Skip to main content
SpringerLink
Log in

Modelling of stick–slip behaviour in a Girling brake using network simulation method

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The stick–slip model of a Girling brake is composed of nonlinear and coupled differential equations that reproduce the friction occurring in this mechanical system. The brake is equivalent to a body sliding on a belt. The problem is very interesting since the possible solutions, which are very sensitive to the parameters of the system, show a chaotic behaviour. In this contribution, the model, which is designed following network method rules, is explained in detail and runs on standard electrical circuit simulation software to provide the displacement and the velocity of the sliding body and the phase planes. In comparison with other models, the considered system does not include dampers to get a more unstable behaviour. Furthermore, a suitable selection of parameters is implemented to reduce the computational time.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Blau, P.J.: Friction Science and Technology. Marcel Dekker, New York (1986)

    Google Scholar 

  2. Dowson, D.: History of Tribology. Professional Engineering Publishers, London (1998)

  3. Perry, S.S.: Progress in the pursuit in the fundamentals of tribology. Tribol. Lett. 10(1–2), 1–4 (2001)

    Article  Google Scholar 

  4. Blau, P.J.: The significance and use of the friction coefficient. Tribol. Int. 34, 585–591 (2001)

    Article  Google Scholar 

  5. Müser, M.H.: Theory and simulation of friction and lubrication. Lect. Notes Phys. 704, 65 (2006)

    Article  Google Scholar 

  6. Hess, D.P., Soom, A.: Friction at a lubricated line contact operating at oscillating sliding velocities. J. Tribol. Trans. ASME 112(1), 147–152 (1990)

    Article  Google Scholar 

  7. Armstrong-Hélouvry, B.: Control Mach. Frict. Kluwer Academic Press, Hingham (1991)

    Book  MATH  Google Scholar 

  8. Courtney-Pratt, J.S., Eisner, E.: The effect of a tangential force on the contact of metallic bodies. Proc. R. Soc. A 238, 529–550 (1956)

    Article  Google Scholar 

  9. Liang, J.W., Feeny, B.F.: A comparison between direct and indirect friction measurements in a forced oscillator. J. Appl. Mech. Trans. ASME 65(3), 783–786 (1998)

    Article  Google Scholar 

  10. Powell, J., Wiercigroch, M.: Influence of non-reversible Coulomb characteristics on the response of a harmonically excited linear oscillator. Mach. Vib. 1(2), 94–104 (1992)

    Google Scholar 

  11. Johannes, V.I., Green, M.A., Brockley, C.A.: Role of rate of application of tangential force in determining static friction coefficient. Wear 24(3), 381–385 (1973)

    Article  Google Scholar 

  12. Rabinowicz, E.J.: The nature of the static and kinetic coefficients of friction. J. Appl. Phys. 22(11), 1373–1379 (1951)

    Article  Google Scholar 

  13. Stribeck, R.: Fundamental characteristics of the friction bearing and the roller bearing. Zeitschrift des Vereines Deutscher Ingenieure 46(46), 1341–1348 (1902)

    Google Scholar 

  14. Wojewoda, J., Stefanski, A., Wiercigroch, M., Kapitaniak, T.: Estimation of Lyapunov exponents for a system with sensitive friction model. Arch. Appl. Mech. 79(6–7), 667–677 (2009)

    Article  MATH  Google Scholar 

  15. Coulomb, C.A.: Théorie des machines simples. Mémoire de Mathémathiques et de Physique de l’Academie royale des Sciences 10, 161–182 (1785)

  16. Stefanski, A., Wojewoda, J., Wiercigroch, M., Kapitaniak, T.: Chaos caused by non-reversible dry friction. Chaos Solitons Fract. 16(5), 661–664 (2003)

    Article  MATH  Google Scholar 

  17. Awrejcewicz, J., Olejnik, P.: Analysis of dynamic system with various friction laws. Appl. Mech. Rev. 58(6), 389–411 (2005)

    Article  Google Scholar 

  18. Fu, W.P., Fang, Z.D., Zhao, Z.G.: Periodic solutions and harmonic analysis of an anti-lock brake system with piecewise-nonlinearity. J. Sound Vib. 246(3), 543–550 (2001)

    Article  Google Scholar 

  19. Choi, H.S., Lou, J.Y.K.: Nonlinear behaviour and chaotic motions of an sdof system with piecewise non-linear stiffness. Int. J. Non-linear Mech. 26(5), 461–473 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  20. Fling, R.T., Fenton, R.E.: A describing-function approach to antiskid design. IEEE Trans. Veh. Technol. 30(3), 134–144 (1981)

    Article  Google Scholar 

  21. Perko, L.: Differential Equations and Dynamical Systems. Springer, New York (1991)

  22. Yeh, E.C., Day, G.C.: Parametric study of antiskid brake systems using Poincaré map concept. Int. J. Veh. Des. 13, 210–232 (1992)

