The eccentric wheel train in Hiscox no. 978 uses a fixed, inclined elliptical bevel wheel A, a carrier C, and a carried shaft H joining wheels B and D. D drives wheel F and output shaft E. The source describes irregular, reversing output. The exploded animation identifies these parts; the calculator explores a declared effective-ratio model because the source provides no dimensions or tooth law.
Eccentric Bevel-Wheel Train · Conditional Ratio Calculator
Identify the fixed wheel and carried compound shaft. Explore signed output speed using the stated effective-ratio law.
Equation Used
- Fixed A and carrier-mounted compound shaft H follow the source arrangement.
- Periodic ratio law is a selected reconstruction assumption.
- Constant input speed and ideal kinematics.
- Exploded gear bodies do not depict tooth contact.
Signed results distinguish forward motion, reversal and net drift.
A fixed wheel and a carried compound shaft
Wheel A stays fixed at an inclination that lets B clear F. Carrier C is attached to the input crank shaft. B and D are locked to shaft H, which the carrier takes around the main axis. D drives F and output E.
The animation separates the gear bodies so that the fixed wheel, carrier, shared shaft and output are visible. It is an exploded assembly schematic, not a tooth-contact simulation. The two compound wheels share their shaft phase; A does not rotate.
Explore output reversal and drift
Change carrier speed, noncircular ratio parameter and final-stage effective ratio. The calculator reports signed speed limits, mean speed and net angle per input revolution. Positive and negative speeds indicate opposing directions. At a final-stage ratio of 1 the chosen law produces a rocking output; changing that ratio adds a net rotational drift.
The parameter ε controls a selected periodic ratio law. It is not a measured historical wheel eccentricity, bevel angle or tooth count. These results can support a conceptual motion study, but cannot specify a gear to manufacture.
The declared effective-ratio model
Let θ be carrier angle, n its speed in rpm, ε the selected ratio parameter and k the effective final-stage ratio. Set q(θ)=(1−ε²)/(1+ε²+2ε cos θ), and define the continuous phase Φ by dΦ/dθ=q and Φ(0)=0. Φ advances one full turn for each carrier revolution.
The selected output relation is α=θ−kΦ, so signed output speed is n(1−kq). Over one cycle q ranges from (1−ε)/(1+ε) to (1+ε)/(1−ε). Mean output speed is n(1−k); net angle is 360(1−k) degrees per input revolution.
For k=1, peak-to-peak rocking angle is 4 asin ε in radians. For other k, the output drifts and the calculator does not label it as a bounded oscillation. Reversal occurs only when the signed minimum is negative and the signed maximum is positive.
This is an explicit ellipse-inspired effective-ratio assumption. It is not a derived meshing law for the historical inclined bevel wheels. The illustration supplies insufficient data for that synthesis.
20 rpm, ε=0.25 and ratio 1
The selected compound-to-carrier speed ratio ranges from 0.6 to 1.666667. At 20 rpm input, signed output speed ranges from −13.333 to +8 rpm. Mean speed and net angle per carrier revolution are zero; peak-to-peak output swing is about 57.910°.
Setting ε to zero and k to 1 makes the output stationary. At k=0.5 or 1.5 the mean output becomes +10 or −10 rpm respectively, while the variable part still follows the selected law.
What this reconstruction establishes
The source establishes which wheel is fixed, the inclined arrangement, the compound shaft and the reversing output. It does not give pitch surfaces, tooth counts, shaft offsets, load ratings or the exact variable-ratio function. Those missing details are not inferred from the appearance of the drawing.
The calculator and dials agree with the stated analytical model, while the separated gear bodies explain the assembly. Contact geometry, conjugate tooth design, interference, bearing loads and efficiency require a separate engineered design. The former positive-only offset-radius formula did not represent reversing output and has been replaced.
Eccentric wheel-train questions
Why can speed be negative?
The compound action can reverse the output relative to its initial direction. Signed speed makes that visible.
Does the fixed wheel rotate?
No. A stays fixed while C carries shaft H and wheels B and D around it.
Are the displayed teeth intended to mesh?
No. This is an exploded assembly view. Gear-body marks identify rotation, not manufactured tooth flanks.
Is this the exact historical speed curve?
No. It is a selected, clearly stated effective-ratio law. Recovering the historical curve requires additional gear geometry.
Primary reference
Gardner D. Hiscox, Mechanical Movements, Powers, Devices and Appliances, no. 978, printed page 247: inclined fixed elliptical bevel wheel and carried compound shaft.
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