Study of drilling systems plays important role in drilling industry. During the operation mode these systems experience different kinds of vibration, which may cause malfunctioning (i.e. dissipation of kinetic energy, noise, excessive wear, machine parts premature failure etc). In this article the most common type of vibrations is considered - torsional vibrations. The two mass model of a drilling system and modified version of it, supplemented by equations of induction motor, are studied. Both systems experience so-called hidden oscillations. It is extremely difficult to analyze such hidden oscillations since they cannot be found with the help of standard analytical procedures of trajectory modeling in the equilibrium state domain. Hidden oscillations correspond to torsional vibrations in real systems, thus they may cause breakdowns.