Drilling Systems: Stability and Hidden Oscillations

  • M. A. Kiseleva
  • N. V. KuznetsovEmail author
  • G. A. Leonov
  • P. Neittaanmäki
Part of the Nonlinear Systems and Complexity book series (NSCH, volume 6)


There are many mathematical models of drilling systems Despite, huge efforts in constructing models that would allow for precise analysis, drilling systems, still experience breakdowns. Due to complexity of systems, engineers mostly use numerical analysis, which may lead to unreliable results. Nowadays, advances in computer engineering allow for simulations of complex dynamical systems in order to obtain information on the behavior of their trajectories. However, this simple approach based on construction of trajectories using numerical integration of differential equations describing dynamical systems turned out to be quite limited for investigation of stability and oscillations of these systems. This issue is very crucial in applied research; for example, as stated in Lauvdal et al. (Proceedings of the IEEE control and decision conference, 1997) the following phrase: “Since stability in simulations does not imply stability of the physical control system (an example is the crash of the YF22) stronger theoretical understanding is required”. In this work, firstly a mathematical model of a drilling system developed by a group of scientists from the University of Eindhoven will be considered. Then a mathematical model of a drilling system with perfectly rigid drill-string actuated by induction motor will be analytically and numerically studied. A modification of the first two models will be considered and it will be shown that even in such simple models of drilling systems complex effects such as hidden oscillations may appear, which are hard to find by standard computational procedures.


