The effects of the tension on the yarn dynamics

Abstract

Yarn unwinding is essential in many textile processes. To be able to describe yarn unwinding, we had to develop firstly a general equation of motion for the yarn in the balloon. We also had to take into account residual tension of the yarn in the interior of the package. In this contribution, we therefore complemented the usual treatment of yarn unwinding by deducing from the general equation of motion of yarn during unwinding an equation for the tension in yarn and by deducing residual tension from this equation. Instead of perturbation theory, we used quasi-stationary approximation. Using approximations, we will derive the tension equation in the unwinding yarn, which describes the relevant physical phenomena. In this way, we will get a yarn unwinding model that is well mathematically defined and can be solved with the tools of numerical mathematics.

Keywords:

• Quasi-stationary approximation
• yarn tension
• tension in balloon
• residual tension

1. Introduction

The theory of yarn unwinding from a stationary package in its current form was developed in the last century, first with the fundamental work of D.G. Padfield in which she modified Mack’s equations for the balloon (Mack, 1953; Padfield, 1956) and added a term that describes the systemic Coriolis force. In her work (Kothari & Leaf, 1979a; Padfield, 1958), she obtained a solution for a single balloon in yarn unwinding under stationary conditions. It also obtained a solution for calculating the shapes of multiple balloons with a winding angle greater than zero for cylindrical and conical packages. The theory was re-derived by Kothari and Leaf (Fraser et al., 1992; Kothari & Leaf, 1979b) by including the influence of the gravitational force and the tangential component of the air resistance force. Based on numerical calculations, they found that the influence of these two forces can be neglected. In the following, the development of the theory was significantly influenced by Fraser et al (Batra & Fraser, 2015; Clark et al., 1998, 2001; Fraser, 1992, 1993; Fraser et al., 1992; Fraser & Stump, 1998; Stump & Fraser, 2000). In his work, he used perturbation theory and proved that time dependence can be eliminate from the equations of motion for the balloon and found that with stretchable yarn the tension in the balloon is less than its diameter. This effect is found to be small for the range of elastic constants encountered in conventional yarns. Kong et al (Kong et al., 1999) carried out extensive experimental research in the late 1990s and the results proved that the theoretical findings were correct. This opens the way for Mathematical modeling in the search for optimal yarn movement in textile processes. In our previous work (Praček et al., 2012, 2016), we presented how computer simulation allows us to predict the dynamic properties of the yarn and the mechanical tension in it. Therefore, it is a useful tool in package design, which was also confirmed by Pei et al (Ronggen et al., 2022) in their last work. Both theories were developed in parallel, the balloon theory as well as the theory of the movement of the yarn sliding over the surface of the package before it lifts from the package and forms a balloon.

In this paper, the consideration of yarn unwinding from packages is further supplemented by developing the equation of the tension in yarn. The theory stems from the fact that in 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.

2. The equation of motion for yarn

There are numerous situations where the use of accelerated observation frames is more convenient that of non-accelerated observation frames, in spite of the addition of the fictitious forces. In weather prediction it is, for example, necessary to solve a complex system of differential equations in a reference frame which is fixed to the Earth. Since the Earth is not an inertial reference frame (due to its rotation), it is crucial to take into account the Coriolis and centrifugal forces. Particularly important is the Coriolis force which makes the low-pressure systems spin in the counter-clockwise direction in the Northern hemisphere.

Accelerated observation frames are also useful in studying the yarn unwinding. It may be used, for example, when working within the quasi-stationary approximation to describe the yarn unwinding from packages where layers have a large number of yarn loops. On the other hand, when the yarn is being unwound from packages with a small number of loops, the angular velocity of the yarn unwinding is not well defined (constant), thus the problem may just as well be described within a fixed inertial reference frame.

Let us have a non-accelerated observation frame R and an accelerated observation frame R', which rotates relative to the non-accelerated frame with an angular velocity ω. The angular velocity is not constant in both direction and magnitude. We consider the motion of a short segment of yarn defined by the vector r = r(s,t) in both frames (Praček et al., 2012, 2016) (see Figure 1): (1) $$\mathbf{r}(s,t)=r(s,t){\mathbf{e}}_r(\theta (s,t),t)+z(s,t){\mathbf{e}}_z(t).$$(1)

Figure 1. Mechanical setup in over end yarn unwinding from cylindrical package.

