Mathematical Analysis of the Effect of a Shock Sub on the Longitudinal Vibrations of an Oilwell Drill String
Authors: Kreisle Leonardt F. , Vance John M. 
Published: December 2007

Abstract
Shock subs have been in use for several years in oilwell drill strings to reduce longitudinal and torsional vibrations. In this paper a mathematical investigation is made into the longitudinal vibrations of a drill string, with and without a shock sub. The Laplace transformation method of solution is used, with the inverse transformation being accomplished by the theory of residues and facilitated by the use of a digital computer. The solution curves are mathematically exact at each computed point, and confirm favorable field reports on the use of the sub.

Introduction
Several years ago a device known as a shock sub was introduced to the oiltool field. Its purpose is the reduction of drill string vibrations, both longitudinal and torsional. Reports from the field indicate that the shock sub successfully performs this function, at least to a degree and under some conditions. A mathematical analysis of the longitudinal vibrations of an idealized drill string, with and without a shock sub, makes possible a determination of the precise mechanism of the action of the sub so that its effect can be quantified and best conditions established for optimum performance. Several authors have mathematically performance. Several authors have mathematically analyzed drill string vibrations due to various sources, but have not considered the effect of a shock sub. The shock sub is effectively a spring with internal damping and is located as a segment of the drill string directly above the bit and below the drill collars. Garrett published a paper giving experimental observations at the top of the drill string, both with and without the shock sub installed. He noted that the predominant observed frequencies of longitudinal vibration are three times the rotational frequency of the drill string. Photographs are included showing bottom-hole Photographs are included showing bottom-hole patterns with a three-lobe shape around the area patterns with a three-lobe shape around the area swept by the bit. This lobe pattern was formed by a tricone roller bit. It is believed by some investigators that the number of cones on the bit determines the number of lobes formed on the bottom of the hole and, consequently, the frequency of the driving force generating the longitudinal vibration in the drill string. The buildup of a bottom-hole lobe pattern is most apparent when drilling hard rock with a roller bit and may not occur at all under other conditions of drilling. In this investigation the bottom-hole lobes are assumed to be the primary cause of longitudinal vibration of the drill string; hence the driving force for longitudinal vibration is assumed to be a sinusoidal displacement function at the drill bit with a frequency of N times the rotational frequency of the drill string, where N is the number of cones on the bit.

THE MATHEMATICAL MODEL
A schematic of the entire system to be described by the mathematical model is shown in Fig. 1. The system is considered to be linear throughout. The internal damping of the shock sub element is neglected in the analysis, although viscous damping of the drill string is included. Bradbury has shown the effect of the tool joints on the undamped vibrations of the drill string to be negligible for the frequency range encountered here, although their effect on the damping coefficient could be considerable. The longitudinal motion along the doll pipe and drill colors is described by the solution to the classical wave equation, linearly damped, subject to the boundary conditions at the top and bottom of the drill string and at the point of junction of the pipe and collars. The wave equations and associated pipe and collars. The wave equations and associated boundary conditions which must be satisfied along the drill string are shown in Fig. 2.
SPEJ
P. 349