Flat Flexural Vibration of Drill-String with an Initial Curvature

Drill-string vibration is one of the major causes for a deteriorated drilling performance. It is crucial to understand the complex vibrational mechanisms experienced by a drilling system in order to better control its functional operation and improve its performance. This work is devoted to modeling of nonlinear dynamics of the drill-string taking into account initial curvature and finiteness of deformations. The drill-string is considered as a one-dimensional rod of a symmetric cross-section compressed by an axial force applied at the top end of the rod. The equation of motion is derived using the theory of finite deformations of V.V. Novozhilov and the Hamilton-Ostrogradsky's variational principle. The initial curvature is modelled as a random shape that can be approximated by a finite series of smooth functions. All numerical calculations are carried out in the environment of symbolic mathematical computations-Wolfram Mathematica. The obtained results allow to study the influence of initial curvature on excited flexural vibrations in the drill-string in order to improve its performance and avoid of severe and destructive oscillations.