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In this paper, the concept of a model reference adaptive control of a sensorless induction motor (IM) drive with elastic joint is proposed. An adaptive speed controller uses fuzzy neural network equipped with an additional option for online tuning of its chosen parameters. A sliding-mode neuro-fuzzy controller is used as the speed controller, whose connective weights are trained online according to the error between the estimated motor speed and the speed given by the reference model. The speed of the vector-controlled IM is estimated using the MRASCC rotor speed and a flux estimator. Such a control structure is proposed to damp torsional vibrations in a two-mass system in an effective way. It is shown that torsional oscillations can be successfully suppressed in the proposed control structure, using only one basic feedback from the motor speed given by the proposed speed estimator. Simulation results are verified by experimental tests over a wide range of motor speed and drive parameter changes.