Abstract
Helical blades are increasingly used in the designs of forage harvesting equipment. They demonstrate better cutting characteristics compared to traditional flat ones, but their manufacture is complicated by the non-developable surface of the surface. The problem of creating an accurate flat workpiece for such knives necessitates the need for a mathematical description of their geometry. The purpose of this study was to determine an analytical method for constructing a flat workpiece for a helical knife, considering the minimum resistance during plastic deformation of the workpiece. To achieve this goal, differential geometry methods were used, in particular, vector analysis of helical surfaces, construction of a Frenet trihedron, and analysis of the first quadratic form of the surface. It was established that the working surface of the knife is a straight open helicoid, which can be bent into a surface of revolution without changing the first quadratic form. Parametric equations of bending of the knife surface using a variable parameter describing the process of transforming the helicoid into a one-sheeted hyperboloid of revolution were constructed. It is proven that the latter is approximated with high accuracy by a truncated cone, the sweep of which is determined by the design parameters of the knife. Formulas for calculating the geometric dimensions of the sweep from the known parameters of the knife, in particular the radii of the bases and the height of the truncated cone, are obtained. It is shown that the length of the blade arc and the central angle that outlines the workpiece enable precise determination of its shape. The practical value of the study consists in the creation of an effective method for constructing the most accurate flat workpiece for the manufacture of a helical knife, which allows minimising the resistance during forming, reducing the labor intensity, and increasing the accuracy of manufacturing parts of grinding drums