Optimisation of a developable surface model passing through a helical curve with variable pitch

Insufficient consideration of the developable helical surfaces in engineering practice complicates their manufacture and design, especially for variable pitch surfaces. The purpose of the article was to develop an algorithm for designing a helical surface of variable pitch and its mathematical implementation. For this purpose, the methods of differential geometry of curved lines and surfaces were used, as well as the MatLab software environment for computing, data analysis, visualisation and development of algorithms for constructing surfaces based on the results obtained. The basis for the surface construction was a spiral line of variable pitch, which can be specified by various dependencies. The task was to draw a set of rectilinear surface components through this spiral line with a vertical axis, provided that it was a developable surface. An additional condition was that these lines must be inclined at a constant angle to the horizontal plane, i.e. the receptive surface must be a surface of equal inclination of the lines. Usually, the unfolding surface is defined by a spatial curve – an edge of inverse. The set of straight-line tangents to the back edge forms the developable surface. However, in practical problems, it is important to ensure that the scan surface passes through a given curve, for example, a helical line. It has been established that a set of reamer surfaces of the same slope of the constituent parts with different specified angles can be drawn through a given helical line. It was proved that it is easy to obtain a surface compartment bounded by two coaxial cylinders, one of which has a given helical line. The results of the study can be used to improve the technology of manufacturing screws in agriculture, food, mining and construction industries