The elliptical gear is defined by a pitch curve (centrode) as a standard ellipse. A significant kinematic difference exists in circular gears rotating around an eccentric axis: to maintain a constant center distance E while rolling without slip, a pair of identical elliptical gears must rotate about one of their foci, O1 or O2, rather than their geometric centers O, as with circular gears. This requirement stems from a geometric property of the ellipse that for any point on the pitch curve, the sum of the distances to the two foci is constant and equal to the major axis length 2 * a. When the rotation centers are placed at the foci, the sum of the instantaneous radii of the two ellipses, r1 + r2, equals the axial distance E = r1(φ) + r2(ψ), φ and ψ being the rotational angles of the two ellipses. For identical elliptical gears where one revolution of the driver corresponds to one revolution of the driven gear, the center distance is E = 2 * a, where a is the major axis length.

In polar coordinates, the elliptical centrode rotating about its focus is r(θ) = p / (1 – e * cos(θ)) with p = a * (1 - e2), a is the semi-major axis, and e = c / a, the eccentricity, and c is the distance from the ellipse center to the focal point. This rotation about the focal point results in a single, smooth speed fluctuation cycle per revolution, Figure 5. With the eccentricity chosen, the ratio spread is 10.1515 / 0.0985 = 103. The dynamics in a real-world application will be challenging. The momentary ratio course is a function of a / b (major to minor half axis), as shown below for one of the horizontal axes, Figure 6. For a / b = 1, we have a circle; the ratio is then constant at I = 1.00, see the left front edge of the color plot. In dashed red, the current design is shown. As the ellipse gets slimmer and slimmer (a / b increasing), the ratio spread goes towards several orders of magnitude (note that the vertical axis is logarithmic). Oval Gears Rotate Around Their Center While a elliptical gears rotating around the focal points result in one speed cycle per revolution, industrial applications may require multiple speed cycles. This requirement led to the development of the oval gear, which is technically a modified elliptical gear characterized by multiple lobes (in our case, just two). Oval gears rotate about their geometric centers O. The polar equation for the oval gear centrode is r1(φ) = p / (1 + e * cos (2 * N)). A two-lobed oval gear (N = 2) will produce two ratio cycles per revolution of the driving gear, Figure 7. This symmetry ensures that the gears are balanced, a critical factor for the high-speed rotation. For oval gears, if the ratio between maximum radius Rmax = max(r1(φ)) and minimum radius Rmin = min(r1(φ) ) is too high, the centrode turns partially concave, Figure 8. Concave centrodes are more difficult to manufacture; they require a shaping cutter-type tool (or e.g., wire erosion) as opposed to a rack-type tool. The centrode remains convex if the oval gear has radii that fulfill the condition Rmin > Rmax * (1-2 / N2), or, with N = 2, Rmax / Rmin < 2.0. In Figure 10, the geometry for Rmax / Rmin = 2.0 is shown. If the ratio exceeds this limit, the centrode has concave areas, Figure 8.

Gear Generation with a Rack-Type Tool Tooth geometry is generated with a rack-type topping tool, Figure 11. The cutter (manufacturing profile shift is applied for backlash) has a straight reference line that rolls on the centrode without slip. The contact point is the momentary center of rotation for the rack. The movement of the rack equals the arc length of the centrode when the contact point travels from the start position to the current position. For noncircular gears to operate continuously, the circumference L of the centrode must be exactly equal to an integer number of teeth, z, times the pitch, L = m * z, where m is the module chosen to satisfy this condition. Elliptical gears require an odd number of teeth. This ensures that if a tooth is centered on, say, the left major half-axis, a gap is present on the right major half-axis. Into this gap, the tooth on the left major half-axis of the mating gear will fit. Note that the major axes are aligned at the start. For oval gears, the number of teeth must be even but not divisible by four. This ensures there are two teeth on the major axes and two gaps on the minor axes, again fulfilling the meshing condition, as at the start of the mesh, the gear major axes are arranged perpendicularly.

Calculations were implemented using Python in the Google Colab environment.