CONTROLLABILITY AND STABILITY CAPACITY OF A ROLL PAIR MOTION RESEARCH

The process of deriving the equation of the perturbed motion of a roll pair during the processing of leather material is discussed in the article, taking into account the influence of dynamic factors. It is shown that one of the reasons for the unstable stress state on the contact surfaces in the roller mechanism are dynamic factors arising from inaccuracies in the manufacture of its individual parts, assembly defects, and the occurrence of an oscillatory process in the roller mechanism, as well as due to the non­uniform thickness of the processed material at its gripping during starting and stopping the machine. Methods for determining optimal controls are shown; they provide asymptotic stability of the unperturbed motion of a roll pair and the torque applied to the upper roll as a function of generalized coordinates. It is shown that the width of the contact strip of the clamp, which depends on the radii of the rolls and the hardness of the coatings, has a significant effect on the efficiency of the rolls. The larger the shaft radius, the lower the actual pressure per unit contact area. It is shown that the squeezing efficiency increases with the improvement of the conditions for the removal of the squeezable liquid from the rolls (with their horizontal arrangement), with an increase in its temperature and a decrease in viscosity. Efficiency decreases with increasing material speed and thickness. It is shown that the width of the contact strip of the clamp, which depends on the radii of the shafts and the stiffness of the coatings, has a significant impact on the efficiency of the shafts. The larger the shaft radius, the lower the actual pressure per unit contact area.


Introduction
Roller mechanisms are widely used due to the simplicity of design, the possibility of a continuous technological operation when processing the material and the combination of several functions. The variety of operations performed by roller mechanisms has not made it possible to create a unified system for their design, due to the difference in technological tasks and phenomena occurring in the contact zone of the rollers.
Most roller mechanisms perform two or more functions at the same time: active and auxiliary ones. The latter most often refers to the feeding the leather material; the squeezing mechanism can be wringing, pressing and deforming one. Now consider the dynamic factors affecting the operation of the roll pair. One of the reasons for the unstable stress state on the contact surfaces in the roller mechanism is a dynamic factor arising from inaccuracies in the manufacture of its individual parts, assembly defects, and the occurrence of an oscillatory process in the roller mechanism, as well as due to the non-uniform thickness of the processed material, during its capture, machine start and stop operations.
The reason for the occurrence of an oscillatory process in the roller mechanism may be inaccuracies in the manufacture of machine parts. Let, for example, the geometric axis of the roller do not coincide with its axis of rotation. Then there is a kinematic excitation of harmonic nature, which leads to vibrations and resonance in the system. Vibrations in the roller mechanism are transmitted to the machine frame and require a definition of the inertial and elastic characteristics of its elements [1,2].

Engineering
With the effective use of a biocatalytic modifier, the stiffness index of the semifinished leather product is reduced by two times compared to the semifinished leather product developed according to the existing technology, while maintaining its mechanical strength [3]. It should be noted that this paper does not consider the removal of excess fluid from the semifinished leather product.
The articles [4][5][6] present the results of studies on the control of technological parameters of the operation to extract excess liquid from multilayer moisturesaturated fibrous materials using leather as an example. Mathematical dependences of the amount of moisture removed for each layer of wet leather on the feed rate between the rotating working rollers and their pressing force were obtained.
However, a critical analysis of the operation of the experimental roller stand used in these works shows that when processing a multilayer package of wet leathers, the control of their feed rate between the rotating working shafts is of great importance. Consequently, the greater the thickness of the leather package, the correspondingly lower the feed rate should be, but the authors of these works did not take this into account when conducting experiments. From this it should be noted that by controlling the feed rate of processed wet skins between the working shafts, it would be possible to achieve a significant increase in the quality of their processing.
There are publications devoted to the development and improvement of leather machines. In [7], the results of studies on determining the ratio of forces in the process of feeding a semifinished leather product to the working area of a multioperational machine by a conveying device are presented, and in [8] a solution to the problems of ensuring the stable motion of the feeding mechanism of multi-operation machines is proposed. These studies were carried out for manual work, where the issues of process control and the stability of the movement of the roll pair, depending on the uneven thickness of the processed material, were not considered.
The study in [9] is devoted to the solution of contact interaction in the roll pair. Mathematical models of the contact stresses distribution patterns are developed. Research is underway aimed at solving the problems of using toothlever differential gears in technological roller machines [10,11]. However, these works do not take into account the dynamic factors of the process of contact interaction in roller pairs with a leather semi-finished product.