Let the velocity in the non-accelerated observation frame R be denoted as $\mathrm{v}=\dot{\mathrm{r}}-V\partial \mathrm{r}/\partial s,$ and the velocity in the accelerated observation frame R′ as ${\mathbf{v}}_{\mathit{\text{re}}\mathbf{l}}=\dot{r}{\mathbf{e}}_r+r\dot{\theta }{\mathbf{e}}_{\theta }+\dot{z}{\mathbf{e}}_z.$ Within the time δt, the yarn moves in the non-accelerated observation frame R by vδt, which must equal v_relδt. In other words, the displacement of the yarn in accelerated observation frame R′ and (ω(t)δt) × r′ the displacement of the yarn due to the rotation of accelerated observation frame R'. We get (2) $$\text{v}={\mathrm{v}}_{\text{rel}}+\omega \times \mathrm{r}-V\frac{\partial \mathrm{r}}{\partial s}.$$(2)

Where:

v_rel = The relative velocity of a selected point according to the rotating coordinate system

ω × r = The circular velocity of the point P around the axis z

-V∂r/∂s = The velocity of dragging the yarn through the guide.

Partial time derivatives in non-accelerated observation frame and in accelerated observation frame are related by the following general formula (Goldstein et al., 2002): (3) $${\left(\frac{\partial }{\partial t}\right)}_R={\left(\frac{\partial }{\partial t}\right)}_{R\prime }+\omega \times .$$(3)

The operator expression (3) becomes meaningful only when applied to some (vectorial) function p. It is necessary to write p to the right of each term, obtaining (4) $${\left(\frac{\partial \mathbf{p}}{\partial t}\right)}_R={\left(\frac{\partial \mathbf{p}}{\partial t}\right)}_{R\prime }+\omega \times \mathbf{p}.$$(4)

If this rule is applied to the velocity Equation (2)(2) $$\text{v}={\mathrm{v}}_{\text{rel}}+\omega \times \mathrm{r}-V\frac{\partial \mathrm{r}}{\partial s}.$$(2) , we obtain the accelerations a (Giancoli, 2009; Praček et al., 2016): (5) $$\mathbf{a}=D^2\mathbf{r}+2\omega \times (D\mathbf{r})+\omega \times (\omega \times \mathbf{r})+\dot{\omega }\times \mathbf{r}$$(5)

Where:

D = The operator of the total time derivative which follows the motion of the point inside the spinning coordinate system, D = ∂/∂t|_r,θ,z-V∂/∂

The transverse dimension of the yarn is usually neglected in description of yarn during unwinding from packages. On the other hand, when the yarn is being unwound from packages with a small number of loops, the angular velocity of the yarn unwinding is not well defined (constant), thus the problem may just as well be described within a non-accelerated observation frame. The equation of motion is written as (Goldstein et al., 2002; Praček et al., 2016): (6) $$\rho \mathbf{a}=\frac{\partial }{\partial s}\left(T\frac{\partial \mathbf{r}}{\partial s}\right)+\mathbf{f}.$$(6)

Where:

f = The air drag for that part of the yarn that forms the balloon (or the force of friction for the part of yarn between unwinding point (U_P) and lift-off point on the package, which is sliding on lower layers of yarn)

T = The yarn tension, and the force is obviously directed along the yarn.

Rotating cylindrical system is also useful in studying the yarn unwinding. It may be used, for example, when working within the quasi-stationary approximation to describe the yarn unwinding from packages where layers have a large number of yarn loops. Using the expression for acceleration (5), the equation of motion can be written in an accelerated observation frame. We get (Praček et al., 2016): (7) $$\rho \left(D^2\mathbf{r}+2\mathit{\omega }\times D\mathbf{r}+\mathit{\omega }\times \left(\mathit{\omega }\times \mathbf{r}\right)+\dot{\mathit{\omega }}\times \mathbf{r}\right)=\frac{\partial }{\partial s}\left(\mathrm{T}\frac{\partial \mathbf{r}}{\partial s}\right)+\mathbf{f}.$$(7)

In an accelerated observation frame, the yarn is accelerated not only by the true external forces F, but also by the fictional forces. These are the Coriolis, centrifugal and Euler’s forces.

