# 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  December 2020 Early access  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 and Control Theory, 2020, 9 (4) : 915-934. doi: 10.3934/eect.2020042
##### References:
 [1] S. Adly and D. Goeleven, A stability theory for second-order nonsmooth dynamical systems with applications to friction problems, J. Math. Pures Appl., 83 (2004), 17-51.  doi: 10.1016/S0021-7824(03)00071-0. [2] S. Adly, A Variational Approach to Nonsmooth Dynamics. Applications in Unilateral Mechanics and Electronics, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-68658-5. [3] L. X. Ahn, Dynamics of Mechanical Systems with Coulomb Friction, Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-540-36516-7. [4] G. Amontons, On the Resistance Originating in Machines, Proceedings of the French Royal Academy of Sciences, 1699, 206–222. [5] S. Anderson, A. Söderberg and S. Björklund, Friction models for sliding dry, boundary and mixed lubricated contacts, Tribology International, 40 (2007), 580-587.  doi: 10.1016/j.triboint.2005.11.014. [6] B. Armstrong-Hélouvry, Control of Machines with Friction, Kluwer Academic Publishers, Springer, Boston, MA, 1991. doi: 10.1007/978-1-4615-3972-8. [7] K. J. Ǻström, Control of systems with friction, Proceedings of the Fourth International Conferences on Motion and Vibration Control, (1998), 25–32. [8] P. A. Bliman and M. Sorine, Easy-to-use Realistic Dry Friction Models for Automatic Control, Proc. of 3rd European Control Conference, Rome, Italy, 1995, 3788–3794. [9] L. C. Bo and D. Pavelescu, The friction-speed relation and its influence on the critical velocity of stick-slip motion, Wear, 82 (1982), 277-289.  doi: 10.1016/0043-1648(82)90223-X. [10] H. Brézis, Problémes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168. [11] C. A. Coulomb, Théorie des machines simples, en ayant egard au frottement de leurs parties, et a la roideur dews cordages, Mem. Math Phys., Paris, (1785), 161–332. [12] L. da Vinci, The Notebooks of Leonardo Da Vinci (Ed. J. P. Richter), Dover Pub. Inc., New York, 1970. [13] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), 263-294.  doi: 10.1137/1034050. [14] D. Goeleven, Complementarity and Variational Inequalities in Electronics, Academic Press, London, 2017. [15] M. Jean and J. J. Moreau, Unilateraly and dry friction in the dynamics of rigid body collections, Proc. Contact Mechanics Int. Symp., (1992), 31–48. [16] D. P. Hess and A. Soom, Friction at a Lubricated Line Contact Operating at Oscillating Sliding Velocities, Journal of Tribology, 112 (1990), 147-152.  doi: 10.1115/1.2920220. [17] D. Karnopp, Computer simulation of stick-slip friction in mechanical dynamic systems, J. Dyn. Sys. Meas. Control., 107 (1985), 100-103.  doi: 10.1115/1.3140698. [18] T. Kato, Accretive operators and nonlinear evolutions equations in banach spaces, Nonlinear Functional Analysis, 18 (1970), 138-161. [19] M. Kidouche and R. Riane, On the design of proportional integral observer for a rotary drilling system, 8th CHAOS Conference Proceedings, Henri Poincaré Institute, (2015), 1–12. [20] R. I. Lein and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Vol. 18, Lecture Notes in Applied and Computational Mechanics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-44398-8. [21] Y. F. Liu, J. Li, Z. M. Zhang, X. H. Hu and W. J. Zhang, Experimental comparison of five friction models on the same test-bed of the micro stick-slip motion system, Mech. Sci., 6 (2015), 15-28.  doi: 10.5194/ms-6-15-2015. [22] S. E. Lyshevski, Electromechanical Systems and Devices, CRC Press, Boca Raton, 2008.  