Skip to main content
SpringerLink
Log in

Experimental analysis of the influence of damping on the resonance behavior of a spherical pendulum

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

Abstract

This study describes the experimental and numerical dynamic analysis of a kinematically excited spherical pendulum. The stability of the response in the vertical plane was analyzed in the theoretically predicted auto-parametric resonance domain. Three different types of the resonance domain were investigated the properties of which depended significantly on the dynamic parameters of the pendulum and the excitation amplitude. A mathematical model was used to represent the nonlinear characteristics of the pendulum, which includes the asymmetrical damping. A special frame was developed to carry out the experiments, which contained the pendulum supported by the Cardan joint and two magnetic units attached to the supporting axes of rotation, and this was able to reproduce linear viscous damping for both of the principal response components. The stability analysis of the system was compared with the numerical solution of the governing equations and experimental observation. The most significant practical outcomes for designers are also summarized.

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.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12

Similar content being viewed by others

References

  1. den Hartog, J.P.: Mechanical Vibrations. McGraw-Hill, New York (1956)

    MATH  Google Scholar 

  2. Korenev, B.G., Reznikov, L.M.: Dynamic Absorbers of Vibrations: Theory and Technical Applications. Nauka, Moscow (1988)

    Google Scholar 

  3. Lee, C.L., Chen, Y.T., Chung, L.L., Wang, Y.P.: Optimal design theories and applications of tuned mass dampers. Eng. Struct. 28(1), 43–53 (2006)

    Article  Google Scholar 

  4. He, M., Ma, R., Lin, Z.: Design of structural vibration control of a tall steel TV tower under wind load. Int. J. Steel Struct. 7(1), 85–92 (2007)

    Google Scholar 

  5. Chen, X., Kareem, A.: Efficacy of tuned mass dampers for bridge flutter control. J. Struct. Eng. ASCE 129(10), 1291–1300 (2003)

    Article  Google Scholar 

  6. Giaccu, G.F., Caracoglia, L.: Effects of modeling nonlinearity in cross-ties on the dynamics of simplified in-plane cable networks. Struct. Contr. Health Monit. 193, 348–369 (2012)

  7. Yalla, S.K., Kareem, A.: Optimum absorber parameters for tuned liquid column dampers. J. Struct. Eng. ASCE 126(8), 906–915 (2000)

    Article  Google Scholar 

  8. Zuo, L., Nayfeh, S.A.: Minimax optimization of multi-degree-of-freedom tuned-mass dampers. J. Sound Vibr. 272(3–5), 893–908 (2004)

    Article  Google Scholar 

  9. Kareem, A., Kline, S.: Performance of multiple mass dampers under random loading. J. Struct. Eng. ASCE 121(2), 1291–1300 (1999)

    Google Scholar 

  10. Náprstek, J., Pirner, M.: Non-linear behaviour and dynamic stability of a vibration spherical absorber. In: Smyth A. et al. (ed) Proceedings of the 15th ASCE Engineering Mechanics Division Conference vol. 150. Columbia University, New York paper (2002)

  11. Náprstek, J., Fischer, C., Pirner, M., Fischer, O.: Non-linear model of a ball vibration absorber. In: Papadrakakis M., Fragiadakis, M., Plevris, V. (eds) Computational Methods in Earthquake Engineering, vol. 2, pp. 381–396. Springer, Dordrecht (2013)

  12. Haxton, R.S., Barr, A.D.S.: The auto-parametric vibration absorber. J. App. Mech. ASME 94, 119–125 (1974)

    Google Scholar 

  13. Tondl, A.: To the analysis of auto-parametric systems. Zeit. Angew. Math. Mech. 77(6), 407–418 (1997)

    Article  MATH  Google Scholar 

  14. Náprstek, J., Fischer, C.: Auto-parametric semi-trivial and post-critical response of a spherical pendulum damper. Comp. Struct. 87(19–20), 1204–1215 (2009)

    Article  Google Scholar 

  15. Nabergoj, R., Tondl, A., Virag, Z.: Auto-parametric resonance in an externally excited system. Chaos Sol. Fract. 4(2), 263–273 (1994)

    Article  MATH  Google Scholar 

  16. Hatwal, H., Mallik, A.K., Ghosh, A.: Forced non-linear oscillations of an auto-parametric system. J. App. Mech. ASME 50, 657–662 (1983)

