The conventional ring-spinning process has been used for over a century, being one of the most used processes in the textile industry. However, this process has some disadvantages at high spindle speeds, such as friction in the traveler/ring components that generates heat, resulting in yarn breakages or lower-quality production. New technological advances like superconducting magnetic bearings (SMB) systems have been implemented to face the limitations in the spindle speed. The SMBs use superconductors and permanent magnets (PM) to achieve magnetic levitation, decreasing in this way drastically the friction and enabling spindle speeds up to 50 000 rpm. Previous studies have investigated SMB dynamics, and it has been proved that the mathematical model for a six-dimensional motion can be reduced into a second-order ordinary system in matrix form that consists of mass, stiffness, and damping matrices, but due to the freedom given by the magnetic levitation, there is the presence of oscillations in the permanent magnet due to external forces. A recent study implemented an Eddy Current Damper (ECD) based on copper rings that increase the damping coefficients, resulting in the reduction of oscillations in the permanent magnet. This work analyzes the frequency response of the PM ring in an SMB system with ECDs of varying copper thicknesses, considering all motion directions. Theoretical results show that tilting modes exhibit frequency splitting due to gyroscopic effects. Experimental tests using laser sensors validated the model, and parametric identification was performed to estimate stiffness, damping, and external torque contributions.