###### Abstract
[27] analyzed the possibility of application of magnetorheological dampers with the aim to avoid irregular or even chaotic vibrations of rotors. The particular attention in the field of rotor dynamics is paid to the cases when the rotor increases or decreases the speed of its rotation during passing critical revolutions and to the attenuation of the induced transient vibration. The up-to-now presented solutions are aimed at tuning the stiffness of the support elements. [PERSON] et. al. [28] were engaged in the experimental investigation of vibration attenuation of a rotor passing through the critical speed. Their effort was aimed at the development of a variable stiffness damping support with high damping and large variable stiffness. [PERSON] and [PERSON] [29,30] developed a model of a rotor, the shaft of which was represented by a slender beam. Their work was concentrated on investigation of the influence of the intentional change in the support stiffness on the rotor vibration in the resonance area. They dealt with the study of this physical phenomenon, but not with the design solution. The experience obtained from the performed studies showed that efficiency of this manipulation depended not only on the magnitude and rate of the stiffness change, but also on the moment of time when the tuning started before reaching the critical rotational speed. Optimization of the technological or operation parameters of rotating machines is needed to meet rising demands put on their design or accomplish the technological processes. [PERSON] et al. [31] and [PERSON] et al. [32] reported on the optimization of the start-up procedure of large steam turbines, focusing on heat parameters. [PERSON] and [PERSON] [33], [PERSON] and [PERSON] [34], and [PERSON] et al. [35] dealt with balancing the rotors based on the utilization of different optimization methods. [PERSON] et al. [36] used an optimization technique for increasing the efficiency of vibration attenuation of an aero-engine rotor. This article deals with minimization of vibration of a flexible accelerating rotor when passing the resonance area. Unlike the mentioned publications, the vibration attenuation is controlled by means of changing damping in the magnetorheological squeeze film dampers mounted in the rotor supports. The research work is focused on determination of the damping parameters by application of an optimization method. The solution leads to solving an unconstrained optimization problem with the implicit objective function. The presented study is computational, and it is based on theoretical and experimental research. The approach is new and original. It provides a new idea to the designers and mechanical engineers regarding how to minimize amplitude of the rotor vibration when passing the critical speed. This idea supported by the results of simulations is contributive for the designers and engineers working in the field of rotors, and it has a great potential for its application, and it leads to increase in the utility value of rotating machines. In addition, the article reports on a new possibility of usage of magnetorheological squeeze film dampers in the field of rotating machines.