An invariant approach for gear generation with supplemental motions

In some recent papers a new vector approach to the theory of gearing has been proposed by the present authors. However the formulation there presented was limited to fixed or, at most, translating axes. The present paper presents the non-trivial extension to the case of gear generation with supplemental spatial motions (helical motion, tilt motion, etc.), particularly interesting for gear generation with modern free-form cutting machines. The proposed formulation is invariant with respect to reference systems, and therefore does not need any of them. As a consequence, notwithstanding the complexity of the supplemental motions, rather simple and compact equations are obtained, making the implementation in a computer code a fairly straightforward task. As an example of application of the proposed method, the mathematical description of the face-milling process is presented.