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The main factors affecting the elastic deformation of the rolling mill include roll bending, memorial archway deformation, roll flattening, deformation of other components, and compression deformation of the hydraulic cylinder and the liquid inside the cylinder. Among these, the first three factors account for the vast majority of the rolling mill's elasticity, with roll deformation accounting for the largest proportion. Roll bending and roll flattening can be solved using the theory of roll elastic deformation. For the roll deformation of a common four-high rolling mill, the following assumptions are made:

① The rolling process is axisymmetric, so only half of the work roll and backup roll need to be analyzed;

② The flattening between rollers and the flattening between the work roller and the rolled piece are analyzed using Nakashima's modified semi-infinite body theory;

③ The rolled piece is isotropic and incompressible;

④ During the rolling process, the entire length of the work roll and backup roll maintains full contact. Since the pressure is directly applied to the backup roll bearing seat through the screw down or hydraulic cylinder, the bending and vertical movement of the backup roll axis corresponding to the force point reflect the change in roll gap position. The change amount is denoted by Ys, as shown in equation (1), Ys = Yd + Ywb-f0 + Yws-f0 ·(1) where Yd is the deflection deformation of the backup roll axis; Ywb-f0 is the center flattening amount between rolls; Yws-f0 is the center flattening amount of the work roll

Deformation L1 and L2 at the position of the backup roll during screw-down — the positions of the backup roll axis before and after deformation; P — the force on the screw-down screw; Y f0 — the displacement of the backup roll axis; Yr — the deflection of the backup roll axis. This is shown in the force model of the rolling process for a conventional rolling mill. By calculating the elastic deformation of the roll using the influence function method, the lateral distribution of contact pressure between the work roll and the backup roll, the lateral distribution of contact pressure between the work roll and the rolling piece, and the lateral distribution of exit plate thickness can be determined. By changing the work roll radius Rw, backup roll radius Rb, work roll crown Cw0, backup roll crown Cb0, rolling piece width Ws, and the magnitude of the rolling force F0, the effects of these factors on the elastic deformation of the roll can be obtained.