Drilling system Induction motor Hidden oscillation Simulation 


  1. 1.
    Al-Bender F, Lampaert V, Swevers J (2004) Modeling of dry sliding friction dynamics: From heuristic models to physically motivated models and back. Chaos 14(2):446–460CrossRefGoogle Scholar
  2. 2.
    Bragin VO, Vagaitsev VI, Kuznetsov NV, Leonov GA (2011) Algorithms for finding hidden oscillations in nonlinear systems. The Aizerman and Kalman conjectures and Chua’s circuits. J Comput Syst Sci Int 50(4):511–543, DOI 10.1134/S106423071104006XMathSciNetzbMATHGoogle Scholar
  3. 3.
    Brett J (1992) Genesis of torsional drillstring vibrations. SPE Drilling Eng 7(3):168–174Google Scholar
  4. 4.
    Brockley C, Cameron R, Potter A (1967) Friction-induced vibrations. ASME J Lubricat Technol 89:101–108CrossRefGoogle Scholar
  5. 5.
    de Bruin J, Doris A, van de Wouw N, Heemels W, Nijmeijer H (2009) Control of mechanical motion systems with non-collocation of actuation and friction: a Popov criterion approach for input-to-state stability and set-valued nonlinearities. Automatica 45(2):405–415Google Scholar
  6. 6.
    Filippov AF (1988) Differential equations with discontinuous right-hand sides. Kluwer, DordrechtCrossRefzbMATHGoogle Scholar
  7. 7.
    Gubar’ NA (1961) Investigation of a piecewise linear dynamical system with three parameters. J Appl Math Mech 25(6):1011–1023MathSciNetGoogle Scholar
  8. 8.
    Horbeek J, Birch W (1995) In: Proceedings of the society of petroleum engineers offshore, Europe, pp 43–51Google Scholar
  9. 9.
    Ibrahim R (1994) Friction-induced vibration, chatter, squeal, and chaos: dynamics and modeling. Appl Mech Rev: ASME 47(7):227–253CrossRefGoogle Scholar
  10. 10.
    Ivanov-Smolensky A (1980) Electrical machines. Energiya, MoscowGoogle Scholar
  11. 11.
    Jansen J (1991) Non-linear rotor dynamics as applied to oilwell drillstring vibrations. J Sound Vibration 147(1):115–135CrossRefGoogle Scholar
  12. 12.
    Kiseleva MA, Kuznetsov NV, Leonov GA, Neittaanmäki P (2012) Drilling systems failures and hidden oscillations. In: IEEE 4th international conference on nonlinear science and complexity, NSC 2012 – Proceedings, pp 109–112, DOI 10.1109/NSC.2012.6304736
  13. 13.
    Kondrat’eva N, Leonov G, Rodjukov F, Shepeljavyj A (2001) Nonlocal analysis of differential equation of induction motors. Technische Mechanik 21(1):75–86Google Scholar
  14. 14.
    Kreuzer E, Kust O (1996) Analyse selbsterregter drehschwingugnen in torsionsstäben. ZAMM – J Appl Math Mech 76(10):547–557CrossRefzbMATHGoogle Scholar
  15. 15.
    Kuznetsov N, Kuznetsova O, Leonov G, Vagaitsev V (2013) Informatics in control, automation and robotics. Lecture notes in electrical engineering, vol 174, Part 4, Chap. Analytical-numerical localization of hidden attractor in electrical Chua’s circuit. Springer, Berlin, pp 149–158. DOI 10.1007/978-3-642-31353-0∖_11Google Scholar
  16. 16.
    Kuznetsov NV, Leonov GA, Vagaitsev VI (2010) Analytical-numerical method for attractor localization of generalized Chua’s system. IFAC Proc Vol (IFAC-PapersOnline) 4(1):29–33, DOI 10.3182/20100826-3-TR-4016.00009Google Scholar
  17. 17.
    Kuznetsov NV, Leonov GA, Seledzhi SM (2011) Hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1):2506–2510, DOI 10.3182/20110828-6-IT-1002.03316Google Scholar
  18. 18.
    Lauvdal T, Murray R, Fossen T (1997) Stabilization of integrator chains in the presence of magnitude and rate saturations: a gain scheduling approach. In: Proceedings of the 36th IEEE Conference on Decision and Control, Vol. 4, pp 4404–4005, DOI 10.1109/CDC.1997.652491Google Scholar
  19. 19.
    Leine R (2000) Bifurcations in discontinuous mechanical systems of filippov-type. Ph.D. thesis, Eindhoven University of Technology, The NetherlandsGoogle Scholar
  20. 20.
    Leine R, Campen DV, Keultjes W (2003) Stick-slip whirl interraction in drillstring dynamics. ASME J Vibrat Acoustics 124Google Scholar
  21. 21.
    Leonov G, Kuznetsov N (2013) IWCFTA2012 Keynote Speech I - Hidden attractors in dynamical systems: From hidden oscillation in Hilbert-Kolmogorov, Aizerman and Kalman problems to hidden chaotic attractor in Chua circuits. In: 2012 Fifth International Workshop on Chaos-Fractals theories and applications (IWCFTA), pp XV–XVII, DOI 10.1109/IWCFTA.2012.8Google Scholar
  22. 22.
