A MODE METHOD FOR AN ELASTIC ROD IN A FLUID PUMPING SYSTEM

A rod pumping system, as used to lift oil to the surface in non-flowin g wells, is analyzed by describing longitudinal rod stretch vibrations as a sum of fixed-free modes. The rod oscillates vertically, driven a t the fixed end by a constant speed motor through a four-bar mechanism . Equilibrium is used to derive the partial differential equation of r od upstroke and downstroke motion. The partial differential equation i s reduced by modal analysis (and aided by a convenient transformation to simplify an inhomogeneous boundary condition at the plunger on the upstroke) to a set of piecewise linear (and hence non-linear) and unco upled ordinary differential equations. Based on response studies, a on e-mode representation is found to capture most of the rod string stret ch at practical operating speeds, and was used to investigate the resp onse with dimensional and non-dimensionalized equations at various cra nk speeds, crank lengths and damping rates.