A mode method for an elastic rod in a fluid pumping system

Abstract

A rod pumping system, as used to lift oil to the surface in non-flowing wells, is analyzed by describing longitudinal rod stretch vibrations as a sum of fixed-free modes. The rod oscillates vertically, driven at the fixed end by a constant speed motor through a four-bar mechanism. Equilibrium is used to derive the partial differential equation of rod upstroke and downstroke motion. The partial differential equation is 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 uncoupled ordinary differential equations. Based on response studies, a one-mode representation is found to capture most of the rod string stretch at practical operating speeds, and was used to investigate the response with dimensional and non-dimensionalized equations at various crank speeds, crank lengths and damping rates.