A general mechanical formulation of two-dimensional bristle friction models with a focus on the LuGre model and its dynamics

Friction is an important phenomenon in the modeling of mechanical systems. To represent the dynamic properties of friction, bristle models such as the LuGre model are used mostly in the one-dimensional case. Since friction is a nonlinear phenomenon and the respective frictional degrees of freedom influence each other, independent modeling in the spatial dimensions is physically inadmissible. The two-dimensional description of friction in bristle models proves to be complex and has already been considered in some works, but without intuitive illustration and interpretation. In this paper, an intuitive derivation of the general bristle friction model is presented, summarizing validated existing model derivations. In addition, a geometric interpretation is given using a proposed mechanical equivalence model. Subsequently, all model parameters are discussed, which highlights the potential for extensions and the benefits of flexible parametrization. Based on the general form of the bristle friction model, the special case of the two-dimensional LuGre model is derived. Furthermore, the derived two-dimensional LuGre model is applied in a simple, two-dimensional friction system and the static and dynamic properties are visualized and discussed.