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This article investigates the movement trajectory of the cotton particle from Point D of belt transporter of the saw gin camera to Point B of the saw cylinder. In other words, the forces acting on the cotton particle and the equations of their interactions are developed, where, the differential equation of the law of movement of the cotton particle “belt + saw” is made according to the d’Alembert principle. Arc-BD of cotton particle gives the length of the arc of their fall on the saw teeth. The maximum separation of fibers and the quality of separation of fiber from seeds, as well as the time (t) of rapid separation of fiber from seeds, depend on the length of the arc-BD, because the critical length of the arc-BD makes it possible to reduce the multi-layering of the falling cotton particle on the saw teeth. In this scientific research work, patented by the new working camera of the saw gin, the trajectories of the movement of the cotton particles along the arc-DB, as well as the trajectory of the move in the direction of the grate are studied. The results of the study show that the number of fibers increases along the height of the teeth of the saw with one touch. This leads to a decrease in the average stay of seeds in the working chamber of the gin. As a result, the efficiency of the saw gin is increased.

It is known that the production of high-quality cotton fiber, necessary for the textile industry is of great importance. The main process equipment for the production of high quality cotton fiber is the saw gin machine [^{2} in the working chamber without an accelerator, due to the significant increase in lateral conditions, the seed chamber stops rotating and the ginning process is completely stopped. G. I. Boldinsky [

Above mentioned issue has not been fully resolved. The laws of motion of the fly on the ginning drum have been studied in below [

P 0 = m g ; (1)

F x = K x V x ( t ) ; (2)

F y = K y V y ( t ) ; (3)

We will compose a differential equation of the law of movement of the cotton particles “grate + saw” according to the d’Alаmber principle (

Fixed Cartesian coordinate system—XOY will place on the left end of the grate.

{ m d V x d t = − F x ; m d V y d y = − F y + F 0 ; (4)

Here: m—mass of the cotton particle;

K_{X}, K_{Y}—aerodynamic drag coefficient;

V_{0}—motion speed of belt;

α —the angle of the guide bar on the grate;

F X = F V X ; F Y = K V Y —aerodynamic drag forces;

P 0 = m g —Cotton particle gravity;

V_{X}, V_{Y}—horizontal and vertical components of the cotton particle speed;

{ d V x d t = − ω 2 V x ; d V y d t = − ω 2 V y + g ω 2 ; (5)

ω 2 = k m ; (6)

Having integrated Equation (5), we determine the velocities V_{X}, V_{Y} of the cotton particle on the coordinate axis [

Initial conditions:

| t = 0 ; V x ( 0 ) = V 0 cos α ; V y ( 0 ) = 0 ; (7)

{ V x = e − ω 2 t C 1 ; V y = e − ω 2 t + g ω 2 t + C 2 ; (8)

From the initial conditions we find С_{1}, С_{2};

C 1 = V 0 cos α ; C 2 = 1

{ V x = V 0 ⋅ cos α ⋅ e − ω 2 t ; V y = e − ω 2 t + g ω 2 t − 1 ; (9)

From the law of motion:

{ X ˙ ( t ) = V x ; Y ˙ ( t ) = V y ; (10)

We determine the trajectory of the cotton particle by integrating Equation (10).

{ X ( t ) = − V 0 cos α ω 2 e − ω 2 t + C ¯ 1 ; Y ( t ) = − 1 ω 2 e − ω 2 t + 1 2 ω 2 t 2 − t + C ¯ 2 ; (11)

From the initial conditions: | t = 0 ; X ( 0 ) = 0 ; Y ( 0 ) = 0 ; the system will have the following expression:

C ¯ 1 = V 0 cos α ω 2 ; C ¯ 2 = 1 ω 2 ; (12)

So, the trajectory of the cotton particle will be as follows:

{ X ( t ) = V 0 cos α ω 2 ( 1 − e − ω 2 t ) ; Y ( t ) = 1 ω 2 ( 1 − e − ω 2 t ) + q 2 ω 2 t 2 − t ; (13)

It is known that in this process the determination of the points of incidence B (X_{1}; Y_{1}) of cotton particles on the circumference of the saw is important [

Regarding the coordinate system-ХОY, the saw circumference equation will have the following form:

( X − X 0 ) 2 + ( Y − Y 0 ) 2 = R 2 ; (14)

Here:

X 0 = ( R + l + r + δ ) cos α ; (15)

Y 0 = ( R + l + r + δ ) sin α ; (16)

R—saw radius;

r—tape disk radius;

l—length of the guide bar;

δ —distance between saw teeth and guide bar;

tg β = Y 1 X 1 —angle of impact of the cotton particle on the saw teeth.

Define the arc BD-distance of the incidence of the cotton particle on the saw teeth. To do this, determine the position of points B and D: if, α = 0 ,

{ X ( t ) = V 0 ω 2 = ( 1 − e − ω 2 t ) ; Y ( t ) = 1 ω 2 ( 1 − e − ω 2 t ) + q 2 ω 2 t 2 − t ; ( X 1 − X 2 ) 2 + ( Y 1 − Y 2 ) 2 = R 2 ; (17)

The solution of Equation (17) is given by the coordinates of points B (X_{1}; Y_{1}). If α ≠ 0 ,

{ X ( t ) = V 0 cos α ω 2 ( 1 − e − ω 2 t ) ; Y ( t ) = 1 ω 2 ( 1 − e − ω 2 t ) + q 2 ω 2 t 2 − t ; ( X 1 − X 2 ) 2 + ( Y 1 − Y 2 ) 2 = R 2 ; (18)

The solution to system (18) is given by the coordinates of the points D (X_{2}; Y_{2}).

