Mathematics and Skis

In contrary to Herbert, who is an avid cross-country skier, I try to avoid those damgerous activities (UnRisk them, so to say.)

Anyway, in the late 1980s, the world production of cross-country skis was almost exclusively in Austria, and we (i.e. the Industrial Mathematics Institute, where I did my PhD) had a project with one of the Austrian ski producers.

At that time, cross-country skis were frequently produced from wood. In the above figure, the ski consists of three layers, which, in the course of the production process, are laid upon the adjustable metal basis shape (here: the bottom sine wave) an then glued together. The glue hardens at a certain temperatue (around 90 centigrade) which is achieved during a heating and pressing procedure.

As the different layers have different thermal elongation properties, the final shape of the ski differs from the basis shape (this is the same effect utilized in bimatallic strip thermometers). The question to us was "How can we adjust the basis shape in such a way that the final ski (at snow temperature) matches a prescribed bending shape?" A classical inverse problem.

The forward problem would then be: Given the basis shape, calculate the bending of the ski.

My solution approach for the forward problem was the following:
Assume that each layer is a (thermoelastic) beam. Put layer 1 (the yellow one) on the basis shape. Calculate the top curve of layer 1 by doing some enveloping. The top curve of layer one is then the basis shape for the red layer 2. And so on.

Close the press and heat it. As long as the glue has not hardened, assume that the layers can slide against each other without friction. Each layer should assume the position with minmal deformation energy. At gluing temperature fix the layers immediately against their neighbours.

Remove temperature and press and find the snow shape of the ski by minimizing the combined deformation energy of the glued layers.

My next post will discuss the success of this approach.

For those who cannot wait:
Andreas Binder: Adjustment of a ski--pressing--machine, ECMI Newsletter, No. 7, 1990