ideal engaged travel s = 10 × π × 4 = 125.664 mm.
- Sector sweep β = 360 × 10/26 = 138.462°.
The reference video gives 163.3 mm as the total stroke of its assembly. The 125.664 mm result here is the distance assigned to ten tooth pitches, not a reproduction of that full stroke. Entry and exit contact geometry would be needed to establish complete travel.
Each ideal dwell spans 180° − β = 41.538° of sector rotation. Increase k to 13 and β becomes 180°, so that dwell vanishes. Doubling module to 8 mm instead doubles pitch, radius and engaged travel while leaving the angles unchanged.
Motion and handoff limits
Alternating sectors provide reversal while the input shaft continues rotating. With fewer than 13 retained teeth, they also provide intervals without drive. An external restraint must hold the rack if the load would move it during those intervals.
In the model, rack speed changes abruptly between rolling speed and zero. At the no-dwell setting, its direction reverses abruptly. Those ideal transitions do not predict the forces in a real mechanism. End-tooth engagement, backlash and inertia require separate analysis.
A slider-crank instead reverses its slider continuously, but has no finite dwell without additional features. A controlled linear drive can use a selected motion schedule; its performance then depends on the actuator and controller.
Rack and sector questions
Why does the rack reverse if both sectors turn the same way?
The sectors contact opposite sides of the rack. Counter-clockwise rotation at the upper sector’s lower contact drives the rack right; the same rotation at the lower sector’s upper contact drives it left.
What holds the rack during a dwell?
Missing teeth interrupt the drive but do not lock the rack. The animation assumes that an external restraint holds it while neither sector is engaged. A real mechanism needs a suitable holding arrangement if its load would otherwise move the rack.
Does kπm give the complete machine stroke?
It gives the rolling distance assigned to k tooth pitches in this ideal model. At module 4 mm and k = 10, that is 125.66 mm. The reference video states 163.3 mm for its complete assembly. Entry and exit contact geometry are not calculated here, so this result should not be used as that assembly’s total stroke.
Why are pitch marks shown instead of complete teeth?
The marks identify tooth spacing and the portions of each sector that engage the rack. They are not cutting profiles. Tooth shape, end relief, interference and contact ratio require a separate gear design.
Why does changing module leave the drawing about the same size?
The drawing fits the mechanism into a fixed display area. Module changes the calculated physical pitch, radius and travel, but the displayed gear circles retain their screen size. The millimetre readouts describe the selected dimensions.
What happens at 13 retained teeth?
Thirteen is half of the completed 26-tooth gear. Each sector then occupies half a revolution, leaving no dwell interval in this ideal schedule. The instantaneous handoff is a geometric simplification, not a prediction of impact or a proven tooth-entry design.
Primary mechanism reference
- Nguyen Duc Thang, Gear rack drive for linear reciprocating motion 2 — blue input shaft, double-sided pink rack, two alternating green sector gears, 26 completed teeth, 16 removed teeth, 10 retained teeth and module 4 mm.
- Carnegie Mellon University, Chapter 7: Gears — pitch-circle rolling and rack-and-pinion geometry.
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