Equation of motion 7 expressed in terms of the dimensionless quantities (Praček et al., 2012). (8) $$\begin{array}{c}\overline{\mathbf{r}}=\mathbf{r}=/c, \overline{r}=r/c, \overline{z}=z/c,\overline{s}=s/c,\\ \overline{t}=t/\tau =\omega t, \overline{\mathbf{v}}=\mathbf{v}/\mathit{V}, {\overline{\mathbf{v}}}_{{}_n}={\mathbf{v}}_n/\mathit{V},\\ \overline{\mathbf{f}}=\frac{\mathbf{f}c}{\rho V^2}, \overline{\mathbf{n}}=\frac{\mathbf{n}c}{\rho V^2}, \overline{T}=\frac{T}{\rho V^2}.\end{array}$$(8)

Transforming the equation of motion into the dimensionless form we get (Praček et al., 2012): (9) $${\overline{D}}^2\overline{\mathbf{r}}\mathrm{\hspace{0.17em}+\hspace{0.17em}}2\Omega \times \overline{D}\overline{\mathbf{r}}+\Omega \times \left(\Omega \times \overline{\mathbf{r}}\right)+\Omega \frac{\partial \Omega }{\partial \overline{t}}\times \overline{\mathbf{r}}=\frac{\partial }{\partial \overline{s}}\left(\overline{T}\frac{\partial \overline{\mathbf{r}}}{\partial \overline{s}}\right)+\overline{\mathbf{f}}$$(9)

Where:

$\overline{D}$ = The dimensionless differential operator, $\overline{D}=\Omega \frac{\partial }{\partial \overline{t}}-\frac{\partial }{\partial \overline{s}},$

Ω = The dimensionless angular velocity, Ω = cω/V

3. Transition to the quasi-stationary approximation

The yarn unwinds from a package with a high angular velocity. During one period the yarn makes a rotational body with one or more ‘bellies’.

Let r(s) describe the yarn in the rotating coordinate system. In the resting coordinate system, the movement of the curve r(s) can be decomposed into two movements with different characteristic times. The first movement is the rotation of the curve around the Z-axis, this movement has a short characteristic time (whose order of magnitude is 2π/ω, where ω is the angular velocity). This is the time of unwinding of one loop of the yarn. The second movement is the change of shape of the balloon. This movement has a long characteristic time (whose order of magnitude is the time of unwinding of one layer). Such a decomposition is suitable mostly for packages (or layers) with a large number of loops, i.e. mostly for precisely wound packages.

Assuming that one layer has around a hundred loops, the two typical times differ for two orders of magnitude, since we are dealing with two movements in two very different timeframes. It makes sense to explicitly use this decomposition in our equations.

In such circumstances, rotating coordinate system the shape of the rotational body changes very slowly and we can use a quasi-stationary approximation applies. Therefore, in the first approximation, all time dependence can be transferred to boundary conditions, while in the equation of motion, time dependence can be neglected. Fraser found a solution using perturbation theory (7). He found that with an appropriate choice of dimensionless variables, the equation of motion can be written in a form in which the time derivatives are multiplied by a small parameter. Fraser estimated that, in the first approximation, we can neglected the time derivatives. He proved that the zero-order equation is equivalent to the stationary motion of the yarn, in which all time dependence passes to the boundary conditions. Furthermore, it turns out that at order zero of the theory the dimensionless angular velocity is equal to one. This means that the solution is only consistent with a winding angle equals to zero. Perturbation theory is a more reasonable approach from the point of view of linear algebra, but from the physical-technical point of view, the first simplification is completely satisfactory. It also allows for greater generality, so we will decide to use the quasi-stationary approximation. In the equation of motion (9), we set all time derivatives to zero, we get the quasi-stationary equation of motion can be written in the form; (10) $$\frac{{\partial }^2\mathbf{r}}{\partial s^2}-2\Omega \times \frac{\partial \mathbf{r}}{\partial s}+\Omega \times \left(\Omega \times \mathbf{r}\right)=\frac{\partial }{\partial s}\left(T\frac{\partial \mathbf{r}}{\partial s}\right)+\mathbf{f}.$$(10)

