Necessary and sufficient conditions for passivity of the LuGre friction model

Friction is a nonlinear phenomenon difficult to describe analytically, To capture its effect in mechanical systems, a bristle-based dynamical model, known as the LuGre model; was recently proposed in the literature. It is difficult to assess whether this (or any other) mathematical model constitutes a bona fide friction model, It should, however, reflect the dissipative nature of friction, which mathematically translates into the requirement of defining a passive operator from velocity to friction force, In this note, we give necessary and sufficient conditions for this property to hold for the LuGre model. The conditions are expressed in terms of a simple algebraic inequality involving the parameters of the model, If this inequality does not hold, we construct an input signal that generates a periodic orbit along which the passivity inequality is violated.