Toward computerized determination of envelope to family of parametric planar curves

This paper introduces an original analytical procedure for the computation of the planar curve in meshing with a general two dimensional curve. Considering a planar gear mechanism with a constant transmission ratio, in this work a general expression of the explicit solution of the equation of meshing is presented. Besides, an original method to evaluate the undercutting conditions, based on the explicit solution of the equation of meshing, is shown. Taking into account a generating curve in homogeneous coordinates and the transformation matrixes which define the relative motion between the profiles, the solution of the equation of meshing and consequently the profile conjugated to the generating curve can be directly found, by a simple coordinate transformation. The explicit solution is expressed with the general conventions of the theory of gearing. Once the equation of the conjugated curve is obtained, a direct method for the evaluation of the undercutting condition can be applied. The proposed method allows to implement effective and affordable computer codes for the computation of conjugate planar profiles.