Linear theory for lateral displacements of a metal strip in a tandem cold-rolling mill with asymmetries

Asymmetries in the state of a cold-rolling mill cause lateral displacements of the metal strip, which sometimes result in a major breakdown of the mill. Non-linear model equations, previously employed, are reduced to a linear form via a first-order expansion of the plastic reduction equations, appropriate for small displacements. The continuity equations, normally applied at the entry and exit of a stand, are replaced by simpler equations applied on the neutral line. These simplifications provide explicit formulae for equilibrium lateral displacements, which lead to more general conclusions. Moreover, these formulae do not involve the compliance and reaction of the mill, but relate the displacements to strip conditions and roll-gap asymmetries alone. A strong coupling between the stands and a strong upstream influence are demonstrated. The results agree with that of a non-linear model and with data from a laboratory mill. Increasing entry tension to the first stand, up to moderate levels, has the beneficial effect of reducing the size of the lateral displacements. However, the model predicts that abnormally high strip tension or low friction can lead to divergent displacements, which would crash the mill. The results offer the potential for better control and should be of interest to mill operators.