Three-parameter bifurcation scenarios in a minimal mechanical model with LuGre friction

The LuGre friction model is among the most frequently used models in multibody dynamical applications. It is common that the steady-state conditions of the model are described by a specific shape of the Stribeck characteristic, which is influenced by the Stribeck exponent (or shape factor) that depend on the properties of the contacting surfaces. The paper discusses the bifurcations in the case of the LuGre model with the three typical shapes of the Stribeck curve. To capture the essence of the nonlinear effects associated with friction, we consider a minimal mechanical model: a block sliding on a horizontal surface, which is subjected to a constant external force and the LuGre friction force with a viscous damping term. Despite the simplicity of the mechanical model, several equilibria of different types and bifurcations – including saddle-node and Hopf bifurcation – are pointed out. Various bifurcation scenarios are examined revealing rich phase-space structures with several coexisting equilibria and limit cycles. The paper highlights that one must be aware when choosing the parameters of the LuGre model – including the Stribeck exponent – since the bifurcation map can change significantly depending on the shape of the Stribeck characteristic.