Analysis of research literature indicates that the structure of yarn depends on technological and kinematic factors. Structure control, i.e., ways to reduce twist loss and twist distribution along the yarn, and the methods of density regulation are not considered. It is a big problem as characteristics and quality of yarn depend on it. We consider the change in torsion quantity on a ring spinning machine. Formed yarn gets one twist per one turn of the runner. The number of coiled yarn twists changes depending on coil diameter. So, with an increase in diameter of winding, a larger twist occurred. Practice has shown that when twisting the yarn into a coil with a larger and smaller diameters, the difference is about 1% . Twist unevenness also occurs due to vertical movement of the ring bar at a certain distance. So, for example, when lowering the ring strip, a shorter yarn is wound on the package because a part of its length is not wound due to increasing cylinder height. Therefore, in practice, it is used the average twist obtained when the number of torsions in any section of yarn divided by its length. In this case, appearance of twist irregularities is not taken into account, as it is very important for uniform yarn structure. In well-known papers [2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17], the roughness in twist is not analyzed, i.e., the conditions for formation of structure, on which the quality indicators of yarn obtained by the spinning machine are largely dependent, are not taken into account. As noted above, the difference in the number of yarn twist segments leads to appearance of twists in short sections when the annular strip moves up and down. Uniformity of twist distribution in this case depends on the spinning method, i.e., a twist along the yarn is distributed differently. Investigations of these issues are described in papers [2, 5,6,7,8,9,10]. On yarn twist distribution at different spinning methods, the positive results were obtained. A number of works are about the study of twist distribution in OE yarn [11,12,13,14,15]. The issue of twist distribution in the modified yarn was also studied . In all papers, the process of twist wave distribution along the yarn is considered. In this case, the distribution of shear deformations occurs both along the axis of the yarn and along the radial direction. For twist distribution along the yarn, the structural unevenness is important. Twist distribution has been sufficiently investigated, but, as a rule, a structurally uniform and homogeneous product is considered, while the structure of the product is not considered. Therefore, twist distribution with such shortcomings is investigated in this paper.
In paper [5, 6] considered twist distribution of the yarn with structural unevenness where applied the generalized law for Hooke's anisotropic medium, according to which the correspondences between the components of the stress and strain tensors in ax symmetric coordinates are represented by five elastic constants .
We denote the angular displacement of yarn arbitrary section by
We denote the angular displacement of yarn arbitrary section by
In Eq. (2), we see that deformation shifts
Eq. (4) describes the distribution of twisting waves along the yarn with a structurally heterogeneous property. In this case, the twist changes both along the axis of the yarn and along the radial direction. We consider a yarn with radius
For yarn axis, the displacement roundedness condition is applied:
Yarn initial cross section
Full angular speed of the yarn in a cross section
The solution for Eq. (5) satisfying the boundary conditions of Eqs (6) and (7) can be represented as follows:
The solution of Eq. (11) is represented through the first-order Bessel function,
Using the condition of orthogonality
Eq. (12) for known values of characteristic numbers
The solution of Eq. (14) is represented as follows:
We consider the case
Thus, the angular movement of the yarn between sections
As seen in Eq. (19), the values
If we do not take into account the twist changes in radial direction, then from Eq. (14), we obtain the expressions for the wave numbers as given in the study by Xu et al .
Moreover, the wave numbers
Figure 2 presented the curves of
The analysis of curves shows that Doppler effect for waves distributing along the yarns is held for each fluctuation form of the yarn by its thickness. In this case, the Doppler effect for the backward wave is stronger than that for the direct one. This is due to the processes of wave's reflection from the axis of the yarn.
The conditions for distribution of twist waves for a moving yarn with a constant speed are studied, and the parameters of Doppler effect for waves that are distributed along the yarn are determined. It was found that the Doppler effect for the backward wave is stronger than that for the direct one for 3.5 times. This is due to the processes of wave's reflection from yarn axis.
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