This work is devoted to the investigation of mathematical models of drilling systems described by ordinary differential equations. Here, we continue the study done by the researchers from Eindhoven where the two-mass mathematical model of a drilling system has been investigated (Mihajlovic et al. J. Dyn. Syst. Meas. Control 126(4): 709-720, 2004; de Bruin et al. Automatica 45(2): 405-415, 2009). The modified version of this model, which takes into account a full description of an induction motor, is studied. It is shown that such complex effects as hidden oscillations may appear in these kinds of systems. These effects may lead to drill string failures and breakdowns.