Non-circular gear

Ignoring the gear teeth for the moment (i.e. assuming the gear teeth are very small), let ${\displaystyle r_{1}(\theta _{1})}$ be the radius of the first gear wheel as a function of angle from the axis of rotation ${\displaystyle \theta _{1}}$, and let ${\displaystyle r_{2}(\theta _{2})}$ be the radius of the second gear wheel as a function of angle from its axis of rotation ${\displaystyle \theta _{2}}$. If the axles remain fixed, the distance between the axles is also fixed:[3]

${\displaystyle r_{1}(\theta _{1})+r_{2}(\theta _{2})=a\,}$

Assuming that the point of contact lies on the line connecting the axles, in order for the gears to touch without slipping, the velocity of each wheel must be equal at the point of contact and perpendicular to the line connecting the axles, which implies that:[3]

${\displaystyle r_{1}\,d\theta _{1}=r_{2}\,d\theta _{2}}$

Each wheel must be cyclic in its angular coordinates. If the shape of the first wheel is known, the shape of the second can often be found using the above equations. If the relationship between the angles is specified, the shapes of both wheels can often be determined analytically as well.[3]

It is more convenient to use the circular variable ${\displaystyle z=e^{i\theta }}$ when analyzing this problem. Assuming the radius of the first gear wheel is known as a function of z, and using the relationship ${\displaystyle dz=iz\,d\theta }$, the above two equations can be combined to yield the differential equation:

${\displaystyle {\frac {dz_{2}}{z_{2}}}={\frac {r_{1}(z_{1})}{a-r_{1}(z_{1})}}\,{\frac {dz_{1}}{z_{1}}}}$

where ${\displaystyle z_{1}}$ and ${\displaystyle z_{2}}$ describe the rotation of the first and second gears respectively. This equation can be formally solved as:

${\displaystyle \ln(z_{2})=\ln(K)+\int {\frac {r_{1}(z_{1})}{a-r_{1}(z_{1})}}\,{\frac {dz_{1}}{z_{1}}}}$

where ${\displaystyle \ln(K)}$ is a constant of integration.

• Noncircular Gears: Design and Generation by Faydor L. Litvin, Alfonso Fuentes-Aznar, Ignacio Gonzalez-Perez, and Kenichi Hayasaka

• Historic Video of Non-Circular Gears on YouTube
• The Eye of an Artist
• Kinematic Models for Design Digital Library (KMODDL)
• The Gear Oscillator at the Wayback Machine (archived 2013-06-05)
• Laczik- Involute Profile of Non-Circular Gears
• "Gear geometry and applied theory" by Faydor L. Litvin and Alfonso Fuentes
• A paper on designing of non-circular gears
• Maurice Lacroix Masterpiece Regulateur Roue Carree. Squaring the Circle.