Skip to main content
SpringerLink
Log in

Surrogate model approach for investigating the stability of a friction-induced oscillator of Duffing’s type

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

Abstract

Parametric studies are required to detect instability regimes of dynamic systems. This prediction can be computationally demanding as it requires a fine exploration of large parametric space due to the disrupted mechanical behavior. In this paper, an efficient surrogate strategy is proposed to investigate the behavior of an oscillator of Duffing’s type in combination with an elasto-plastic friction force model. Relevant quantities of interest are discussed. Sticking time is considered using a machine learning technique based on Gaussian processes called kriging. The largest Lyapunov exponent is considered as an efficient indicator of chaotic motion. This indicator is estimated using a perturbation method. A dedicated adaptive kriging strategy for classification called MiVor is utilized and appears to be highly proficient in order to detect instabilities over the parametric space and can furthermore be used for complex response surfaces in multi-dimensional parametric domains.

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
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19
Fig. 20

Similar content being viewed by others

References

  1. Rabinowicz, E.: Stick and slip. Sci. Am. 194(5), 109–119 (1956)

    Google Scholar 

  2. Barton, D., Blackwood, A.: Braking 2004: Vehicle Braking and Chassis Control, vol. 6. Wiley, New York (2004)

    Google Scholar 

  3. Ashraf, N., Bryant, D., Fieldhouse, J.D.: Investigation of stick–slip vibration in a commercial vehicle brake assembly. Int. J. Acoust. Vib. 22(3), 326–333 (2017)

    Google Scholar 

  4. Owen, W.S., Croft, E.A.: The reduction of stick–slip friction in hydraulic actuators. IEEE/ASME Trans. Mechatron. 8(3), 362–371 (2003)

    Google Scholar 

  5. Wu, Q., Luo, S., Qu, T., Yang, X.: Comparisons of draft gear damping mechanisms. Veh. Syst. Dyn. 55(4), 501–516 (2017)

    Google Scholar 

  6. Rubio, D., San Andres, L.: Structural stiffness, dry friction coefficient, and equivalent viscous damping in a bump-type foil gas bearing. J. Eng. Gas Turbines Power 129(2), 494–502 (2007)

    Google Scholar 

  7. Jiménez, M., Bielsa, J., Rodríguez, R., Bernad, C.: Two FEM approaches for the prediction and quantification of “stick–slip” phenomena on rubber–metal sliding contacts. In: IUTAM Symposium on Computational Methods in Contact Mechanics, pp. 291–309. Springer, Berlin (2007)

  8. Galvanetto, U., Bishop, S.: Dynamics of a simple damped oscillator undergoing stick–slip vibrations. Meccanica 34(5), 337–347 (1999)

    MathSciNet  MATH  Google Scholar 

  9. Hinrichs, N., Oestreich, M., Popp, K.: Dynamics of oscillators with impact and friction. Chaos Solitons Fractals 8(4), 535–558 (1997)

    MATH  Google Scholar 

  10. Stelter, P.: Stick–slip vibrations and chaos. Philos. Trans. R. Soc. Lond. A 332(1624), 89–105 (1990)

    MATH  Google Scholar 

  11. Devarajan, K., Balaram, B.: Analytical approximations for stick–slip amplitudes and frequency of duffing oscillator. J. Comput. Nonlinear Dyn. 12(4), 044501 (2017)

    Google Scholar 

  12. Awrejcewicz, J., Sendkowski, D.: Stick–slip chaos detection in coupled oscillators with friction. Int. J. Solids Struct. 42(21–22), 5669–5682 (2005)

    MathSciNet  MATH  Google Scholar 

  13. Santhosh, B., Narayanan, S., Padmanabhan, C.: Discontinuity induced bifurcations in nonlinear systems. Procedia IUTAM 19, 219–227 (2016)

    Google Scholar 

  14. Balcerzak, M., Dabrowski, A., Stefański, A., Wojewoda, J.: Spectrum of Lyapunov exponents in non-smooth systems evaluated using orthogonal perturbation vectors. In: MATEC Web of Conferences, vol. 148, p. 10003. EDP Sciences, Les Ulis (2018)

    Google Scholar 

  15. Pikunov, D., Stefanski, A.: Numerical analysis of the friction-induced oscillator of Duffing’s type with modified LuGre friction model. J. Sound Vib. 440, 23–33 (2019)

    Google Scholar 

  16. Lima, R., Sampaio, R.: Stick-mode duration of a dry-friction oscillator with an uncertain model. J. Sound Vib. 353, 259–271 (2015)

    Google Scholar 

  17. Grassberger, P., Procaccia, I.: Characterization of strange attractors. Phys. Rev. Lett. 50(5), 346 (1983)

    MathSciNet  MATH  Google Scholar 

  18. Benetti, G., Galgani, L., Strelcyn, J.: Kolmogorov entropy and numerical experiments. Phys. Rev. A 14(6), 2338 (1976)

    Google Scholar 

  19. Kocarev, L., Szczepanski, J., Amigó, J., Tomovski, I.: Discrete chaos-i: theory. IEEE Trans. Circuits Syst. I Regul. Pap. 53(6), 1300–1309 (2006)

