This historical six-bevel train combines a fixed-axis reversing pair with a moving-pinion differential. The input drives the through-shaft and hollow sleeve at equal, opposite speeds. The through-shaft also carries the final pinion. For the equal gears and connections shown, the output turns three times per input revolution. Its historical name is “doubling speed”; the shaft and sleeve connections shown here give a three-to-one ratio.
Six-Gear Bevel Train Interactive Calculator
Follow all six gears and the reversing sleeve. The carrier moves forward while the sleeve turns backward, giving three output turns per input turn for the connections shown.
Equation Used
- Six equal bevel gears and common even tooth count.
- G and carrier A fixed to through-shaft F.
- Both C gears fixed to a freely rotating hollow sleeve.
- Output E free on F; planet D carried by A.
The illustrated shaft and sleeve connections give a three-to-one speed ratio. Equal gears, rigid shafts and simplified tooth ribs; no backlash or strength calculation.
Six gears, a through-shaft and a reversing sleeve
Input bevel B meshes with G on the through-shaft F and with the first gear of hollow sleeve C. Those two coaxial members turn oppositely. The sleeve’s second gear is rigidly connected to its first gear. Carrier A is fixed to F and carries planet D around the final pair. Its radial stud is exposed in this cutaway; the surrounding historical frame is omitted for clarity. Output E runs freely around F.
The animation shows two equal-gear groups sharing one shaft line: B, G and the first C gear at the right; the second C gear, orbiting D and output E at the left. Colors follow the physical members: blue through-shaft and carrier, green reversing sleeve, purple orbiting pinion, and gold output.
All six gears have equal 45-degree pitch cones and a common even tooth count. Tooth ribs show phasing but are simplified illustrations, not manufactured tooth surfaces.
Use the calculator to follow each member
Change the input and watch the carrier, sleeve, pinion and output together. The signed sleeve result is negative because it turns opposite the through-shaft when viewed from the same end. The pinion result is its spin relative to the moving carrier, not its total angular velocity in space.
Change the common tooth count to see different tooth spacing without changing the speed ratio. Setting the input to zero stops the mechanism. Playback is slowed on one shared time scale so none of the members moves independently of the gear relation.
How the reversing sleeve produces three-to-one
Choose the through-shaft direction as positive and let the input speed magnitude be n. The first group gives ωF=+n and ωC=−n. Frame A is fixed to F, so ωA=+n.
In the second group, equal side gears satisfy (ωE−ωA)=−(ωC−ωA). Thus ωE=2ωA−ωC=2n−(−n)=3n. Relative to the carrier, sleeve C turns at −2n, so planet D spins at magnitude 2n about its stud. The same equations apply to revolutions over an interval.
Brown movement 226 and Hiscox figure 939 use the name doubler. In the model shown here, the input gear drives the sleeve backward while the carrier moves forward, so E = 2A − C gives three-to-one. Holding the sleeve fixed would instead give two-to-one.
Example: one input turn
With 24 teeth on every gear, one input revolution gives one carrier revolution, minus one sleeve revolution, two planet spins relative to the carrier, and three output revolutions. Tooth pitch is 15°.
At 60 input rpm, the corresponding values are +60 carrier rpm, −60 sleeve rpm, 120 rpm pinion spin relative to the carrier, and +180 output rpm. Doubling every gear’s tooth count retains those speeds and halves tooth pitch.
What this model calculates
This is a rigid, ideal kinematic model. The four meshes follow equal pitch-line velocities, with the carrier and sleeve shown separately. It does not calculate backlash from a single linear clearance, gear strength, bearing life, efficiency or maximum usable speed.
The view omits bearing housings and the surrounding carrier frame, exposing its radial stud to show the gear contacts. The original historical drawing defines the connections, not production dimensions. Do not use the displayed ribs as a machining profile.
Six-bevel train questions
Why does a page called a doubler show three-to-one?
The historical name is retained for identification. With the published connections, the sleeve turns backward while the carrier turns forward, so the equal-gear differential equation gives three-to-one.
Are C’s two gears free to turn independently?
No. Both are fixed to one hollow sleeve around the through-shaft.
Which part carries D?
Frame A, fixed to shaft F. Pinion D both orbits with that frame and spins about its radial stud.
Can I choose different tooth counts for the two sides?
This calculator uses equal gears. Changing individual counts would require a different pitch-cone geometry and ratio calculation.
Construction and calculation references
Brown, movement 226, and Hiscox, Mechanical Movements, Powers and Devices, printed page 239, figure 939 (PDF page 249), provide the six-gear connections and historical doubling claim.
J. W. Kamman, Intermediate Dynamics: Differential Gear Set, derives equal-side-gear velocities relative to a carrier. For the shaft and sleeve connections shown here, that relation gives a three-to-one output.
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