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Stochastic synchronization of rotating parametric pendulums

  • Nonlinear Dynamics and Control of Composites for Smart Engi design
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Abstract

In this paper synchronization of two pendulums mounted on a mutual elastic single degree-of-freedom base is examined. The response of the pendulums is considered when their base is externally excited by a random phase sinusoidal force, thus leading to stochastic parametric excitation of the pendulums. The target is for the pendulums to establish and preserve rotary response since this study is motivated by a recently proposed ocean wave energy extraction concept where the heaving motion of waves excites a pendulum’s hinge point. Since the wave bobbing motion is random the system’s excitation is modelled as a narrow-band stochastic process. Mounting two pendulums on the same elastic base creates a coupling between them through their interaction with the base, providing a path for energy exchange between them. The dynamic response of the pendulums is numerically investigated with respect to establishment of rotations as well as identification of synchronization with the pendulums characteristics spanning along non-identical parameters.

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References

  1. Nayfeh AH, Mook DT (1979) Nonlinear oscillations. Wiley, New York

    MATH  Google Scholar 

  2. Clifford MJ, Bishop SR (1996) Locating oscillatory orbits of the parametrically-excited pendulum. J Aust Math Soc Ser B Appl Math 37:309–319

    Article  MATH  MathSciNet  Google Scholar 

  3. Bishop SR, Garira W (2003) Oscillatory orbits of the parametrically excited pendulum. Int J Bifurcat Chaos 13(10):2949–2958

    Article  MATH  MathSciNet  Google Scholar 

  4. Bishop SR, Sofroniou A, Shi P (2005) Symmetry-breaking in the response of the parametrically excited pendulum. Chaos Solitons Fractals 25:257–264

    Article  ADS  MATH  Google Scholar 

  5. Szemplinska-Stupnicka W, Tyrkiel E (2002) The oscillation-rotation attractors in the forced pendulum and their peculiar properties. Int J Bifurcat Chaos 12(01):159–168

    Article  MATH  MathSciNet  Google Scholar 

  6. Koch BP, Leven RW (1985) Subharmonic and homoclinic bifurcations in a parametrically forced pendulum. Physica 16D:1–13

    ADS  MathSciNet  Google Scholar 

  7. Clifford MJ, Bishop SR (1994) Approximating the escape zone for the parametrically excited pendulum. J Sound Vib 172(4):572–576

    Article  ADS  MATH  MathSciNet  Google Scholar 

  8. Xu X, Wiercigroch M (2007) Approximate analytical solutions for oscillatory and rotational motion of a parametric pendulum. Nonlinear Dyn 47:311–320

    Article  MATH  MathSciNet  Google Scholar 

  9. Clifford MJ, Bishop SR (1995) Rotating periodic orbits of the parametrically excited pendulum. Phys Lett 201A:191–196

    Article  ADS  MathSciNet  Google Scholar 

  10. Garira W, Bishop SR (2003) Rotating solutions of the parametrically excited pendulum. J Sound Vib 263:233–239

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Xu X, Wiercigroch M, Cartmell MP (2005) Rotating orbits of a parametrically-excited pendulum. Chaos Solitons Fractals 23:1537–1548

    Article  ADS  MATH  Google Scholar 

  12. Lenci S, Pavlovskaia E, Rega G, Wiercigroch M (2008) Rotating solutions and stability of parametric pendulum by perturbation method. J Sound Vib 310:243–259

    Article  ADS  Google Scholar 

  13. Leven RW, Koch BP (1981) Chaotic behaviour of a parametrically exited damped pendulum. Phys Lett A 86:71–74

    Article  ADS  Google Scholar 

  14. De Paula AS, Savi MA, Pavlovskaia E, Wiercigroch M (2012) Bifurcation control of a parametric pendulum. Int J Bifurcat Chaos 22(5):1250111

    Article  Google Scholar 

  15. Yao M, Zhang W (2014) Multi-pulse chaotic motions of high-dimension nonlinear system for a laminated composite piezoelectric rectangular plate. Meccanica 49(2):365–392

    Article  MathSciNet  Google Scholar 

  16. Amer Y, Hegazy U (2012) Chaotic vibration and resonance phenomena in a parametrically excited string-beam coupled system. Meccanica 47(4):969–984

    Article  MATH  MathSciNet  Google Scholar 

  17. Belyakov AO (2011) On rotational solutions for elliptically excited pendulum. Phys lett A 375:2524–2530

    Article  ADS  MATH  MathSciNet  Google Scholar 

  18. Pavlovskaia E, Horton B, Wiercigroch M, Lenci S, Rega G (2011) Approximate rotational solutions of pendulum under combined vertical and horizontal excitation. Int J Bifurcat Chaos 22(5):1250100

    Article  Google Scholar 

  19. Horton B, Sieber J, Thompson JMT, Wiercigroch M (2011) Dynamics of the nearly parametric pendulum. Int J Non Linear Mech 46:436–442

    Article  Google Scholar 

  20. Yabuno H, Miura M, Aoshima N (2004) Bifurcation in an inverted pendulum with tilted high-frequency excitation: analytical and experimental investigations on the symmetry-breaking of the bifurcation. J Sound Vib 273:493–513

    Article  ADS  Google Scholar 

  21. Sartorelli C, Lacarbonara W (2012) Parametric resonances in a base-excited double pendulum. Nonlinear Dyn 69:1679–1692

