ANALYTICAL SOLUTION OF THE PROBLEM OF DYNAMIC SYNTHESIS OF A SIX-LINK STRAIGHT-LINE CONVERTING MECHANISM OF THE SUCKER- ROD PUMPING DRIVE

Technological machines for various purposes are widely used in technology. In these machines, the transfer of energy from the engine to the working body is carried out through the mechanisms of levers. Modern technology increases the power of such machines, which leads to an increase in the speed of movement of the working bodies of the machines. In addition, inertial loads increase sharply in mechanisms, and the problem of equilibrium is of particular importance. The level of vibration of the machine depends on the quality of balancing, as well as its performance, reliability and accuracy of work, and the quality of technological processes. There is a static and dynamic balance of the mechanism; their elimination in the designed mechanism will correspond to its static and dynamic balancing. Depending on the degree of balance, an exact or approximate solution (balancing) can be obtained. Traditionally, the criterion for accurate static balancing of the mechanism is the condition that the main vector of inertia of its links is equal to zero


Introduction
Technological machines for various purposes are widely used in technology. In these machines, the transfer of energy from the engine to the working body is carried out through the mechanisms of levers. Modern technology increases the power of such machines, which leads to an increase in the speed of movement of the working bodies of the machines. In addition, inertial loads increase sharply in mechanisms, and the problem of equilibrium is of particular importance. The level of vibration of the machine depends on the quality of balancing, as well as its performance, reliability and accuracy of work, and the quality of technological processes.
There is a static and dynamic balance of the mechanism; their elimination in the designed mechanism will correspond to its static and dynamic balancing. Depending on the degree of balance, an exact or approximate solution (balancing) can be obtained.
Traditionally, the criterion for accurate static balancing of the mechanism is the condition that the main vector of inertia of its links is equal to zero 0, which corresponds to the immobility of the general mass center of the mechanism. With precise dynamic balancing, simultaneously with the abovementioned condition, it is also required to zero the main moment of the inertial forces of the links, i. e.
If it was possible for the mechanism in some way to satisfy the conditions of exact balancing, then these conditions will persist for any law of motion of the input link and, therefore, the balance of the mechanism (both static and dynamic) becomes an intrinsic quality of the mechanism [1].
Approximate balancing of the mechanism can be considered as an approximation to the precise one, if some secondary conditions can be neglected when solving a specific problem.
The tasks associated with reducing the swinging moment, the load on the gearbox and the efforts on the crank pins, which cause shocks in the drive mechanism of the rocker machine, do not have an analytical solution.
This allows us to assert that it is expedient to analytically solve the problem of optimal dynamic balancing of the converting mechanism of sucker-rod pumping units using the analytical method to reduce the required engine power.
The goal of balancing is to cancel or reduce variable dynamic responses. When solving the problem of balancing by adding counterweights, the method of selecting the value of the correction factor was used. Therefore, an analytical solution to the problem in this direction is relevant.
In practice, according to the results of the analytical solution of the problem of optimal balancing of the converting mechanism of sucker-rod pumping units, the location and the weight of the counterweight are determined, which provides the minimum swing moment to the foundation.

