Preface

The following case study considers how tension varies in space and time in a continuous sheet of paper (commonly called a web) running at high speed through a newspaper press. The model will consider two different parts of the problem in isolation. First, those portions of the paper that are travelling over various rollers are treated, and second, the spans of paper between the rollers are considered. Modelling the first problem requires basic mechanics and considers the forces within the paper and the friction acting against the roller. Practical limiting cases result in a simple ordinary differential equation for the tension as a function of position. When we consider the spans of paper, the model is again mechanical in nature but requires modelling of the stretch within the paper. For practical situations, the resulting model is again an ordinary differential equation which is nowtime-dependent. The two models are then combined to show how rollers interact and how the tension changes along the paper's length. The mathematics used is relatively simple; the emphasis is on generating a model which is as simple as possible, but which mimics the physical situation. The problem is well suited to mathematical modelling courses because of the wide variety of different aspects of a printing machine that can be considered, and also because the mathematical models that emerge from the modelling process are similar to classical mechanics problems.

The models presented here will require some knowledge of basic mechanics. Hooke's law for a linear elastic material, conservation of mass, and conservation of linear and angular momentum give us the evolution equations of the systems.