Study on the conditions of material capture by pairs of rolls of squeezing machines

. This paper is devoted to the analysis of the condition for material capture with pairs of rolls of squeezing machines. Calculation formulas for the parameters of roll pairs of squeezing machines are given, such as: contact angles, minimum distance between the rolls, the thickness of the processed material after squeezing. It has been established that the capture of material in the roll pairs of squeezing machines is determined by the center distance between the rolls, the thickness of the processed material and the coefficients of friction of the rolls on the processed material. It was determined that the displacement of the upper roll relative to the lower roll does not affect the values the nip angles.


Introduction
One of the most economical and versatile types of machines used in many industries, including construction, when performing various technological processes, are roller machines.Compliance of roller machines with certain technological requirements involves the use of various designs of both working bodies (pairs of rolls) and auxiliary mechanisms.A special group in the technology of mechanical processing of materials in roller machines is the process of roller squeezing of wet materials.Based on the analysis of the design of roller wringers of the textile, light and pulp and paper industries [1][2][3] and the features of the roll wringing of wet materials, we select the scheme of the studied roll pair of squeezing machines.(Fig. 1), in which the upper roll is mixed relative to the lower roll towards the movement of the processed material to a distance  , which is determined by the angle  .Rolls with different diameters are coated with materials with different stiffnesses.The upper roll is free, and the lower one is driven.The layer of material is fed in such a way that the line, which is a continuation of its front end, passes through the axis of rotation of the upper roll since no additional external forces are required to conduct the gripping [4].In mathematical modeling of the squeezing process, it is required to solve the problems of contact interaction and moisture filtration.In works [5][6][7][8][9][10][11][12][13][14][15][16][17][18][19], the problems of contact interaction and moisture filtration during the pressing process were solved.The boundary conditions of these problems are determined by the condition of material capture by pairs of rolls [20].The conditions for capturing the material by pairs of rolls, in turn, depend on the deformation properties of the processed material [19,20].There are quite a lot of works devoted to the analysis of the conditions of capture in symmetric [21] and asymmetric [22][23][24][25][26][27] roll pairs of squeezing machines.For further development of theoretical concepts, let us analyze the conditions for the capture of material in the investigated roll pair of squeezing machines.

Resultative Methods
Let the material layer touch the rolls in section According to [28], in triangle ( 2 Substituting the values of the sides of triangle If we accept assumption , 2 2 sin 1 1

  
then we obtain an approximate formula to calculate the angle 1 The accuracy of the approximate formula (3) is sufficient.Comparative calculations using formulas ( 2 Formulas ( 3) and ( 4) are used to calculate the angles of capture at the moment of contact between the processed material and pairs of rolls.Next, we evaluate the capture angles 1

 and 2
 taking into account the forces acting on the material.
At the points of contact from the side of the rolls, normal pressure forces and friction forces , F F on the material (Fig. 1).
The capture conditions depend on the ratio of the retracting and retracting forces, as well as the equation for the balance of forces along the ordinate axis [23]: , 0 From the force diagram in Fig. 1, we find For the friction force 1 T , we apply the Amonton-Coulomb friction law , where , the coefficient and angle of friction of the lower roll during gripping, and for the friction force 2 T -we apply where , F and 2 T into inequality (8) and equality ( 9), taking into account expression (7), we have 0 cos From equality (11), we determine 2 Q and substitute it into inequality (10).After transformations, we find the capture condition for the considered roll pair: Using formula (1), from triangle  (11), from inequality (10) we obtain In most cases, the rollers on roller machines are mounted in ball bearings, where the amount of friction is small.As a result, the friction force 2 F can be ignored in comparison with other forces acting on the roll [23].
Then the gripping condition is Transforming inequality (14) and assuming that 0 By condition (17), knowing the radii and friction angles of the rolls, it is possible to determine the minimum distance (a gap) between the rolls to be set to grip a layer of material of a given thickness: .) ( 2 At the balance of the indicated forces, the process of contact interaction in the two-roll module passes into a steadystate process.Now let us analyze the contact angles for a steady-state process.Let the layer of material in the steady process pass from section , where  h is the distance between the rolls in the steady process.With (17), we obtain Similarly to formulas (3), (4), and (17), we obtain  19) are used to calculate the grip and exit angles for a steady-state process.From expression (18), we find expressions for calculating the distance between the rolls .) ( 2 Taking into account expressions ( 16) and ( 21), we find the value of the upper roll lift: .) ( 2) )( ( Taking into account equality (20), by analogy with formula (21), we have .) Equating the right-hand sides of formulas ( 22) and ( 23), we find expressions for calculating the thickness of the processed material after pressing: .)

Conclusions
1.The formulas for calculating the parameters of the roll of squeezing machines, such as contact angles and minimum distances between the rolls during gripping and in a steady-state process, and the thickness of the processed material after squeezing, were determined.2. In most cases, the rollers on roller machines are mounted in ball bearings, where the amount of friction is small.As a result, the friction force can be ignored in comparison with other forces acting on the roll.3. It has been established that the capture of material in the roll pairs of squeezing machines is determined by the center distance between the rolls, the thickness of the processed material and the coefficients of friction of the rolls on the processed material.The results of calculations using formulas (3) and (4) made it possible to reveal the following: − with an increase in 2 R , 1  and a decrease in 1 R , 1 h , the angle 1  and a decrease in 1 R , 1  lead to an increase in the angle 2  ; − the displacement of the upper roll relative to the lower roll does not affect the values of the nip angles.

1 ) 1  2 Fig. 1 .
Fig. 1.Scheme of a two-roller module of a squeezing machine after simple transformations, we obtain .
formula (3), we find the formula for calculating the angle 2

2 f  2 
are the coefficient and angle of friction of the upper roll during gripping;  ш r is the upper roll neck radius;  ш T is the resultant of friction forces in the neck of the upper roll.Let us substitute the values of 1 2); this is accompanied by the rise of the upper roll to distance 1 1

Fig. 2 . 1 
Fig. 2. Scheme of a two-roll module in a steady state Let, under a steady process, the nip angles (in section

1  1 С
is the friction angle of the lower roll at point .Formulas (