Quasi-analytical Describing Function for the Karnopp Friction Model

A quasi-analytical Describing Function (DF) for the Karnopp stick-slip friction model is presented. The proposed DF has additional phase lag at low frequencies in comparison to the DF of the Coulomb friction model. The new DF is more appropriate to describe quasi-linear stick-slip friction behavior in the frequency domain, and to predict limit cycling. The additional phase lag is shown to be dictated by the static-to-dynamic friction ratio, and it is generally limited by a constant value of about 32.5 deg. This fact can facilitate robust control design to avoid low-frequency limit cycling. The reliability of the DF was experimentally verified by the common sine-scan technique which is based on direct computation of the first Fourier coefficients. The quasi-analytical form of the new DF allows for rapid computation, and could therefore be of use in real-time control algorithms.