Brown movement 269 produces alternate rotation from a translating rack frame. An upper toothed run turns a fixed-axis spur gear one way; a later lower run turns it back. Calculate the ideal rotation, pitch length and travel time for equal opposing rack runs.
Opposed Rack-Run Calculator
Enter a declared spur-gear tooth count, equal teeth per upper/lower rack run, common module and frame speed. The worksheet reports ideal rotation magnitude, rack-run pitch length and travel time. It does not design transition teeth, a locking device or a practical continuous return path.
Equation Used
- One frame translates in one shown direction past a fixed-axis spur gear.
- The upper and lower rack runs have equal declared whole-tooth pitch counts and common module with the gear.
- The gear engages one rack run at a time; opposite-side contact reverses angular sign.
- Transition teeth, frame-loop closure, contact ratio, locking, backlash, profile shift, force, torque, inertia, impact, wear, bearings, guide loads, speed rating and safety are excluded.
Two rack runs act on opposite sides of the same gear. Their equal tooth counts produce equal rotations in opposite directions. Playback timing is illustrative; the entered frame speed affects the calculated travel time.
One translating frame presents rack runs on opposite sides
Brown movement 269 shows a frame that continues translating rectilinearly. A toothed run above the spur gear rotates it in one direction; a later toothed run below the gear acts at the opposite contact point and reverses rotation.
The historical animation notes that some transition teeth were altered to avoid jamming. This viewer marks the transition as schematic and does not claim a manufacturable tooth handoff.
Alternate rotation from a translating frame
The arrangement illustrates alternate gear rotation while the rack frame continues moving in one direction. Equal tooth counts return the gear to its initial angle after both runs; unequal counts would give a net change, which this equal-run calculator does not model.
Equal pitch counts on opposite contact sides
The upper and lower angular increments have opposite signs. Equal rack-run tooth counts therefore return the ideal gear to its starting angle after the pair.
Declared equal-run example
For a 24-tooth spur gear, eight teeth per rack run and module 2 mm, each run spans 50.27 mm of pitch length and turns the gear 120°. At a declared 100 mm/s frame speed, each rack-run pitch length passes in 0.503 s. The equal lower run then returns the ideal gear by 120°.
Transition geometry and practical limits
The full spur gear meets each rack run separately. The transition between runs requires tooth geometry that avoids interference, and a toothless interval alone does not hold the gear in position. The calculator describes ideal pitch motion; it does not design transition teeth, a holding device or a continuous frame-return path.
Mutilated rack-frame questions
Does a smooth rack gap automatically lock the gear?
No.
Why does the lower run reverse the gear?
It contacts the opposite side of the same gear while the frame keeps translating in the same direction.
Are the transition teeth ready to manufacture?
No. Brown's animation explicitly notes tooth alterations to prevent jamming.
Mutilated rack-frame references
- Brown, 507 Mechanical Movements, movement 269: historical scan and caption showing continued frame translation, upper/lower mutilated racks and alternate spur-gear rotation; the animation notes altered transition teeth.
- US508387A: primary patent describing a toothed sector working between two racks so opposite-rack engagement reverses rotary motion.
- Cornell scan of 507 Mechanical Movements: public-domain source for Brown's historical plate and caption.
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