Roll Shape Design for Foil Rolling of a Four-High Mill

Abstract

In this paper, a mathematical model for a four‐high mill is proposed to analyze the elastic deformation of its back‐up and work rolls, including their respective axial deflections and surface flatness. The contact pressure between the back‐up and work rolls and that between the work roll and foil are functions of the roll’s position in the axial direction. In this analytical model, the rolls and foil are divided into many small regions and finite difference or matrix methods are used to derive the deflections of the work and back‐up rolls as a function of the force density in the foil width direction. A MATLAB program is developed to solve the simultaneous equations of the rolls’ deflections and flatness and to calculate the pressure distributions by iteration. A camber shape on the surfaces of the back‐up and work rolls is designed to counteract the rolls’ elastic deformation and to make the rolled foil as flat as possible. A thickness variation within 1 μm can be obtained for a foil of 50 μm thick and 40 mm wide under reduction of 32% using a tungsten carbide (WC) work roll with a camber radius of 150 m.