In this work, the finite-element method (FEM) is used to develop the governing equation of motion of the working roll of a four-high rolling mill and to study its vibration due to different process parameters. The working roll is modeled as an Euler Bernoulli beam by taking beam elements with vertical displacement and slope as the nodal degrees-of-freedom in the finite-element formulation. The bearings at the ends of the working rolls are modeled using spring elements. To calculate the forces acting on the working roll, the interaction between the working roll and the backup roll is modeled by using the work roll submodel, and the interaction between the working roll and the sheet is modeled by using the roll bite submodel (Lin et al., 2003, “On Characteristics and Mechanism of Rolling Instability and Chatter,” ASME J. Manuf. Sci. Eng., 125(4), pp. 778–786). Nodal displacements and velocities are obtained by using the Newmark Beta method after solving the governing equation of motion of the working roll. The transient and steady-state variation of roll gap, exit thickness profile, exit stress, and sheet force along the length of the strip have been found for different bearing stiffnesses and widths of the strip. By using this model, one can predict the shape of the outcoming strip profile and exit stress variation which will be useful to avoid many defects, such as edge buckling or center buckling in rolling processes.

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