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Gulamov Azamat Eshankulovich, doctor of science in Technical Tashkent institute of textile and light industry E-mail: daminov.asror86@gmail.com Eshmirzayev Alisher Pardayevich, senior lecturer,
Tashkent institute of textile and light industry
Mardonov Botir, doctor of science, professor, Tashkent institute of textile and light industry
Bobotov Ulugbek, senior lecturer,
Tashkent institute of textile and light industry
SIMULATION OF THE PROCESS OF ONE-DIMENSIONAL MOTION IN FULL SUBMERGED COCOON IN AN AQUEOUS MEDIUM
Abstract: The article presents the results of an analysis of the motion of a submerged cocoon in the process of winding an elastic thread on a reel in a viscous medium. A model of a completely submerged cocoon is considered, which moves vertically upward in an aquatic environment.
Keywords: raw materials, silkworm breeding, cocoon, deformation, Kinematic scheme.
Cocoons are bodies of various shapes (an ellipsoid, two spheres with a hyperboloid of rotation that unites them into a single body), the envelope ofwhich consists of fibroin (silk) threads of a silkworm (or silkworm) glued together with seri-cin and densely packed in a strong system that protects the pupa from External influences, allowing it to breathe and develop into a butterfly.
The density of the shell and pupae is higher than the density ofwater (up to 1.2 kg/dm2), but the air inside the cocoon makes its average density much less than the water density so that the dry cocoons possess high buoyancy.
When the cocoons are smeared, sericin is partially removed (digested) from the shell and water enters the cocoon volume, pushing out air from there. In connection with this, the bulk density of the cocoon increases, tending, in the limit, to a value exceeding unity, since the density of the fiber and the pupa is certainly greater than unity.
In the absence of external forces, the cocoon immersed in the aquatic environment is in a state of equilibrium under the action of the gravitational force of the buoyancy force of
Fig.1. The scheme of the action of external and elastic forces on the cocoon
We believe that the filament from the surface of the cocoon descends continuously, and the contact of the thread with the cocoon takes place through the elastic element. The upper end of the element makes movement according to the law and is conjugated to a segment of a deformable filament
Archimedes.
Let us study the process of winding an elastic thread on a reel that is unwound from the surface of the cocoon. In this case, the cocoon is immersed in a viscous medium and makes a vertical movement in it (Fig. 1).
moving with the reel according to the law. We replace the segment deformed by an elastic element with a stiffness coefficient (- Young's modulus of filament,- the cross-sectional area and the length of the filament). We also denote by the stiffness coefficient of the connection of the yarn with the cocoon converging from the surface. The equation of equilibrium at the conjugation point of two elastic elements is written
[ki (x 1 (t) - h(t) ] - k0 [x o (t) - Xi(t) ] = 0 From where we determine the displacement of the end of the elastic element
kn
kg + k
x Jt)
The coefficient of elasticity of the bond k1 of the yarn with the cocoon depends on the degree of wetting by its aqueous medium and determines the degree of fixation of the converging thread to the surface of the cocoon. During the removal process, continuous filament removal takes place and by the poet this coefficient in the general case is a function of the difference by moving the lower end of the wound and moving the cocoon. The form of this function is determined experimentally. To simplify the problem, we assume that the binding force of the filament to the surface of the cocoon varies periodically, at the initial instant of time this force reaches a maximum value, then, as a result of sericin dissolution, the process of continuous separation of the filament from the cocoon surface takes place and the resistance decreases to a minimum k1 = K min at time t = tpr and again increases to the maximum. Such a pattern of variation of the coefficient k1 in time, for example, can be represented by the formula
k1 = k1(t) = (kmax - kmJ(l + COS^ I t„p )I2 + kmm (1) Here the values, and can be determined by experience. Until the start ofwinding, we assume that the cocoon immersed in the liquid medium is in static equilibrium.
Taking the cocoon as an ellipsoid of rotation, we consider the case when the cocoon occupies a vertical position in the aquatic environment, and its upper point is at a depth from the surface of the water (Fig. 1)
We believe that the cocoon is acted upon by the force of weight (the cocoon mass), the buoyancy force and the elastic contact force of the cocoon with the end of the thread as a result of the removal of the product from the cocoon surface and the viscous friction force proportional to the speed of the cocoon. Denote by moving the center of the cocoon.
