Autoparametric Vibrations of a Nonlinear System with a Pendulum and Magnetorheological Damping

  • Jerzy Warminski
  • Krzysztof Kecik
Part of the Solid Mechanics and Its Applications book series (SMIA, volume 181)


The chapter deals with autoparametric vibrations of a system composed of a nonlinear oscillator with an attached pendulum. Dynamics of the mechanical structure is studied analytically around the principal parametric resonance region, numerically and experimentally for a wide range of parameters. The influence of damping, nonlinear stiffness (hard and soft), amplitude and frequency of excitation on the system’s behaviour is analysed in details. The obtained results show that the pendulum can be applied as a dynamical absorber. However, for selected parameters, near the main parametric resonance, instability, which transits the pendulum to chaotic oscillations or to a full rotation, occurs. Therefore, the application of a magnetorheological (MR) damper and a nonlinear spring is proposed to improve the dynamics and to control the response online. Periodic vibrations, chaotic motions or a full rotation of the pendulum obtained numerically are confirmed by the experiment. The chaotic nature of motion is determined from real signals by the attractor reconstruction and the recurrence plot calculation. The results show that the semi-active suspension may reduce dangerous motion and it also allows to maintain the pendulum at a given attractor or to jump to another one.


Autoparametric vibrations control stability chaos parametric resonance magnetorheological damping nonlinearity basins of attraction 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Acheson, D.J.: A pendulum theorem. Proc. R. Soc. Lond. A (443) (1993)Google Scholar
  2. 2.
    Acheson, D.J.: Upside-down pendulums. Nature 336, 215–216 (1993)CrossRefGoogle Scholar
  3. 3.
    Anxin, G., Xu, Y.L., Hui, L.: Dynamic performance of cable-stayed bridge tower with multi-stage pendulum mass damper under wind excitations - I: Analytical. Earthquake Engineering and Engineering Vibration 6(3), 295–306 (2007)CrossRefGoogle Scholar
  4. 4.
    Areemit, N., Warnttchai, P.: Vibration suppression of a 90-m tall steel stack by using tuned mass damper. In: The Eighth East Asia-Pacific Conference on Structural Engineering and Constrution. December 5-7, Nanyang Technological University, Singapore (2001); paper 1316Google Scholar
  5. 5.
    Bajaj, K., Chang, S.I., Johnson: Amplitude modulated dynamics of a resonantly excited autoparametric two degree-of -freedom system. Nonlinear Dynamics 5, 433–457 (1994)CrossRefGoogle Scholar
  6. 6.
    Bishop, S.R., Cliford, M.J.: Rotating periodic orbits of the parametrically excited pendulum. Physical Letter A 201, 191–196 (1995)zbMATHCrossRefGoogle Scholar
  7. 7.
    Cartmell, M.P., Lawson, J.: Performance enhancement of an autoparametric vibration ab-sorber by means of computer control. Journal of Sound and Vibration 177(2), 173–195 (1994)zbMATHCrossRefGoogle Scholar
  8. 8.
    Cartmell, M.P., Roberts, J.W.: Simultaneous Combination Resonances in an Autoparametrically Resonant System. Journal of Sound and Vibration 123(1), 81–100 (1988)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Eckmann, J.P., Kampshort, S.O., Ruelle, D.: Recurrence Plot of Dynamical Systems. Euro-physics Letters 4, 973–977 (1987)CrossRefGoogle Scholar
  10. 10.
    Fraser, A.M., Swinney, H.L.: Independent coordinates for strange attractors from mutual information. Physical Review A 33, 1134–1140 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Hatwal, H., Mallik, A.K., Ghos, A.: Forced nonlinear oscillations of an autoparametric System- Part1: Periodic Responses. Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers 50, 657–662 (1983)zbMATHGoogle Scholar
  12. 12.
    Hatwal, H., Mallik, A.K., Ghosh, A.: Forced nonlinear oscillations of an autoparametric system. Journal of Applied Mechanics, Transactions of the American Society of Mechanical Engineers, Part 2. Chaotic responses 50, 663–668 (1983)zbMATHGoogle Scholar
  13. 13.
    Horton, B.W., Wiercigroch, M.: Effects of Heave Excitation on Rotations of a Pendulum for Wave Energy Extraction. In: IUTAM Bookseries, IUTAM Symposium on Fluid-Structure Interaction in Ocean Engineering, vol. 8, pp. 117–128 (2008)Google Scholar
  14. 14.
    Jumani, S.S.: An invertible open-loop nonlinear dynamic temperature dependent MR damper model. Master of Science (2010)Google Scholar
  15. 15.
    Kamath, G.M., Hurt, M.K., Wereley, N.M.: Analysis and testing of Bingham plastic behavior in semi-active electrorheological fluid dampers. Smart Materials and Structures 5(5), 576–590 (1996)CrossRefGoogle Scholar
  16. 16.
    Kecik, K.: Regular and chaotic vibrations of a nonlinear mechanical system with a pendulum. PhD Thesis, Lublin University of Technology (2009)Google Scholar
  17. 17.
    Kecik, K., Warminski, J.: Analysis of chaotic and regular motions of an autoparametric system by recurrence plots applications. Vibrations in Physical Systems 24, 221–226 (2010)Google Scholar
  18. 18.
