Hassan Zohoor

Drill strings are subjected to complex coupled dynamics. Therefore, accurate dynamic modeling, which can represent the physical behavior of real drill strings, is of great importance for system analysis and control. The most widely used dynamic models for such systems are the lumped element models, which neglect the system distributed feature. In this paper, a dynamic model called neutral-type time delay model is modified to investigate the coupled axial–torsional vibrations in drill strings. This model is derived directly from the distributed parameter model by employing the d'Alembert method. Coupling of axial and torsional vibration modes occurs in the bit–rock interface. For the first time, the neutral-type time delay model is combined with a bit–rock interaction model that regards cutting process in addition to frictional contact. Moreover, mistakes made in some of the related previous studies are corrected. The resulting equations of motion are in terms of neutral-type delay differential equations with two constant delays, related to the oscillatory behavior of the system, and a state-dependent delay, induced by the bit–rock interaction. Illustrative simulation results are presented for a representative drill string, which demonstrates intense axial and torsional vibrations that may lead to system failure without a controller.

Using Hamilton’s principle, exact equations of motion for non-linear planar and spatial Euler–Bernoulli beams are derived. In the existing non-linear Euler–Bernoulli beam formulations, some elastic terms are dropped by differentiation from the incomplete Green–Lagrange strain tensor followed by negligible elastic deformations of cross-sectional frame. On the other hand, in this article, the exact strain field concerning considerable elastic deformations of cross-sectional frame is used as a source in differentiations. As a result, the achieved closed-form equations are exact and more accurate than formerly reported equations in the literature. Moreover, the applicable dynamic model of inextensional beams which is fully accurate, yet simple has been shown. The planar and inextensional dynamic models have been compared with the existing dynamic models in the literature, and the proposed dynamic models demonstrate significant improvements in the numerical results. Finally, experiments on the carbon fibre rods verify the model presented for inextensional beams.