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<p style="text-align: center;"><img src="/ueditor/php/upload/image/20260724/1784884399115385.png" title="1784884399115385.png" alt="1.png"/></p><p><span style="font-size: 14px; font-family: arial, helvetica, sans-serif;">Are noncircular, elliptical, and oval gears all the same thing? Noncircular Gears Noncircular gears are found as early as the sketches of Leonardo da Vinci in his Codice Atlantico, documenting his work from 1478 to 1519. The gear community is, of course, familiar with this; refer to, e.g., Aaron Fagan’s Addendum column “Noncircular Gears: The Unicorn of Machine Technology” in the June 2022 issue of Gear Technology. Their use is limited; they are a niche component due to their complexity of design and manufacturing. The advent of servo motors solved the problem of controlled motion in a more general approach, reducing the need for noncircular gears further. Still, they remain high-performing mechanical solutions. The dedicated book by Faydor L. Litvin et al, Noncircular Gears: Design and Generation (Cambridge University Press, 2009), is widely known and gives a dense overview of the topic. In the context of this paper, we will limit the observations to convex elliptic, Figure 2 and oval gears Figure 3. Gears 1 and 2 have the same properties. These two types of noncircular gears transform motion between parallel axes, where a continuously changing transmission ratio results within a single revolution. Unlike linkage mechanisms but similar to cam-follower mechanisms, elliptical and oval gears provide a compact arrangement for delivering periodic speed variations, making them suitable in e.g., packaging systems, textile machines, flow meters, or heavy press machines, with a slow working stroke and a fast return stroke. This document focuses on the differences between elliptical gears and their oval cousins. In extremis, both are transformed into circular gears, Figure 1.</span></p><p><span style="font-size: 14px; font-family: arial, helvetica, sans-serif;">The defining feature of noncircular transmissions is the momentary (instantaneous) ratio i. In circular gears, the ratio i is constant and may be determined from the number of teeth, Figure 4. In elliptical, Figure 5, and oval gears, Figure 6, it is a continuous function of the polar angle of rotation φ: i(φ) = ω2 / ω1 = r1(φ) / r2(ψ). With the sum E = r1 + r2 = constant, the ratio is a function of the driver’s radius alone i(φ) = r1(φ)/(E - r1(φ)).</span></p>