doi: 10.1201/9781420069754. [23] J. J. Moreau, La notion du surpotentiel et les liaisons unilatérales on elastostatique, C. R. Acad. Sci. Paris Ser. A-B, 267 (1968), A954–A957. [24] J. J. Moreau, Dynamique des Systémes à Liaisons unilatérales avec Frottement sec Éventuel; Essais Numériques, Tech. Rep., Montpellier, France, 1986. [25] J. J. Moreau and P. D. Panagiotopoulos, Non-Smooth Mechanics and Applications, Vol. 302, CISM International Centre for Mechanical Sciences. Courses and Lectures, Springer-Verlag, Vienna, 1988. doi: 10.1007/978-3-7091-2624-0. [26] A. J. Morin, New friction experiments carried out at Metz in 1831-1833, Proceedings of the French Royal Academy of Sciences, 4 (1833), 1-128. [27] H. Olsson, Control Systems with Friction, Department of Automatic Control, Lund Institute of Technology (LTH), Lund, 1996. [28] H. Olsson, K. J. Aström, C. Canudas de Wit, M. Göfvert and P. Lischinsky, Friction models and friction compensation, European Journal of Control, 4 (1998), 176-195. [29] P. D. Panagiotopoulos, Nonconvex superpotentials in the sense of F. H. Clarke and applications, Mech. Res. Comm., 8 (1981), 335-340.  doi: 10.1016/0093-6413(81)90064-1. [30] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [31] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, 1978. [32] V. L. Popov, Contact Mechanics and Friction. Physical Principles and Applications, Springer, Berlin, Heidelberg, 2010. [33] M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact. Variational Methods, Springer-Verlag, 2004. [34] O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil, Phil. Trans. R. Soc., 177 (1886), 157-234. [35] R. Stribeck, Die Wesentlichen Eigenschaften der Gleit-und Rollenlager, Springer, 1903. [36] E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.

show all references

##### References:
 [1] S. Adly and D. Goeleven, A stability theory for second-order nonsmooth dynamical systems with applications to friction problems, J. Math. Pures Appl., 83 (2004), 17-51.  doi: 10.1016/S0021-7824(03)00071-0. [2] S. Adly, A Variational Approach to Nonsmooth Dynamics. Applications in Unilateral Mechanics and Electronics, SpringerBriefs in Mathematics, Springer, Cham, 2017. doi: 10.1007/978-3-319-68658-5. [3] L. X. Ahn, Dynamics of Mechanical Systems with Coulomb Friction, Foundations of Engineering Mechanics, Springer-Verlag, Berlin, 2003. doi: 10.1007/978-3-540-36516-7. [4] G. Amontons, On the Resistance Originating in Machines, Proceedings of the French Royal Academy of Sciences, 1699, 206–222. [5] S. Anderson, A. Söderberg and S. Björklund, Friction models for sliding dry, boundary and mixed lubricated contacts, Tribology International, 40 (2007), 580-587.  doi: 10.1016/j.triboint.2005.11.014. [6] B. Armstrong-Hélouvry, Control of Machines with Friction, Kluwer Academic Publishers, Springer, Boston, MA, 1991. doi: 10.1007/978-1-4615-3972-8. [7] K. J. Ǻström, Control of systems with friction, Proceedings of the Fourth International Conferences on Motion and Vibration Control, (1998), 25–32. [8] P. A. Bliman and M. Sorine, Easy-to-use Realistic Dry Friction Models for Automatic Control, Proc. of 3rd European Control Conference, Rome, Italy, 1995, 3788–3794. [9] L. C. Bo and D. Pavelescu, The friction-speed relation and its influence on the critical velocity of stick-slip motion, Wear, 82 (1982), 277-289.  doi: 10.1016/0043-1648(82)90223-X. [10] H. Brézis, Problémes unilatéraux, J. Math. Pures Appl., 51 (1972), 1-168. [11] C. A. Coulomb, Théorie des machines simples, en ayant egard au frottement de leurs parties, et a la roideur dews cordages, Mem. Math Phys., Paris, (1785), 161–332. [12] L. da Vinci, The Notebooks of Leonardo Da Vinci (Ed. J. P. Richter), Dover Pub. Inc., New York, 1970. [13] A. Dontchev and F. Lempio, Difference methods for differential inclusions: A survey, SIAM Rev., 34 (1992), 263-294.  