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Yarn unwinding from a package is a key problem in many textile processes, such as weft insertion and warping. Stability of the unwinding has a direct influence on the efficiency of the entire textile process and the quality of the final product. The quality of the yarn is numerically expressed mainly in terms of mechanical quantities. In the unwinding process, viscoelastic properties are the most important. They depend on how the yarn is stressed. The quality of the yarn that is being unwound should not be reduced, unless this reduction does not significantly lower the quality of the fabric. We strive to achieve as large warping and weaving speeds as possible; therefore, our aim is to improve the theory of cross-wound package unwinding and to find the necessary modifications of the yarn unwinding process. The goal of our contribution is to state the equations of motion that describe the unwinding yarn.

SummaryIn the production of fabric, the unwinding of thread occurs in the warping and weft insertion processes. In order to achieve low and constant tension of thread or yarn it is necessary to optimize the process of unwinding. Computer simulations are now in use for this purpose, so it is important to obtain a mathematical description of yarn motion. This article is devoted to the derivation of boundary conditions that considerably affect the form of the balloon. In this way, a mathematically well defined model of yarn unwinding will be obtained which could be solved by using the tools of numerical mathematics. The unwinding of yarn from an optimally designed package can be simulated and this knowledge can be used to find an optimal design of packages.

In this paper we discuss the general equation of motion for yarn in a rotating coordinate system. This equation is often used to describe the motion of yarn that is unwinding from packages. The rotating coordinate system is non-inertial and the equation of motion therefore contains fictitious forces. We comment on the physical significance of fictitious forces that appear in a non-inertial frame and we devote particular attention to a less known Euler force that only appears in non-uniformly rotating frames. We show that this force should be taken into account when the unwinding point is near the edges of the package, when the quasi-stationary approximation is not valid because the angular velocity is changing with time. The additional force has an influence on the yarn dynamics in this transient regime, where the movement of yarn becomes complex and can lead to yarn slipping and even breaking.