New Investigation on the Geometry of the Contact in Gear Generation

Based on a recently developed geometric approach to the theory of gearing that does not make use of any reference sys- tems [1], this paper presents some useful relations between the geometric properties of the enveloping surface and those of its envelope. Treating vectors as such, that is without expressing their components in any reference systems, it is possible to ob- tain compact expressions for the coefficients of the first and sec- ond fundamental forms of the envelope surface. These coeffi- cients show to be central in the determination of the contact ma- trix between mating surfaces. Moreover, since this approach is coordinate free, it is valid regardless of the reference frame ac- tually employed to perform calculations and allows a, hopefully, clearer understanding of the roles played by the intrinsic geomet- ric properties of the enveloping surface, the relative position of the gear axes and the gear ratio.