Calculation of energy-power parameters of roll mechanisms

. Calculation formulas for determining the energy-power parameters of roll mechanisms, such as the forces created by the roll gripping devices, the torque, and the power required to rotate the rolls were derived in the study. It was revealed that the radii of the rolls and the thickness of the processed material have the greatest influence on the energy-power parameters. Reducing the roll radius and material thickness makes it possible to reduce the gripping force related to a decrease in the length of the contact zone. As the radius of the roll decreases, the torque decreases as well due to the decrease in the radius of action of the resulting friction force.


Introduction
Technologies for processing materials using roll mechanisms have become widespread in industry and vehicles.Roll mechanisms determine the efficiency of the technological process.They affect the environmental safety of the enterprise since many technological processes performed using roller machines are associated with the problem of wastewater disposal.
To perform calculations when designing a roller machine, it is necessary to know the energy-power parameters of its roll mechanisms.These parameters are the forces created by the roll gripping devices, the torque, and the power required to rotate the rolls.To create an economical design of a roller machine, it is necessary that the energy-power parameters of the roll mechanisms be as minimal as possible, while ensuring the required performance of the technological process.
Formulas for calculating the energy-power parameters of roller machines, obtained at present [1][2][3], have insufficient accuracy and a certain complexity.They do not make it possible to establish the parameters with the accuracy required in the technology.Formulas obtained on the basis of experimental studies [4][5][6] are aimed at certain objects and therefore they are used mainly for their intended purpose.

Materials and methods
This study is devoted to determining the calculation formulas for the energy-power parameters of roll mechanisms.The possibility of determining these formulas has been prepared to a large extent by earlier studies on mathematical modeling of the shape of the roll contact curves and the patterns of distribution of contact and hydraulic forces .
Due to the stationary operating conditions of roll mechanisms and the balance of their links, it is appropriate to use mainly static methods for power calculations [29].
Let us consider the scheme of a symmetrical roll mechanism with processed material of thickness 1  and drive rolls of the same radius R coated with cloth materials of the same characteristics and thickness H (Figure 1).In a steady-state process [29], each roll is acted upon by the pressure force of the gripping device Q  , the response of roll supports F  , the moment of friction in roll supports fr M , the torque rot M , the elementary forces of normal pressure dN and friction dT , acting throughout the contact zone of the rolls.
Since the two-roll module under consideration is symmetrical, we will determine the power parameters for any of them, for example, for the top roll.The roll contact curve consists of compression and recovery zones.

Results and discussion
Considering the roll in equilibrium under the action of applied forces, we obtain: , where are the main moments of normal forces and friction forces,  y y T N , are the projections of the main normal and shear forces onto the Oy axis, equal to the sum of the projections of forces of the first and second zones onto the Oy axis, that is: , from expression (1) we find , or, differentiating, we obtain: .
The moduli of elementary normal forces are expressed as [29]: where  1 1 , t n are the normal and shear stresses distributed over the first zone of the contact curve.
By projecting elementary forces onto the Oy axis, and transforming, considering expressions , cos , where  1  is the angle between force 1 dN and radius 1 r , we obtain: , ) sin cos ( According to [12], the contact curve for the considered roll mechanism is described by the following expression: Hence, we have: . sin cos cos 1 where , 2  -is the indicator that determines the ratio of deformation rates of contacting bodies. Considering equalities ( 6) and ( 7), from expressions (5) we determine: . sin ) 1 ( In [13], it was stated that a parabolic law of distribution of normal stresses corresponds to the roll mechanism under consideration; the law is described by the dependence of the following form: is the maximum value of normal stresses.Based on this, we approximate the patterns of distribution of normal stresses along the contact curve by the following formulas: ), ( where  are the contact angles.After substituting expressions 1 n from equation (10) into equalities (9), we obtain: .
From here we have After integrating these expressions, substituting limits and some transformations, we have Thus, the first part of the gripping force due to normal forces is determined.To determine the second part of the gripping force caused by shear forces, it is necessary to develop a friction stress model that would relate shear and normal stresses.
One of the friction stress models widely used in engineering practice, including the study of roll mechanisms, is the so-called Amonton-Coulomb law.However, in [3,1,27], it was stated that Amonton-Coulomb's law is valid only in slip zones.Other models of friction stress are also known [13,11].Research conducted in [1,2,26], has established that for roll mechanisms the most appropriate model is the one developed in [29]; according to this model, there is a relationship between contact stresses of the form: Considering expressions ( 6) and (7), this dependence for the roll mechanism under study takes the following form: . cos 1 Taking into account dependencies (10) and ( 13), from equality (9) we obtain .
From here, we have:

Rn dT y
After integrating these expressions and substituting the limits, after some transformations, we have Substituting expressions ( 12) and ( 13) into equalities (3), we obtain: Similarly, we find: Therefore, the pressing force of the roll gripping device has the following form: Pressure Q is linear pressure (load intensity).It is known [5,6] that technological processes performed by roller machines are determined not by linear pressure, but by specific pressure sp Q .Specific pressure is defined as the ratio of linear pressure to the length of the roll contact curve.The length of the compression zone is determined by the following formula: This formula is transformed taking into account expression From expressions ( 6) and ( 7), we find or assuming that After substituting these expressions into equalities (17) and integrating, we obtain: With formulas ( 16) and ( 19), we determine Defining 2 sp Q , in a similar way, we obtain: The moments of elementary normal and shear forces relative to the pole are determined by the following expressions [8,17]: , The main moment in the first zone of the roll is determined by integrating these expressions: Substituting expression 1 t from ( 13) and considering equalities ( 6) and (10), we obtain: After integrating these expressions and substituting limits, and some transformations, we have: Having determined, similar to expression (22), the main moment of the second zone of the roll, we find the main moment of all elementary forces relative to the pole of the roll: From equality (1), it follows that M M M fr rot
In roll mechanisms, the friction moment fr M in the roller supports depends on the type of roll transmission mechanism.With a known moment of resistance to roll rotation rot M , we can find the power N required to rotate the rolls using formula [30]: where   is the angular velocity of the roll,  i is the total gear ratio of the drive mechanism;   is the overall efficiency of the drive mechanism.

Conclusions
Thus, calculation formulas ( 16), (18), and (21) were obtained to determine the energy-power parameters of roll mechanisms; using these formulas, the values of the parameters were determined with the accuracy required in the technology.
Analysis of the formulas obtained showed that the radii of the rolls and the thickness of the processed material have the greatest impact on the energy-power parameters.Reducing the roll radius and material thickness makes it possible to reduce the gripping force, which is related to a decrease in the length of the contact zone.As the radius of the roll becomes less, the torque decreases as well due to the decrease in the radius of action of the resulting friction force.