Materials and methods
The aim of the study is to determine the optimal controls that ensure the asymptotic stability of the unperturbed motion of the roll pair and the torque applied to the upper roll as a function of generalized coordinates.
Roll pairs are widely used in many industries. When studying the dynamics of a roll pair, it is necessary to proceed from the forces acting on the rolls during operation. The magnitude and direction of the acting forces during the capture of the processed material and in the steady state are different. They also depend on many parameters and factors, i.e. from the diameters of the rolls, which may be equal or different, from the kinematic connection between the rolls, which may be rigid or one of the rolls is free, which will rotate due to fiction; from the installation of rolls that can be installed horizontally or obliquely, one above the other with the location of their axes of rotation on a vertical or inclined plane, also vertically; at the same time, to create a clamp between the rolls, the upper, lower or both rolls can be movable; swaths can be with hard or elastic coatings, which can be moisturepermeable or impervious, depending on the technology, combinations of them can be selected. In addition, one of the rolls or both rolls may be composite. In all cases, the action of force, movement must be continuous and stable. Therefore, it is important to ensure the stability of the process of pressing leather materials.
To determine the value of the basic parameters for controlling the motion of a roll pair, let's first compose a differential equation of motion in the Lagrange form with holonomic servo constraints, obtained in [12]: is the kinetic energy of the system; are the generalized forces referred to coordinates q j ; are the known functions of time and q j .
In the center of the fixed roll, let's install a fixed system of coordinates Oxyz, and the x axis is directed along the roll axis, and the y axis in the direction of leather motion and the z axis is directed vertically upwards. The origin of the moving coordinate system O 1 x 1 y 1 z 1 is set to the center of mass of the moving roller. Let's direct the x 1 and y 1 axes parallel to the x and у axes, respectively, and let's direct the z axis along the z axis.
To determine the positions of the fixed roll, semifinished leather product and the upper movable roll, five independent parameters must be set. Let's take the following values as gene ralized coordinates:  where ϑ M and ϑ N are the linear velocities of the point lying on the rim of the movable and fixed rolls ( Fig. 1). In order for condition (2) to be fulfilled, the velocity of a point lying on a movable roll relative to the x 1 axis must be zero. In addition, to fulfill condition (2) at each time point (if the system moves from a state of rest), the following conditions should be fulfilled: Original Research Article: full paper (2022), «EUREKA: Physics and Engineering» Number 3 Engineering Since angular velocities of the movable and fixed rolls, and the leather feed rate are small, the fulfillment of condition (2) could be done using special equipment.
Let's compose the equations of motion (1) taking into account relation (3) obtained in [13]: The resulting equations (4)-(6), together with the constraint equations (2), completely describe the motion of the squeezing machine shown in Fig. 1.