Literature review and problem statement
In [2], the general foundations of the structural analysis of mechanisms, features of kinematic analysis using Lagrange variables and dynamic analysis based on the energy model of mechanics with the fulfillment of the law of conservation of energy on any elements and for the mechanism as a whole for any time interval are given. But there were still unresolved issues related to balancing. The paper [3] presents an optimization method for finding the link shapes for a dynamically balanced flat four-link mechanism. The force of shock and the moment of shock, arising in the mechanism due to inertia, are minimized due to the optimal distribution of the masses of the links. It is shown that using cubic B-spline curves, the link shapes are found and an optimization problem is formulated to minimize the percentage error in the resulting link inertia values, in which the control points of the B-spline curve are taken as design variables. Since in the paper the dynamic balancing of the mechanism is achieved due to the optimal distribution of the masses of the links, an analytical solution to the problem is not given.
The paper [4] considers the mechanism of balancing the upper roll significantly affecting the accuracy of rolling and the operating conditions of the pressure device. Balancing mechanism calculations are usually limited to the case of static balancing for some middle position of the mechanism. The dependence of the rebalancing coefficient on the position of the mechanism has been investigated; a refined calculation of the balancing device was carried out taking into account the friction forces and the dynamic operating conditions of the mechanism.
However, the mechanism operates in a dynamic mode, which means that large masses are accelerated and decelerated in short periods of time, and, in addition, friction forces in kinematic pairs have a significant effect on the mechanism operation. But the analytical approach to solving balancing problems is not considered.
The paper [5] presents the research results of the developed method of graphic synthesis, which allows determining the initial values of free parameters of the considered crank-slider mechanisms of the 3 rd class, realizing the required cyclogram with an approximate height, the output link of which moves along the guide. Analytical expressions are obtained for calculating the parameters of the kinematic diagram of the mechanism. Multi-link crank-link mechanisms are used in crank presses. But there were still unresolved issues related to balancing the mechanisms of the 3 rd class.
The paper [6] is devoted to the study of the peculiarities of using the analytical method of kinematic analysis of lever mechanisms. It is shown how to use a number of analytical methods based on obtaining formal mathematical expressions describing the position functions in the form of functions of the angles of rotation of the movable links or in the form of functions of movement of the characteristic points of the mechanism for the mathematical modeling of multi-link lever mechanisms. However, questions related to kinetostatic analysis and balancing of multi-link linkage mechanisms remained unresolved.
In [7], a dynamic system simulation is presented, compiled in Visual Basic V6.0, and the correctness and practicality of the simulation are verified by field measurements.
To study the complex forces acting in a system, static and dynamic models of the injection pump, hermetic piston, and production pump are set up from the bottom up using a mechanical method, and the solution is found using the difference method. The drive mechanism is not considered here, as the research is carried out by a mechanical method.
The paper [8] shows that the inherently balanced communication architecture, based on four rods, is only part of a complete or grandiose four-rod based inherently balanced communication architecture that is presented here. It is shown that this architecture includes all related theories and they all depend on the principal vectors. Various new balanced relationships are also displayed. Here the problem of mechanism balancing is solved due to the balanced communication architecture and the analytical method for solving the problem is not considered.
The work [9] shows the modes of the frequency-elastic drive for the pumping system to reduce the peak loads on the polished rod and the total energy consumption. It is shown that the variable-frequency drive mode is a software solution for variable-speed drive systems that can be applied in the controller and does not require any hardware settings. The new drive mode adjusts the reference frequency transmitted by the controller to the frequency converter based on the actual power requirements. But the power consumption of the sucker-rod pumping drive mechanism is not given. The reason for this may be objective difficulties associated with the analytical solution of the problem of dynamic analysis and synthesis.
Based on the analysis of the kinematics, dynamics and running characteristics of the beam pumping unit, in [10] a full-fledged mathematical model of the engine, pumping unit, sucker rod and oil pump was created. The system of differential equations for pumping out the pumping unit used the method of cyclic iterations to solve the problem of strong adhesion between the engine, pumping unit, sucker rod and pump. The model is confirmed by experimental data on production pumping wells.
But there are still unresolved issues related to the reduction of the swinging moment, the load on the gearbox and the efforts on the crank fingers, which cause shocks in the drive mechanism of the rocking machine.
All this allows us to assert that it is expedient to conduct a study on the optimal dynamic balancing of the converting mechanism of sucker-rod pumping units by the analytical method to reduce the required engine power and its uniform load per cycle of movement and to determine the optimal values of the counterweight weight.

The aim and objectives of the study
The aim of the study is to solve the problem of optimal dynamic balancing of the six-link converting mechanism of sucker-rod pumping units by the analytical method to reduce the required engine power and uniform its load during the movement cycle and to determine the optimal values l П =OL of the counterweight weight G П and the distance from the crank axis at which the minimum peak value of the balancing moment on the crank shaft is ensured. This will allow optimal placement of the counterweight to counterbalance the conversion mechanism of the sucker-rod pumping unit.
To achieve the aim, the following objectives were set: -by selecting the value of the correction factor k, to solve the problem of balancing using a computer model of dynamics; -to analytically solve the balancing problem, that is, find the location of the counterweight on the third link of the six-link converting mechanism and for which the principle of possible displacement is used; -to analytically solve the problem of combined balancing, that is, placing the counterweight on the third link and on the crank of the six-link converting mechanism.

