A Study on Spatial Cycloid Gearing

Understanding the geometry of gears with skew axes is a complex task, hard to grasp and to visualize. However, due to Study’s Principle of Transference, the geometric treatment based on dual vectors can be readily derived from that of the spherical case. This paper is based on Martin Disteli’s work and on the authors’ previous results where Camus’ concept of an auxiliary curve is extended to the case of skew gears. We focus on the spatial analogue of the following case of cycloid bevel gears: When the auxiliary curve is specified as a pole tangent, we obtain ’pathologic’ spherical involute gears with vanishing pressure angle. The profiles are always penetrating at the meshing point because of G2-contact. In view of the Camus Theorem, the spatial analogue of the pole tangent is a skew orthogonal helicoid Π4 as auxiliary surface. Its axis lies on the cylindroid and is normal to the instant screw axis (ISA). Under the roll-sliding of Π4 along the axodes Π2 and Π3 of the gears, any generator g of Π4 traces a pair of conjugate flanks Φ2,Φ3 with permanent line contact. Again, these flanks are not realizable because of the reasons below: (1) When g coincides with the ISA, the singular lines of the two flanks come together. At each point of g the two flanks share the tangent plane, but in the case of external gears the surfaces open toward opposite sides. (2) We face the spatial analogue of a spherical G2-contact, which surprisingly does not mean a G2contact at all points of g but only at a single point combined with a mutual penetration of the flanks Φ2 and Φ3. However, when instead of a line g a plane Φ4 is attached to the right helicoid Π4, the envelopes of Φ4 under the roll-sliding of Π4 along Π2 and Π3 are torses that serve as conjugate tooth flanks Φ2,Φ3 with a permanent line contact. So far, it seems that these flanks,Φ2 andΦ3, are geometrically feasible. This is a possible spatial generalization of octoidal gears or even of planar involute gears.