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ISSN:2164-6457 (print)
ISSN:2164-6473 (online)
Journal of Applied Nonlinear Dynamics
Miguel A. F. Sanjuan (editor), Albert C.J. Luo (editor)
Miguel A. F. Sanjuan (editor)

Department of Physics, Universidad Rey Juan Carlos, 28933 Mostoles, Madrid, Spain

Email: miguel.sanjuan@urjc.es

Albert C.J. Luo (editor)

Department of Mechanical and Industrial Engineering, Southern Illinois University Ed-wardsville, IL 62026-1805, USA

Fax: +1 618 650 2555 Email: aluo@siue.edu

Stability Boundaries of Period-1 Rotation for a Pendulum Under Combined Vertical and Horizontal Excitation

Journal of Applied Nonlinear Dynamics 2(2) (2013) 103--126 | DOI:10.5890/JAND.2013.04.001

B. Horton$^{1}$, S. Lenci$^{2}$, E. Pavlovskaia$^{1}$, F. Romeo$^{3}$, G. Rega$^{3}$, M. Wiercigroch$^{1}$

$^{1}$ Centre for Applied Dynamics Research, School of Engineering, University of Aberdeen, Kings College, Aberdeen, AB24 3UE, UK

$^{2}$ Università Politecnica delle Marche, Dipartimento di Ingegneria Civile, Edile e Architettura (DICEA), Via Brecce Bianche, 60131 Ancona, Italy

$^{3}$ Sapienza Università di Roma, Dipartimento di Ingegneria Strutturale e Geotecnica (DISG), Via A. Gramsci 53, 00197 Roma, Italy

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Abstract

The aim of this work is to study the dynamics of pendulum driven through its pivot moving in both horizontal and vertical directions. It expands the results obtained for the parametric pendulum by Lenci et al. to two other cases, i.e. the elliptically excited pendulum and the pendulum, with an inclined rectilinear base motion (the tilted pendulum). Here we derive approximate analytical expressions representing the position of the saddle-node bifurcation associated with period-1 rotations in the excitation amplitude/frequency plane in the presence of damping by using the perturbation method proposed by Lenci et al. This includes development of a procedure for deducing expressions for the period doubling, creating a pair of stable period-2 rotational attractors. The obtained approximations are plotted on the excitation parameters plane and compared with numerical results. Simple Padé approximations for the analytical expressions relating to the position of the saddle-node bifurcation are also obtained.

