Approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure

A systematic procedure is explained for finding both stable and unstable periodic stick-slip vibrations of autonomous dynamic systems with dry friction. The motivation for this work is the torsional stick-slip vibration that occurs in oil exploration drill strings. In the procedure, the discontinuous friction forces are approximated by smooth functions. Using the simple shooting method and a path-following algorithm, branches of periodic solutions are computed for a changing design variable. Both one- and two-degree-of-freedom autonomous block-on-belt models and a one-degree-of-freedom autonomous drill string model are tested using the procedure.