This calculator converts standard metric pinion module and tooth count to pitch diameter and pitch-line speed, then divides that ideal travel rate by an independently entered stroke. The animation sketches a vertically shifting internal rack; its transition teeth and dynamics are not solved.
Internal Rack Pinion Reciprocator Interactive Calculator
Calculate ideal pinion pitch diameter and straight-rack speed, then compare that travel rate with an entered one-way stroke.
Equation Used
- Standard metric pitch relation d=mz and pure rolling at the pitch line during straight-rack engagement.
- Entered rpm is constant and produces ideal travel pi d per pinion revolution.
- Entered stroke is usable one-way travel and all ideal travel contributes to it.
- The inner rack's vertical handoff is illustrated, but its distance and time are omitted from the rate outputs.
- No handoff tooth form, stop/guide dimensions, conjugate engagement transfer, interference, backlash, pressure angle, undercut, acceleration/jerk, carriage mass, torque, loads, strength, wear, lubrication, drive or safety model.
The rate outputs assume calculated pitch-line travel is used continuously over each full stroke. The illustrated rack shifts vertically at each stroke end before the opposite side drives; a real handoff takes time. Rates are nominal comparisons, not achievable throughput.
Sliding internal rack and alternating engagement
In the illustrated arrangement, a rectangular carriage carries an internal rack that can move a short distance vertically. The continuously rotating pinion drives one toothed side during a straight traverse; at the end, the inner rack shifts so the opposite side engages and the carriage reverses without reversing the pinion.
The animation shows the two straight-drive states and two fixed-end handoffs. It does not claim exact transition tooth, guide or stop geometry.
Use as a preliminary kinematic comparison
The outputs can check module/tooth-count units and compare ideal travel rate with an entered stroke. They cannot establish a buildable track, actual reversal motion, throughput, loads or service life.
Nominal rate assumption
d=mz and v=pi d n. With v in mm/min and stroke S in mm, v/S is nominal one-way traverses per minute; dividing by two gives nominal out-and-back cycles.
This assumes continuous useful distance accumulation and ignores transition geometry and time, so it is not a dynamic reciprocator solution.
Worked example
For module 2 mm, 16 teeth and 100 rpm, ideal pitch diameter is 32 mm and pitch-line speed is 167.55 mm/s. Dividing ideal travel by a 250 mm stroke gives 40.21 nominal one-way traverses/min or 20.11 nominal round trips/min.
What the nominal rate omits
An actual reciprocator needs compatible tooth forms and a controlled transfer between its two rack sides. Reversal time, acceleration, carriage mass and guide friction affect the achievable rate and load. Those effects are absent here, so the nominal traverses per minute should not be treated as production throughput.
Questions
Why does the inner rack move vertically?
That shift lets the continuously rotating pinion engage the opposite toothed side and reverse the carriage.
Is nominal cycle rate achievable?
Not necessarily. Handoff time, acceleration, control and losses are omitted.
Can module and tooth count size the drive?
No. Torque, contact stress, undercut, face width and material are absent.
References & Further Reading
- Wikipedia contributors. Rack and pinion. Wikipedia
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