ABSTRACT
The purpose of this chapter is to introduce the mathematical background and physical concepts for the study of friction-induced vibrations, instabilities, and self-organization. These three phenomena are inter-related. Linear vibrations and waves caused by friction can become unstable under certain circumstances. In that case, the amplitude of vibration may grow unlimitedly within the linear vibration and stability analysis. However, the linear models are based on the assumption that the amplitudes of the vibrations are small. With the growing amplitude, this assumption will not be valid after a certain time, and the vibrations will become non-linear. Non-linear vibrations are complex motions and can often lead to limit cycles such as, in the case of friction, the stick-slip motion. The stick-slip is an organized pattern in the temporal and/or spatial domain.
