Mechanical Modelling of Drill-Strings

Vibrations in long torsional strings result in spatio-temporal dynamics. These vibrations disturb the accuracy of the drilling process and may cause damage due to fatigue. The high costs of drillings are the main motivation for the suppression of vibrations of the drill–string. In this paper a model for the drill string is proposed. In order to describe the dynamics of the drill–string appropriately but as simple as possible the model has to comprise bending vibrations, self–excited torsional vibrations as well as the coupling between them. The aim of the model with only few degrees of freedom is the development of an optimal control technique for the active damping of drill–string vibrations. The string is modeled as a one–dimensional continuum. Two sets of variables are proposed for the formulation of kinematics: one is suitable for the description of the kinetic energy the other for the description of the elastic energy. Differential geometry yield equations which express one sets of variables in terms of the other. These equations can be linearised. In order to justify the linearisation, an analysis of the parameters is made. Since the Lagrangian can be written in a more compact form than the equations of motion, the Lagrangian is investigated: All variables are normalised and the order of magnitude of the terms are compared. The simplified Lagrangian yields the equations of motion which are solved approximately using Galerkin's method. Forces due to the interaction between the fluid and the structure are considered as well as those due to contact with the borehole.