Elliptical Spur Gears: Animation and Variable-Speed Calculator

Two fixed shafts and a moving pitch contact

The input gear turns steadily. Its longer and shorter pitch radii successively meet the other gear, causing the output to speed up and slow down. The output turns in the opposite direction.

The shafts are at the foci of equal ellipses. For shaft spacing C, each ellipse has semi-major axis a=C/2 and semi-minor axis b=a√(1−e²). The geometric centre of each gear therefore travels around its fixed shaft as the gear rotates.

The dashed curves are the pitch ellipses. The green point is their instantaneous rolling contact, not the complete tooth-contact path. The coloured radial segments meet at that point and always add to the shaft spacing.

Explore a variable-speed gear pair

Use eccentricity to change the amount of speed variation and input revolution time to set its rate. Shaft spacing scales the gears and both pitch radii without changing the speed ratio. The initial angle chooses the starting position.

The live output speed and both pitch radii follow the rolling relationship of the focal ellipse pair shown in the animation.

Pitch geometry determines the speed

Let θ be the input angle, measured from its long-radius position. The input pitch radius is r1=a(1−e²)/(1−e cos θ). The output pitch radius is r2=2a−r1. Equal velocities at the pitch contact require the output-speed magnitude divided by input speed to be r1/r2.

Thus the displayed speed multiplier is (1−e²)/(1+e²−2e cos θ). Output rpm is this multiplier times 60/T. Its direction is opposite to the input.

A continuous output angle over one input revolution is β=−2 atan2((1+e) sin(θ/2),(1−e) cos(θ/2)). It runs from 0 to −2π. Both gears complete one revolution per cycle, although their instantaneous speeds differ.

The maximum multiplier is (1+e)/(1−e), and the minimum is its reciprocal. This focal ellipse pair has one fast interval and one slow interval per revolution, not two of each.

Default pair: 100 mm shaft spacing

At e=0.3, C=100 mm and T=6 s, the input turns at 10 rpm. At the initial 0° position, r1=65 mm and r2=35 mm. The output therefore turns at approximately 18.57 rpm in the opposite direction.

At 180°, the radii exchange values and the output slows to about 5.38 rpm. Reducing eccentricity reduces this variation. Changing C to 200 mm doubles both radii but leaves the speed multiplier unchanged.

What the tooth illustration represents

The outline uses 47 equally spaced positions measured along the pitch ellipse, with flanks generated from a rolling 20° rack. Small flank clearance and relieved roots make the engagement visible. Tip and root transitions are illustrative and are not a cutter-ready tooth definition.

The calculation is a kinematic model. It does not calculate contact stress, torque capacity, backlash under load, undercutting limits or manufacturing tolerances. Those depend on a complete tooth and material specification. Tolerances, wear and machine suitability depend on the specified gears, materials and operating loads.

KISSsoft describes the general rolling-pitch-line and rack-generation principles used for noncircular gears. The particular focal-ellipse equations and displayed example are given above so the calculated result can be checked directly.

Elliptical spur-gear questions

Why are the shafts off centre?

They are at the ellipse foci. Mounting these same shapes at their geometric centres would not produce the displayed fixed-spacing conjugate pair.

Why does a larger input radius make the output faster?

The output contact radius becomes smaller while the two pitch-contact velocities remain equal.

Does the output make more revolutions?

No. It completes one opposite revolution per input revolution, passing through faster and slower intervals.

Can these outlines be sent directly to a manufacturer?

No. They are a visual tooth-envelope illustration, not a production gear specification.