4. A derivation of component-wise equations of motion

We also find the equations for the tension at the point where the yarn rises from the package surface and forms a balloon. The radius-vector can be written as: (11) $$\mathbf{r}\left(r,\theta ,z\right)=r\left(s\right){\mathbf{e}}_r\left(\theta \left(s\right)\right)+z\left(s\right){\mathbf{e}}_z.$$(11)

The derivatives over s of the radius vector are then. (12) $${\mathbf{r}}^{\prime }=\left[\begin{array}{c}{r}^{\prime }\\ r{\theta }^{\prime }\\ z^{\prime }\end{array}\right]{\mathbf{r}}^{″}=\left[\begin{array}{c}{r}^{″}-r{{\theta }^{\prime }}^2\\ 2r^{\prime }{\theta }^{\prime }+r{\theta }^{″}\\ z^{″}\end{array}\right]$$(12)

We will also need the results obtained using the rules for calculating vector products (13) $$\Omega \times \mathbf{r}=\Omega \left[\begin{array}{c}0\\ r\\ 0\end{array}\right],\Omega \times {\mathbf{r}}^{\prime }=\Omega \left[\begin{array}{c}-r{\theta }^{\prime }\\ r^{\prime }\\ 0\end{array}\right],\Omega \times \left(\Omega \times \mathbf{r}\right)={\Omega }^2\left[\begin{array}{c}-r\\ 0\\ 0\end{array}\right]$$(13)

The velocity is (14) $$\mathbf{v}=-{\mathbf{r}}^{\prime }+\Omega \times \mathbf{r}=\left[\begin{array}{c}-r^{\prime }\\ -r{\theta }^{\prime }+\Omega r\\ -z^{\prime }\end{array}\right]$$(14)

The normal component of velocity (15) $$v_n=\Omega r\sqrt{{r^{\prime }}^2+{z^{\prime }}^2}$$(15)

The air resistance force is (16) $$\mathbf{ }{\mathbf{f}}_u=\frac{1}{16}\left[\begin{array}{c}{p}_0\Omega r^2{r}^{\prime }{\theta }^{\prime }{v}_n\\ -p_0{v}^3{}_n/\left(\Omega r\right)\\ p_0\Omega r^2{\theta }^{\prime }{z}^{\prime }{v}_n\end{array}\right]$$(16)

Now consider that $\partial /\partial s(T\partial \mathbf{r}/\partial s)=T^{\prime }{\mathbf{r}}^{\prime }+T{\mathbf{r}}^{″}$ and write the equation of motion in the form (17) $$\left(1-T\right){\mathbf{r}}^{″}-2\Omega \times {\mathbf{r}}^{\prime }+\Omega \times \left(\Omega \times \mathbf{r}\right)=T^{\prime }{\mathbf{r}}^{\prime }+\mathbf{f}.$$(17)

The motion equation for yarn can be obtained from Equation (17)(17) $$\left(1-T\right){\mathbf{r}}^{″}-2\Omega \times {\mathbf{r}}^{\prime }+\Omega \times \left(\Omega \times \mathbf{r}\right)=T^{\prime }{\mathbf{r}}^{\prime }+\mathbf{f}.$$(17) by substituting r with (11) and f with (16) and the Ω terms with (13). The components of this equation are as follows: (18) $$\left(1-T\right)\left(r^{″}-r{{\theta }^{\prime }}^2\right)+2\Omega r{\theta }^{\prime }-{\Omega }^{2}r=T^{\prime }{r}^{\prime }+\frac{1}{16}\Omega p_0{r}^2{r}^{\prime }{\theta }^{\prime }{v}_n$$(18) (19) $$\left(1-T\right)\left(2r^{\prime }{\theta }^{\prime }+r{\theta }^{″}\right)-2\Omega r^{\prime }=T^{\prime }r{\theta }^{\prime }-\frac{1}{16}{p}_0{v}_n{}^3/\left(r\Omega \right)$$(19) (20) $$\left(1-T\right)z^{″}=T^{\prime }{z}^{\prime }+\frac{1}{16}\Omega p_0{r}^2{\theta }^{\prime }{z}^{\prime }{v}_n.$$(20)

We also have a inextensibility condition: (21) $${r^{\prime }}^2+r^2{{\theta }^{\prime }}^2+{z^{\prime }}^2=1.$$(21)

Therefore we have four Equations (18)–(21) for four variables: r,θ,z and T where T is as in picture 2. The problem is thus mathematically well-defined. Since the tension T is an unknown quantity we would like to find out whether it can be easily expressed with other quantities.