Research materials and methods
In [1], the movement of a six-link hinge-lever mechanism was investigated and the following problems were solved: -the problem of kinematic analysis of a six-link rectilinear guiding converting mechanism; defining the functions of the mechanism positions; -the problem of the power analysis of a six-link rectilinear guiding converting mechanism; -determination of the reaction force in kinematic pairs. Also, a computer model for studying the kinematics and kinetostatics of the six-link rectilinear guiding mechanism of the SRP drive has been developed.
Based on these studies, we will solve analytically the problem of optimal dynamic balancing of the six-link converting mechanism of the sucker-rod pump drive.
The converting mechanism of the pumping unit, shown in Fig. 1, is a class II mechanism, which consists of a crank -1, a double-drive group (2, 3) FCO, also attached to it double-drive group (4, 5) ABC. The working point is the suspension point D of the rod string.
The crank is affected at the point S 1 in the center of mass of the crank by G 1 -the weight of the crank and M D -engine torque, the connecting rod is affected in the center of mass of the connecting rod by G 2 -the weight of the connecting rod. The third link is affected at the point S 3 by the force G 3 -the weight of the third link. The connecting rod 4 is affected in the center of mass of the connecting rod by G 4the weight of the connecting rod. G 5 -the weight of the fifth link operates on the 5th link at the center of mass, as well as load P (the weight of the rod string and pumped liquids) in the suspension point of the rod string D.
The purpose of the balancing task is to minimize input torque M on the crank shaft. To do this, one needs to properly pick up the mass of the counterbalance and the distance of the center of the counterweight from the axis of rotation. In the case of rotary balancing, the counterweight is set on the crank. And in the case of combined balancing, a second counterweight is added, which is installed on the balancer.
The distance of the center of the counterweight mass from the axis of crank rotation is defined in the first approximation as: where H E -length of the rod string, P up , P down -loads at the point of rod suspension at the motion up and down, G Π -total weight of counterweights; k -correcting coefficient that is manually entered by the user until the two peak values of torque М on the crank shaft will be equal.
Let's use the well-known principle of possible movements According to the principle of possible movements, the power of these forces should be zero. Let's write this down for our problem: Here, V i are the velocities of the corresponding points of gravity forces application; ω i -angular speed of the crank; М -torque on the crank shaft.
In this section, we consider an analytical method for solving this problem. The starting point is the principle of possible displacement, which is written in the form of expression (5).
To find the unknown variable x 1 at which the minimum value of the balancing moment is achieved at full rotation of the crank, two methods are proposed.
2) From the obtained expression, we find M i , expressing it in terms of the remaining values of the powers: Let us introduce the following notations: and rewrite the expression (8) 1 .
Then the problem is reduced to finding the minimum of the S function depending on the х 1 variable ( ) Thus, such an х 1 value is sought, at which the minimum of the root-mean-square value of torque М is reached.
As you know, in order to reach the minimum of a function, it is necessary that its first derivative be equal to zero, that is 2) The essence of the second method is as follows.
Consider the problem of balancing, when the counterweight on the third link is displaced by the α ∏ angle (Fig. 2). Fig. 2 We rewrite the expression (11), with variables and notations introduced for each i-th position of the mechanism, i=1, …, N.
We find such an x 1 value, at which the minimum of the root-mean-square value of torque М is reached.
From the necessary condition for the minimum of the S function, we obtain two equations with two unknowns 3) Now we are solving the problem of combined balancing, when the counterweight is placed not only on the 3rd link, but also on the crank. Let's try to solve analytically the problem of optimal dynamic balancing in general form. In addition, we assume that the counterweight on the crank is displaced by the third link angle and the counterweight on the third link is displaced by the α Π angle (Fig. 3).
For the corresponding terms in the expression for the principle of possible displacements (21), we have ( ) cos cos cos sin sin .
Let us introduce variables Then (16) takes the form:  .
4) Assuming that M i =М * =const, we introduce a new variable x 5 =M * and rewrite (28) as follows: The left part of (30) we designate as ∆ i and then, according to the synthesis condition, we find x 1 -x 5 values, at which ∆ i approaches zero. For what, we will minimize the mean square value of the S function. From the minimum condition, we get five equations with five unknowns.
Thus, the unknown parameters x 1 -x 4 and torque value x 4 are determined, where M i =М * =const is considered as a constant parameter.

1. Results of solving the balancing problem using a computer model of dynamics
By choosing the value of the correction factor k, the mass of the counterweight and the distance of the center of the counterweight from the axis of rotation were selected. The distance of the center of mass of the counterweight from the axis of rotation of the crank is determined in the first approximation by (1). In the 2nd column of Table 1, the balancing results are obtained through the correction factor, which adjusts the distance of the center of mass of the counterweight from the axis of rotation and is entered manually until the two peak values of shaft torque M become equal. Moore values change -is the torque shown in Fig. 4, a. Also, changes in the values of torque M after determining the place of the counterweight are shown in Fig. 4, b, c.

