Rigorous Methods and Numerical Results for Dry Friction Problems

It is the aim of the present section to first explain some mathematically rigorous methods that can be used to analyze dry friction problems, or non-smooth systems in general. The first concept, Lyapunov exponents, is well-accepted for smooth systems to predict the long-term behaviour, and the existence of at least one positive Lyapunov exponent is often used as the definition of “chaos”. Then in section 9.2 we will turn to a different issue, namely to verify that some system has undergone a bifurcation when some external parameter has been varied. Often a good indicator for bifurcations appearing is that some “index” changes when the parameter crosses a critical value. We will introduce the so-called Conley index and explain how it applies to detect such bifurcations. Since usually the dynamical behaviour will be quite complicated, often only numerical analysis will yield satisfactory results. In section 9.3 we will describe some phenomena observed in a single-mass friction oscillator which are due to the fact that trajectories can collide with discontinuity surfaces. Finally, Section 9-4 contains some conclusions. Most of the material is taken from [13].