Let us multiply the Equation (18)(18) $$\left(1-T\right)\left(r^{″}-r{{\theta }^{\prime }}^2\right)+2\Omega r{\theta }^{\prime }-{\Omega }^{2}r=T^{\prime }{r}^{\prime }+\frac{1}{16}\Omega p_0{r}^2{r}^{\prime }{\theta }^{\prime }{v}_n$$(18) with r′, the Equation (19)(19) $$\left(1-T\right)\left(2r^{\prime }{\theta }^{\prime }+r{\theta }^{″}\right)-2\Omega r^{\prime }=T^{\prime }r{\theta }^{\prime }-\frac{1}{16}{p}_0{v}_n{}^3/\left(r\Omega \right)$$(19) with rθ′ and the Equation (20)(20) $$\left(1-T\right)z^{″}=T^{\prime }{z}^{\prime }+\frac{1}{16}\Omega p_0{r}^2{\theta }^{\prime }{z}^{\prime }{v}_n.$$(20) with z′ to get: (22) $$\begin{array}{c}\left(1-T\right)\left[r^{″}{r}^{\prime }+r^{\prime }r{{\theta }^{\prime }}^2+r^2{\theta }^{\prime }{\theta }^{″}+z^{\prime }{z}^{″}\right]-r{r}^{\prime }{\Omega }^2=\\ T^{\prime }\left({r^{\prime }}^2+r^2{{\theta }^{\prime }}^2+{z^{\prime }}^2\right)+\frac{1}{16}{p}_0{v}_n\Omega \left[r^2{r^{\prime }}^2{\theta }^2-{\theta }^{\prime }{v}_n{}^2/{\Omega }^2+r^2{\theta }^{\prime }{z^{\prime }}^2\right].\end{array}$$(22)

The expression between the brackets on the right-hand side of the equation is equal to 0 which can be proved by the following computation: (23) $$\begin{array}{l}\left(r^{\prime }{r}^{″}\right)+\left(z^{\prime }{z}^{″}\right)+\left(r^{\prime }r{{\theta }^{\prime }}^2+r^2{\theta }^{\prime }{\theta }^{″}\right)={\left[\frac{1}{2}\left({r^{\prime }}^2\right)+\frac{1}{2}{\left({z^{\prime }}^2\right)}^2+\frac{1}{2}{\left(r^2{{\theta }^{\prime }}^2\right)}^{\prime }\right]}^{\prime }\\ =\frac{1}{2}{\left({r^{\prime }}^2+{z^{\prime }}^2+r^2{{\theta }^{\prime }}^2\right)}^{\prime }\\ =\frac{1}{2}{\left(1\right)}^{\prime }=0\end{array}$$(23)

By introducing the normal component of velocity v²_n (and using $v_n=\Omega r\sqrt{{r^{\prime }}^2+{z^{\prime }}^2}$) we can rewrite Equation (22)(22) $$\begin{array}{c}\left(1-T\right)\left[r^{″}{r}^{\prime }+r^{\prime }r{{\theta }^{\prime }}^2+r^2{\theta }^{\prime }{\theta }^{″}+z^{\prime }{z}^{″}\right]-r{r}^{\prime }{\Omega }^2=\\ T^{\prime }\left({r^{\prime }}^2+r^2{{\theta }^{\prime }}^2+{z^{\prime }}^2\right)+\frac{1}{16}{p}_0{v}_n\Omega \left[r^2{r^{\prime }}^2{\theta }^2-{\theta }^{\prime }{v}_n{}^2/{\Omega }^2+r^2{\theta }^{\prime }{z^{\prime }}^2\right].\end{array}$$(22) as (24) $$T^{\prime }=-{\Omega }^{2}r{r}^{\prime }.$$(24)

This equation can also be written as (25) $$T=T_0-\frac{1}{2}{\Omega }^2{r}^2.$$(25)

The quantity $T_0$ indicates the tension in the yarn passing through the guide (see pictures 1).

