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1DETERMINING THE THREAD TENSION DEPENDING ON THE SHAPE OF THE SURFACE OF THE FALSE TWISTING
DEVICE
1Ergashev Z.N., 2Sayidmuradov M.M.
Researcher, Namangan Engineering and Technology Institute
2Candidate of Technical Sciences, Associate Professor, Namangan Engineering and
Technological Institute https://doi.org/10.5281/zenodo.13891463
Abstract. Improving rotor-mechanical spinning machines that produce high-quality yarn using the rotor-mechanical method, in order to improve the quality of products produced in the textile and light industry on a global scale and reduce its cost, requires the introduction of methods and means to eliminate factors that negatively affect the quality of products in the process of spinning yarn. The article theoretically examines the factors influencing the quality indicators of semi-finished products and yarn, the optimal performance of technological processes and the development of yarn with a low twist factor.
Keywords: yarn, rotor spinning method, false twist, yarn quality parameters, yarn models, geometric relationship, flat cut hypothesis, tensile forces, Lagrangian coordinate, equilibrium of ideal elastic yarn.
INTORDUCTION. One of the most important and urgent issues is to determine the factors affecting the quality indicators of semi-finished products and yarn, as well as the optimal indicators of technological processes in the production of yarn by the pneumomechanical spinning method. Based on the results of the study of the state of new thread forming methods and the development conditions, and based on theoretical analysis, we will consider the issue of researching the dynamics of a false twisting die that develops a thread with a low twist coefficient.
As one of the most important processes in the textile industry, a lot of attention has always been paid to the process of twisting the product, especially in non-stabilized operating modes. Thread twisting in non-stabilized operating modes in pneumomechanical spinning [1] and yarn dynamics [2,3,4,5] have not yet been fully resolved.
SETTING THE ISSUE. The following basic restrictions and simplifications are introduced for the construction of thread models [4]:
1) geometric limitation - a thread, when its axis is a straight line, is considered as a cylindrical or prismatic body whose length is many times larger than the dimensions of its cross section;
2) hypothesis of integrity - we conditionally believe that the composition of the thread is made of a whole different substance or of thin and long, whole fibers that are located in the form of helical lines in relation to its axis when the screwdriver is deformed;
3) hypothesis of flat shears - conditionally we assume that the normal cross-section of the thread before deformation is flat relative to its axis and remains unchanged during normal or oblique deformation;
4) we consider the thread to be an anisotropic material - the physical and mechanical properties of the thread are different depending on the length and thickness of the thread; we
assume that the deformation characteristics of the thread are the same in the longitudinal direction (along the axis of the thread), and not deformed in the cross section;
5) We do not take into account the effect of the inertial force of the mass, the force of gravity and the force of air resistance.
Theoretical part. We obtain the equilibrium equation of an ideal elastic string in a stationary force field [4,5]. We take a string of length l, which is in equilibrium under the influence of a distributed force with a tension F and tension forces T at the ends (Fig. 1). We understand the ratio of the force per unit length of the thread as tension. On the string, we specify the beginning of calculation of arc Lagrangian coordinate s and the positive direction of calculation.
Fig. 1. Schematic and model of balancing under the action of distributed force and tension
forces
In the string in equilibrium, we isolate the segment AB of length As and discard the remaining segments. To maintain the equilibrium state of part As, we apply tension forces T0 and Tl in the experimental directions to points A and B. Thus, the equal acting force and tension forces T0 and Tl with the value FAm=Fp.As will be the active forces acting externally on the element As of the thread.
At each point of the element there are also internal forces T and T" =-T'.
We write the equilibrium equation for the As element of the thread, assuming that Tl = T0 + AT:
T0+Tl+ FAm = 0 (1)
If we bring the acting forces to the length of the corresponding part of the thread, then
aL = -FaJ± (2)
As As v '
If we move to the limit As ^ 0, then we get the following equation:
(3)
Here, j - the linear density of the product.
We write equation (3) in the following form:
1 ^ + F=0 (4)
^ ds v y
The tension T is always in the direction of the test, that is, t:
T = TT
If we pass the natural axes with centers t, n, b at the starting point of the thread element, then we can write the following [71]:
dT d(Tr) dT dr ds ds ds ds
From point kinematics, it is known that, like the velocity of two points at the boundary of the plane, the tension, passing through three infinitely close points, lies in a mutually intersecting plane, where:
dr 1 ds p
where p is the radius of curvature of the thread element.
