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A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction

  • * Corresponding author: Samir Adly

    * Corresponding author: Samir Adly 

Dedicated to 70th birthday of Professor Meir Shillor.

Abstract / Introduction Full Text(HTML) Figure(10) / Table(5) Related Papers Cited by
  • In this paper, we show how the approach of nonsmooth dynamical systems can be used to develop a suitable method for the modelling of a rotary oil drilling system with friction. We study different kinds of frictions and analyse the mathematical properties of the involved dynamical systems. We show that using a general Stribeck model for the frictional contact, we can formulate the rotary drilling system as a well-posed evolution variational inequality. Several numerical simulations are also given to illustrate both the model and the theoretical results.

    Mathematics Subject Classification: 34A60, 46N10, 49J52, 70F40.

    Citation:

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  • Figure 1.  General friction model as in (6) and the function $ \varphi $ as in (7)

    Figure 2.  Coulomb friction model

    Figure 3.  Stribeck friction model

    Figure 4.  Stiction model

    Figure 5.  Consistent stiction model as in (6) with $ \varphi_+(x) = F_C +(F_S-F_C)e^{-\frac{x}{v_s}} $

    Figure 6.  Oil drilling rig illustration-1. Mud tank, 2. Shale shakers, 3. Suction line (mud pump), 4. Mud pump, 5. Motor or power source, 6. Vibrating hose, 7. Draw-works (winch), 8. Standpipe 9. Kelly hose, 10. Goose-neck, 11. Traveling block, 12. Drill line, 13. Crown block 14. Derrick-Author: Tosaka-Attribution 3.0 Unported (CC BY 3.0) - https://creativecommons.org/licenses/by/3.0/deed.en (https://commons.wikimedia.org/wiki/File:Oil_Rig_NT.PNG)

    Figure 7.  Rotary drilling system

    Figure 8.  Graph of the function $ V(t) $ in Table 3

    Figure 9.  Numerical solution of the evolution variational inequality (19) with the initial conditions given in Table 5

    Figure 10.  Numerical solution of the evolution variational inequality (23) with the initial conditions given in Table 5

    Table 1.  Parameters

    $ J_1 $ $ 999.35 \; (kg.m^2) $
    $ J_2 $ $ 127.27\; (kg.m^2) $
    $ d_1 $ $ 51.38\; (N.m.s/rad) $
    $ d_2 $ $ 39.79 \;(N.m.s) $
    $ k $ $ 481.29 \; (N.m/rad) $
    $ R $ $ 0.01 \;(\Omega) $
    $ L $ $ 0.005 \;(H) $
    $ K_M $ $ 6 \;(N.m/A) $
    $ N $ $ 7.20 $
    $ K = NK_M $ $ 43.20 \;(N.m/A) $
    $ E $ $ 130 \;(MJ/m^3) $
    $ \delta $ $ 0.64 \times 10^{-3}\;(m/rad) $
    $ R_B $ $ 0.10 \; m $
    $ \mu_C $ $ 0.4 $
    $ \mu_S $ $ 0.6 $
     | Show Table
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    Table 2.  Empirical coefficients

    $ \sigma $ $ 1 $
    $ \omega_s $ $ 10^{-3} $ $ (rad/s) $
     | Show Table
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    Table 3.  Motor voltage. Augmentation of DC motor voltage from 125 (V) to 150 (V) at $ t = 30 \; (s) $ (see Figure 8)

     | Show Table
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    Table 4.  Weight-On-Bit and corresponding friction torques

    $ W $ $ 15 000 \; (kg) $
    $ T_C = \frac{1}{2}\mu_CR_BW $ $ 300 \; (kg.m) $
    $ T_S = \frac{1}{2}\mu_SR_BW $ $ 450 \; (kg.m) $
    $ T_{{\rm CUT}} = \frac{1}{2} \delta R_B^2E $ $ 4.16\, 10^{-4}\; (MJ/rad) $
     | Show Table
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    Table 5.  Initial conditions for problems (19) and (23)

     | Show Table
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