# American Institute of Mathematical Sciences

December  2020, 9(4): 915-934. doi: 10.3934/eect.2020042

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

 1 Laboratoire XLIM, Université de Limoges, 87060 Limoges, France 2 Laboratoire PIMENT, Université de La Réunion, 97400 Saint-Denis, France

Dedicated to 70th birthday of Professor Meir Shillor.

Received  August 2019 Revised  November 2019 Published  March 2020

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.

Citation: Samir Adly, Daniel Goeleven. A nonsmooth approach for the modelling of a mechanical rotary drilling system with friction. Evolution Equations & Control Theory, 2020, 9 (4) : 915-934. doi: 10.3934/eect.2020042
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##### References:
General friction model as in (6) and the function $\varphi$ as in (7)
Coulomb friction model
Stribeck friction model
Stiction model
Consistent stiction model as in (6) with $\varphi_+(x) = F_C +(F_S-F_C)e^{-\frac{x}{v_s}}$
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)
Rotary drilling system
Graph of the function $V(t)$ in Table 3
Numerical solution of the evolution variational inequality (19) with the initial conditions given in Table 5
Numerical solution of the evolution variational inequality (23) with the initial conditions given in Table 5
Parameters
 $J_1$ $999.35 \; (kg.m^2)$ $J_2$ $127.27\; (kg.m^2)$ $d_1$ $51.38\; (N.m.s/rad)$ $d_2$ $39.79 \;(N.m.s)$ $k$ $481.29 \; (N.m/rad)$ $R$ $0.01 \;(\Omega)$ $L$ $0.005 \;(H)$ $K_M$ $6 \;(N.m/A)$ $N$ $7.20$ $K = NK_M$ $43.20 \;(N.m/A)$ $E$ $130 \;(MJ/m^3)$ $\delta$ $0.64 \times 10^{-3}\;(m/rad)$ $R_B$ $0.10 \; m$ $\mu_C$ $0.4$ $\mu_S$ $0.6$
 $J_1$ $999.35 \; (kg.m^2)$ $J_2$ $127.27\; (kg.m^2)$ $d_1$ $51.38\; (N.m.s/rad)$ $d_2$ $39.79 \;(N.m.s)$ $k$ $481.29 \; (N.m/rad)$ $R$ $0.01 \;(\Omega)$ $L$ $0.005 \;(H)$ $K_M$ $6 \;(N.m/A)$ $N$ $7.20$ $K = NK_M$ $43.20 \;(N.m/A)$ $E$ $130 \;(MJ/m^3)$ $\delta$ $0.64 \times 10^{-3}\;(m/rad)$ $R_B$ $0.10 \; m$ $\mu_C$ $0.4$ $\mu_S$ $0.6$
Empirical coefficients
 $\sigma$ $1$ $\omega_s$ $10^{-3}$ $(rad/s)$
 $\sigma$ $1$ $\omega_s$ $10^{-3}$ $(rad/s)$
Motor voltage. Augmentation of DC motor voltage from 125 (V) to 150 (V) at $t = 30 \; (s)$ (see Figure 8)
Weight-On-Bit and corresponding friction torques
 $W$ $15 000 \; (kg)$ $T_C = \frac{1}{2}\mu_CR_BW$ $300 \; (kg.m)$ $T_S = \frac{1}{2}\mu_SR_BW$ $450 \; (kg.m)$ $T_{{\rm CUT}} = \frac{1}{2} \delta R_B^2E$ $4.16\, 10^{-4}\; (MJ/rad)$
 $W$ $15 000 \; (kg)$ $T_C = \frac{1}{2}\mu_CR_BW$ $300 \; (kg.m)$ $T_S = \frac{1}{2}\mu_SR_BW$ $450 \; (kg.m)$ $T_{{\rm CUT}} = \frac{1}{2} \delta R_B^2E$ $4.16\, 10^{-4}\; (MJ/rad)$
Initial conditions for problems (19) and (23)
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