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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 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.

The finite element (FE) approach constitutes an essential methodology when modelling the elastic properties of structures in various research disciplines such as structural mechanics, engine dynamics and so on. Because of increased accuracy requirements, the FE method results in discretized models, which are described by higher order ordinary differential equations, or, in FE terms, by a large number of degrees of freedom (DoF). In this regard, the application of an additional methodology, referred to as the model order reduction (MOR) or DoF condensation, is rather compulsory. Herein, a reduced dimension set of ordinary differential equations is generated, i.e. the initially large number of DoF is condensed, while aiming to keep the dynamics of the original model as intact as possible. In the commercially available FE software tools, the static and the component mode syntheses (CMS) are the only available integrated condensation methods. The latter represents the state of the art generating well-correlated reduced order models (ROMs), which can be further utilized for FE or multi-body systems simulations. Taking into consideration the information loss of the CMS, which is introduced by its part-static nature, the improved CMS (ICMS) method is proposed.Here the algorithmic scheme of the standard CMS is adopted, which is qualitatively improved by adequately considering the advantageous characteristics of another MOR approach, the so-called improved reduction system method. The ICMS results in better correlated reduced order models in comparison to all the aforementioned methods, while preserving the required structural properties of the original FE model.

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Various research areas in the field of vehicle modelling, structural mechanics, engine dynamics, microelectromechanical systems (MEMS), etc. require the utilization of both multibody system formalism (MBS) and finite element method (FEM) in order to sufficiently capture the model's dynamics. The FEM-MBS coupling is accomplished by reducing the dimension of the FE-modelled part and then importing it into an MBScode for further simulation. When using commercial FEM (Nastran, ANSYS, etc.) as well as MBS (SIMPACK) software packages the necessary standard input data (SID) file is needed for the coupling procedure (FEMBS interface). A problem arises by the restriction that both commercial FEM and MBS codes support only two condensation methods (Guyan reduction and component mode synthesis (CMS)), thus disabling the direct application of any other reduction approach (e.g. from the field of control theory) that actually could be better. In this article, the theoretical background of an implemented FEM-MBS interface (MORPACK) is presented allowing the application of any kind of reduction method for FE-modelled structures and furthermore their import (Ritz approximation) into SIMPACK via the SID file generation. A benchmark problem (UIC60-rail) is used in order to capture in SIMPACK the discrepancy between the standardized CMS and the Krylov subspace method (KSM), as one of the alternatives offered by the interface.

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