Opposed conical wheels can turn a uniform input into a varying output speed, as illustrated in Brown’s mechanism 37. The left pinion has teeth extending across its full face; the right cone carries spirally arranged studs. The shafts are parallel, and the contact radii change along the opposed tapers.
Spiral-Stud Conical Wheel · Local Ratio and Pass Timing
Explore one selected spiral pass between opposed cones on parallel shafts. Compare local speed ratio, accumulated input rotation and phased stud stations.
Equation Used
- Radii add to constant shaft spacing.
- Axial contact position increases linearly with output phase in this selected spiral.
- Input speed is uniform.
- Stud stations match successive pinion pitches under the ideal local law.
- No load-sharing, tolerance or seamless-cycle claim.
The opposed cones vary output speed by changing the contact radii along their parallel shafts.
A changing contact radius
The cones taper in opposite directions. At a selected height, the left pinion radius and right stud-wheel radius add to the shaft spacing. A contact near the large end of the pinion meets the small end of the stud wheel; farther along the face, their proportions reverse.
The long pinion teeth can meet studs at different heights. The studs follow one spiral around the right cone, with their angular positions timed to meet successive pinion teeth. As the illustrated contact station moves along the taper, the ideal local ratio changes.
The animation follows one selected pass from the small to the large stud-wheel radius. It pauses and resets at the end. That reset is a preview action, not a driven reversal or proof of a seamless repeating gear cycle. The source does not supply the tooth profiles or spiral-end handoff needed to establish continuous meshing.
Compare a selected variable-ratio pass
Use the small and large pitch radii to compare the range of ideal output speeds. Input speed sets the time scale. Pinion tooth count sets angular tooth pitch and how many matching stud stations can be placed in the selected pass.
This model illustrates a variable-motion arrangement over one spiral pass. It does not establish a rated industrial application, permissible speed, load-sharing count or manufacturing fit. Assessing contact forces or torque ripple requires tooth profiles and loading data beyond this local speed calculation.
Local rolling law and accumulated input angle
Let small radius be a, large radius b and shaft spacing C=a+b. For stud-wheel angle φ from 0 to 2π during one selected spiral pass, choose a linear axial sweep z=φ/(2π). The local stud-wheel radius is r₂=a+(b−a)z and pinion radius is r₁=C−r₂.
Equal ideal pitch-line speed gives n₂/n₁=r₁/r₂, with opposite rotation. Thus output speed starts at n₁b/a and ends at n₁a/b. This is a local kinematic relation, not a synthesized conjugate tooth profile.
Put k=(b−a)/(2π). Accumulated input-angle magnitude is Ψ(φ)=(C/k)ln[b/(b−kφ)]−φ. For uniform input, the animation inverts this monotonic relation to obtain output position. Input turns per pass are Ψ(2π)/(2π), and pass time is 60 times that value divided by input rpm.
For Z pinion teeth, successive stud stations satisfy Ψ(φⱼ)=2πj/Z. Their angular spacing is therefore generally unequal. The number of stations in this bounded pass is floor[ZΨ(2π)/(2π)]+1, including the initial station. Their axial coordinates follow φⱼ/(2π).
25 mm and 100 mm pitch radii
At 20 rpm input, the local model starts at 80 rpm output and ends at 5 rpm. The selected output revolution requires about 1.3105 input revolutions and takes about 3.9315 seconds.
With 12 full-width pinion teeth, pitch is 30° and the selected spiral contains 16 phased stud stations. Doubling input speed doubles both limiting output speeds and halves pass time, without changing the geometric phase relationship.
Changing tooth count changes the phased stud layout, not the ideal radius ratio. Whether discrete studs and actual tooth flanks maintain contact between those stations requires additional design information.
Limits of the local speed model
The arrangement uses opposed tapers, parallel shafts, full-width teeth and a spiral stud sequence. It enforces the selected radius law and accumulated phase relationship.
It does not calculate tooth stress, contact pressure, friction, backlash, interference or a continuous spiral-end handoff. No claim is made that two or more studs always share load. The displayed dimensions are selected pitch geometry, not manufacturing dimensions.
Spiral-stud wheel questions
Are the shafts at 90°?
Not in Brown 37. The drawing shows parallel shafts and opposed conical tapers.
Why does the output slow down?
The selected contact moves toward a smaller pinion radius and a larger stud-wheel radius.
Why are the studs not equally spaced in angle?
The chosen local rolling law must advance the pinion by one tooth pitch between successive stud stations.
Why does the animation reset?
It illustrates one bounded spiral pass. The source does not specify how to join the spiral ends into a seamless working cycle.
Does this predict torque ripple?
No. Actual contact and loading data would be needed.
Primary reference
Henry T. Brown, Five Hundred and Seven Mechanical Movements, no. 37: uniform-to-variable rotation, full-face pinion teeth and spirally arranged studs on a conical wheel. The calculator assumes the stated local rolling law, selected pitch radii and timing for a single spiral pass.
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