A dynamic model for coupled torsional and bending vibrations of drillstrings is presented. The bit/formation interactions are assumed to be related to the bit motion, leading to a set of highly non-linear equations with a potential for self-excited behaviour. Impacts with the borehole wall are accounted for by a consistent contact model which is capable of capturing rolling both with and without slip of the drill collars along the borehole wall. Simulation results show that the non-linear coupling significantly affects the response. Due to the combined parametric and forcing excitations and the effect of strong non-linearities, the response is not simple or easily predictable. At certain frequencies, there is significant energy transfer between the two modes of vibrations. The simulation results agree well with laboratory and field observations when the so called stick-slip vibrations occur. This study suggests a new mechanism for this behaviour which was shown to be due to static friction or speed dependent torque on bit. © 1998 Academic Press.