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Research Papers

# A New Dynamic Model of Coupled Axial–Torsional Vibration of a Drill String for Investigation on the Length Increment Effect on Stick–Slip Instability

[+] Author and Article Information

Department of Mechanical Engineering,
Amirkabir University of Technology,
424 Hafez Avenue,
Tehran 15875-4413, Iran

Department of Mechanical Engineering,
Amirkabir University of Technology,
424 Hafez Avenue,
Tehran 15875-4413, Iran
e-mail: baktiari@aut.ac.ir

1Corresponding author.

Contributed by the Technical Committee on Vibration and Sound of ASME for publication in the JOURNAL OF VIBRATION AND ACOUSTICS. Manuscript received June 20, 2016; final manuscript received July 6, 2017; published online August 17, 2017. Assoc. Editor: Mahmoud Hussein.

J. Vib. Acoust 139(6), 061016 (Aug 17, 2017) (9 pages) Paper No: VIB-16-1311; doi: 10.1115/1.4037299 History: Received June 20, 2016; Revised July 06, 2017

## Abstract

In this paper, a new model is proposed to study the coupled axial–torsional vibration of the drill string. It is assumed that rotary table angular speed is constant and equals to the nominal angular speed of the drill string. In addition, axial displacement of any point on the drill string is considered to be as the sum of rigid-body motion and elastic vibrations. The depth of cut is defined using instantaneous dynamic states instead of using the delayed model as presented in previous researches. A velocity-weakening function is introduced for modeling the behavior of the frictional component of the torque-on-bit (TOB) with respect to the bit angular speed. After discretizing vibration equations, stability analysis of the system is investigated by linearizing the nonlinear system around its steady-state response point. Considering nominal weight-on-bit (WOB) ($W0$) and nominal rotational speed ($Ω$) as the input parameters of the drilling, variation of maximum allowable value of ($W0$) is presented with respect to variation of $Ω$ . It is shown that the maximum allowable value of $W0$ has an increasing–decreasing behavior with respect to $Ω$. The effect of drill string upper and lower part lengths is studied on the stability of the system, and practical results are presented both in the condition that $W0$ is constant and in the condition that the hook upward force is constant. It is shown that by increasing the drill string length, the system is more exposed to instability, and this must be considered in regulating the input parameters of drilling.

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## References

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## Figures

Fig. 1

Schematic of the rotational drilling rig [2]

Fig. 2

Stable drilling area in the plane of W0 and Ω

Fig. 3

Time history of the bit angular speed for point B (W0=24 kN and Ω=20 rad/s )

Fig. 4

Time history of the bit angular speed for point A (W0=25 kN and Ω=20 rad/s)

Fig. 5

Comparison of the stability analysis with the results of Liu et al. [16] (L1 = 1000 m and L2 = 200 m)

Fig. 6

Maximum allowable W0 versus Ω and L1 (L2 = 100 m)

Fig. 7

Variation of the allowable angular speed versus L1 and L2

Fig. 8

Comparison of two modes of drilling; H0 constant and W0 constant (L2 = 125 m)

Fig. 9

Variation of axial bit speed versus L1; comparison for different values of L2 (W0 is maximum)

Fig. 10

Variation of steady-state axial speed versus L1 and L2 (W0 is optimum)

Fig. 11

Variation of steady-state axial speed versus L1 for different values of L2 (W0 is optimum)

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