J Jaap Molenaar

Mathematical modeling and dimensional analysis

We present in §2 a general introduction on mathematical modeling. Irrespective of the case under consideration, the process of mathematical modeling has a general structure. See also references [1,–,6]. This structure comprises a sequence of stages followed again and again. An important ingredient of the modeling process is dimensional analysis . In §3 the main ideas of this technique are presented.

The Optimal Collar

We deal with the design of a part of certain packaging machines. In these machines the packaging material (paper or plastic sheet) is unrolled from a horizontal cylinder and folded against the inner side of a vertical, hollow cylinder. During this folding process, the sheet passes over a curved surface, the so-called ‘collar’ or ‘shoulder’. In Fig. 1 the geometry is drawn. After a piece of sheet has been positioned inside the vertical cylinder, it is sealed at the bottom and the side and filled by dropping the product to be packed from above into the newly formed bag. Then, the bag is drawn downwards, sealed at the top and cut off. This technique allows for packaging at high speed (hundreds of bags per minute), but is sensitive to disturbances. The curvature of the shoulder should be such, that the sheet is nowhere stretched or torn. In the literature a mathematical description of possible surfaces is given by Mot [1] and Culpin [2]. The former author constructs the shoulder out of pieces of a plane and a cone.