A Spatial Version of Octoidal Gears Via the Generalized Camus Theorem

Understanding the geometry of gears with skew axes is a highly demanding task, which can be eased by invoking Study's Principle of Transference. By means of this principle, spherical geometry can be readily ported into its spatial counterpart using dual algebra. This paper is based on Martin Disteli's work and on the authors' previous results, where Camus' auxiliary curve is extended to the case of skew gears. We focus on the spatial analog of one particular case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain “pathologic” spherical involute gears; the profiles are always interpenetrating at the meshing point because of G2-contact. The spatial analog of the pole tangent, a skew orthogonal helicoid, leads to G2-contact at a single point combined with an interpenetration of the flanks. However, when instead of a line a plane is attached to the right helicoid, the envelopes of this plane under the roll-sliding of the auxiliary surface (AS) along the axodes are developable ruled surfaces. These serve as conjugate tooth flanks with a permanent line contact. Our results show that these flanks are geometrically sound, which should lead to a generalization of octoidal bevel gears, or even of bevel gears carrying teeth designed with the exact spherical involute, to the spatial case, i.e., for gears with skew axes.