Mathematical model for two-dimensional dry friction modified by dither

Abstract

A new dynamic two-dimensional friction model is developed that is based on the bristle theory. Actually, it is the Reset Integrator Model converted into a two-dimensional space. Usually, two-dimensional friction models are in fact one-dimensional models that are rotated into a slip velocity direction. However, this common approach cannot be applied to the bristle model. That is why the idea of a two-dimensional bristle is presented. The bristle’s deformation is described using polar coordinates. The carried-out numerical simulation of a planar oscillator has proved that the new model correctly captures the mechanism of smoothing dry friction by dither applied in both a perpendicular and co-linear way regarding body velocity. Furthermore, the introduced mathematical model captures two-dimensional stick-slip behaviour. Cartesian slip velocity components are the only inputs to the model. In addition, our proposed model allows one to describe friction anisotropy using bristle parameters. The paper contains the results of an experimental verification of the new friction model, conducted with a special laboratory rig employed to investigate a two-dimensional motion in the presence of dither as well as to validate our numerical results.

A review of friction damping modeling and testing

“…This approach works for static friction models but cannot be physically justified for dynamic models. Effectively, the recent work of Wijata et al [60] mentions that it cannot be applied to models with an internal state. As stated earlier, models which incorporate the bristle deformation analogy such as LuGre and its elastoplastic extension have an internal state.…”

“…As stated earlier, models which incorporate the bristle deformation analogy such as LuGre and its elastoplastic extension have an internal state. Wijata et al [60] extend the reset integrator model of friction, which is bristle based, to work for plane motion. They do so by changing the state variable to a state vector having radial and rotational components, which represent strain and rotation of the bristle, respectively.…”

“…They provide good experimental correlations for stickslip transition in 2D for displacements that are, however, much larger than typical presliding displacement. The relative recency and consequent absence of adoption by other authors of the approach by Wijata et al [60] indicate that care must be taken before assuming it can be easily implemented into any type of friction modeling problem. Furthermore and at the least, numerical instabilities as those pointed out by Barber and Wang [5] may require to be dealt with prior to attempting such an implementation into generic multibody codes.…”