Research Concerning the Influence of Balancing the Conventional Pumping Units on their Dynamics

The paper analyses the influence of the rotary balancing and of the combined balancing (rotary and oscillating) on the variation of the motor torque at the crankshaft and on the variation of the connection forces acting in the joints of the mechanism of the conventional pumping units. For this purpose a computer program has been developed and a series of results in the case of a C-640D-305120 pumping unit are presented.


Introduction
It is well known that for a good functioning of the sucker rod pumping installations the pumping units from their component must be correctly balanced. Balancing the pumping units represents the operation by which it is followed that the maximum values of the motor torque at the crankshaft corresponding to the ascending and descending strokes of the sucker rod column are equal [1,2]. The balancing of the conventional pumping units is achieved by adding counterweights in the extension of the cranks in the case of rotary balancing or by adding counterweights both in the extension of the cranks and in the extension of the rocker in the case of combined balancing (rotary and oscillating).
In the paper is analyzed the influence of the rotary balancing and of the combined balancing on the variation of the motor torque at the crankshaft and on the variation of the connection forces acting in the joints of the mechanism of the conventional pumping units. Starting from the calculation methodology a computer program has been developed which allows performing a comparative analysis of the influence of the rotary balancing and of the combined balancing on the variation of the mentioned parameters.

Experimental part
For establishing the values of the connection forces acting on the joints of the pumping unit mechanism and of the motor torque at the crankshaft were analyzed several records concerning the variation of the force at the polished rod in the case of a C-640D-305-120 pumping unit to check that this one is in a stable operating condition. In Figure 1 is represented such a variation curve corresponding to the stroke 85. The processing of the records was done with the Total Well Management program [3].

Determining the connecting forces and the motor torque
Dynamics of the sucker rod pumping installations represents a topical field of research, some significant results regarding the dynamics of the conventional pumping units and of the sucker rod column being presented in [4][5][6][7][8][9][10].
In Figure 2 where is presented the mechanism of a conventional pumping unit the following notations were used:    In Figure 3 and Figure 4 were represented the load schemes of the connecting rods and the rocker, which make up dyad (2-3) and of the cranks. are the weight forces of the counterweights mounted on the cranks and on the rocker, respectively. Also, in Figure 3 and Figure 4 were represented the projections of the connection forces acting in the component bearings: 12 2F from the two crank pin bearings acting on the connecting rods and From the equation of equilibrium in moments on the whole dyad in relation to point A (Figure 3), it results: Considering only the component 2 of the dyad (Figure 3), from the equilibrium equation in moments in relation to point B results: From the equilibrium equations in forces on the directions of the x and y axes in the case of the cranks (Figure 4) by considering that:  The coordinates of the different points that appear in the equations of equilibrium 2,3 and 5 are determined following the positional analysis of the pumping units mechanism [11][12][13]. For this, the method of projection of closed and independent vector contours can be used [14,15].
Solving this system of equations it results the angles 2 and 3, and then knowing these angles we can determine the coordinates of any point on the mechanism of the pumping unit.

Results and discussions
Starting from the calculation methodology presented a computer program has been developed using Maple programming environment. It is presented below a series of results of the simulations realized in the case of a pumping installation serviced by a C-640D-305-120 pumping unit produced by Lufkin [17] for which are known: , respectively. A way to calculate these angles is presented in [16].
The working angular speed of the cranks is: rot/min 6 . 6 . The simulations were performed considering the two balancing cases: rotary and combined (rotary and oscillating). For the case of rotary balancing it was considered that the counterweights of total mass equal to 4808 kg are mounted on the cranks so that OA' = 1.6 m (Figure 2). In the case of combined balancing it was considered that the counterweights mounted on the cranks are of total mass equal to 2404 kg and OA' = 2.032 m and the counterweights mounted on the rocker are of total mass equal to 2500 kg and BE = 1.05 m (Figure 2).
In the following figures are represented the variations on a cinematic cycle beginning with the angle corresponding to the beginning of the ascending stroke of the sucker rod column of the values of the connection forces:   Figure 10 shows that the maximum values of the motor torque Mm corresponding to the ascending and descending strokes of the sucker rod column are very close, which shows that both balancing possibilities (rotary and combined) lead to a correct operation of the pumping installation. Moreover, the balancing solutions proposed for the analysis lead to close values of the motor torque Mm, with the observation that in the case of combined balancing there is a distribution of the mass of the counterweights on the structure of the pumping unit mechanism. This distribution of the mass of the counterweights leads to obtaining lower loads of the joints of the pumping unit as can be seen in Figures 6÷9.

Conclusions
In the paper has been analyzed the mechanism of conventional pumping units for determining the motor torque at the crankshaft and the connecting forces acting in the component joints. The simulation program developed using Maple programming environment allowed performing a comparative analysis of the influence of the rotary balancing and of the combined balancing on the variation of the motor torque at the crankshaft and on the variation of the connection forces acting in the joints. The obtained results highlighted that both balancing possibilities (rotary and combined) lead to a correct operation of the pumping installation, but the distribution of the mass of the counterweights corresponding the combined balancing leads to obtaining lower loads of the bearings of the pumping unit.