Energy losses and wear in the support elements of high-speed rotating machines can be reduced by mounting the rotors in superconducting bearings. Their operation is based on a stable magnetic levitation utilizing interaction between permanent magnets connected to the shaft journal and cooled superconducting material of the second kind embedded in the bearing housing. Vibration attenuation of the rotating part is offered by application of contactless electrodynamic dampers working on the principle of eddy currents. The research conducted was focused on analysis of dynamical behavior of horizontal rotors supported by superconducting bearings. The frozen mirror image method was used to determine magnetic forces acting between the superconducting material and the permanent magnets. The performed research showed that the superconducting bearings have the needed load capacity and the electrodynamic dampers sufficient efficiency.