    Kennel, M.B., Brown, R., Abarbanel, H.D.I.: Determining embedding dimension for phase space reconstruction using a geometrical construction. Physics Review A 45, 3403–3411 (1992)CrossRefGoogle Scholar
  19. 19.
    Kodba, S., Perc, M., Marhl, M.: Detecting chaos from a time series. European Journal of Physics 26, 205–215 (2005)CrossRefGoogle Scholar
  20. 20.
    Kromulski, J., Kazimierczak, J.: Damping of vibrations with using magnetorheological fluid devices. Journal of Research and Applications in Agricultural Engineering 51(3), 47–49 (2006)Google Scholar
  21. 21.
    Lee, W.K., Hsu, C.S.: A global analysis of an harmonically excited spring-pendulum system with internal resonance. Journal of Sound and Vibration 171, 335–359 (1994)MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Marwan, N., Romano, M.C., Thiel, M., Kurths, J.: Recurrence plots for the analysis of complex systems. Physics Reports 438, 237–329 (2007)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Nagasaka, I., Ishida, Y., Ishii, T., Okada, T., Koyoma, T.: Vibration suppresion of helicopter blades by pendulum absorbers. Analytical and experimental investigations in case of rigid -body mode. Transactions of the Japan Society of Mechanical Engineers 73(725), 129–137 (2007)Google Scholar
  24. 24.
    Nayfeh, A.H., Zavodney, L.D.: Experimental observation of amplitude and phase modulated responses of two internally coupled oscillators to a harmonic excitation. Journal of Applied Mechanics 10, 706–710 (1988)CrossRefGoogle Scholar
  25. 25.
    Sado, D.: Energy Transfer in Nonlinearly Coupled Systems with Two Degrees of Freedom (Przenoszenie energii w nieliniowo sprzonych ukadach o dwch stopniach swobody). In: Oficyna Wydawnicza Politechniki Warszawskiej, Prace Naukowe, Mechanika, Warszawa, p. 166 (1997)Google Scholar
  26. 26.
    Sanjuan, M.A.F.: The effect of nonlinear damping on the universal escape oscillator. International Journal of Bifurcation and Chaos 9, 735–744 (1999)zbMATHCrossRefGoogle Scholar
  27. 27.
    Sapinski, B., Snanima, J., Maslanka, M., Ros, M.: Factility for testing magneto-rheological damping system for cable vibrations. Mechanics 25(3), 135–142 (2006)Google Scholar
  28. 28.
    Song, Y., Sato, H., Iwata, Y., Komatsuzaki, T.: The response of a dynamic vibration absorber system with a parametrically excited pendulum. Journal of Sound and Vibration 259, 747–759 (2003)CrossRefGoogle Scholar
  29. 29.
    Spencer, B.F., Sain, M.K.: Controlling Buildings: A new frontier in feedback. Special Issue of the IEEE Control Systems Magazine on Emerging Technology 17(6), 19–35 (1997)CrossRefGoogle Scholar
  30. 30.
    Stephenson, A.: On a new type of dynamical stability. Manchester Memoirs 8, 1–10 (1908)Google Scholar
  31. 31.
    Tang, D., Gavin, H., Dwell, E.: Study of airfoil gust response alleviation using on electro-magnetic dry friction damper. Part1: Theory. Journal of Sound and Vibration 269, 853–874 Google Scholar
  32. 32.
    Tel, T., Grulz, M.: Chaotic Dynamics. An introduction based on classical mechanics. Cambridge University Press, Cambridge (2006)zbMATHGoogle Scholar
  33. 33.
    Thomas, K.I., Ambika, G.: Occurrence of stable modes in a pendulum with cubic damping. Pramana-Journal of Physics 59, 445–456 (2002)CrossRefGoogle Scholar
  34. 34.
    Tondl, A., Ruijgork, T., Verhulst, F., Nabergoj, R.: Autoparametric resonance in mechanical system. Cambridge University Press, New York (2000)Google Scholar
  35. 35.
    Vyas, A., Bajaj, K.: Dynamics of Autoparametric Vibration Absorbers Using Multiple Pendulums. Journal od Sound and Vibration 246, 115–135Google Scholar
  36. 36.
    Warminski, J., Kecik, K.: Autoparametric Vibration of a nonlinear systems with pendulum. Mathematical Problems in Engineering. Article ID 80705 (2005)Google Scholar
  37. 37.
    Warminski, J., Kecik, K.: Instabilities in the main parametric resonance area of mechanical system with a pendulum. Journal of Sound Vibration 332, 612–628 (2009)CrossRefGoogle Scholar
  38. 38.
    Warminski, J., Kecik, K.: Regular and chaotic motions of an autoparametric real pendulum system with the use of a MR damper. In: Modeling, Simulation and Control of Nonlinear Engineering Dynamical Systems, pp. 267–276. Springer, Heidelberg (2009)Google Scholar
  39. 39.
    Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining Lyapunov exponent from a time series. Physica D 16, 285–317 (1985)MathSciNetzbMATHCrossRefGoogle Scholar
  40. 40.
    Xu, X., Wiercigroch, M., Cartmell, M.P.: Rotating orbits of a parametrically excited pendulum. Chaos, Solitons and Fractals 23(5), 1537–1548 (2005)zbMATHGoogle Scholar
  41. 41.
    Xu, X., Wiercigroch, M.: Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dynamics 47, 311–320 (2006)MathSciNetCrossRefGoogle Scholar
  42. 42.
    Yabuno, K., Endo, Y., Aoshima, N.: Stabilization of 1/3-order subharmonic resonance using an autoparametric vibration absorber. Journal of Vibration and Acoustics 121, 309–315 (1999)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Jerzy Warminski
    • 1
  • Krzysztof Kecik
    • 1
  1. 1.Department of Applied MechanicsLublin University of TechnologyLublinPoland

Personalised recommendations