An eccentric circular-gear train produces a changing output speed even when its input shaft turns uniformly. In Hiscox figure 950, gear B is fixed eccentrically to driver A. Pinion C moves on two pivoted arms to stay meshed with B and output gear D. The input and output shafts remain fixed while the idler moves. The animation and calculator use the same illustrative geometry.
Irregular Circular Motion Interactive Calculator
Change the eccentric offset, input period and selected input angle. The animation follows the complete cycle; the numerical tiles describe the selected angle and sampled cycle limits. Gear sizes and fixed shaft locations are illustrative, not dimensions taken from the historical drawing.
Equation Used
The two large gears have 45 teeth and 45 mm pitch radius; the idler has 25 teeth and 25 mm pitch radius. Module is 2 mm. Each center-distance arm is 70 mm. Input shaft A=(40,45) mm and output shaft D=(−55,−35) mm in the drawing plane. α is the direction from B to C and β from C to D. The ratio includes the motion of these arms. Cycle limits are sampled every 0.5°; no loads, losses or backlash are calculated.
Three circular gears and two moving arms
Driver A rotates about a fixed shaft. The center of gear B is offset from that shaft, so B both spins and moves around A. One arm connects the centers of B and idler C; a second connects C to the fixed center of D. Their lengths hold the two gear meshes at the sum of the corresponding pitch radii.
C must move as B orbits. That carrier motion changes the relationship between the gears’ absolute angular speeds. D turns continuously in this example, but speeds up and slows down during each input revolution. The large background disc represents the driver carrying B; the outlined gears and connecting arms occupy separate planes.
A geometric way to vary rotary speed
A varying rotary speed can be useful where part of a machine cycle needs to be faster than another part. The arrangement demonstrates that circular gears can produce such variation when their centers move on constrained arms.
The required motion law and mechanical loads determine whether this arrangement suits an application. The example does not specify a production machine, torque capacity, operating speed limit or manufacturing tolerance.
Include the carrier motion in the speed ratio
Let θ be the input angle and also the spin angle of B. For an external mesh on an arm whose angle is α, ideal rolling gives rBωB + rCωC = (rB+rC)α̇. For the second mesh, rCωC + rDωD = (rC+rD)β̇. Eliminating the idler spin gives rDωD = rBωB + (rC+rD)β̇ − (rB+rC)α̇.
Here rB=rD=45 mm and rC=25 mm, so dividing by input speed gives the displayed ratio. The arm derivatives are found by differentiating the two fixed center-distance constraints. This includes both gear spin and center movement; applying an ordinary fixed-center tooth ratio alone would miss the speed variation.
Input speed is 60/T RPM. Multiplying it by the instantaneous ratio gives D’s speed at the selected input angle. The animation starts from that angle and continues through the cycle; Pause and the Animation slider inspect its moving pose.
Example: 10 mm offset and a four-second input turn
With 10 mm eccentric offset, a 4 s input period gives 15 RPM at A. At the selected input angle of 0°, the model gives an output/input speed ratio of about 1.40429 and an output speed of 21.06 RPM.
Sampling the complete input revolution at 0.5° intervals gives ratios of approximately 0.49895 to 1.50105, or 7.48 to 22.52 RPM. D still completes one revolution per input revolution in this equal-large-gear example. Its changing speed, rather than a changing average turn count, is the useful distinction.
What the illustration establishes
The model preserves both 70 mm arm lengths, the fixed input and output shafts, and the rolling relationships at the gear pitch circles. Gear outlines use an illustrative 20° involute form. The historical figure establishes the arrangement, not these dimensions or tooth details.
A real design also needs bearing support, clearance between parts in adjacent planes, tooth-root geometry, lubrication and load analysis. Those are not supplied by the animation. The offset range stays within the illustrated linkage’s continuous-motion geometry; it is not an operating limit for actual equipment.
Frequently asked questions
Are these elliptical gears?
No. This example follows Hiscox figure 950, which uses circular gears. Its variable speed comes from the eccentric motion of B and the constrained motion of C.
Which centers stay fixed?
The driver shaft A and output center D stay fixed. Gear B orbits A, and idler C moves while remaining 70 mm from both B and D.
What happens at zero offset?
B becomes concentric with A. The gear centers stop moving, and equal-size gears B and D give a constant 1:1 output/input speed ratio.
Does the output reverse or dwell?
Not over the offsets offered in this illustrative model. The minimum ratio remains positive, so D continues in the input direction.
Why does the moving ratio differ from the numerical tile?
The drawing follows the moving animation angle. The tiles report the angle selected by the Calculated input angle control. Pause and scrub the animation to inspect its cycle.
Are the fast and slow ratios exact extrema?
They are numerical samples at 0.5° input-angle intervals. The instantaneous ratio uses the differentiated geometric constraints; the limit search has finite angular resolution.
Can this calculator size the gears or bearings?
No. It models rigid-body motion and ideal rolling. It does not calculate torque, tooth stress, bearing reactions, wear or service life.
Mechanism references
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