Nonlinear Dynamics

, Volume 70, Issue 2, pp 1017–1035 | Cite as

Drill-string vibration analysis using non-smooth dynamics approach

  • Sandor Divenyi
  • Marcelo A. Savi
  • Marian Wiercigroch
  • Ekaterina Pavlovskaia
Original Paper


In this work, we model and analyse drill-string vibrations. A special attention is paid to stick-slip and bit-bounce behaviours that are normally treated as the non-smooth dynamics. A two degrees-of-freedom lumped parameters model is employed to account for axial and torsional vibrations based on the model proposed by Christoforou and Yigit (J. Sound Vibr. 267:1029–1045, 2003). The coupling between both vibration modes is through contact with the formation, where the axial force is the catalyst to generate a resistive torque. The forces and torques are defined according the contact or non-contact scenarios, establishing a non-smooth system. Besides, the dry friction between the formation and the drill-bit introduces the other non-smoothness of the system. Here we adopt smoothened governing equations which are advantageous in terms of mathematical description and numerical analysis. Our studies have shown that the mathematical model is capable of predicting a full range of dynamic responses including the stick-slip and drill bit-bounce. A global analysis shows different scenarios related to parameter changes allowing to develop an in depth understanding of the drill-string dynamics and define critical behaviours of the system.


Non-linear dynamics Stick-slip Bit-bounce Drill-string Non-smooth 



The authors would like to acknowledge the support of the Brazilian Research Agencies CNPq, CAPES, and FAPERJ and through the INCT-EIE (National Institute of Science and Technology—Smart Structures in Engineering) the CNPq and FAPEMIG. The Air Force Office of Scientific Research (AFOSR) is also acknowledged.


  1. 1.
    Aguiar, R.R., Weber, H.I.: Mathematical modeling and experimental investigation of an embedded vibro-impact system. Nonlinear Dyn. 65, 317–334 (2011) CrossRefGoogle Scholar
  2. 2.
    Andreaus, U., Casini, P.: Dynamics of friction oscillators excited by a moving base and/or driving force. J. Sound Vib. 245(4), 685–699 (2001) CrossRefGoogle Scholar
  3. 3.
    Blazejczyk-Okolewska, B., Kapitaniak, T.: Dynamics of impact oscillator with dry friction. Chaos Solitons Fractals 7(9), 1455–1459 (1996) CrossRefGoogle Scholar
  4. 4.
    Blazejczyk-Okolewska, B., Kapitaniak, T.: Co-existing attractors of impact oscillator. Chaos Solitons Fractals 9(8), 1439–1443 (1998) zbMATHCrossRefGoogle Scholar
  5. 5.
    Christoforou, A.P., Yigit, A.S.: Fully coupled vibrations of actively controlled drillstrings. J. Sound Vib. 267, 1029–1045 (2003) CrossRefGoogle Scholar
  6. 6.
    Divenyi, S., Savi, M.A., Franca, L.F.P., Weber, H.I.: Nonlinear dynamics and chaos in systems with discontinuous support. Shock Vib. 13(4/5), 315–326 (2006) Google Scholar
  7. 7.
    Divenyi, S., Savi, M.A., Weber, H.I., Franca, L.F.P.: Experimental investigation of an oscillator with discontinuous support considering different system aspects. Chaos Solitons Fractals 38(3), 685–695 (2008) CrossRefGoogle Scholar
  8. 8.
    Franca, L.F.P., Weber, H.I.: Experimental and numerical study of a new resonance hammer drilling model with drift. Chaos Solitons Fractals 21, 789–801 (2004) zbMATHCrossRefGoogle Scholar
  9. 9.
    Hinrichs, N., Oestreich, M., Popp, K.: On the modelling of friction oscillators. J. Sound Vib. 216(3), 435–459 (1998) CrossRefGoogle Scholar
  10. 10.
    Ing, J., Pavlovskaia, E., Wiercigroch, M., Banerjee, S.: Experimental study of impact oscillator with one-sided elastic constraint. Philos. Trans. R. Soc. Lond. A 366, 679–704 (2008) zbMATHCrossRefGoogle Scholar
  11. 11.
    Leine, R.I.: Bifurcations in discontinuous mechanical systems of Filippov-type. Ph.D. thesis, Technische Universiteit Eindhoven (2000) Google Scholar
  12. 12.
    Maistrenko, Y., Kapitaniak, T., Szuminski, P.: Locally and globally riddled basins in two coupled piecewise-linear maps. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 56(6), 6393–6399 (1997) MathSciNetCrossRefGoogle Scholar
  13. 13.
    Pavlovskaia, E., Wiercigroch, M., Grebogi, C.: Modeling of an impact system with a drift. Phys. Rev. E, Stat. Nonlinear Soft Matter Phys. 64, 056224 (2001) CrossRefGoogle Scholar
  14. 14.
    Savi, M.A., Divenyi, S., Franca, L.F.P., Weber, H.I.: Numerical and experimental investigations of the nonlinear dynamics and chaos in non-smooth systems with discontinuous support. J. Sound Vib. 301, 59–73 (2007) CrossRefGoogle Scholar
  15. 15.
    Silveira, M., Wiercigroch, M.: Low dimensional models for stick-slip vibration of drill-strings. J. Phys. Conf. Ser. 181, 012056 (2009) CrossRefGoogle Scholar
  16. 16.
    Spanos, P.D., Sengupta, A.K., Cunningham, R.A., Paslay, P.R.: Modelling of roller cone bit lift-off dynamics in rotary drilling. J. Energy Resour. Technol. 117, 197–207 (1995) CrossRefGoogle Scholar
  17. 17.
    Wiercigroch, M.: Modelling of dynamical systems with motion dependent discontinuities. Chaos Solitons Fractals 11, 2429–2442 (2000) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2012

Authors and Affiliations

  • Sandor Divenyi
    • 1
  • Marcelo A. Savi
    • 1
  • Marian Wiercigroch
    • 2
  • Ekaterina Pavlovskaia
    • 2
  1. 1.COPPE, Department of Mechanical EngineeringUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil
  2. 2.Centre for Applied Dynamics Research, School of EngineeringUniversity of AberdeenAberdeenUK

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