    Article  MathSciNet  Google Scholar 

  22. Rivas-Cambero I, Sausedo-Solorio J (2012) Dynamics of the shift in resonance frequency in a triple pendulum. Meccanica 47(4):835–844

    Article  MATH  MathSciNet  Google Scholar 

  23. Leven RW, Pompe B, Wilke C, Koch BP (1985) Experiments on periodic and chaotic motions of a parametrically forced pendulum. Physica D 16(3):371–384

    Article  MATH  MathSciNet  Google Scholar 

  24. Xu X, Pavlovskaia E, Wiercigroch M, Romeo F, Lenci S (2007) Dynamic interactions between parametric pendulum and electro-dynamical shaker. ZAMM 87:172–186

    Article  MATH  MathSciNet  Google Scholar 

  25. Lenci S, Brocchini M, Lorenzoni C (2012) Experimental rotations of a pendulum on water waves. J Comput Nonlinear Dyn 7(1):011007

    Article  Google Scholar 

  26. Kozi P, Pavlovi R, Janevski G, Golubovi Z (2010) Influence of the mode number on the stochastic stability regions of the elastic beam. Meccanica 45(4):553–565

    Article  MathSciNet  Google Scholar 

  27. Yurchenko D, Naess A, Alevras P (2013) Pendulum’s rotational motion governed by a stochastic Mathieu equation. Probab Eng Mech 31:12–18

    Article  Google Scholar 

  28. Alevras P, Yurchenko D (2013) Stochastic rotational response of a parametric pendulum coupled with an SDOF system. Probab Eng Mech. doi:10.1016/j.probengmech.2013.10.008

  29. Yurchenko D, Alevras P (2013) Stochastic dynamics of a parametrically base excited rotating pendulum. Procedia IUTAM 6:160–168

    Article  Google Scholar 

  30. Kozubovskaya IG, Khrisanov SM (1982) Random parametric resonance. Ukr Math J 34(4):350–357

    Article  MathSciNet  Google Scholar 

  31. Wedig WV (1990) Invariant measures and Lyapunov exponents for generalised parameter fluctuations. Struct Saf 8:13–25

    Article  Google Scholar 

  32. Pierson WJ, Moskowitz L (1964) A proposed spectral form for fully developed wind seas based on the similarity theory of A. Kitaigorodskii. J Geophys Res 69:5181–5190

    Article  ADS  Google Scholar 

  33. Kecik K, Mitura A, Sado D, Warminski J (2014) Magnetorheological damping and semi-active control of an autoparametric vibration absorber. Meccanica. doi:10.1007/s11012-014-9892-2

    Google Scholar 

  34. Blekhman II (1988) Synchronization in science and technology. ASME press, New York

    Google Scholar 

  35. Pecora LM, Carroll TL (1990) Synchronization in chaotic systems. Phys Rev Lett 64:821–824

    Article  ADS  MATH  MathSciNet  Google Scholar 

  36. Boccaleti S, Kurths J, Osipov G, Valladares DL, Zhou CS (2002) The synchronization of chaotic systems. Phys Rep 366:1–101

    Article  ADS  MathSciNet  Google Scholar 

  37. Rosenblum M, Pikovsky A (2003) Synchronization: from pendulum clocks to chaotic lasers and chemical oscillators. Contemp Phys 44:401–416

    Article  ADS  Google Scholar 

  38. Marcheggiani L, Lenci S (2010) On a model for the pedestrians-induced lateral vibrations of footbridges. Meccanica 45(4):531–551

    Article  MATH  Google Scholar 

  39. Kapitaniak M, Czolczynski K, Perlikowski P, Stefanski A, Kapitaniak T (2014) Synchronous states of slowly rotating pendula. Phys Rep. doi:10.1016/j.physrep.2014.02.008

  40. Czolzynski K, Perlikowski P, Stefanski A, Kapitaniak T (2012) Synchronization of slowly rotating pendulums. Int J Bifurcat Chaos 22(05):1250128

    Article  Google Scholar 

  41. Strzalko J, Grabski J, Wojewoda J, Wiercigroch M, Kapitaniak T (2012) Synchronous rotation of the set of double pendula: experimental observations. Chaos 22(4):047503

    Article  ADS  Google Scholar 

  42. Lei Y, Xu W, Shen J, Fang T (2006) Global synchronization of two parametrically excited systems using active control. Chaos Solitons Fractals 28(2):428–436

    Article  ADS  MATH  MathSciNet  Google Scholar 

  43. Stratonovich RL (1963) Topics in the theory of random noise, vol 2. Gordon and Breach, New York

    Google Scholar 

  44. Neiman A, Silchenko A, Anishchenko V, Schimansky-Geier L (1998) Stochastic resonance: noise-enhanced phase coherence. Phys Rev E 58:7118–7125

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Institute of Mechanical, Process and Energy Engineering, Heriot-Watt University, Edinburgh, EH14 4AS, UK

    Panagiotis Alevras & Daniil Yurchenko

  2. Department of Mathematical Sciences, NTNU, 7491, Trondheim, Norway

    Arvid Naess

Authors
  1. Panagiotis Alevras
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  2. Daniil Yurchenko
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  3. Arvid Naess
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Correspondence to Panagiotis Alevras.

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The second author is also known as Iourtchenko.

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Alevras, P., Yurchenko, D. & Naess, A. Stochastic synchronization of rotating parametric pendulums. Meccanica 49, 1945–1954 (2014). https://doi.org/10.1007/s11012-014-9955-4

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  • DOI: https://doi.org/10.1007/s11012-014-9955-4

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