doi: 10.1137/1034050. [14] D. Goeleven, Complementarity and Variational Inequalities in Electronics, Academic Press, London, 2017. [15] M. Jean and J. J. Moreau, Unilateraly and dry friction in the dynamics of rigid body collections, Proc. Contact Mechanics Int. Symp., (1992), 31–48. [16] D. P. Hess and A. Soom, Friction at a Lubricated Line Contact Operating at Oscillating Sliding Velocities, Journal of Tribology, 112 (1990), 147-152.  doi: 10.1115/1.2920220. [17] D. Karnopp, Computer simulation of stick-slip friction in mechanical dynamic systems, J. Dyn. Sys. Meas. Control., 107 (1985), 100-103.  doi: 10.1115/1.3140698. [18] T. Kato, Accretive operators and nonlinear evolutions equations in banach spaces, Nonlinear Functional Analysis, 18 (1970), 138-161. [19] M. Kidouche and R. Riane, On the design of proportional integral observer for a rotary drilling system, 8th CHAOS Conference Proceedings, Henri Poincaré Institute, (2015), 1–12. [20] R. I. Lein and H. Nijmeijer, Dynamics and Bifurcations of Non-smooth Mechanical Systems, Vol. 18, Lecture Notes in Applied and Computational Mechanics, Springer-Verlag, Berlin, 2004. doi: 10.1007/978-3-540-44398-8. [21] Y. F. Liu, J. Li, Z. M. Zhang, X. H. Hu and W. J. Zhang, Experimental comparison of five friction models on the same test-bed of the micro stick-slip motion system, Mech. Sci., 6 (2015), 15-28.  doi: 10.5194/ms-6-15-2015. [22] S. E. Lyshevski, Electromechanical Systems and Devices, CRC Press, Boca Raton, 2008.  doi: 10.1201/9781420069754. [23] J. J. Moreau, La notion du surpotentiel et les liaisons unilatérales on elastostatique, C. R. Acad. Sci. Paris Ser. A-B, 267 (1968), A954–A957. [24] J. J. Moreau, Dynamique des Systémes à Liaisons unilatérales avec Frottement sec Éventuel; Essais Numériques, Tech. Rep., Montpellier, France, 1986. [25] J. J. Moreau and P. D. Panagiotopoulos, Non-Smooth Mechanics and Applications, Vol. 302, CISM International Centre for Mechanical Sciences. Courses and Lectures, Springer-Verlag, Vienna, 1988. doi: 10.1007/978-3-7091-2624-0. [26] A. J. Morin, New friction experiments carried out at Metz in 1831-1833, Proceedings of the French Royal Academy of Sciences, 4 (1833), 1-128. [27] H. Olsson, Control Systems with Friction, Department of Automatic Control, Lund Institute of Technology (LTH), Lund, 1996. [28] H. Olsson, K. J. Aström, C. Canudas de Wit, M. Göfvert and P. Lischinsky, Friction models and friction compensation, European Journal of Control, 4 (1998), 176-195. [29] P. D. Panagiotopoulos, Nonconvex superpotentials in the sense of F. H. Clarke and applications, Mech. Res. Comm., 8 (1981), 335-340.  doi: 10.1016/0093-6413(81)90064-1. [30] P. D. Panagiotopoulos, Hemivariational Inequalities. Applications in Mechanics and Engineering, Springer-Verlag, Berlin, 1993. doi: 10.1007/978-3-642-51677-1. [31] D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Sijthoff and Noordhoff International Publishers, Alphen aan den Rijn, 1978. [32] V. L. Popov, Contact Mechanics and Friction. Physical Principles and Applications, Springer, Berlin, Heidelberg, 2010. [33] M. Shillor, M. Sofonea and J. J. Telega, Models and Analysis of Quasistatic Contact. Variational Methods, Springer-Verlag, 2004. [34] O. Reynolds, On the theory of lubrication and its application to Mr. Beauchamp Tower's experiments, including an experimental determination of the viscosity of olive oil, Phil. Trans. R. Soc., 177 (1886), 157-234. [35] R. Stribeck, Die Wesentlichen Eigenschaften der Gleit-und Rollenlager, Springer, 1903. [36] E. Zeidler, Nonlinear Functional Analysis and its Applications, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-0985-0.
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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