Recursive determination of conjugate profiles in planar motion with application to gear design

A general procedure for determining the envelope of the outline of planar rigid figures is presented which allows both local and finite undercutting to be taken into account. A suitable parametrization of the enveloped curves, together with the insertion of cusps wherever undercutting occurs, enables the procedure to be applied recursively while maintaining high accuracy in describing the resulting envelopes. The proposed procedure provides the basis for an original validation method regarding parallel-axis spur and helical gears. The method takes into account possible undercutting during hobbing or shaping, considers the desired amount of meshing clearance over the entire tooth profile, and is able to detect meshing interference. Further, it provides a criterion in choosing the proper cutter profile. A numerical example showing the application of the method is discussed at the end of the paper.