2. Results of analytical solution of the balancing problem
Unknown parameter of the counterweight. Further, the value of the unknown parameter is determined, at which the minimum of the root-mean-square value of torque M is achieved. From the condition of the minimum of the function, an equation is obtained to determine the location of the counterweight.
Solving the problem of optimal balancing, an expression was obtained that allows one to determine the mass and location of the counterweight along the 3rd link of the six-link converting mechanism of the rocking machine.  The balancing problem is also analytically solved, when the counterweight on the third link is displaced by an angle and the unknown variables are 1 cos , The analytical solution of the problem for det 0 A ≠ is obtained as follows 1 , The 2 nd column of Table 2 gives the result of the analytical method where the value of G П l П is found, at which the minimum of the torque М value is reached.
The 3 rd column of Table 2 gives the result of the analytical method for solving the balancing problem, when the counterweight on the third link is displaced by an angle.
Also, assuming that the torque M i =М * =const, we introduce a new variable x 5 =M * .
Similarly, the analytical solution of the problem for det 0 A ≠ is as follows 1 , Changes in the values of torque M after determining the locations of the counterweights are shown in Fig. 4, d.

Discussion of the results of solving the problem of balancing the six-link drive mechanism of the sucker-rod pumping unit
The results confirm that by the analytical solution of the problem, it is possible to more accurately determine the location of the counterweight when balancing the converting mechanism of the sucker-rod pumping unit.
As can be seen from Table 1, the result obtained when solving the balancing problem by selecting the value of the correction factor k, comparing two peak values of torque M on the crank shaft per cycle of movement of the mechanism, was also obtained when solving the balancing problem by the analytical method.
This proves the correctness of the analytical expressions obtained for the parameters of the counterweight of the transmission mechanism of sucker-rod pumping units.
Also, with the combined balancing method, the value of the maximum torque M and the value of the maximum power are reduced by 20 % than when the counterweight is placed on the third link of the converting mechanism, as well as when the value of the maximum torque is determined through the correction factor k. This shows the optimality of the combined balancing of the six-link converting mechanism of the SRP drive, which allows reducing the swinging moment, load on the gearbox and efforts on the crank pins, which cause impacts.
As you can see, the combined balancing significantly reduces the swinging moment on the foundation of the converting mechanism of sucker-rod pumping units, thereby ensuring optimal balancing of the mechanism. Therefore, the obtained analytical solutions of the balancing problem can be widely applied in practice.
The disadvantage of the combined balancing is the increased crank bearing response due to the weight of the counterweight.
In the future, a multicriteria synthesis of the six-link converting mechanism of the SRP drive will be carried out to reduce the upper strut and improve the angle of motion transmission. It is also required to reduce the values of the reactions of hinges and supports.

Conclusions
1. The mass of the counterweight and the distance of the center of the counterweight from the axis of rotation when solving the problem, by choosing the value of the correction factor k are m 3 =478 kg and l 3 =0.525 m. Maximum torque value M max =8.260 kNm.
2. An analytical solution to the problem of optimal dynamic balancing of the six-link converting mechanism of the sucker-rod pumping drive is obtained in various formulations. In the case when the counterweight is placed along the 3rd link, the following values are obtained m 3 =479 kg and l 3 =0.653 m, the value of the maximum torque M max =8.412 kNm. In the case when the counterweight on the third link is displaced by the angle, m 3 =479 kg and l 3 =0.653, the value of the maximum torque M max =8.365 kNm. Comparison of the results obtained shows that the value of the maximum torque and the location of the counterweight have insignificant differences when solving by choosing the value of the correction factor and when solving by an analytical method. 3. A system of equations is obtained for solving the problem of combined balancing, that is, when the counterweight is placed on the third link, displaced by an angle and on the crank, displaced by an angle. The following values are obtained m 3 =478 kg and l 3 =0.844 m, m 1 =400 kg, l 1 =0.334, the value of the maximum torque M max =6.605 kNm. The results show that with the combined balancing method, the value of the maximum torque M and the value of the maximum power are reduced by 20 % than when the counterweight is placed on the third link of the sucker-rod pumping drive converter mechanism.