Generalization Process of the Integrated Mathematical Model Created for the Development of the Production Geometry of Complicated Surfaces

Computer modelling of technical constructions is increasingly carried out using software that includes more detailed knowledge, which requires an increase in the level as well as an expansion of the scope of the geometric knowledge. A significant part of motion transmission mechanisms are worm drive pairs, for which the separation of the parts dealing with the theoretical and practical problems found in the literature can be experienced in numerous instances. Due to the different technical features, in many cases the helical surfaces are not designed and manufactured in a geometrically correct way, or the best solution is not the compulsory chosen. The geometric model describing the production process of the worm surfaces provides the basis for examining the deviation between the surface mathematically determined by the designer and the surface produced. An integrated mathematical kinematic model was developed for the production geometrical analysis of the elements of cylindrical and conical worm gear drive pairs for machining with a traditional thread grinding machine, which causes a serious pitch fluctuating error among several other problems in the case of machining the conical worm. Modelling of the production process of surfaces and the simultaneous study of the manufacturing errors is basically performed with the toolbox of descriptive geometry, including the use of the projective invariants. Knowing the inheritance of the invariants of projective geometry, the aim was the mathematical generalization of the integrated model and the creation of a projective relationship between the reference surfaces of conical and cylindrical spiral surfaces. As a result, the improved constructive geometric model was created, in which the method of analytically creating the projective geometric relationship between the reference surfaces of conical and cylindrical helicoid surfaces has been described for the first time in this article. This procedure is considered the most important result of the present article. Another significance of the further development presented is that during production of the conical helicoid surface, the thread pitch fluctuation has been eliminated. The results obtained, consisting of an improved geometric model, lead to a new geometry of the technological environment regarding the relative position of the cutting tool and the workpiece as well as the relative motion between them.