Shabnam Tashakori Department of Mechanical Engineering, Sharif University of Technology, Tehran 1136511155, Iran e-mail: shabnam.tashakori@mech.sharif.edu Gholamreza Vossoughi 1 Department of Mechanical Engineering, Sharif University of Technology, Tehran 1136511155, Iran e-mail: vossough@sharif.edu Hassan Zohoor Department of Mechanical Engineering, Sharif University of Technology, Tehran 1136511155, Iran e-mail: zohoor@sharif.edu Ehsan Azadi Yazdi Department of Mechanical Engineering, Shiraz University, Shiraz 7193616548, Iran e-mail: ehsanazadi@shirazu.ac.ir Modification of the Infinite- Dimensional Neutral-Type Time-Delay Dynamic Model for the Coupled Axial–Torsional Vibrations in Drill Strings With a Drag Bit 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 fea- ture. 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. Cou- pling 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. [DOI: 10.1115/1.4043147] Keywords: drill sting, neutral-type time-delay dynamic model, coupled axial–torsional vibrations, bit-bounce phenomenon, stick-slip oscillations, drag bit 1 Introduction A drill string is one of the components of a drill rig used to explore and extract oil and gas. Figure 1 illustrates a schematic view of a rotational drill rig. This system is rotated by an electrical drive, placed at the rotary table, and it is subjected to a tensile force, exerted by the hoisting system. On the other extremity, the drill bit is subjected to a compressive load and a resistive torque, known as weight-on-bit (WOB) and torque-on-bit (TOB), respec- tively, applied from the formation. Drill strings experience different vibrations during drilling, namely, axial (bit-bouncing), torsional (stick-slip), and transverse (whirl motion) vibrations [2], which lead to fatigue, performance deterioration, decrement of the rate of penetration, and even sys- tem failure. Therefore, studying the root causes of these unwanted vibrations is of great importance, which needs accurate and com- prehensive dynamic modeling. Existing drilling models can be categorized into three classifica- tions [3]: lumped element models [4–7], distributed parameter models [8,9], and neutral-type time delay model (for simplicity, we call it neutral-type time delay (NTD) model) [10–13]. The NTD model obtained directly from the distributed parameter model, with negligible damping along the drill string, by employ- ing the d’Alembert method [3,12]. This model provides a distrib- uted setting to study the coupled axial–torsional vibrations of the drill string, in terms of neutral-type delay differential equations. There are several modeling approaches for the bit–rock interac- tion. Most regard the bit–rock interface like a frictional encounter with a (locally) velocity weakening effect such as Ref. [4]. How- ever, no velocity-weakening effect has been disclosed in experi- ments using single cutters. Revised models are proposed in Refs. [14–19]. In Ref. [15], a discrete model considering the axial and Fig. 1 Schematic drilling structure [1] 1 Corresponding author. Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received August 10, 2018; final manuscript received March 5, 2019; published online June 23, 2020. Assoc. Editor: Katrin Ellermann. Journal of Computational and Nonlinear Dynamics AUGUST 2020, Vol. 15 / 081006-1 Copyright V C 2020 by ASME Downloaded from http://asmedigitalcollection.asme.org/computationalnonlinear/article-pdf/15/8/081006/6544918/cnd_015_08_081006.pdf by Eindhoven University of Technology user on 01 November 2021