Unwinding of yarn from stationary helically wound cylindrical packages is of fundamental importance in many textile processes. From the mathematical point of view, the unwinding yarn is a complex non-linear system with multiple solutions and bifurcation phenomena. The yarn is described as a flexible yet non-extensible uniform string moving through space under the action of the air drag and takes the shape of a "balloon". In this paper we derive a very general system of partial differential equations that describe the motion of this string during unwinding. We provide a physical interpretation of individual terms in the equations of motion with a particular emphasis on virtual forces which appear in the rotating coordinate system. We show that, in addition to the centrifugal and Coriolis forces, a third virtual force needs to be taken into account in rotating coordinate systems with time-varying angular velocity.

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.

We derive a system of coupled nonlinear differential equations that govern the motion of yarn in general. The equations are written in a (non-uniformly) rotating observation frame and are thus appropriate for description of over-end unwinding of yarn from stationary packages. We comment on physical significance of virtual forces that appear in a non-inertial frame and we devote particular attention to a lesser known 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, and 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.

Abstract-Yarn unwinding from a package is important in many textile processes. The stability of the unwinding process has a direct influence on the efficiency of the process and on the quality of the end product. During the unwinding, the tension is oscillating. This is especially noticeable in over-end unwinding from a static package, where the yarn is being withdrawn with a high velocity in the direction of the package axis. The optimal form of the package allows an optimal shape of the yarn balloon and low and steady tension even at very high unwinding velocities.The purpose of this work is to write down the equations that describe the motion of yarn during unwinding and to construct a mathematical model whichwould permit to simulate the process of unwinding.

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.