Skip to main content
SpringerLink
Log in

Sub-harmonic resonant solutions of a harmonically excited dry friction oscillator

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

Abstract

Special sub-harmonic solutions of a harmonically forced dry-friction oscillator are analysed. Although the typical non-sticking solutions are stable and symmetric, a continuum of possible asymmetric, marginally stable solutions exist at excitation frequencies Ω = 1/2n. We determine the explicit form of the one-parameter family of these solutions, and give the conditions under which our formulae are valid. The stability of the solutions is examined in the third-order approximation. Finally, our analytical results are checked by numerical simulations.

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.

Similar content being viewed by others

References

  1. Kato, S., Yamaguchi, K., Matsubayashi, T.: Stick-slip motion of machine tool slideway. ASME J. Eng. Ind. 96(2), 557–566 (1974)

    Google Scholar 

  2. Dahl, P.R.: Measurement of solid friction parameters of ball bearings. In: Proceedings of 6th Annual Symposium on Incremental Motion, Control Systems and Devices, Illinois, Urbana-Champaign (1977)

  3. Karnopp, D.: Computer simulation of stick–slip friction in mechanical dynamics systems. ASME J. Dyn. Syst. Meas. Contr. 107(1), 100–103 (1985)

    Article  Google Scholar 

  4. Ibrahim, R.A.: Friction-induced vibration, chatter, squeal, and chaos, Part II: dynamics and modeling. Appl. Mech. Rev. 47(7), 227–253 (1994)

    Article  Google Scholar 

  5. Dankowicz, H.: On the modeling of dynamic friction phenomena. Z. Angew. Math. Mech. 79(6), 399–409 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berger, E.J.: Friction modeling for dynamic system simulation. ASME Appl. Mech. Rev. 55(6), 535–577 (2002)

    Article  Google Scholar 

  7. Den Hartog, J.P.: Forced vibrations with combined Coulomb and viscous damping. Trans. ASME 53, 107–115 (1930)

    Google Scholar 

  8. Pratt, T.K., Williams, R.: Non-linear analysis of stick/slip motion. J. Sound Vib. 74(4), 531–542 (1981)

    Article  Google Scholar 

  9. Shaw, S.W.: On the dynamic response of a system with dry friction. J. Sound Vib. 108(2), 305–325 (1986)

    Article  Google Scholar 

  10. Feeny, B., Moon, F.C.: Chaos in a forced dry-friction oscillator: experiments and numerical modelling. J. Sound Vib. 170, 303–323 (1994)

    Article  MATH  Google Scholar 

  11. Awrejcewicz, J.: Chaos in simple mechanical systems with friction. J. Sound Vib. 109(1), 178–180 (1986)

    Article  MathSciNet  Google Scholar 

  12. Popp, K., Stelter, P.: Stick-slip vibrations and chaos. Philos. Trans. R. Soc. Lond. A 332, 89–105 (1990)

    Article  MATH  Google Scholar 

  13. Narayanan, S., Jayaraman, K.: Chaotic vibration in a non-linear oscillator with Coulomb damping. J. Sound Vib. 146(1), 17–31 (1991)

    Article  Google Scholar 

  14. Natsiavas, S.: Stability of piecewise linear oscillators with viscous and dry friction damping. J. Sound Vib. 217, 507–522 (1998)

    Article  MathSciNet  Google Scholar 

  15. Filippov, A.F.: Differential equations with discontinuous right-hand side. Am. Math. Soc. Trans., Ser. 2 42, 199–231 (1964)

    Google Scholar 

  16. Awrejcewicz, J., Lamarque, C.-H.: Bifurcation and Chaos in Nonsmooth Mechanical Systems. World Scientific, Singapore (2003)

  17. Deimling, K.: Multivalued Differential Equations. de Gruyter Series in Nonlinear Analysis and Applications, Vol. 1. Berlin, New York (1992)

  18. Csernák, G., Stépán, G.: On the periodic response of a harmonically excited dry friction oscillator. J. Sound Vib. 295, 649–658 (2006)

    Article  Google Scholar 

  19. Blazejczyk-Okolewska, B., Kapitaniak, T.: Dynamics of impact oscillator with dry friction. Chaos Solitons Fractals 7, 1455–1459 (1996)

    Article  Google Scholar 

  20. Dankowicz, H., Nordmark, A.B.: On the origin and bifurcations of stick-slip oscillations. Physica. D 136, 280–302 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  21. Cheng, G., Zu, J.W.: A numerical study of a dry friction oscillator with parametric and external excitations. J. Sound Vib. 287, 329–342 (2005)

    Article  Google Scholar 

  22. Hong, H.-K., Liu, C.-S.: Coulomb friction oscillator: modelling and responses to harmonic loads and base excitations. J. Sound Vib. 229(5), 1171–1192 (2000)

    Article  Google Scholar 

  23. Hong, H.-K., Liu, C.-S.: Non-sticking oscillation formulae for Coulomb friction under harmonic loading. J. Sound Vib. 244(5), 883–898 (2001)

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. HAS-BUTE Research Group on Dynamics of Machines and Vehicles, Department of Applied Mechanics, Budapest University of Technology and Economics, H-1521, Budapest, Hungary

    G. Csernák & G. Stépán

  2. Department of Mechanical Engineering, Michigan State University, East Lansing, MI, 48824, USA

    S. W. Shaw

Authors
  1. G. Csernák
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Stépán
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. W. Shaw
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to G. Csernák.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Csernák, G., Stépán, G. & Shaw, S.W. Sub-harmonic resonant solutions of a harmonically excited dry friction oscillator. Nonlinear Dyn 50, 93–109 (2007). https://doi.org/10.1007/s11071-006-9145-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-006-9145-6

Keywords

Search

Navigation