¯sip vary when body i rotates. The expanded from of Eq. ͑1͒ is shown in Eq. ͑3͒. If the orien-
tation of the body at any instant of time is known, then the global position of any point on the body can be determined from Eq. ͑3͒. The vector of coordinates for body i is defined by the vector qi = ͓rT , ␾͔iT = ͓x , y , ␾͔iT, where r and ␾ represent the body translation and rotation, respectively. The total vector of coordinates for this
Introduction
Artificial lift by beam pumping is used extensively on most wells in the U.S. and in the other parts of the world. In the U.S., they represent the majority of the total wells placed on artificial lift. This makes the study of beam pumping units’ kinematics more challenging in terms of its optimization. Several studies have been presented in the past on the kinematic analysis of pumping units. The first kinematic analysis method ͓1͔ focused on the analysis of polished rod position, velocity, and acceleration as a function of crank angle.
e-mail: jamesflea@
Petroleum Engineering Department, Texas Tech University,
214 8th and Canton Avenue, Lubbock, TX 79409-3111
A Computational Method for Planar Kinematic Analysis of Beam Pumping Units
tively, and their relation is given by
rip = ri + AiSiЈp = ri + ¯sip
͑1͒
ͫ ͬ cos ␾ − sin ␾
Ai = sin ␾ cos ␾ i
͑2͒+
␰
p i
cos
␾i
−
␩
p i
sin
␾i
yip
=
yi
+
␰
p i
sin
␾i
Contributed by the Petroleum Division of ASME for publication in the JOURNAL OF ENERGY RESOURCES TECHNOLOGY. Manuscript received April 26, 2004; final manuscript received May 21, 2007. Review conducted by G. Robello Samuel.
−
␩
p i
cos
␾i
͑3͒
where ri is the vector containing the global components of a point on the body i, Ai is the transformation matrix defined by Eq. ͑2͒,
and ¯sip is the global x-y component of vector SiЈp. The elements of
A generalized computational method for planar kinematic analysis of pumping units is presented in this study. In this method, a local coordinate system is assigned to each body with respect to a fixed global coordinate system. The position of each point in a body is determined by specifying the global translational coordinates of the local coordinate system origin and its rotational angle relative to the global coordinate system. Constraint equations of motion are developed using the vector of coordinates of the connected bodies. These equations are solved to yield the position, velocity, and acceleration of the individual linkages at each instance of time. Both rotational and translational types of joints are considered in the analysis. The translational joint analysis is not discussed in this paper as they are not applicable for beam pumping units. This method can be used as an effective tool for pumping unit design and optimization. An example is provided to show the application of this method. ͓DOI: 10.1115/1.2790981͔
Ramkamal Bhagavatula
e-mail: ramkamal.bhagavatula@
Olu A. Fashesan
e-mail: olu.fasesan@
Lloyd R. Heinze
e-mail: lloyd.heinze@
James F. Lea
From a review of the kinematic analysis methods presented so far, it is observed that all the methods focused on the development of various analytical equations based on specific pumping unit geometry. Most equations have very rigorous and convoluted solutions based on their formulation procedure. In this paper, a new method of kinematic analysis is presented, which is a generalized method that can be used for kinematic analysis of any pumping unit. This method does not depend on specific unit geometry and can be used for analyzing any multibody closed loop mechanism. The kinematic parameters such as position, velocity, and acceleration at any point on linkage can be easily evaluated, which help in better understanding of linkage motion, and ultimately serve as a modern tool for pumping unit design optimization.
Cartesian Coordinate Method
In Fig. 1, body i of any arbitrary shape is defined in a plane and a local ␰i-␩i coordinate system defined within it. The local coordinate system is referenced with respect to a fixed global x-y coordinate system ͓4͔. The local position of any point p on the
The polished rod position was represented by a complex mathematical equation and the velocity and acceleration were derived by differentiating the position equation. From the geometry of the pumping unit, another analytical solution ͓2͔ was developed that related the polished rod position as a function of crank angle. This was developed mainly to be used as input for the solution of the wave equation with the finite difference method. An analysis procedure was later presented that analyzed the pumping unit geometry based on synthesis of a four-bar linkage ͓3͔. By representing vectors with complex numbers, the polished rod position, velocity, and acceleration equations were derived. This method calculates the position, angular velocity, and angular accelerations of all the linkages of a pumping unit.