Then the equations of motion of the cocoon in an aqueous medium under the action of the above forces are written in the form
k
mh = ki (t)--°—[x o (t) - h] - S(h)^h - mg + P (2)
ko + ki(t)
mg = P = PmSV = 2Рж&
(3)
We assume that in each moment of time the force of the weight of the cocoon and the buoyancy force according to Archimedes' law are balanced; The equality
_2nR2R /3 + nzl (R1 - z0) + +nR2(R -z1 -(R3 - z3)/3R2)
Formula (3) establishes a relationship between the geometric parameters of a cocoon, modeled by a compound ellipsoid of revolution, with a known weight.
The wetting area of the cocoon in equation (2) depends on the position of the cocoon in the aqueous medium, and for a shallow depth of immersion, we determine by formulas S = 2Sn при 0 < h < h0 (4)
S = S0(0) + S0(h -h0) при h0 < h < 2RX - z0 + h0 (5)
S = S0(0)-
■S0(4Rr
-2z0 -h + h0) при
2R1 - z0 + h0 < h < 4R1 - 2z0 + h0
(6)
The speed of the thread when winding it on reel, we assume that the maximum speed of the thread is reached by contacting it with the vertices of the polygon, the smallest - when it contacts the sides of the polygon. In this case, the velocity distribution along the hexagon surface will be non-uniform, which leads to a periodic change in the velocity of the converging filament in time.
Let us determine the speed of the thread running at the time at the apex of an isosceles triangle, where, (Fig. 1). At the moment of time, the side of the triangle rotates by an angle (is the number ofrevolutions ofthe reel). In this case, from (Fig. 1), we determine the point velocity for the instants of time (7)
For the instants of time, we have
Figure 1. Kinematic scheme of motion of the point of winding of a thread along the surface of a polygon
xi =
Then this process is repeated and the speed of the thread at the point is determined by the formulas
v = v3 = v0cos(4p -a>t) при 3в / с< t < 4ft / со, v = v4 = v0 cos(wt -4в) при 4в / с< t < 5в / с, (7) v = vn_ = v0cos(nfi -a>t) при (n - 1)в / co< t < nfl / со, v = vn = v0 cos(ne - cot) при nfi / ax t < (n + 1)в / со, n = 4,5,6,,,, (8) The length of the winding thread is determined by integrating the velocity expressions (5) - (8). To simplify the calculations, we subsequently assume the winding speed constant and equal, then the movement of the end of the thread is determined by the formula
Figure 2 shows the graphical dependence of the linear speed of the hexagon reel on time (curve 2). The straight line 1 corresponds to the average winding speed of the filament. In the calculations it is accepted. The calculations were carried out for times corresponding to one complete rotation of the reel, equal to, the average velocity is
Equation (8), taking (9) into account, reduces to the form
mil = kl(t ) [■
k0x0 (t)
- h] - S(h)vh
(9)
k0 + kl(t )
Equation (9) is integrated numerically with the initial
conditions
The calculations were performed when the geometric equality was fulfilled and for the values of the parameters:,,,, (medium water).
Figure 2. Change of linear speed of hexagon reel from time
In (Figure 2) are graphs of the dependence of the upward movement of the cocoon on time for different values, and. Calculations were carried out for two reel types, with the total length of the reel being wound (Fig. 3-5) shows the curves for a hex reel, where its average linear velocity will be. For moments of time corresponding to 1000 turns of the reel.
Figure 3. The graphs of the dependence of the displacement of the top of the cocoon center on time for the values of approximations z0 = 0.005 m, R1 = 0.0075 m centers of ellipses and the time duration
Figure 4. The graphs of the dependence of the displacement of the top of the cocoon center on time for the values of approximations z0 = 0.005 m, R1 = 0.0075 m centers of ellipses and the time duration
Figure 5. Graphs of the dependence of the displacement of the top of the cocoon center on time for the values of approximations z0 = 0.005 m, R1 = 0.0075 m centers of ellipses and the time duration tpr = 5 sec.
From the analysis of graphs it follows that when leaving the aquatic environment due to variability, the coupling coefficient with the cocoon surface begins to oscillate with a variable amplitude. The time of the decrease in the elastic coupling coefficient plays an important role in the law ofcocoon motion. As the value of this time increases, the frequency of the cocoon's oscillations decreases, the cocoon moves towards the aquatic environment.
This shows that the character of the cocoon oscillation essentially depends on the frequency of the change in the coefficient of rigidity of the elastic connection of the filament with the cocoon. With short ranges of variation in the coefficient, the oscillations are of a high-frequency nature, and with an increase in this range, the oscillation is characterized by a long period.
References:
1. Bronstein I. N., Semendyaev K. A. Handbook on mathematics for engineers and students of universities.- M.: Science. Main edition of physical and mathematical literature. 1981.- 230 p.
