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Tension in the yarn and its oscillations during the over-end unwinding of the yarn from stationary packages depend on the unwinding speed, the shape and the winding type of the package, the air drag coefficient, and also the coefficient of friction between the yarn and the package. The yarn does not leave the surface package immediately at the unwinding point. Instead, it first slides on the surface and then lifts off to form the balloon. The problem of simulating the unwinding process can be split into two smaller subproblems: the first task is to describe the motion of the yarn in the balloon; the second one is to solve the sliding motion. In spite of the seemingly complex form of the equations, they can be partially analytically solved as we show in the paper.

We study the motion of yarn modelled as a one-dimensional inelastic string. In textile production, the yarn is being withdrawn from cross-wound packages in warping and weft insertion. During unwinding, there appear forces in the yarn that are approximately proportional to the square of the unwinding velocity. The yarn tension is not constant, but it oscillates within some interval. This is especially noticeable in over-end unwinding from a static cross-wound package. Even when the yarn is not strongly stressed, so that the tension never exceeds a few percent of the breaking strength, the yarn can still break sometimes. The production process requires as large warping and weaving speeds as possible; therefore, it is necessary to improve our understanding of the cross-wound package unwinding and to find the necessary modifications of the yarn unwinding process. In addition to empirical tests, it has proved useful to study yarn unwinding by mathematical modelling and computer simulations. We state the equati...

Mathematical modeling can be used to simulate the unwinding of yarn from packages of different shapes. This method can be applied to design packages that can sustain high unwinding velocities at low and steady tension in the yarn. In the case of conic packages the angular velocity of unwinding depends not only on the winding angle as is the case for cylindric packages, but also on the apex angle. We will show that the dimensionless angular velocity depends very little on the apex angle. The apex angle, however, also determines the effective radius of the package at the lift-off point, therefore the angular velocity can be proportionally higher. We will compare unwinding from a cylindrical and a conic package with equal smallest radius and show that unwinding from the conic package is faster due to higher average radius of the package at the lift-off point.

Stability of the yarn unwinding directly affects the efficiency of the textile production process and the quality of the final product. A package with an optimal shape will result in an optimal shape of the balloon. In addition, the yarn tension will be small and steady even at high unwinding velocity. Computer modeling is a valuable tool in the search for the optimal package shape. We demonstrate a mathematical model for simulating the unwinding from cylindrical and conic packages. We show how the winding angle and the apex angle influence the angular velocity of the yarn during the unwinding. Since the centrifugal forces on the yarn in the balloon depend on the angular velocity, this velocity has a large influence on the tension that we wish to reduce.