Set-valued anisotropic dry friction laws: formulation, experimental verification and instability phenomenon

Abstract

Many technical applications, such as brakes and metal forming processes, are affected by anisotropic frictional behavior, where the magnitude and the direction of the friction force are dependent on the sliding direction. Existing dry friction laws do not sufficiently describe all relevant macroscopic aspects of anisotropic friction, and the influence on the dynamics of mechanical systems is largely unknown. Furthermore, previous experimental work on anisotropic friction is limited and the fact that the friction force is not always acting parallel to the sliding direction is often neglected. In this paper, an anisotropic dry friction law with the capability to describe the nonsmooth behavior of stick and slip and allowing for non-convex but star-shaped sets of admissible friction forces is formulated using tools from convex analysis. The formulation of the friction law as normal cone inclusion enables the direct implementation in numerical time-stepping schemes. The stability of systems with anisotropic friction is studied and an eigenvalue analysis reveals that the anisotropic friction law is in theory capable of causing anisotropic friction-induced instability. In addition, experimental setups for detailed investigations of the frictional behavior are described. The measurements reveal complex shaped force reservoirs and confirm the validity of the presented friction law. Finally, it is shown that the presented friction law leads to a more accurate prediction of the motion of nonsmooth mechanical systems.