    MathSciNet  MATH  Google Scholar 

  20. Awrejcewicz, J., Krysko, A., Erofeev, N., Dobriyan, V., Barulina, M., Krysko, V.: Quantifying chaos by various computational methods. Part 1: simple systems. Entropy 20(3), 175 (2018)

    MathSciNet  Google Scholar 

  21. Kleijnen, J.: Regression and kriging metamodels with their experimental designs in simulation: a review. Eur. J. Oper. Res. 256(1), 1–16 (2017)

    MathSciNet  MATH  Google Scholar 

  22. Kleijnen, J.: Kriging metamodeling in simulation: a review. Eur. J. Oper. Res. 192(3), 707–716 (2009)

    MathSciNet  MATH  Google Scholar 

  23. Williams, C.K.I, Rasmussen, C.E.: Gaussian processes for regression. In: Advances in Neural Information Processing Systems, vol. 8, pp. 514–520. MIT Press (1996)

  24. Clarke, S., Griebsch, J., Simpson, T.: Analysis of support vector regression for approximation of complex engineering analyses. J. Mech. Des. 127(6), 1077–1087 (2005)

    Google Scholar 

  25. Park, J., Sandberg, I.: Universal approximation using radial-basis-function networks. Neural Comput. 3(2), 246–257 (1991)

    Google Scholar 

  26. Krige, D.: A statistical approach to some basic mine valuation problems on the witwatersrand. J. S. Afr. Inst. Min. Metall. 52(6), 119–139 (1951)

    Google Scholar 

  27. Sacks, J., Welch, W., Mitchell, T., Wynn, H.: Design and analysis of computer experiments. Stat. Sci. 4, 409–423 (1989)

    MathSciNet  MATH  Google Scholar 

  28. Jiang, P., Zhang, Y., Zhou, Q., Shao, X., Hu, J., Shu, L.: An adaptive sampling strategy for kriging metamodel based on Delaunay triangulation and topsis. Appl. Intell. 48, 1–13 (2017)

    Google Scholar 

  29. Van Beers, W., Kleijnen, J.: Kriging for interpolation in random simulation. J. Oper. Res. Soc. 54(3), 255–262 (2003)

    MATH  Google Scholar 

  30. Pennestrì, E., Rossi, V., Salvini, P., Valentini, P.: Review and comparison of dry friction force models. Nonlinear Dyn. 83(4), 1785–1801 (2016)

    MATH  Google Scholar 

  31. Olsson, H., Åström, K., De Wit, C., Gäfvert, M., Lischinsky, P.: Friction models and friction compensation. Eur. J. Control 4(3), 176–195 (1998)

    MATH  Google Scholar 

  32. Marques, F., Flores, P., Claro, J.C., Lankarani, H.: A survey and comparison of several friction force models for dynamic analysis of multibody mechanical systems. Nonlinear Dyn. 86(3), 1407–1443 (2016)

    MathSciNet  Google Scholar 

  33. Dupont, P., Hayward, V., Armstrong, B., Altpeter, F.: Single state elastoplastic friction models. IEEE Trans. Autom. Control 47(5), 787–792 (2002)

    MathSciNet  MATH  Google Scholar 

  34. Prandtl, L.: Spannungsverteilung in plastischen Körpern. In: Proceedings of the 1st International Congress on Applied Mechanics, pp. 43–54 (1924)

  35. Dupont, P., Armstrong, B., Hayward, V: Elasto-plastic friction model: contact compliance and stiction. In American Control Conference, 2000. Proceedings of the 2000, vol. 2, pp. 1072–1077. IEEE (2000)

  36. De Wit, C., Olsson, H., Astrom, K., Lischinsky, P.: A new model for control of systems with friction. IEEE Trans. Autom. Control 40(3), 419–425 (1995)

    MathSciNet  MATH  Google Scholar 

  37. Townsend, W., Salisbury, Jr.: The effect of Coulomb friction and stiction on force control. In Proceedings. 1987 IEEE International Conference on Robotics and Automation, vol. 4, pp. 883–889. IEEE (1987)

  38. Stribeck, R.: Die wesentlichen Eigenschaften der Gleit-und Rollenlager. Zeitschrift des Vereines Deutscher Ingenieure 46, 1341–1348 (1902)

    Google Scholar 

  39. Oseledec, V.: A multiplicative ergodic theorem. Liapunov characteristic number for dynamical systems. Trans. Mosc. Math. Soc. 19, 197–231 (1968)

    MathSciNet  Google Scholar 

  40. Rosenstein, M., Collins, J., De Luca, C.: A practical method for calculating largest Lyapunov exponents from small data sets. Physica D 65(1–2), 117–134 (1993)

    MathSciNet  MATH  Google Scholar 

  41. Shimada, I., Nagashima, T.: A numerical approach to ergodic problem of dissipative dynamical systems. Prog. Theor. Phys. 61(6), 1605–1616 (1979)