5. Simulation model

During unwinding yarn, the unwinding point moves slowly along the package surface, so we do not need the Initial Conditions at all. The first boundary condition is that the yarn passes through the guide at the origin, which is written as r (s = 0) = 0, or equivalently r (0) = 0, θ (0) = 0, z (0) = 0). At the lift-off point, the yarn must be continuous and must not be broken. The following are the conditions for continuity: (26) $$\begin{array}{c}r(s=s_{\mathit{\text{Lp}}}^{+})=r(s=s_{\mathit{\text{Lp}}}^{-})\\ r^{\prime }(s=s_{\mathit{\text{Lp}}}^{+})=r^{\prime }(s=s_{\mathit{\text{Lp}}}^{-})\end{array}$$(26)

The index + or − indicates the point just behind or just before the lifting point. In the lift-off point the yarn is tangential to the package. We can then write: (27) $$r^{\prime }(s=s_{\mathit{\text{Lp}}}^{})=0$$(27)

The non dimensionless boundary condition at the lift-off point becomes [14]. (28) $$\overline{r}\left({\mathrm{s}}_{\text{Lp}},\mathrm{t}\right)=1$$(28)

Inserting condition (28) into Equation (25)(25) $$T=T_0-\frac{1}{2}{\Omega }^2{r}^2.$$(25) the tension at the Lp is then equal to (29) $$T=T_0-\frac{1}{2}{\Omega }^2.$$(29)

The expression (29) tells us that tension in the yarn T_o passing through the guide in the given point is reduced from the residual yarn tension in the package to the value of tension T in the balloon. If we write T_R for the residual tension in the yarn. We get (30) $$T=T_0-T_R.$$(30)

Where (31) $$T_R=\frac{{\Omega }^2}{2}$$(31)

When the yarn is being unwound from a cylindrical package, the angular velocity of the yarn forming the balloon depends on three parameters: the package radius, the unwinding velocity, and the winding angle. Particularly important is the last parameter, since it is closely related to the oscillations of the tension in the yarn (Praček et al., 2012): (32) $$\omega =\frac{2\pi }{t}=\frac{V\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}\varphi }{c(1-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\varphi )}$$(32)

The dimensionless angular velocity can obviously be expressed as (Praček et al., 2012): (33) $$\Omega =\frac{\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}\varphi }{1-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}\varphi }$$(33)

In a recent paper (Praček et al., 2012), we developed a function:

ƒ(t) = sign(sin t)|sin t|^1/40 (34)

which would permit to simulate the process of unwinding. In our simulation, we calculate the winding angle ϕ₀ using this function, ϕ(t) = ϕ₀ƒ(t). We get (35) $$\Omega (\varphi (t))=\frac{\hspace{0.17em}\mathrm{\text{cos}}\hspace{0.17em}(\varphi (t))}{1-\hspace{0.17em}\mathrm{\text{sin}}\hspace{0.17em}(\varphi (t))}$$(35) and finally we obtain the simulation model of residual tension in the yarn using Equation (31)(31) $$T_R=\frac{{\Omega }^2}{2}$$(31) we can write it like (36) $$T_R(\varphi (t))=\frac{{\Omega }^2(\varphi (t))}{2}$$(36)

6. Practical part

The tension in the yarn passing through the guide is unknown at this point. It will have to be obtained experimentally. The expression (25) is very interesting. We derived the expression defining the tension in the yarn which tells us that tension in the yarn in the given point depends directly only on the centrifugal force and not on the Coriolis force or the air resistance force [15]. The larger the distance between the yarn and the axis the larger is the centrifugal force. Thus the tension is the largest on the axis when r = 0, therefore on the guide itself (Figure 2) we get. (37) $$T=\text{To}$$(37)

Figure 2. Tension in the yarn passing through the guide and winding angle during unwinding.

The yarn guide is the most suitable area for performing experimental tension measurements. In the following, when we use the word 'yarn tension’, we refer to the tension on the eyelet and denote it by T. When deriving this model, we neglected: the length of the package, the coefficient of friction between the yarn and package, the coefficient of air resistance and the winding angle.

