One dimensional self-alignment of a mass using stick-slip vibrations

In this paper, one-dimensional self-alignment of a rigid mass via stick-slip vibrations is studied. The mass is situated on a table, which has a prescribed periodic motion. Friction is exploited as the mechanism to move the mass in a desired direction and to stop and self-align the mass at a desired end position with the smallest possible positioning error. In the modeling and analysis of the system, theory of discontinuous dynamical systems is used. Analytic solutions can be derived for a model based on Coulomb friction and an intuitively chosen table accelerationprofile, which allows for a classification of different possible types of motion. Next, near thedesired end position, the Coulomb friction coefficient is increased (e.g. by changing the roughness of the table surface) in order to stop the mass. In the transition area from low friction to high friction coefficient, it is shown that, under certain conditions, accumulation of the mass to a unique end position occurs. This behavior can be studied analytically and a mapping is given for subsequent stick positions.