Automated synthesis of surface generation equations with application to gearing

A new method for automation of synthesis of generation equations of a machined surface for the general case of machining is proposed. The concepts describing an interrelation among the topological, geometric and kinematic features of the generating system are introduced. The synthesis of the relations among the parameters of the generating process is performed by the following operations: (1) the form-shaping (FS) function (i.e. position vector-function of the machine tool) is synthesized; (2) the redundant kinematic parameters within the FS function are elucidated and linked through analysis of the rank of the gradient of the FS function; (3) an enveloping order of the generation process is calculated through analysis of the number of degrees of freedom; and (4), depending on the enveloping order and the dimensionality of the set of generating points of the cutting tool, six standard representations of enveloping equations are formed. The intermediate analytical procedures are automated using the 'Mathematica' system. The setup of the machine tool for grinding a helical surface of the K-worm is computed as a real-world application.