We measured how the tension at the eyelet depends on the angular velocity of unwinding. We measured the unwinding speed, package radius and tension of yarn as we unwound yarn from parallel cylindrical packages (Table 1). We used a balloon limiter that limits the balloon radius. Figure 3 shows the setup of the analysis system. The yarn is unwound with the help of the lesson drive at a transport speed of up to 200 meters per minute, through the guide which is also the end of the balloon limiter, which will reduce the tension in the yarn during unwinding. The balloon limiter consisted of eight Teflon rings, the largest had a diameter of 25.3 cm and then (in the direction toward the guide) 23.1 cm, 18.5 cm, 15 cm, 12.3 cm, 8 cm, 4.5 and 1.6 cm. The leader is 3.6 cm away from the smallest ring, and the distance between the rings is 5.5 cm. We installed the package between the rings. The guide just behind the smallest ring and the yarn guide in front of the unwinding drive are separated by 6 cm, and between them we installed a tension sensor (SCHMIDT control instruments, model DTMX-200) to measure the tension in the yarn.

Figure 3. System for unwinding yarn from package.

7. Results of unwinding simulation

In our simulation, we calculate the winding angle using function 24, then we determine the corresponding dimensionless angular velocity, and finally we obtain an approximation for the tension in the unwinding process using data from Table 2.

The graphs show numerical simulations of unwinding from cylindrical cross-wound packages. We show the time dependence of the residual yarn tension T_R (black line) at the unwinding point, the tension at the eyelet T_o (red line) and the tension in the balloon T (green line), see Figures 4 and 5.

Figure 4. Variation of the tension T₀, T_R and T during the unwinding of the yarn from a cylindrical package for a range of winding angles. The parameters are V = 1000 m/min, c = 100 mm, ϕ₀ = 5°–25°. Note: The quantity t is the duration of the transition from the unwinding in the forward direction to the backward direction (the time during which one loop of yarn is unwound) t = 2π/ω.

Figure 5. Variation of the tension T₀, T_R and T during the unwinding of the yarn from a cylindrical package for a range of winding angles. The parameters are V = 1250 m/min, c = 100 mm, ϕ₀ = 5°−25°.

Figures 4 and 5 show the dependence of the tension oscillations on the unwinding speed v = 1000–1250 m/min and winding angles from 5° to 25° in three different areas of the unwinding yarn, namely, the tension measurement area in the yarn—in the guide T_o, the area of the rotary body of the yarn—in the balloon T and the area of movement of the yarn along the package—at the lift-off point T_R. For the first two areas, the tension oscillations are noticeably higher with higher unwinding speed and packages with a large winding angle, but for the third area, the tension oscillations are considered to be inappropriately small compared to the first two areas. This happens because during unwinding we are dealing with two different movements of the yarn, the first two areas are conditioned by the movement of the yarn in the balloon and are subject to real and virtual (dynamic) external forces, and the third area is conditioned by the movement of the yarn along the package at the lift-off point. The winding angle ϕ₀ is approximately constant as the lift-off point moves up or down the package, but its sign changes abruptly at the package edges.

Therefore, the movement of the yarn on the package surface and in the part of the balloon near the lifting point is very confused. At the edge, various undesirable phenomena can occur: the yarn can slip off the winding or the winding layer can dry out. The description of these transients is beyond the scope of our model. A careful analysis shows that, the yarn 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, which means that the shape of the yarn changes very slowly and the tension oscillation is not affected by external dynamic forces, but only by internal forces. The differences are largest for the winding angle of ϕ₀ = 25°, where the difference at unwinding velocity of V = 1250 m/min equals 25 cN. At winding angle of ϕ₀ = 15° this difference is still 12 cN (see Tables 3 and 4). This means that at both speeds, for the same values of the angles, the same values for the residual tension are obtained for all five winding angles. Which undoubtedly proves that the residual stress has absolutely no kinematic origin. Therefore, the third region is not always negligible, but can have very significant effects as the winding angle increases.

