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This paper presents a mathematical model to predict the distribution of yarn tension and the balloon shape as a function of spindle speed in the ring spinning process. The dynamic yarn path from the delivery rollers to the winding point on the cop has been described with a non-linear differential equation system. These equations have been integrated with a Runge–Kutta method using MATLAB software. Since the numerical solution of the equations strongly depends on initial values, an algorithm of sensitivity analysis has been developed to predict the right choice of initial values in order to find a stable solution. For model validation purposes, the yarn tension has been measured between delivery rollers and yarn guide. Furthermore, a high-speed camera has been used to capture the balloon shape at different spindle angular velocities in order to compare the theoretically determined balloon shape with the one that actually occurs on the machine.

The most common measuring method to characterise the dynamic yarn path in the ring spinning process is to measure the yarn tension, where the yarn path is almost straight. However, it is much more complex to measure the yarn tension at the other positions, for example, between the yarn guide and traveller (balloon zone) and between the traveller and winding point of the cop (winding zone), as the yarn rotates continuously around the spindle axis. In this paper, two new methods of yarn tension measurement in the balloon zone are proposed. In the first method, the balloon shape was first recorded with a high speed camera. The balloon tension was then calculated by comparing the yarn strain (occurring in the balloon zone) measured by a digital image analysis program with the stress-strain curve of the yarn produced. In the second method, the radial forces of the rotating balloon were measured by using modified measurement techniques for measurement of yarn tension. Moreover a customised sensor was developed to measure the winding tension between the traveller and cop. The values measured were validated with a theoretical model and a good correlation between the measured and theoretical values could be revealed.

Twist plays an important role to impart tensile strength in yarn during the spinning process. In the most widely used ring-spinning machine for short staple yarn production, a combination of ring and traveler is used for inserting twist and winding the yarn on cops. The main limitation of this twisting mechanism is the friction between the ring and traveler, which generates heat at higher speed and limits the productivity. This limitation can be overcome by the implementation of a magnetic bearing system based on superconducting technology, which replaces completely the existing ring/traveler system of the ring-spinning machine. This superconducting magnet bearing consists of a circular superconductor and permanent magnet ring. After cooling the superconductor below its transition temperature, the permanent magnet ring levitates and is free to rotate above the superconductor ring according to the principles of superconducting levitation and pinning. Thus the superconducting magnetic bearing (SMB) ensures a friction-free operation during spinning and allows one to increase spindle speed and productivity drastically. The yarn properties using the SMB system have also been investigated and they remain nearly identical to those of conventional ring yarns.

The new concept of a superconducting magnetic bearing (SMB) system can be implemented as a twisting element instead of the existing one in a ring spinning machine, thus overcoming one of its main frictional limitations. In the SMB, a permanent magnet (PM) ring rotates freely above the superconducting ring due to the levitation forces. The revolution of the PM ring imparts twists similarly to the traveler in the existing twisting system. In this paper, the forces acting on the dynamic yarn path resulting from this new technology are investigated and described with a mathematical model. The equation of yarn movement between the delivery rollers and the PM ring is integrated with the Runge-Kutta method using MATLAB. Thus, the developed model can estimate the yarn tension and balloon form according to different spindle speeds considering the dynamic behavior of the permanent magnet of the SMB system. To validate the model, the important relevant process parameters, such as the yarn tension, are measured at different regions of the yarn path, and the balloon forms are recorded during spinning with the SMB system using a high speed camera.Keywords mathematical modeling, yarn tension, balloon form, ring spinning, superconducting magnetic bearingIn the existing ring spinning process, the frictional heat generated in the ring/traveler system causes damage to both the twisting element and the yarn structure. 1 The traveler is not allowed to rotate at more than 50 m/s, especially in the case of man-made fibers, due to their melting, caused by the high friction-induced heating, which limits productivity. 2,3 The friction-free superconducting magnet bearing (SMB) eliminates this restriction and thus allows increase of the spindle speed much higher than with existing spinning machines. In our previous work, different concepts of SMB system have been presented, and a suitable one has been successfully integrated in a ring spinning tester. 4 The SMB system comprises of two rings, a magnetic element of Neodymium Iron Boron (NdFeB) with a yarn guide attached to it, and a high temperature superconductor (SC) from YBCO (YBa 2 Cu 3 O 7-x ) chemical compounds. The superconductor (SC) ring is cooled down below its critical temperature at a fixed distance from the PM ring. The PM ring levitates above the SC ring according to the principle of levitation. During the spinning process, the yarn (wound onto the bobbin) rotates the PM ring, instead of the traveler. The patented concept of the SMB system ensures a smooth running of the spinning process for significantly higher productivity with similar yarn properties to the conventional process. 5