The Role of the Orthogonal Helicoid in the Generation of the Tooth Flanks of Involute-Gear Pairs With Skew Axes

Camus' concept of auxiliary surface (AS) is extended to the case of involute gears with skew axes. In the case at hand, we show that the AS is an orthogonal helicoid whose axis (a) lies in the cylindroid and (b) is normal to the instant screw axis of one gear with respect to its meshing counterpart; in general, the helicoid axis is skew with respect to the latter. According to the spatial version of Camus' Theorem, any line or surface attached to the AS, in particular any line $L$ of AS itself, can be chosen to generate a pair of conjugate flanks with line contact. While the pair of conjugate flanks is geometrically feasible, as they always share a line of contact and the tangent plane at each point of this line, they even have the same curvature, G²-continuity, when $L$ coincides with the instant screw axis (ISA). This means that the two surfaces penetrate each other, at the same common line. The outcome is that the surfaces are not realizable as tooth flanks. Nevertheless, this is a fundamental step toward the synthesis of the flanks of involute gears with skew axes. In fact, the above-mentioned interpenetration between the tooth flanks can be avoided by choosing a smooth surface attached to the AS, instead of a line of the AS itself, which can give, in particular, the spatial version of octoidal bevel gears, when a planar surface is chosen.