This page corrects the former factor-of-two labeling. One common pinion makes two opposed equal-pitch racks move equal distances in opposite directions; counter-rack travel is S, while their relative separation changes by 2S.
Ideal Equal-and-Opposite Rack Motion
Enter a declared one-way rack stroke, pinion pitch diameter and one-way stroke rate. The worksheet calculates ideal no-slip kinematics only and removes the unsupported rod-hunt output.
Equation Used
- Two straight equal-pitch racks mesh on diametrically opposite sides of one rigid pinion.
- The meshes are ideal, no-slip and free of backlash.
- S is the declared one-way travel magnitude of rack 1; rack 2 moves the same magnitude oppositely.
- D is the operative pinion pitch diameter and each pinion revolution corresponds to pi D of rack travel.
- The declared stroke rate counts one-way strokes per minute.
- On-screen rack travel is normalized for visibility; pinion phase follows the declared S/D ratio.
- Tooth geometry, guides, backlash, compliance, loads, acceleration, power, strength, wear, manufacture, selection, operation and safety are excluded.
Corrects counter-rack travel from 2S to S, labels 2S as relative separation change, removes the unsupported b/2 rod-hunt output, and bounds all results to ideal kinematics.
One pinion reverses the second rack's direction
US Patent 4,630,701 states that two parallel racks connected by a gear move equal distances in opposite directions. EP3126238 likewise places racks on diametrically opposite sides of pinions to obtain opposite motion.
The animation preserves that constraint: both meshes remain connected and rack displacements are equal and opposite.
Use this only as ideal common-pinion kinematics
The worksheet distinguishes counter-rack travel magnitude from the change in separation between the two racks and reports ideal pinion rotation.
It cannot size, guide, load, synchronize or validate a real mechanism.
Equal and opposite pitch-line motion
The 2S quantity is relative separation change, not travel of the counter rack.
A transparent ideal example
For S=600 mm, D=120 mm and f_s=30 one-way strokes/min, counter-rack travel magnitude is 600 mm, relative separation changes by 1200 mm, the pinion turns 1.592 rev/stroke and its ideal speed is 47.7 rpm.
Ideal pitch-line motion omits real guidance and mesh effects
A practical arrangement requires rack guides, correct common pitch, tooth geometry, backlash control, bearings, stiffness and load analysis.
The former 2S counter-rack error, b/2 rod-hunt claim, application, maintenance and troubleshooting statements are removed or bounded.
Opposed-rack questions
Does the second rack move twice as far?
No; it moves the same distance oppositely.
Where does 2S appear?
In the change of relative separation.
Is pinion rpm a load rating?
No.
Is backlash calculated?
No.
Opposed-rack references
- US Patent 4,630,701: equal rack movement in opposite directions through a gear between parallel racks.
- European Patent Application EP3126238A1: opposed parallel racks meshing at diametrically opposite pinion sides.
- MIT Dynamics and Control II: rack linear velocity proportional to pinion angular velocity.
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