This work is devoted to the investigation of oscillations in a drilling system using an induction motor with wound rotor as a drive. It is motivated by the problems of drilling rig failures in the oil and gas industry. The study is based on a modified version of the mathematical model of a drilling rig proposed by scientists from the Eindhoven University of Technology. The model of drilling rig developed in this work takes into account full description of the rotor geometry of induction motor. It is shown that such complex effects as hidden oscillations may appear in this kind of systems. To damper these oscillations a control strategy based on changing the external additional resistance in the rotor circuit is suggested.