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Preface
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 |
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.
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. |








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