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 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

The productivity of the conventional ring spinning process is currently limited by the frictional heat that occurs in the ring/traveler twisting system. In the framework of a fundamental research project from the German Research Foundation (DFG), the levitation principle of superconducting magnetic bearing (SMB) was implemented as a twisting element in order to eliminate the frictional problem and thus aim, at least, to double the productivity. A mathematical model of the dynamic yarn path has already been presented considering the friction free SMB system up to an angular spindle speed of 25,000 r.p.m. In this paper, the existing theoretical model, which was developed up to 25,000 r.p.m, was further modified considering the balloon control ring and yarn elasticity at a higher angular spindle speed, such as 50,000 r.p.m. The model was solved numerically using the RUNGE-KUTTA method. With this model, it is possible to estimate the yarn tension distribution and balloon form considering the above-mentioned parameters. The model established was further validated by comparing the yarn tension and balloon forms predicted with measured ones up to an angular spindle speed of 15,000 r.p.m in a ring spinning tester based on superconducting magnetic bearing.

Edge detection is one of the most fundamental necessities in image processing. Usally, edge detection algorithms are based on integer order differentiation operators. In many applications it is essential to perform a robust edge detection also to noisy input image data with low SNR as well. Thereby, integer based differentiation operators are often not leading to sufficient detection results. For this purpose an edge detector based on fractional order differentiation is introduced, which can significantly improve the detection performance to noisy images. Furthermore, a real application scenario of fractional order based edge detection is given within a modular railway track measurement system.