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.

During the yarn unwinding from a stationary package, the yarn slides on the surface of the package before it lifts off to form a balloon. The point where the yarn begins to slide is known as the unwinding point, while the point where the yarn lifts off from the surface is known as the lift-off point. On this section of the yarn, that is, between the unwinding point and the lift-off point, the tension in the yarn drops from its value in the balloon (at the lift-off point) to its residual value, defined as the tension of the yarn inside the package. The equations of motion which govern the motion of the yarn are known: we have established them in Section

The problem of yarn motion on the package surface during the unwinding can be treated in analogy with the motion of the yarn forming the balloon between the lift-off point and the eyelet, through which the yarn is being pulled.

The yarn is being withdrawn with velocity

Mechanical setup in over-end yarn unwinding from cylindrical package.

The general equation of motion for the yarn was derived and justified in one of the previous works [

The position vector

There is a friction between the package and the yarn which is sliding on its surface before it lifts off to form the balloon. The yarn is exerting a normal force on the package (i.e., a force perpendicular to the package surface, thus in radial direction). This force is not known a priori, but must be determined as part of the solution to the full problem. The simplest expression of the friction law states that the friction force is proportional to the normal component of the force. The coefficient of proportionality is known as the coefficient of friction

The quantity

The force of friction between the package surface and the yarn.

When the yarn slides on the surface, it thus experiences the normal force

The friction law is at best a rough approximation to a more complex real behavior. In reality, the coefficient of friction depends in a complicated way on the sliding velocity [

Equation (

Strictly speaking, the yarn undergoes sliding motion on the package surface only when the unwinding point is at a certain distance away from the package edges. In such circumstances, the conditions are quasi-stationary: in the rotating coordinate frame the yarn only slowly changes its form. For this reason, in the first approximation the time dependence can be fully described by time-variable boundary conditions, while the time-derivative terms in the equation of motion can be neglected:

When the yarn slides on the package surface, its motion effectively occurs within a two-dimensional subspace. This fact can be taken into account in (

The radius vector to a point on the surface of a cylinder can be expressed as (compare with equation

The cylindrical coordinate system.

The arc-length derivatives of the radius vector are computed using the relations

Equation (

Equation (

After introducing the dimensionless angular velocity

(a) The surface of the cylinder is cut along the long edge and the surface is flattened. (b) The flattened surface is a plane with axes

Equation (

As can be easily verified, we thus obtain

Exponentiating the expression we had obtained and rearranging it slightly, we obtain

Therefore at the lift-off point

In the section of yarn which slides on the surface and experiences friction from the lower layers, the tension decreases from the value at the lift-off point to the residual value. At the same time, the angle

Equation (

The decrease of the tension along the yarn is proportional to the coefficient of friction, as expected. The larger the coefficient of friction is, the shorter is the sliding segment of the yarn. The derivative is also proportional to the angle

We have shown how the equation of motion on the package surface can be obtained from the general equation of yarn motion by considering the force of friction. The external force has two components: the normal force of the package surface and the force of friction. We have described the conditions for the validity of the quasi-stationary approximation which was then used to simplify the equation of motion to a two-dimensional problem. We have also shown that the simplification of the equation of motion for the sliding motion of the yarn to a two-dimensional problem makes it possible to establish the main conclusions analytically. We have shown how the section of the yarn which slides on the package surface makes it possible that the tension in the yarn reduces to its residual yarn and how this is related to the form of the sliding yarn. More accurate solutions of the problem can, however, only be obtained using a full numerical solution of the equations using the shooting method [