    MathSciNet  MATH  Google Scholar 

  42. Benettin, G., Galgani, L., Giorgilli, A., Strelcyn, J.: Lyapunov characteristic exponents for smooth dynamical systems and for Hamiltonian systems; a method for computing all of them. Part 1: theory. Meccanica 15(1), 9–20 (1980)

    MATH  Google Scholar 

  43. Kantz, H.: A robust method to estimate the maximal Lyapunov exponent of a time series. Phys. Lett. A 185(1), 77–87 (1994)

    Google Scholar 

  44. Wolf, A.: Quantifying chaos with Lyapunov exponents. Chaos 16, 285–317 (1986)

    Google Scholar 

  45. Molaie, M., Jafari, S., Sprott, J., Golpayegani, S.: Simple chaotic flows with one stable equilibrium. Int. J. Bifurc. Chaos 23(11), 1350188 (2013)

    MathSciNet  MATH  Google Scholar 

  46. Matheron, G.: Principles of geostatistics. Econ. Geol. 58(8), 1246–1266 (1963)

    Google Scholar 

  47. Stein, A., Corsten, L.C.A.: Universal kriging and cokriging as a regression procedure. Biometrics 47, 575–587 (1991)

    Google Scholar 

  48. Handcock, M., Stein, M.: A Bayesian analysis of kriging. Technometrics 35(4), 403–410 (1993)

    Google Scholar 

  49. Matérn, B.: Spatial variation: Meddelanden fran statens skogsforskningsinstitut. Lect. Notes Stat. 36, 21 (1960)

    Google Scholar 

  50. Dubourg, V., Sudret, B., Deheeger, F.: Metamodel-based importance sampling for structural reliability analysis. Probab. Eng. Mech. 33, 47–57 (2013)

    Google Scholar 

  51. Bouhlel, M., Martins, J.: Gradient-enhanced kriging for high-dimensional problems. Eng. Comput. 35(1), 157–173 (2019)

    Google Scholar 

  52. Toal, D., Bressloff, N., Keane, A., Holden, C.: The development of a hybridized particle swarm for kriging hyperparameter tuning. Eng. Optim. 43(6), 675–699 (2011)

    Google Scholar 

  53. Santner, T., Williams, B., Notz, W.: The Design and Analysis of Computer Experiments. Springer, Berlin (2013)

    MATH  Google Scholar 

  54. Jones, D., Schonlau, M., Welch, W.: Efficient global optimization of expensive black-box functions. J. Glob. Optim. 13(4), 455–492 (1998)

    MathSciNet  MATH  Google Scholar 

  55. Turner, C.J., Crawford, R.H., Campbell, M.I.: Multidimensional sequential sampling for NURBs-based metamodel development. Eng. Comput. 23(3), 155–174 (2007)

    Google Scholar 

  56. Singh, P., Deschrijver, D., Dhaene, T.: A balanced sequential design strategy for global surrogate modeling. In: Simulation Conference (WSC), 2013 Winter, pp. 2172–2179. IEEE (2013)

  57. Liu, H., Cai, J., Ong, Y.: An adaptive sampling approach for kriging metamodeling by maximizing expected prediction error. Comput. Chem. Eng. 106, 171–182 (2017)

    Google Scholar 

  58. Sundararajan, S., Keerthi, S.S.: Predictive approaches for choosing hyperparameters in Gaussian processes. In Advances in Neural Information Processing Systems, vol. 12, pp. 631–637. MIT Press (2000)

  59. Lam, C.: Sequential adaptive designs in computer experiments for response surface model fit. PhD thesis, The Ohio State University (2008)

  60. Fuhg, J.N., Fau, A.: An innovative adaptive kriging approach for efficient binary classification of mechanical problems (2019). arXiv preprint arXiv:1907.01490

  61. Fuhg, J.N.: Adaptive surrogate models for parametric studies. Master’s thesis, Leibniz Universität Hannover (2019). Arxiv platform https://arxiv.org/abs/1905.05345

  62. Viana, F., Venter, G., Balabanov, V.: An algorithm for fast optimal latin hypercube design of experiments. Int. J. Numer. Methods Eng. 82(2), 135–156 (2010)

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The authors acknowledge the financial support from the Deutsche Forschungsgemeinschaft under Germany’s Excellence Strategy within the Cluster of Excellence PhoenixD (EXC 2122, Project ID 390833453). The results presented in this paper were partially carried out on the cluster system at the Leibniz University of Hannover, Germany.

Author information

Authors and Affiliations

  1. Institute of Continuum Mechanics, Leibniz Universität Hannover, Appelstraße 11, 30167, Hannover, Germany

    Jan N. Fuhg

  2. Institute of Mechanics and Computational Mechanics, Leibniz Universität Hannover, Appelstraße 9A, Hannover, Germany

    Amélie Fau

Authors
  1. Jan N. Fuhg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Amélie Fau
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Jan N. Fuhg.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Fuhg, J.N., Fau, A. Surrogate model approach for investigating the stability of a friction-induced oscillator of Duffing’s type . Nonlinear Dyn 98, 1709–1729 (2019). https://doi.org/10.1007/s11071-019-05281-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05281-2

Keywords

Search

Navigation