    Leonov GA, Kiseleva MA (2012) Analysis of friction-induced limit cycling in an experimental drill-string system. Doklady Phys 57(5):206–209CrossRefGoogle Scholar
  23. 23.
    Leonov GA, Kuznetsov NV (2011) Algorithms for searching for hidden oscillations in the Aizerman and Kalman problems. Doklady Math 84(1):475–481, DOI 10.1134/S1064562411040120MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Leonov GA, Kuznetsov NV (2011) Analytical-numerical methods for investigation of hidden oscillations in nonlinear control systems. IFAC Proc Vol (IFAC-PapersOnline) 18(1): 2494–2505, DOI 10.3182/ 20110828-6-IT-1002.03315Google Scholar
  25. 25.
    Leonov GA, SolovTeva EP (2012) The nonlocal reduction method in analyzing the stability of differential equations of induction machines. Doklady Math 85(3):375–379CrossRefzbMATHGoogle Scholar
  26. 26.
    Leonov GA, Solov’eva EP (2012) On a special type of stability of differential equations for induction machines with double squirrel -cage rotor. Vestnik St Petersburg Univ Math 45(3):128–135MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Leonov GA, Bragin VO, Kuznetsov NV (2010) Algorithm for constructing counterexamples to the Kalman problem. Doklady Math 82(1):540–542, DOI 10.1134/S1064562410040101CrossRefzbMATHGoogle Scholar
  28. 28.
    Leonov GA, Vagaitsev VI, Kuznetsov NV (2010) Algorithm for localizing Chua attractors based on the harmonic linearization method. Doklady Math 82(1):693–696, DOI 10.1134/S1064562410040411MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Leonov GA, Kuznetsov NV, Kuznetsova OA, Seledzhi SM, Vagaitsev VI (2011) Hidden oscillations in dynamical systems. Trans Syst Contl 6(2):54–67Google Scholar
  30. 30.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2011) Localization of hidden Chua’s attractors. Phys Lett A 375(23):2230–2233, DOI 10.1016/j. physleta.2011.04.037MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Leonov GA, Kuznetsov NV, Vagaitsev VI (2012) Hidden attractor in smooth Chua systems. Physica D 241(18):1482–1486, DOI 10.1016/j. physd.2012.05.016MathSciNetCrossRefzbMATHGoogle Scholar
  32. 32.
    Leonov GA, Kuznetsov NV, Yuldahsev MV, Yuldashev RV (2012) Analytical method for computation of phase-detector characteristic. IEEE Trans Circ Syst – II: Express Briefs 59(10):633–647, DOI 10.1109/ TCSII.2012.2213362CrossRefGoogle Scholar
  33. 33.
    Leonov GA, Kuznetsov GV (2013) Hidden attractors in dynamical systems. From hidden oscillations in Hilbert-Kolmogorov, Aizerman, and Kalman problems to hidden chaotic attractors in Chua circuits. Int J Bifurcat Chaos 23(1):1–69, DOI 10.1142/S0218127413300024MathSciNetGoogle Scholar
  34. 34.
    Marino R, Tomei P, Verrelli C (2010) Induction motor control design. Springer, The NetherlandsCrossRefGoogle Scholar
  35. 35.
    Mihajlović N (2005) Torsional and lateral vibrations in flexible rotor systems with friction. Ph.D. dissertation, Eindhoven University of Technology, Eindhoven, NetherlandsGoogle Scholar
  36. 36.
    Mihajlovic N, van Veggel A, van de Wouw N, Nijmeijer H (2004) Analysis of friction-induced limit cycling in an experimental drill-string system. J Dyn Syst Meas Control 126(4):709–720Google Scholar
  37. 37.
    Olsson H (1996) Control systems with friction. Ph.D. thesis, Lund Institute of Technology, SwedenGoogle Scholar
  38. 38.
    Popp K, Stelter P (1990) Stick-slip vibrations and chaos. Philosoph Trans R Soc Lond 332: 89–105CrossRefzbMATHGoogle Scholar
  39. 39.
    Shokir E (2004) A novel pc program for drill string failure detection and prevention before and while drilling specially in new areas. J Oil Gas Bus (1)Google Scholar
  40. 40.
    den Steen LV (2005) Suppressing stick-slip-induced drill-string oscillations: a hyper stability approach. Ph.D. dissertation, University of TwenteGoogle Scholar
  41. 41.
    Yakobovich VA, Leonov GA, Gelig AK (2004) Stability of Stationary Sets in Control Systems with Discontinuous Nonlinearities. World Scientific, SingaporeCrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • M. A. Kiseleva
    • 1
  • N. V. Kuznetsov
    • 1
    • 2
    Email author
  • G. A. Leonov
    • 1
    • 2
  • P. Neittaanmäki
    • 1
    • 2
  1. 1.University of JyväskyläJyväskyläFinland
  2. 2.Saint Petersburg State UniversitySt PetersburgRussia

Personalised recommendations