Taking into account the thickness of the sheet material in the manufacture of a conical disk by bending a flat ring

Abstract

In the manufacture of parts from sheet material is its deformation. For example, conical and spherical tillage discs have different working surfaces. In the first case, the surface is developable, in the second - nondevelopable. The bending of the sheet material into the unfolding surface can be described analytically
using the appropriate theory of differential geometry. It is based on the fact that the length of the lines on the surface and the angles between the lines do not change during bending, and the surface itself has zero thickness. This theory gives fairly accurate results when bending sheet material of small thickness, such as a sheet of paper. Working bodies that carry a certain functional load must have the appropriate thickness of the material. They can be considered as a shell bounded by inner and outer surfaces. When bending such a shell, the theory gives only approximate results. The problem of bending a flat workpiece of a given thickness into a conical disk is considered in the article.
The workpiece in the form of a flat ring has a sharp edge of the outer periphery. It is assumed that after bending the ring into the finished product, the sharpening angle will change. It is necessary to find the angle of sharpening of the workpiece, so that after bending the conical disk acquired a given angle of sharpening. The proposed model of bending is based on the fact that the length of the pointed edge (blade) does not change during bending, and the two surfaces passing through this edge do not change their area, ie they bend as surfaces of zero thickness. A mathematical model describing such bending has been developed. According to the obtained analytical description, the intermediate bending positions of both surfaces with a common line of intersection are constructed. The formula for finding the angle of sharpening of the workpiece for a given angle of sharpening of the finished product is obtained. According to the developed model, the thickness of the sheet on the finished product will be less than the thickness of the flat workpiece. The developed model is one of the possible ones when describing the process of bending sheet material taking into account its thickness.