Determining the geometry of noncircular gears for given transmission function

A pair of noncircular gears can be used to generate a strictly increasing continuous function $\psi(\varphi)$ whose derivative $\psi'(\varphi) = \mathrm{d}\psi(\varphi)/\mathrm{d}\varphi > 0$ is $2\pi/n$-periodic, where $\varphi$ and $\gamma = \psi(\varphi)$ are the angles of the opposite rotation directions of the drive gear and the driven gear, respectively, and $n \in \mathbb{N} \setminus \{0\}$. In this paper, we determine the geometry of both gears for given transmission function $\psi(\varphi)$ when manufacturing with a rack-cutter having fillets. All occurring functions are consistently derived as functions of the drive angle $\varphi$ and the function $\psi$. Throughout the paper, methods of complex algebra, including an external product, are used. An effective algorithm for the calculation of the tooth geometries - in general every tooth has its own shape - is presented which limits the required numerical integrations to a minimum. Simple criteria are developed for checking each tooth flank for undercut. The base curves of both gears are derived, and it is shown that the tooth flanks are indeed the involutes of the corresponding base curve. All formulas for both gears are ready to use.