Arc-BD of cotton particles gives the length of the arc of their incidence on the saw teeth. The maximum separation of cotton particles and the quality of separation of fiber from seeds, as well as the time (t) of rapid separation of fiber from seeds, depends on the length of the arc-BD, because the critical length of the arc-BD makes it possible to reduce the multi-layering of the incidence on the saw teeth.

When separating the fibers from the cotton seeds, their interaction with the teeth of the saws is important. It is known that in order to separate the entire fiber from the cotton seeds, it is necessary to repeatedly interact with the teeth of the saw. Below, we study the mathematical model of the laws of interaction of the cotton particle with the saw teeth of the proposed working chamber of the saw fiber separator.

Model of the mathematical and dynamic action of the teeth of a saw with cotton particle, fiber and seeds is shown in Figures 2-4. The forces acting to the cotton particles:

P = m g —cotton particle gravity; (19)

F т = С х ⋅ Х —the force of attraction of the teeth of the saw acting on cotton fiber;

С_{х}—fiber stretch resistance;

Х—movement of cotton particle in Х axis;

Y—movement of cotton particle in Y axis;

F M = m ( V a − V n sin α ) 2 R —centrifugal force acting on cotton particles; (20)

m—mass of the cotton particle;

V_{a}—Peripheral speed of saw;

V_{n}—flying speed of cotton particle;

I—inertion force;

N—pressure force of cotton particle on saw teeth;

F k = С k ( X − V a ∗ t ) —inter-friction force of cotton particle;

X 0 = V a ⋅ t —movement of the tangential movement of the saw teeth;

β —Angle of incidence of cotton particle relative to the axis DY.

The differential equation of interaction of the system “cotton particle-saw teeth” based on the d’Alаmber principle, relative to the coordinate axis of the XDY is expressed as follows:

{ m x ¨ − F k + F T cos α + P cos β = 0 m y ¨ + F M − N + P sin β − F T sin α = 0 (21)

Here, cos α = X 0 l 0 ; sin α = Y 0 l 0 ; l 0 —the average length of cotton fibers caught in the teeth of the saw;

Х_{0}—distance between teeth caught in fibers;

Y_{0}—teeth height;

Y 0 l 0 —taking into account the minimal ratio, we take sin α ≈ 0 ; then, after replacing the expression of all forces, equation system (21) will take the following form:

{ x ¨ + C x m cos α x − C k m x = − g cos β − C k V a m t y ¨ = N m + g sin β − V a 2 R (22)

or,

{ x ¨ + ω 2 x = + a 1 t + a 2 y ¨ = b 1 (23)

Here:

ω 2 = C x cos α − C k m ; a 1 = − g cos β ; a 2 = − C k V a m (24)

b 1 = N m + g sin β − V а 2 R (25)

Equation (23) is a differential equation of the process of the “cotton particle-saw teeth” system.

Integrating it, we find an analytical expression of the laws of motion X(t), Y(t).

a) x ¨ + ω 2 x = 0 ; (26)

x = sin k t ; x ¨ = − k 2 sin k t ; (27)

− k 2 sin k t + ω 2 sin k t = 0 ; k = ± ω ; (28)

Private solution:

x = c 1 sin ω t + с 2 cos ω t (29)

b) x ¨ + ω 2 x = a 1 t + a 2 ; general solution of the equation:

x ¯ = A t + B → ω 2 A t + ω 2 B = a 1 t + a 2 ; (30)

A = a 1 ω 2 ; B = a 2 ω 2 ; (31)

x ¯ = a 1 ω 2 t + a 2 ω 2 ; (32)

c) Taking into account Equations (29) and (32), the general solution of Equation (23) will take the following form:

x = x ( t ) + x ¯ ( t ) = c 1 sin ω t + c 2 cos ω t + a 1 ω 2 t + a 2 ω 2 (33)

d) y ¨ = b 1 , y → 0 ; (34)

y = b 1 t 2 2 + c 3 t + c 4 (35)

c_{1}, c_{2}, c_{3}, c_{4}—being unknown coefficients, are determined from the initial conditions.

Initial conditions:

t = 0 ; x ( 0 ) = x 0 = 0 ; x ˙ ( 0 ) = V n cos β ; (36)

y ( 0 ) = h ; y ˙ ( 0 ) = V n sin β ; (37)

Here: h—Height of the saw gin teeth.

х ( 0 ) = c 2 + a 2 ω 2 = 0 ; c 2 = − a 2 ω 2 (38)

x ˙ ( 0 ) = ω c 1 + a 1 ω 2 = V n cos β ; (39)

c 1 = V n ω cos β − a 1 ω 2 ; (40)

y ( 0 ) = c 4 = h → c 4 = h ; (41)

y ˙ ( 0 ) = c 3 = V n cos β ; (42)

Therefore, the interaction of the cotton particles with the saw teeth is determined as follows:

{ x ( t ) = ( V n ω cos β − a 1 ω 3 ) ⋅ sin ω t + ( − a 2 ω 2 ) cos ω t + a 1 ω 2 + a 2 ω 2 ; y ( t ) = b 1 t 2 2 + V n cos β t + h ; (43)

It can be seen from the system that oscillations in the form of a sine wave and a cosine wave together with cotton particles and saw teeth occur because of a complex process. Using the formula (43), we can calculate the maximum value of the force Ft of the pulling saw tooth per fiber bundle. In addition, it is possible to determine the change in pressure force (N) of the cotton particles on the saw teeth in time (t).

The authors declare no conflicts of interest regarding the publication of this paper.

Ergashev, J., Akhmedhodjaev, K., Karimov, A., Kayumov, J., Ergasheva, R. and Mahsudov, S. (2019) Studying the Law of the Movement of Cotton Particle on a Saw Cylinder and the Interaction with Saw Teeth. Engineering, 11, 717-725. https://doi.org/10.4236/eng.2019.1110046