MATHEMATICAL MODELING OF THE ELASTIC SHEET METAL BENDING PROCESS USING ROLLERS

Abstract

The article is devoted to solving a relevant scientific and technical problem - the development of a mathematical apparatus for the precise adjustment of rolling equipment in the production of conical parts from sheet metal, taking into account the springback effect. The process of bending sheet material is one of the most common methods of forming in mechanical engineering; however, its accuracy is significantly limited by the properties of the elastic recovery of the workpiece shape after unloading. The paper provides a detailed analysis of the differences between the rolling of cylindrical and conical surfaces, emphasizing the need to coordinate the geometric parameters of the forming rollers and the workpiece blank.

The scientific novelty of the research lies in the application of differential geometry and spherical trigonometry methods to describe the deformation of developable surfaces. The author justifies the use of the "fictitious cone" concept, the parameters of which (fictitious normal curvature and fictitious apex angle) are calculated based on the material's springback index. It is mathematically proven that during bending, the geodesic curvature of a line on the surface remains unchanged, which allowed for the derivation of a system of equations to determine the adjustment parameters of the middle roller.

The article presents the parametric equations of the fictitious cone and proposes a methodology for transitioning from linear to angular dimensions using a unit radius sphere. This enables the determination of the exact spatial position of the upper forming roller through the rotation angle of its axis. The practical significance of the work is confirmed by specific calculation examples for various roller configurations, illustrating the possibility of obtaining identical parts with different equipment settings. The proposed dependencies allow not only for the automation of the design process for blanks in the form of flat rings but also significantly improve the quality of finished products through biased bending. The research results can be implemented at enterprises using plate-bending machines for the production of complex-profile parts.

Key words: mathematical modeling, elastic bending, sheet metal, rolling (plate rolling), conical parts, springback effect, fictitious cone.