Excessive vibrations of the drill strings, e.g., the stick-slip vibration, are the primary cause of premature failures and drilling inefficiencies in oil well drilling. To investigate and suppress such vibrations, this paper studies the dynamics of drill strings using a lumped parameter model, in which both the torsional stick-slip and lateral vibrations are taken into consideration. The friction torque due to the downhole bit-rock interaction, which plays a key role in stick-slip vibration, is modeled as a hysteretic dry friction function. Simulated results of this developed model are shown to have a close qualitative agreement with the field observations in terms of stick-slip vibrations. Afterwards, a sliding mode controller is applied to mitigate the undesired vibrations of drill strings. A good control performance in suppressing the stick-slip phenomenon is demonstrated for the proposed controller. However, numerical simulations also demonstrate that the control action can excite lateral instability in the system, which can result in impacts between the drill collars and the borehole wall due to the large amplitude in lateral vibrations. Thus, a proper choice of the control parameters is essential to suppress the vibrations in the drill strings. The developed lumped parameter model describing the coupled torsional and lateral response in the controlled drill strings presented in this paper can be used to aid in offline tuning of those control variables.

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