The development of superconducting magnetic bearings as the twisting element in ring spinning renews the study of the dynamic model of the ring-spinning process. In this paper, the variational equations governing yarn in the balloon are developed using Hamilton’s principle. The balloon curves are represented by the Bezier function, and the semi-analytical steady-state balloon is obtained by the Galerkin method. The linearized equation of motion is completed by elimination of the nonlinear displacement terms and cancellation of the steady-state terms. The displacement field of the balloon yarn about the steady state is expressed by the Bezier polynomial, and the natural frequencies of the balloon yarn are calculated by the Galerkin method. The semi-analytical solution of the steady-state balloon and the natural frequencies of the balloon yarn are in good agreement with those in the experimental measurement. The influences of rotational speed, balloon height, and yarn count on the natural frequencies of the balloon yarn are investigated. For a realistic parameter set, the possible resonances caused by the change of balloon height, which is due to the up-and-down movement of the ring rail, are discussed in more detail.

The development of superconducting magnetic bearings as the twisting element in ring spinning renews the study of the dynamic model of the ring spinning process. In this paper, the variational equation describing the steady motion of the yarn is developed using Hamilton’s principle, the balloon shape is represented by the Bezier polynomial, the undetermined coefficient is solved by Galerkin method, and the semi-analytical solution of the steady-state balloon shape is obtained. Digital images of the balloon yarn are collected by binocular cameras and a stroboscope, and the balloon curve is reconstructed by stereo vision technology. The calculated balloon curve is in good agreement with the measured balloon curve. The balloon shapes and the radial forces acting on the permanent magnetic ring at different radii of the wind-point on the bobbin are investigated by the semi-analytical model when the spindle is spinning at different speeds. The solution procedures developed can be used to explore the vibration and stability of the system, consisting of the magnetic stiffness, permanent magnetic ring, and balloon yarn in the superconducting magnetic bearing twisting mechanism.