    Google Scholar 

  23. Girling Auto website http://www.girlingauto.com/

  24. Awrejcewicz, J., Olejnik, P.: Numerical and experimental investigations of simple non-linear system modelling a Girling duo-servo brake mechanism. In: Proceedings of DETC’03 ASME, Chicago (2003)

  25. Awrejcewicz, J., Olejnik, P.: Occurrence of stick–slip phenomenon. J. Theor. Appl. Mech. 45(1), 33–40 (2007)

    Google Scholar 

  26. Awrejcewicz, J., Olejnik, P.: Improvement of the Lyapunov exponents computations by extension of time series. In: Proceedings of the 11th World Congress in Mechanism and Machine Science, Tianjin, China (2003)

  27. Alhama, F., Marín, F., Moreno, J.A.: An efficient and reliable model to simulate microscopic mechanical friction in the Frenkel–Kontorova–Tomlinson model. Comput. Phys. Commun. 182, 2314–2325 (2011)

    Article  Google Scholar 

  28. Marín, F., Alhama, F., Moreno, J.A.: Modelling of nanoscale friction using network simulation method. Comput. Mater. Continua 43(1), 1–19 (2014)

    Google Scholar 

  29. Marín, F., Alhama, F., Moreno, J.A.: Modelling of stick-slip behaviour with different hypotheses on friction forces. Int. J. Eng. Sci. 60, 13–24 (2012)

    Article  MathSciNet  Google Scholar 

  30. González-Fernández, C.F., Alhama, F.: Heat transfer and the network simulation method. In: Horno, J. (ed.), Network simulation method, Research signpost, Trivandrum (2001)

  31. Microsim Corporation Fairbanks, PSpice 6.0, Irvine, California (1994)

  32. Nagel, L.W.: SPICE2: A Computer Program to Simulate Semiconductor Circuits. University of California, Berkeley (1975)

  33. Kreith, F.: The CRC Handbook of Thermal Engineering. CRC Press, London (2000)

    MATH  Google Scholar 

  34. Mills, A.F.: Heat Transfer. Prentice Hall, Concord (1998)

  35. González-Fernández, C.F., Alhama, F., López-Sánchez, J.F.: Application of the network method to heat conduction processes with polynomial and potential-exponentially varying thermal properties. Numer. Heat Transf. Part A Appl. 33(5), 549–559 (1998)

    Article  Google Scholar 

  36. Horno, J., González-Fernández, C.F., Hayas, A.: The Network Method for solutions of oscillating reaction–diffusion systems. J. Comput. Phys. 118(2), 310–319 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  37. Soto, A., Alhama, F., González-Fernández, C.F.: An efficient model for solving density driven groundwater flow problems based on the network simulation method. J. Hydrol. 339(1–2), 39–53 (2007)

    Article  Google Scholar 

  38. Zueco, J., Alhama, F.: Inverse determination of heat generation sources in two-dimensional homogeneous solids: application to orthotropic medium. Int. Commun. Heat Mass Transfer 33(1), 49–55 (2006)

    Article  Google Scholar 

  39. Van de Vrande, B.L., Van Campen, D.H., De Kraker, A.: An approximate analysis of dry-friction-induced stick-slip vibrations by a smoothing procedure. Nonlinear Dyn. 19, 157–169 (1999)

  40. Elmer, F.J.: Nonlinear dynamics of dry friction. J. Phys. A Math. Gen. 30(17), 6057–6063 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  41. Andrianov, I.V., Awrejcewicz, J., Kudra, G.: Iterational processes and Padé approximants. Facta Univ. Mech. Autom. Control Robot. 4(17), 279–285 (2005)

  42. Tariku, F.A., Rogers, R.J.: Improved dynamic friction models for simulation of one-dimensional and two-dimensional stick-slip motion. J. Tribol. 123, 661–669 (2001)

    Article  Google Scholar 

  43. Vladimirescu, A.: The SPICE Book. Wiley, New York (1994)

  44. Kielkowski, R.: Inside SPICE. McGraw Hill, New York (1994)

Download references

Acknowledgments

We are grateful to Dr. Parlitz, Full Professor in the University of Göttingen, for his interesting and convenient tips.

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Technical University of Cartagena, Cartagena, Spain

    F. Marín

  2. Department of Applied Physics, Technical University of Cartagena, Cartagena, Spain

    F. Alhama

  3. Department of Mechanical Engineering, Technical University of Cartagena, Cartagena, Spain

    P. A. Meroño & J. A. Moreno

Authors
  1. F. Marín
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. F. Alhama
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. A. Meroño
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. A. Moreno
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to F. Marín.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Marín, F., Alhama, F., Meroño, P.A. et al. Modelling of stick–slip behaviour in a Girling brake using network simulation method. Nonlinear Dyn 84, 153–162 (2016). https://doi.org/10.1007/s11071-015-2312-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-015-2312-x

Keywords

Search

Navigation