This differential-speed arrangement produces two different rotational speeds about one axis. Central pinion A drives idler B on a fixed journal. B then drives internal ring C. A and C turn in opposite directions, with their speed magnitudes set by their tooth counts.
Differential Speed Interactive Calculator
Explore the fixed-journal gear arrangement in Hiscox mechanism 943. Change the central gear and idler tooth counts; the internal ring count follows automatically so both meshes share the same module and center distance. Enter either direction of input speed.
Equation Used
- Same module and pressure angle for all three gears.
- One central gear meshes externally with one idler on a fixed journal.
- The idler meshes internally with a coaxial rotating ring.
- Rigid ideal gearing without slip or backlash.
- Input sign is clockwise positive in the displayed end view.
The idler center is fixed to the frame. The ring and central gear rotate independently about the same axis. This is not the speed-splitting action of a car differential.
A fixed idler between two coaxial gears
The central pinion and ring share an axis but are free to rotate independently. The intervening idler is supported on its own stationary shaft. Its bearing stand stays still while the gear spins; there is no orbiting planet carrier.
The external A–B mesh reverses direction. The internal B–C mesh preserves the idler’s direction, so the ring turns opposite the central input. The enlarged end view exposes both engagements and marks the rotating ring.
Coaxial counter-rotation
Hiscox lists this arrangement as mechanism 943, “Differential speed,” for two gears rotating in different directions about the same shaft axis. It demonstrates how an external mesh followed by an internal mesh can provide counter-rotation.
It should not be confused with an automotive differential, whose moving carrier allows two outputs to change speed relative to each other. Here, fixing the idler journal fixes the input-to-output ratio.
Tooth counts, geometry and speed
For a common module m, pitch radii are mN/2. The distance from the shared central axis to the idler is both rₐ + rᵦ and r꜀ − rᵦ. Equating those distances gives N꜀ = Nₐ + 2Nᵦ. The ring count is therefore calculated rather than chosen independently.
The external mesh satisfies nₐNₐ + nᵦNᵦ = 0. The internal mesh satisfies nᵦNᵦ − n꜀N꜀ = 0. Combining them gives n꜀ = −nₐNₐ/N꜀. The idler count cancels from the speed equation when the ring count is already specified, but changing it here also changes the ring required by the geometry.
The positive reduction figure is N꜀/Nₐ. It remains a geometric ratio at zero speed, while both calculated gear speeds become zero.
24-tooth input and 24-tooth idler
With 24 teeth on A and 24 on B, the internal ring requires 72 teeth. At +120 rpm input, B turns at −120 rpm and C at −40 rpm. The input-to-ring speed magnitude ratio is 3:1.
Reversing the input to −120 rpm makes the idler +120 rpm and the ring +40 rpm. Changing B to 40 teeth requires a 104-tooth ring; the ring then turns at approximately −27.69 rpm for the original +120 rpm input.
What this model establishes
The model establishes the ideal pitch geometry and rotational speed relationships. It does not calculate torque capacity, tooth contact stress, lubrication, efficiency, shaft deflection or bearing loads.
The gear outlines use illustrative involute flanks and simplified roots. A practical assembly also needs independently supported coaxial members, sufficient axial clearance and a fixed idler support. Those construction details require a separate design.
Differential-speed gearing questions
Does the idler travel around the center?
No. Its journal is fixed to the frame; only the gear rotates.
Why does the ring need a particular tooth count?
Both meshes must fit the same center distance. With a common module, the ring needs the central count plus twice the idler count.
Do the input and ring turn in the same direction?
No. The external mesh reverses direction, and the internal mesh preserves that reversed direction.
What does a negative speed mean?
Counterclockwise rotation in the displayed end view. Positive speed is clockwise.
Can this model determine load capacity?
No. It calculates ideal geometry and speed, not a strength or service rating.
Reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, mechanism 943, printed page 240. The original identifies A as the driving pinion, B as the gear on a fixed journal and C as the internal spur gear.
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