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A frequently used technological solution to reduce lateral vibration of rotating machines is placing damping devices between the rotor and its stationary part. To achieve their optimum performance, their damping effect must be controllable. This is offered by magnetorheological squeeze film dampers. Their main parts are two concentric rings between which is a thin layer of magnetorheological oil. The inner ring is coupled with the rotor journal by a rolling element bearing and by a squirrel spring with the damper housing. Lateral oscillation of the rotor squeezes the oil film which produces the damping force. As the flow of magnetorheological oils is affected by a magnetic field, the change of magnetic flux generated in electric coils is used to change the damping effect. The magnetorheological lubricants belong to the class of liquids with a yielding shear stress. Its substance is the resistance againg the flow caused by a chain structure of ferromagnetic particles formed due to acting of a magnetic field. This forming is a process called delayed yielding. The developed mathematical model of the magnetorheological squeeze film damper is based on assumptions of the classical theory of lubrication, the magnetorheological oil is represented by a bilinear material, dependance of the stationary value of the yielding shear stress on magnetic induction is approximated by a power function and its dependence on time by a convolutory integral. The pressure distribution in the full oil film is governed by the Reynolds equation that has been modified to bilinear material. In cavitated areas it is assumed that the pressure remains constant and the yielding shear stress drops to zero. The influence of the delayed yielding of magnetorheological oil on behaviour of the rotor was studied by computational simulations. In the computational model (Fig. 1) the dampers were represented by springs and nonlinear force couplings. As evident from Fig. 2 and 3, the delayed yielding phenomenon has influence on the rotor vibration. The extent and character of this effect depends especially on the delayed yielding time constant characterizing rapidity of the oil response, speed of the rotor rotation, the critical velocity and magnitude of the applied current. The highlights and principal contributions of the carried out research work are the development of the mathematical model of the magnetorheological squeeze film damper based on representing the magnegnetorheological oil by bilinear material, the mathematical description of the delayed yielding phenomenon and its implementation in the mathematical model of the damper, the confirmation of sufficient stability of the developed computational procedures and obtaining new pieces of knowledge on influence of the delayed yielding phenomenon on efficieny of the magnetorheological damping devices.