References

  1. [1]  Blekhman, I.I. (2000), Vibrational Mechanics, World Scientific, Singapore.
  2. [2]  Czolczynski, K., Perlikowski, P., Stefanski, A., and Kapitaniak, T. (2012), Synchronization of slowly rotating pendulums, International Journal of Bifurcation and Chaos, In press.
  3. [3]  Czolczynski, K., Perlikowski, P., Stefanski, A., and Kapitaniak, T. (2012), Synchronization of pendula rotating in dierent directions. Communications in Nonlinear Science and Numerical Simulation, Published online.
  4. [4]  Butikov, E. (1999), The rigid pendulum - an antique but evergreen physical model, European Journal of Physics , 20, 429-441.
  5. [5]  Baker, G.L., Blackburn, J.A., and Smith, H.J.T. (2002), The quantum pendulum: small and large, American Journal of Physics, 70(5), 525-531.
  6. [6]  Stewart, W.C. (1968), Current-Voltage characteristics of Josephson Junctions, Applied Physics Letters, 12(8), 277-280.
  7. [7]  Koch, B.P. and Leven, R.W. (1985), Subharmonic and homoclinic bifurcations in a parametrically forced pendulum, Physica D, 16, 1-13.
  8. [8]  Capecchi, D. and Bishop, S.R. (1994), Periodic oscillations and attracting basins for a parametrically excited pendulum, Dynamics and Stability of Systems, 9(2), 123-143.
  9. [9]  Clifford, M.J. and Bishop, S.R. (1995), Rotating periodic orbits of the parametrically excited pendulum, Physics Letters A, 201, 191-196.
  10. [10]  Garira, W. and Bishop, S.R. (2003), Rotating solutions of the parametrically excited pendulum, Journal of Sound and Vibration, 263, 233-239.
  11. [11]  Szemplinska-Stupnicka, W., Tyrkiel, E., and Zubrzycki, A. (2000), The global bifurcations that lead to transient tumbling chaos in a parametrically driven pendulum, International Journal of Bifurcation and Chaos, 10(9), 2161-2175.
  12. [12]  Szemplinska-Stupnicka, W. and Tyrkiel, E. (2002), The oscillation-rotation attractors in the forced pendulum and their peculiar properties, International Journal of Bifurcation and Chaos, 12, 159-168.
  13. [13]  Xu, X., Wiercigroch, M. and Cartmell, M.P. (2005), Rotating orbits of a parametrically-excited pendulum, Chaos, Solitons and Fractals, 23(5), 1537-1548.
  14. [14]  Xu, X. andWiercigroch, M. (2007), Approximate analytical solutions for oscillatory and rotationalmotion of a parametric pendulum, Nonlinear Dynamics, 47(1-3), 311-320.
  15. [15]  Xu, X., Pavlovskaia, E., Wiercigroch, M., Romeo, F., and Lenci, S. (2007), Dynamic interactions between parametric pendulum and electro-dynamical shaker, ZAMM, 87(2), 172-186.
  16. [16]  Lenci, S., Pavlovskaia, E., Rega, G., and Wiercigroch, M. (2008), Rotating solutions and stability of parametric pendulum by perturbation method, Journal of Sound and Vibration, 310(1-2), 243-259.
  17. [17]  de Paula, A.S., Savi, M.A., Pavlovskaia, E., and Wiercigroch, M. (2012), Bifurcation control of a parametric pendulum, International Journal of Bifurcation and Chaos, 22(5), 1250111 (14 pages).
  18. [18]  Lenci, S. and Rega, G. (2008), Competing dynamic solutions in a parametrically excited pendulum: attractor robustness and basin integrity, ASME Journal of Computational and Nonlinear Dynamics , 3(4), 41010.
  19. [19]  Schmitt, J.M. and Bayly, P.V. (1998), Bifurcations in the mean angle of a Horizontal Shaken Pendulum: Analysis and Experiment, Nonlinear Dynamics, 15, 1-14.
  20. [20]  Van Dooren, R. (1996), Chaos in a pendulum with forced horizontal support motion: a tutorial, Chaos, Solitons and Fractals, 7(1), 77-90.
  21. [21]  Kholostova, O.V. (1995), Some problems of the motion of a pendulum when there are horizontal vibrations of the point of suspension, Journal of Applied Mathematics and Mechanics, 59(4), 553-561.
  22. [22]  Ge, Z.-M. and Lin, T.-N. (2000), Regular and chaotic dynamic analysis and control of chaos of an elliptical pendulum on a vibrating basement, Journal of Sound and Vibration, 230(5), 1045-1068.
  23. [23]  El-Barki, F.A., Ismail, A.I., Shaker, M.O., and Amer, T.S. (1999), On the motion of the Pendulum on an Ellipse, Journal of Applied Mathematics and Mechanics, 79(1), 65-72.
  24. [24]  Mann, B.P. and Koplow, M.A. (2006), Symmetry breaking bifurcations of a parametrically excited pendulum, Nonlinear Dynamics, 46, 427-437.
  25. [25]  Sudor, D. and Bishop, S.R. (1999), Inverted dynamics of a tilted pendulum, European Journal of Mechanics A: Solids, 18, 517-526.
  26. [26]  Yabuno, H., Miura, M., and Aoshima, N. (2004), Bifurcation in an inverted pendulum with tilted highfrequency excitation: analytical and experimental investigations on the symmetry-breaking of the bifurcation, Journal of Sound and Vibration, 273, 479-513.
  27. [27]  Horton, B., Sieber, J., Thompson, J.M.T., andWiercigroch,M. (2011), Dynamics of the nearly parametric pendulum, International Journal of Non-Linear Mechanics, 46(2), 436-442.
  28. [28]  Pavlovskaia, E., Horton, B., Lenci, S., Rega, G., and Wiercigroch, M. (2012), Approximate rotating solutions of pendulum under combined vertical and horizontal excitation at pivot point, International Journal of Bifurcation and Chaos, 22(5), 1250100 (13 pages).
  29. [29]  Belyakov, A.O. (2011), On rotational solutions for elliptically excited pendulum, Physics Letters A, 375(25), 2524-2530.
  30. [30]  Horton, B. (2008), Rotational motion of pendula systems for wave energy extraction, PhD Thesis, Aberdeen University.
  31. [31]  Moon, F.C. (1987), Chaotic Vibrations, John Wiley, New York.
  32. [32]  Clifford, M.J. and Bishop, S.R. (1994), Approximating the escape zone for the parametrically excited pendulum, Journal of Sound and Vibration, 172(4), 572-576.
  33. [33]  Baker, G.A. Jr. (1965), The Theory and Application of The Pade Approximant Method, In: Advances in Theoretical Physics, 1, 1-58, Academic Press, New York.

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