Analytical description of adjustment of rolls for manufacturing parts from elastic sheet material

The object of research is the process of bending sheet material, taking into account its springiness. When manufacturing sheet parts by bending from a completely elastic sheet, its shape is completely restored after the deformation stops, unlike an elastic sheet. Thus, when producing cylindrical parts by drawing between three rolls, the resulting radius of the cylindrical part will be larger than the calculated one. This phenomenon is evaluated by the coefficient of springing – the ratio of the calculated radius to the one obtained after partial expansion. When manufacturing conical parts, this approach cannot be applied, because the value of the radius is variable. The article applies the theory of surface bending from differential geometry. The curvature of the line on the surface has two components - normal and geodesic. When the surface is bent, the normal component changes, while the geodesic component remains unchanged. The magnitude of the normal component depends on the angle between the origin of the cone and its axis. So, for a cone with a base of radius R and an angle of 20°, the normal curvature is 0.94/R, and the geodesic curvature is 0.34/R. For cylindrical parts, the geodesic curvature of the cross-section (circle) is zero, so it is not necessary to take it into account. Usually, adjustment of rolls for the production of conical parts is carried out experimentally. The difference of the proposed approach lies in the elimination of this problem thanks to the decomposition of the curvature of the base of the cone into two components. This allows to calculate the settings of the rolls and thereby reduce their adjustment time. The parameters of the rolls and their mutual placement are calculated for the production of conical parts of the required size, taking into account their springiness. The field of application of the obtained results is the production of parts by bending flat metal sheet blanks