Mathematical modelling of asymmetrical metal rolling processes
This thesis explores opportunities in the mathematical modelling of metal rolling processes, specifically asymmetrical sheet rolling. With the application of control systems in mind, desired mathematical models must make adequate predictions with short computational times. This renders generic numerical approaches inappropriate. Previous analytical models of symmetrical sheet rolling have relied on ad hoc assumptions about the form of the solution. The work within this thesis begins by generalising symmetric asymptotic rolling models: models that make systematic assumptions about the rolling configuration. Using assumptions that apply to cold rolling, these models are generalised to include asymmetries in roll size, roll speed and roll-workpiece friction conditions. The systematic procedure of asymptotic analysis makes this approach flexible to incorporating alternative friction and material models. A further generalisation of a clad-sheet workpiece is presented to illustrate this. Whilst this model was formulated and solved successfully, deterioration of the results for any workpiece inhomogeneity demonstrates the limitations of some of the assumptions used in these two models. Attention is then turned to curvature prediction. A review of workpiece curvature studies shows that contradictions exist in the literature; and complex non-linear relationships are seen to exist between asymmetries, roll geometry and induced curvature. The collated data from the studies reviewed were insufficient to determine these relationships empirically; and neither analytical models, including those developed thus far, nor linear regressions are able to predict these data. Another asymmetric rolling model is developed with alternative asymptotic assumptions, which shows non-linear behaviour over ranges of asymmetries and geometric parameters. While quantitative curvature predictions are not achieved, metrics of mechanisms hypothesised to drive curvature indicate these non-linear curvature trends may be captured with further refinement. Finally, coupling a curved beam model with a curvature predicting rolling model is proposed to model the ring rolling process. Both of these parts are implemented but convergence between them is not yet achieved. By analogy this could be extended with shell theory and a three-dimensional rolling model to model the wheeling process.