Skip to main content
SpringerLink
Log in

Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators

  • Published:
Meccanica Aims and scope Submit manuscript

Abstract

In this paper we consider the problem which can appear at the determination of the dynamical stability of the responses of oscillators with discontinuous or steep derivative of the restoring characteristic obtained in the frequency domain. For that purpose, a simple one degree-of-freedom system with piecewise-linear force-displacement relationship subjected to a harmonic excitation is analysed. Stability of the periodic response obtained in the frequency domain by the incremental harmonic balance method is determined by using the Floquet-Liapounov theorem. Confirmation of the results obtained in the frequency domain is done by comparing with the results obtained in the time domain by the method of piecing the exact solutions. Determination of the dynamical stability can be made more reliable by using the proposed plot of maximum modulus of the eigenvalues of the monodromy matrix in dependence of non-dimensional frequency and the number of harmonics included in the supposed approximate solution.

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

Similar content being viewed by others

References

  1. Choi YS, Noah ST (1988) Forced periodic vibration of unsymmetric piecewise linear systems. J Sound Vib 121:117–126

    Article  ADS  MathSciNet  Google Scholar 

  2. Wong CW, Zhang WS, Lau SL (1991) Periodic forced vibration of unsymmetrical piecewise-linear systems by incremental harmonic balance method. J Sound Vib 149:91–105

    Article  ADS  Google Scholar 

  3. Kahraman A, Blankenship GW (1996) Interactions between commensurate parametric and forcing excitations in a system with clearance. J Sound Vib 194:317–336

    Article  ADS  Google Scholar 

  4. Wolf H, Stegić M (1999) The influence of neglecting small harmonic terms on estimation of dynamical stability of the response of non-linear oscillators. Comput Mech 24:230–237

    Article  MATH  Google Scholar 

  5. Wolf H, Kodvanj J, Bjelovučić-Kopilović S (2004) Effect of smoothing piecewise-linear oscillators on their stability predictions. J Sound Vib 270:917–932

    Article  ADS  Google Scholar 

  6. Wolf H, Terze Z, Sušić A (2004) Dynamical stability of the response of oscillators with discontinuous or steep first derivative of restoring characteristic. Eur J Mech A: Solids 23:1041–1050

    Article  MATH  Google Scholar 

  7. Pierre C, Ferri AA, Dowell EH (1985) Multi-harmonic analysis of dry friction damped systems using an incremental harmonic balance method. J Appl Mech 52:958–964

    MATH  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  9. Galvanetto U (2001) An example of a non-smooth fold bifurcation. Meccanica 36:229–233

    Article  MATH  Google Scholar 

  10. Bogacz R, Ryczek B (2003) Frictional phenomena in dynamical system with two-frequency excitation. Meccanica 38:711–717

    Article  MATH  Google Scholar 

  11. Sorge F (2007) On the frequency behaviour, stability and isolation properties of dry friction oscillators. Meccanica 42:61–75

    Article  MathSciNet  MATH  Google Scholar 

  12. Muller PC (1995) Calculation of Lyapunov exponents for dynamic systems with discontinuities. Chaos Solitons Fractals 5:1671–1681

    Article  MathSciNet  Google Scholar 

  13. Awrejcewicz J (1989) Bifurcation and chaos in simple dynamical systems. World Scientific, Singapore

    MATH  Google Scholar 

  14. Sundararajan P, Noah ST (1998) An algorithm for response and stability of large order non-linear systems—application to rotor systems. J Sound Vib 214:695–723

    Article  ADS  Google Scholar 

  15. Awrejcewicz J, Kudra G, Lamarque CH (2003) Dynamics investigation of three coupled rods with horizontal barrier. Meccanica 38:687–698

    Article  MATH  MathSciNet  Google Scholar 

  16. Li D, Xu J (2004) A method to determine the periodic solution of the non-linear dynamics system. J Sound Vib 275:1–16

    Article  ADS  Google Scholar 

  17. Lau SL, Cheung YK, Wu SW (1983) Incremental harmonic balance method with multiple time scales for aperiodic vibration of nonlinear systems. J Appl Mech 50:871–876

    Article  MATH  MathSciNet  Google Scholar 

  18. Lau SL, Yuen SW (1993) Solution diagram of non-linear dynamic systems by the IHB method. J Sound Vib 167:303–316

    Article  MATH  ADS  Google Scholar 

  19. Leung AYT, Chui SK (1995) Non-linear vibration of coupled duffing oscillators by an improved incremental harmonic balance method. J Sound Vib 181:619–633

    Article  ADS  MathSciNet  Google Scholar 

  20. Szemplinska-Stupnicka W (1990) The behaviour of non-linear vibrating systems. Kluwer Academic, Dordrecht

    Google Scholar 

  21. Awrejcewicz J, Andrianov J, Manevitch L (1998) Asymptotic approach in nonlinear dynamics: new trends and applications. Springer, Berlin

    Google Scholar 

  22. Awrejcewicz J, Lamarque CH (2003) Bifurcation and chaos in nonsmooth mechanical systems. World Scientific, Singapore

    MATH  Google Scholar 

  23. Minorsky N (1962) Nonlinear oscillations. Van Nostrand, Princeton

    MATH  Google Scholar 

  24. Nayfeh AH, Balachandram B (1995) Applied nonlinear dynamics: analytical, computational and experimental methods. Wiley, New York

    MATH  Google Scholar 

  25. D’Souza AF, Garg WH (1974) Advanced dynamics modeling and analysis. Prentice-Hall, Englewood Cliffs

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000, Zagreb, Croatia

    Hinko Wolf & Aleksandar Sušić

  2. Faculty of Graphic Arts, University of Zagreb, Getaldićeva 5, 10000, Zagreb, Croatia

    Dubravko Banić

Authors
  1. Hinko Wolf
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dubravko Banić
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Aleksandar Sušić
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hinko Wolf.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Wolf, H., Banić, D. & Sušić, A. Influence of small harmonic terms on eigenvalues of monodromy matrix of piecewise-linear oscillators. Meccanica 43, 485–494 (2008). https://doi.org/10.1007/s11012-008-9112-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11012-008-9112-z

Keywords

Search

Navigation