We have shown that large tensions are generated during the unwinding of the yarn, so it is desirable to produce such packages in which both the maximum value of the tension in the yarn and the amplitude of the tension oscillations during unwinding are limited. We have proven that increasing the winding angle increases the tension oscillations in the yarn. This winding angle affects the motion of the yarn at the package edges where the winding angle is reversed. Such a rapid change of the winding angle leads to a sudden change of the angular velocity $\mathit{\omega },$ which implies that the angular acceleration $\dot{\mathit{\omega }}$ is large. For this reason, the tension oscillation is also large and it affects the yarn dynamics. The conditions are then not quasi-stationary but transitory, since the type of motion changes. For this reason there might be instabilities of the balloon shape, the yarn can break.

We may thus conclude: by increasing the winding angle the yarn tension oscillations are also increased. This is particularly noticeable at higher unwinding velocity. Furthermore, we find that it is advisable for the winding angle to be below 5°.

8. Conclusion

We have shown that a combination of modeling and theoretical determined relation between the winding angle and the tension of the yarn inside the package can be used to fruition in determining a suitable set of parameters for unwinding from parallel package. From this study, the following conclusions can be drawn.

• Due to residual tension of the yarn in the package, a short segment of yarn slides on the surface of the package and it rubs against it, instead of immediately lifting off in the balloon at the unwinding point. In this part of the yarn the residual tension of the yarn in the package is reduced to the value of tension in the balloon at the lift-off point.

• Unwinding at high residual tension from regular cross-wound packages is impossible for package winding angle up to 20°

• Oscillations of residual tension of the yarn are smaller in packages with small winding angle (parallel-wound packages), but for such packages yarn slips can occur during unwinding.

• Alternatively, the oscillations of residual tension can be reduced if only those layers of yarn that are unwound backwards are wound for winding angle below 10°.

We have discussed the origin of the internal forces in the equation of motion in a non-inertial reference frame using quasi-stationary approximation. These forces are of purely internal origin, since they are a consequence of the stiffness of the winding and they are thereby distinctly different from external and inertial forces. When conditions are quasi-stationary (they do change very slowly with time), the tension oscillation is negligible. This is clearly only true for layers with a large number of loops and sufficiently small winding angle below 5°, that is, for precision wound packages.

Acknowledgements

The author wish to acknowledge asist. prof. Nace Pušnik for technical advice and help. He also obliged to his father, spinning master Ivan Praček from Tekstina Ajdovščina, who tragically deceased.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Table 1. Experimental parameters.

Parameters	Range of values
Yarn type	cotton
Yarn linear density m	1.5 g/m
Yarn titer tex	49.32 tex
Package radius c	105–110 mm
Package winding angle ϕ0	0.5°–5°
Transport speed V	1000–1250 m/min

Table 2. The variation of tension for a range of angular velocities.

Unwinding velocity V / m/min	Radius c / mm	Angular velocity ω / rad/s	Dimensionless angular velocity Ω	Tension T / cN
1000	110	151.5	0.997904191	5.5
1000	108	154.3	0.997868263	6.0
1000	107	155.8	0.998239521	6.2
1250	107	194.7	1.000139222	11.5
1250	106	196.5	0.999951992	12.3
1250	105	198.4	1.000096015	12.6

Table 3. The variation of tension for a range of winding angles.

	Unwinding velocity V = 1000 m/min
	Radius c = 100 mm
ϕ[°]	T0[cN] max	T0[cN] min	TR[cN] max	TR[cN] min	T[cN] max	T[cN] min
5	3.5728	2.5190	0.5955	0.4198	2.9773	2.0992
10	4.2607	2.1123	0.7101	0.3521	3.5506	1.7603
15	5.0949	1.7665	0.8491	0.2944	4.2457	1.4721
20	6.1183	1.4710	1.0197	0.2452	5.0986	1.2258
25	7.3910	1.2177	1.2318	0.2030	6.1591	1.0148

Table 4. The variation of tension for a range of winding angles.

	Unwinding velocity V = 1250 m/min
	Radius c = 100mm
ϕ[°]	T0[cN] max	T0[cN] min	TR[cN] max	TR[cN] min	T[cN] max	T[cN] min
5	12.7629	8.9986	0.5955	0.4198	12.1674	8.9986
10	15.2202	7.5458	0.7101	0.3521	14.5100	7.1938
15	18.2002	6.3103	0.8491	0.2944	17.3511	6.0159
20	21.8562	5.2547	1.0197	0.2452	20.8365	5.0096
25	26.4024	4.3499	1.2318	0.2030	25.1706	4.1470