Results and discussion
Next, let's consider the problems of controllability of technological processes during the pressing of a semi-finished leather product.
When implementing technological processes, the parameters that characterize these processes must change according to certain laws (or be constant).
The purpose of the control is to ensure the necessary velocity between the various nodes of the system or to achieve the maximum power acting to a certain part of the squeezing machine. The processes that must change according to certain laws are velocity and power under the in fluence of external sources, realized by signals. Now let's compose the equation of the perturbed motion of a roll pair. For the unperturbed motion, let's take the motion defined by equations (4)- (6) and assume that these equations have a particular solution [14,15]: Under perturbed motion there are: where x 1 , x 2 and x 3 are the deviations of the perturbed motion from the unperturbed motion. Engineering Then the equations of perturbed motion have the following form: It is easy to see that the righthand sides of these equations vanish under conditions. x 1 = x 2 = x 3 = u 1 = u 2 = u 3 = 0. Expanding the righthand sides into series in powers of x 1 , x 2 , x 3 and restricting ourselves to terms of the first order of smallness, let's obtain the equations of the first approximation: where symbols R 2 and R 3 denote the terms the measurement of which in x x x x , , ,   2 3 3 is higher than the first making the necessary calculations, let's write these equations in normal form. To do this, introducing notation, x 2i-1 = y i , x  The system of equations (11) in a matrix formulation takes the following form or in a short form: where W denotes that the terms the dimension of which in y 1 , y 2 ,..., y 6 and    y y y 1 2 6 , ,..., is higher than the first. Engineering Now consider the matrix [14][15][16][17][18]: Based on the system controllability criterion, i.e. in order for the system described by (12) to be completely controllable on segment [t 1 , t 2 ], it is necessary and sufficient that the rank of matrix K be equal to 6.
Let's now compose matrix K: Given that, the determinant does not change if any linear combination of other rows is added to one of its rows, then: Thus, the rank of matrix K is 6, and, consequently, the system (7)-(9) is completely controllable. Engineering To solve the stabilization problem, let's use the following theorem [19]: Controlled system (12) can always be stabilized with respect to the manifold defined by constraints (2).
Substituting (14) into equations (12), let's obtain the nonlinear equations of perturbed motion: where c a a   Without violating the constancy of coefficients γ ij , let's assume that: Let's construct a characteristic determinant for system (15): Expanding the determinant (17)   , λ Let's construct the following matrix, the socalled Hurwitz matrix [20] from coefficients b 0 , b 1 , ..., b 6 of equation (18) It is known that if for b 0 > 0 all principal diagonal Hurwitz minors Δ 1 , Δ 2 , …, Δ 6 are positive, then the unperturbed motion is asymptotically stable, regardless of the terms higher than the first order of smallness. Therefore, let's determine the coefficients of equation (14) in such a way that the Hurwitz conditions are satisfied [20]: since a 2 > 0, a 9 > 0 and a 10 > 0.   where the values of the coefficients of the control action (14) are determined by the following formulas: Let's substitute the values of the coefficients (20) into equation (14): When designing a squeezing machine, all parameters of the machine must satisfy condition (20). Equations (21) are the sought-for laws of guidance of control parameters u 1 , u 2 , u 3 .
Let's substitute relation (21) into equation (10) and obtain: When determining M a 1 , it is assumed that one roll of the pair (a drive roll), the material being processed and the second roll (a nondrive roll) receive motion due to the force of friction on the contact surfaces.
If both rolls are driven, i. e. are connected with a rigid kinematic constraint and a drive, the material being processed is set in motion due to the friction force on the contact surfaces with the rolls, then it is recommended that the radii of both rolls and the monchons (fabric tires, that absorb and remove moisture well) should be the same. In this case, M a 1 and M a 2 will be equal. Therefore, for the dynamic stability of the movement of a roll pair, the coefficients of equation (14) must satisfy equalities (20).
Without compiling the equation of perturbed motion and the requirement that the condition of asymptotic stability of the process under consideration be satisfied, this method loses its meaning. In order to apply the results obtained in practice, using the initial data (satisfying the conditions of the control function), let's substitute equalities (20) into the equations of motion and, with the exact implementation of the servo constraint, determine the reaction forces of the constraints and the kinematic characteristics of motion.
To develop this method in the future, it is necessary to select servomotors, sensors and create programs that describe all technological processes of machining; to develop a wringing machine operating in automatic mode.

Conclusions
It is substantiated that the primary task that needs to be solved when designing automatic control systems is the construction of a kinematic diagram and mathematical model of the control object. It is shown that the basic aspect in the theory of automatic control is precisely the formulation of a mathematical description of the control object functioning, its properties and relationships, which makes it possible to evaluate (predict) information about the change in the state of the object when external impacts are applied to it.
It is shown that one of the reasons for the unstable stress state on the contact surfaces in the roller mechanism is dynamic factors arising from inaccuracies in the manufacture of its individual parts, assembly defects, the occurrence of an oscillatory process in the roller mechanism, and the nonuniform thickness of the processed material during its capture, starting and stopping the machine.
Optimal controls are determined that ensure the asymptotic stability of the unperturbed motion of the roll pair and the torque applied to the upper roll as a function of generalized coordinates.