Solutions of the equation of meshing for planar gear profiles

Abstract

This paper illustrates the possibility of obtaining an explicit solution for the equation of meshing for any type of regular two-dimensional curve of a planar gearing, in order to compute the conjugate profile of the curve. The analysis of the equation of meshing permits to express its explicit solution when the relative motion between the considered reference systems occurs with a constant transmission ratio (for example internal or external gears and rack). The solution is expressed with the general conventions of the theory of gearing. In this way, when we have a generating curve in homogeneous coordinates, and the transformation matrixes which define the relative motion between the profiles considered, we can directly find the solution of the equation of meshing, and consequently the profile conjugated to the generating one by a simple coordinate transformation. The method here presented can be successfully applied to the profiles of planar gears and rotary machines. © Springer Science+Business Media B.V. 2009.