Parameters of technological machine roll contact curves

. The main parameters of the contact curve of each roll are its shape and length, and the extent of the slip and no-slip zones. Analytical dependencies were obtained that determine the parameters of the contact curves of the rolls of technological machines. It was revealed that at the beginning of the contact zone the skin points are distant from the roll covering point, in the zone at the end of the contact zone it is ahead of it. At the end of the contact zone, the angle between the corresponding points of the skin layer and the roller coating is equal to the modulus of the difference between the lag and advance angles.


Introduction
Roller mechanisms are an integral part of most technological equipment and transport systems.They include roll machines for squeezing out fluid.These machines also have an impact on the environmental safety of enterprises since the squeezing process is directly related to the problem of wastewater disposal.
Technological processes in these mechanisms are the result of the interaction of work rolls with the material being processed.Contact interaction occurs along the contact curves.Therefore, when solving contact problems of the roll squeezing process, the parameters of the contact curves are of particular importance.
The main parameters of the contact curve of each roll are its shape and length, and the extent of the slip and no-slip zones.
In experimental works [8][9][10][11], empirical formulas for the length of contact curves and the extent of sliding and sticking zones in it were obtained.Analysis of the literature showed that there are no theoretical models of the main parameters of contact curves.

Materials and methods
The paper considers the problem of theoretically describing the main parameters of the contact curves of a machine for squeezing leather after dyeing.
According to [1], roll modules of tannery squeezing machines generally have a symmetrical appearance.
We consider a symmetrical roll module, in which a layer of leather of thickness 1  interacts with drive rolls having radius R and an elastic coating made of technical cloth of thickness H . Figure 1 shows the upper part of the roll module relative to the line of symmetry.The interaction of a material with a roll having an elastic coating can be considered by analogy with the rolling of a wheel on deformable soil [24].
The semi-finished leather product after dyeing has a uniform and thin thickness.Therefore, we can assume that it will not recover the deformation in the contact zone.
Based on this, as in the rolling of an elastic wheel on deformable soil, we believe that the roll contact curve consists of two portions -curve-line and straight-line ones.In the curved section, the leather and cloth are compressed, so the contact curve is in the front part of the contact zone.
During the squeezing process, due to the action of reactive forces, the point of maximum deformation of leather is displaced from the line of centers towards the entry of leather into the contact zone [24].Therefore, the straight-line portion is located in the middle and end parts of the contact zone.
According to Figure 1 , 0 where 

Grip angle 1
 is determined by formula , where  f is the coefficient of friction of leather against the cloth surface.
In the theory of wheel rolling, the analytical determination of the contact line is related to the ratio of the deformation rates of contacting bodies.In many publications, the hypothesis that this ratio is constant is accepted [25,26].
Assuming this hypothesis, we obtain

Results and discussion
From Figure 1, it follows that After substituting  and *  into equality (1) and solving with respect to 1 r , we obtain the equation describing the curve-line portion of the roll contact curve

H m 
According to Figure 1, for portion 2, we have The following equality holds (Figure 1) for 0 transforming these equalities, we find an expression that allows us to determine the value of : From here, taking into account condition 0 3   , we can find the condition for angle 2 The length of portion 1 is determined by the following formula: Let us transform formula (6) considering expression Differentiating equation ( 2), we obtain: From expressions ( 2) and ( 8), we determine: We believe that Then from equality (9), we have After substituting these expressions into equality (7) and integrating it, we obtain We expand the logarithmic function into a series and limit ourselves to its first term since 1 ) sin( 3 From Figure 1, it follows that ).sin (sin ) Thus, the length of the roll contact curve is set by the following expression: ). sin (sin On the contact curves of each roll there are lag slip zone A A in the rectilinear part (Figure 1).The value of the relative velocity of the leather lag along the roll contact surface can be written in the following form [27]: or considering expression (10) and where is the velocity of leather,   is the angular velocity of the roll.

Let point 2
A be defined by angle ) ( 5

 
. Then the relative displacement is determined by the formula [28]: .
Let us substitute expression rel v from equation (15) into it, integrate it, and obtain the expression that determines the lag value: The value of angle 5  depends on the value of the neutral angle, which for the roll machine under consideration is defined in portion 1 as [1]: where are the pressure force of gripping devices and the horizontal reaction of the roll supports.
On the other hand, from Figure 1, it follows that , where   is the lag angle.Comparing two expressions tel S , we have The value of the advance angle is determined similarly.It has the following form: ) According to Figure 1, the length of the no-slip zone is determined by the following formula: ), ( rel here, the values of adv S S l , , rel are determined by formulas (13), (18), and (20).

Conclusions
Analytical dependencies were obtained that determine the parameters of the contact curves of the rolls of technological machines.
It was revealed that at the beginning of the contact zone, the points of the skin layer are spaced from the roller covering point by a lag angle, which keeps it in the adhesion zone.After the sticking zone, this point begins to move ahead of the roll covering point.At the end of the contact zone, the angle between the corresponding points of the skin layer and the roller coating is equal to the modulus of the difference between the lag and advance angles.

 3 
are the grip and exit angles,  is the angle dividing sections 1 and 2.