Generation and curvature analysis of conjugate surfaces via a new approach

In the first part, the alternative formulation of the theory of gearing presented in [F. Di Puccio, M. Gabiccini, M. Guiggiani, Alternative formulation of the theory of gearing, Mechanism and Machine Theory 40 (5) (2005) 613–637] is generalized to cope with the case of modified roll and translating axes of the mating surfaces, that is with a setting typical of some computer numerically controlled (CNC) machine. The peculiarity of the proposed approach is that it always deals with vectors as such, thus avoiding the need of a chain of reference systems, necessary when components are used as in the traditional method. It results in a more compact and clearer formulation. In the second part the proposed approach is applied to obtain the normal curvatures and geodesic torsions of two conjugate surfaces in a way that fully exploits the rigid-body relative motions involved in gear generation. In order to prove the ease of use and effectiveness of the proposed approach, the curvature analysis of a spiral bevel pinion generated on a CNC face-milling machine is performed.