In this case:
dT dT T d d p
The vector equation of equilibrium for a string element is written in the following form:
1 (—T + -n) + F = 0 (5)
p\ds p J
By projecting equation (5) onto the natural axes with centers t, n, b we get the equilibrium equations of the string in Lagrangian coordinates:
1dr + Fr = 0,]
V ds L I
1-T- + Fn = 0,\ (6)
v p n I
Fb = 0, }
where Ft , Fn , Fb - F is the projection of force intensity on normal and binormal axes.
From equation (6), we see that at each point of the string, the field force F lies in the plane in which it tries to be fullro We now consider the equilibrium problem of an ideal elastic string on an arbitrary uneven surface. Let's say that the thread is located in an arc on a rotating surface. Here AB is the meridional section of the rotating surface (Fig. 2.).
Fig.2. Scheme of the meridional section of the rotating surface In this case, the thread is under the influence of forces To and Ti at points Mo and M. We do not consider other active forces. We assume that friction is based on Amonton's law:
F < fN,
where F and N are increased friction and normal reaction forces; f - the coefficient of
friction.
A string sliding on a surface is located along a geodesic (shortest curve). In this case, the frictional force is directed towards the string and is at the breaking limit, i.e. the frictional force has a maximum value at all points:
F = fN
Since Ti > To, the fictional force is opposite to the positive direction of effort. In this case, the projection of the tension force to the effort t:
Fr = -fN.
Projection of tension force onto the principal normal n:
Fn = -N
Since the thread is located along the geodesic line, we express the curve of the thread k (k=l/r) in the form of k = k(s) through the natural parameter (the length of the meridian arc s) at each point. For the considered problem, the balance equation of the thread (6) is written as follows [4]:
^ — = -FT = fN, '
ß ds x '
(7)
1
-Tk(s) = -Fn=N
^ >
In equation (7), dividing the first equation by the second, we get the following expression to determine the thread tension:
dT 1
= f
If we separate the variables:
ds Tk(s)
dT
— = f k(s) ds
We integrate both parts of the resulting equation from the point Mo (s=0) to the point M with an arbitrary s coordinate and get the following equation:
T = Cexp(f ^k(s)ds) (8)
Here C=const.
At the Mo point s=0 and T=To. In this case C=To [72] and (8) is written as follows: T = T0exp(fj;;k(s)ds) (9)
The equation (9) obtained on the basis of the methods of differential geometry allows to determine the tension in any cross-section of the string sliding on the surface, located at the limit of equilibrium [4]. To confirm the reliability of the obtained equation (9), we solve this problem using the principle of virtual displacements, which is usually used in the study of physical values [5].
CONCLUSION. For example, a long string is in equilibrium on a rough surface. We isolate the MoM element on the thread (Fig. 2). This element is subjected to forces Ti and T2 applied to points Mo and M. In this case, the thread is located along the geodesic line, because the friction force is directed towards the thread and is at the breaking limit (the friction force Fmp has a maximum value at all points, Fmp = f N, where f - the coefficient of friction, N - the normal pressure force).
If the string is moved to the left by a distance s (Fig. 3), ATs work is done, and its value is equal to the value of the work done by the friction force [4], i.e.:
(T1 + dT) - T1)s = dFmps (10)
d T = dFmp (11)
However dFmp = f dN , here dN - normal compressive force in section MoM: dN = 2Tcos = 2TsinT ~ 2T Y = TdQ (12)
Here 6 - coverage angle.
Fig.3. The scheme for determining the tension in any section of the thread
We put (12) into (10) and get the following equation:
dT = fTd6 (13)
Since the string is located on a geodesic line connecting two given points on the surface, its curvature at each point is k=k(s) and is calculated using the following formula:
k(s)=^ ^ d0 = k (s)ds (14)
From equations (13) and (14) we get the differential equation for T=T(s):
d T = fTk(s)ds (15)
Dividing the variables and integrating both parts of the equation from the point Mo(s=si) to the point M(s=s2), we get the following equations:
fT:T = Cfk(s)ds (16)
lnlTl\!rl=fiSs2ik(s)ds (17)
From the last expression, we find the final equation for calculating the thread tension: T2 = T1exp{fj;;2ik(s)ds} (18)
Formulas (18) and (9), which are generally found by different methods, have a similar and uncomplicated mathematical structure. With the help of these, it is possible to calculate the tension in any section of the string moving along the geodesic lines (plane or spatial) on the guiding surfaces.
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