    Article  MATH  Google Scholar 

  17. Náprstek, J.: Non-linear self-excited random vibrations and stability of an SDOF system with parametric noises. Meccanica 33, 267–277 (1998)

    Article  MATH  Google Scholar 

  18. Warminski, J., Kecik, K.: Auto-parametric vibrations of a non-linear system with pendulum. Math. Prob. Eng. AID 80705, 1–19 (2006)

    Article  Google Scholar 

  19. Tondl, A., Ruijgrok, T., Verhulst, F., Nabergoj, R.: Auto-Parametric Resonance in Mechanical Systems. Cambridge University Press, Cambridge (2000)

    Google Scholar 

  20. Tondl, A., Dohnal, F.: Suppressing flutter vibrations by parametric inertia excitation. J. Appl. Mech. ASME 76(3), 1–7 (2009)

    Google Scholar 

  21. Dohnal, F., Verhulst, F.: Averaging in vibration suppression by parametric stiffness excitation. Nonlinear Dyn. 54, 231–248 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  22. Dohnal, F.: A Contribution to the Mitigation of Transient Vibrations Parametric Anti-Resonance: Theory, Experiment and Interpretation. (Habilitation thesis), TU Darmstadt (2012)

  23. Kovacic, I.: Harmonically excited generalized van der Pol oscillators: entrainment phenomenon. Meccanica 48, 2415–2425 (2013)

  24. Vazquez-Gonzalez, B., Silva-Navarro, G.: Evaluation of the auto-parametric pendulum vibration absorber for a Duffing system. J. Shock Vib. 15(3–4), 355–368 (2008)

  25. Berlioz, A., Dufour, R., Sinha, S.C.: Bifurcation in a non-linear auto-parametric system using experimental and numerical investigations. Nonlinear Dyn. 23, 175–187 (2000)

    Article  MATH  Google Scholar 

  26. Sheheitli, H., Rand, R.H.: Dynamics of a mass ’spring’ pendulum system with vastly different frequencies. Nonlinear Dyn. 70, 25–41 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  27. Miles, J.W.: Stability of forced oscillations of a spherical pendulum. Quart. J. Appl. Math. 20, 21–32 (1962)

    MathSciNet  MATH  Google Scholar 

  28. Miles, J.W.: Resonant motion of spherical pendulum. Physica D 11, 309–323 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  29. Tritton, D.J.: Ordered and chaotic motion of a forced spherical pendulum. Eur. J. Phys. 7, 162–169 (1986)

    Article  Google Scholar 

  30. Gottlieb, O., Habib, G.: Non-linear model-based estimation of quadratic and cubic damping mechanisms governing the dynamics of chaotic spherical pendulum. J. Vib. Control 18(4), 536–547 (2011)

    Article  MathSciNet  Google Scholar 

  31. Pospíšil, S., Fischer, C., Náprstek, J.: Experimental and theoretical analysis of auto-parametric stability of pendulum with viscous dampers. Acta Technica CSAV 56(4), 359–378 (2011)

    Google Scholar 

  32. Meirovitch, L.: Methods of Analytical Dynamics. Dover Publications, Mineola, NY (2003)

    MATH  Google Scholar 

  33. Sofroniou, M., Knapp, R.: Advanced Numerical Differential Equation Solving in Mathematica. Wolfram Research, Inc., Champaign, IL (2008)

    Google Scholar 

  34. Galassi, M., Davies, J., Theiler, J., Gough, B., Jungman, G., Booth, M., Rossi, F.: GNU Scientific library reference manual, 3rd revised edition for GSL version 1.13. Network Theory Ltd. (2009)

Download references

Acknowledgments

The support of the Czech-Taiwan research project GAČR No. 13-24405J and NSC 101WFD0400131 as well as the support of RVO 68378297 is gratefully acknowledged.

Author information

Authors and Affiliations

  1. Institute of Theoretical and Applied Mechanics ASCR, v. v. i., Prosecká 76, Prague 9, Czech Republic

    Stanislav Pospíšil, Cyril Fischer & Jiří Náprstek

Authors
  1. Stanislav Pospíšil
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Cyril Fischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jiří Náprstek
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Stanislav Pospíšil.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Pospíšil, S., Fischer, C. & Náprstek, J. Experimental analysis of the influence of damping on the resonance behavior of a spherical pendulum. Nonlinear Dyn 78, 371–390 (2014). https://doi.org/10.1007/s11071-014-1446-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-014-1446-6

Keywords

Search

Navigation