Superconducting magnetic bearings (SMB) based on a combination of high temperature superconductors and permanent magnets enable the realization of self-stabilized high-speed devices with significantly reduced friction. However, external vibration might couple in the bearing resulting in large amplitude oscillations due to a resonance case. A dedicated eddy current damper (ECD) might be used to eliminate these oscillations for a stable operation. The influence of such damping elements was studied for a frictionless SMB twisting system designed to speed up the conventional ring spinning process. Therefore, conductive copper rings with different thicknesses were implemented at different positions into the bearing setup as ECD. Afterward, the SMB setup was analyzed during acceleration using an array of laser distance sensors to record the displacement of the levitating permanent magnet ring in radial and axial direction, respectively. Simultaneously, a numerical model was developed to investigate the influence of the ECDs on the dynamic and static behavior of the SMB in more detail. It was shown that the simulated damping coefficients are in good agreement with the measured values, which allows further optimization of the ECD with the developed numerical model.