Fixed causality slip-stick friction models for use in simulation of non-linear systems

Slip-stick friction occurs when the relative velocity between sliding surfaces approaches zero and the surfaces become ‘stuck’, requiring a force larger than the sliding friction force to break the surfaces loose, allowing sliding to resume. Mathematically, these physics are an example of ‘ideal switching’ where the velocity is zero and the force is determined by other parts of the system, or the force is set by the friction model (and could be zero), and the velocity is determined by other parts of the system. A switch in an electric circuit is another example. Including ideal switches in an overall physical system model is complicated by the inversion of causality when the switch occurs. In one state the velocity is prescribed and the force is determined, and in the other state the force is prescribed and the velocity is determined. Such causal inversions create formulation and computational problems, and these problems can be quite prohibitive if many switches are part of the model. This paper presents fixed causal models for slip-stick friction that allow a single state space model to be used regardless of the number of switches. Such a development allows simulation of multiple plate brakes and clutches, or ideal rectifiers, using an explicit first-order state space representation. It should be noted that there has been extensive work in the development of models that represent the physics of friction. One such model is the LuGre model [1] where microstructural displacements are modelled. Our intent here is not to extend the physics of slip-stick friction, but rather to reasonably represent the physics while providing a computationally convenient method for including slip-stick friction in overall system models.