# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Prof. Arde
Guran**, Institute for Structronics, University Ottawa,
will give a public talk (followed by a discussion) on **Wed, March 17, 2004**
at **16:00 o'clock** at **HS 10**
on the topic of
"Mathematical Studies of Friction and Contact Mechanics"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series A - General Colloquium:

The intention is to present general information not only to experts, but also to students and guests from outside the mathematical institutes.

### Mathematical Studies of Friction and Contact Mechanics

In everyday life one becomes aware of friction only in such drastic instances as when somebody slips on polished stairs or has to look at the ice on the street from the ditch into which he has fallen. A more detailed look shows that friction results in a nonlinearity which is abundant in nature, machines, structures, transportation systems, and other processes.

The economic losses due to friction and wear have been estimated at 5% of the gross world product. Friction and impact had been a topic of technological attention long before the dawn of science, and still are hot topics in mathematics, physics and engineering research today. As nonlinearity, friction poses challenges to the dynamical systems researcher.

For one, friction is very difficult to model, since the underlying mechanism is not entirely understood. In the case of dry friction the friction characteristic is a set valued mapping which yields discontinuities in the accelerations as functions of time. So, most of the dynamical systems theory for smooth systems is not applicable to a frictional system.

As time progressed, and people yearned to understand machines and nature, the awareness of friction eventually made its way into scientific thought. Friction laws were formulated, and the ideas of dissipation and motion were conceived. It took incredible creativity and ingenuity for the first people of a community to recognize properties of friction, and then implement them in the design of a new machine.

Later, researchers began to question the mechanism of friction. This continues to the present day. In this lecture, we look at the history of the mathematical nonlinearity called friction. We start in prehistoric times when the manifestations of friction caught the attention of the ancients as they developed critical technologies. This lecture is blended with several desktop demonstrations. It ends with a brief review of friction in modern nonlinear dynamics.