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Experimental and numerical investigation of chaotic regions in the triple physical pendulum

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Abstract

The experimental and numerical analysis of triple physical pendulum is performed. The experimental setup of the triple pendulum with the first body externally excited by the square function and the widely used LabView measure-programming system, which is designed especially for measure data processing and acquisition, are described. The mathematical model of the system is then introduced. The parameters of the model are estimated by minimization of the sum of squares of deviations between the signal from the simulation and the signal from the experiment. A good agreement between results from experiment and from simulation is shown in few examples, including periodic as well as chaotic solutions.

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References

  1. Banning, E.J., van der Weele, J.P.: Mode competition in a system of two parametrically driven pendulums: the Hamiltonian case. Physica A220, 485–533 (1995)

    Google Scholar 

  2. Beckert, S., Schock, U., Schulz, C.D., Weidlich, T., Kaiser, F.: Experiments on the bifurcation behavior of a forced nonlinear pendulum. Phys. Lett. A107, 347–350 (1987)

    MathSciNet  Google Scholar 

  3. Blackburn, J.A., Yang, Z.J., Vik, S.: Experimental study of chaos in a driven pendulum. Physica D26, 385–395 (1987)

    Google Scholar 

  4. Heng, H., Doerner, R., Hubinger, B., Martienssen, W.: Approaching nonlinear dynamics by studying the motion of a pendulum. I. Observing trajectories in state space. Int. J. Bif. Chaos 4(4), 751–760 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  5. Starett, J., Tagg, R.: Control of a chaotic parametrically driven pendulum. Phys. Rev. Lett. 74, 1974–1977 (1995)

    Article  Google Scholar 

  6. Piiroinen, D.T., Virgin, L.N., Champneys, A.R.: Chaos and period-adding: experimental and numerical verification of the grazing bifurcation. J. Nonlinear Sci. 14, 384–404 (2004)

    Article  MathSciNet  Google Scholar 

  7. Zhu, Q., Ishitobi, M.: Experimental study of chaos in a driven triple pendulum. J. Sound Vib. 227(1), 230–238 (1999)

    Article  Google Scholar 

  8. Awtar, S., King, N., Allen, T., Bang, I., Hogan, M., Skidmore, D., Craig, K.: Inverted pendulum systems: rotary and arm-driven — a mechatronic system design case study. Mechatronics 12, 357–370 (2002)

    Article  Google Scholar 

  9. Yoshida, K., Sato, K.: Characterization of reverse rotation in chaotic response of mechanical pendulum. Int. J. Non-Linear Mech. 33(5), 819–828 (1998)

    Article  MATH  Google Scholar 

  10. Galan, J., Fraser, W.B., Acheson, D.J., Champneys, A.R.: The parametrically excited upside-down rod: an elastic jointed pendulum model. J. Sound Vib. 280, 359–377 (2005)

    Article  Google Scholar 

  11. Acheson, D.J.: A pendulum theorem. Proc. R. Soc. London A443, 239–245 (1993)

    Google Scholar 

  12. Braun, M.: On some properties of the multiple pendulum. Arch. Appl. Mech. 72, 899–910 (2003)

    MATH  Google Scholar 

  13. Boltežar, M., Hammond, J.K.: Experimental study of the vibrational behavior of a coupled non-linear mechanical system. Mech. Syst. Signal Process. 13(3), 375–394 (1999)

    Article  Google Scholar 

  14. Furata, T., Tawara, T., Okumura, Y., Shimizu, M., Tomiyama, K.: Design and construction of a series of compact humanoid robots and development of biped walk control strategies. Robotics Autonom. Syst. 37, 81–100 (2001)

    Article  Google Scholar 

  15. Selles, R.W., Bussmann, J.B.J., Wagenaar, R.C., Stam, H.J.: Comparing predictive validity of four ballistic swing phase models of human walking. J. Biomech. 34, 1171–1177 (2001)

    Article  Google Scholar 

  16. Donelan, J.M., Kram, R., Kuo, A.D.: Simultaneous positive and negative external mechanical work in human walking. J. Biomech. 35, 117–124 (2002)

    Article  Google Scholar 

  17. Miyashita, T., Ishiguro, H.: Human-like behavior generation based on involuntary motions for humanoid robots. Robotics Autonom. Syst. 48, 203–212 (2004)

    Article  Google Scholar 

  18. Awrejcewicz, J., Kudra, G.: The piston-connecting rod-crankshaft system as a triple physical pendulum with impacts. Int. J. Bif. Chaos 15(7), 2207–2226 (2005)

    Article  Google Scholar 

  19. Sygniewicz, J.: Modeling of Cooperation of the Piston with Piston Rings and Barrel (in Polish). Sci. Bull., 615/149, Technical University of Łódź (1991)

  20. Kudra, G.: Analysis of Bifurcations and Chaos in Triple Physical Pendulum with Impacts (in Polish). PhD thesis, Technical University of Łódź (2002)

  21. Awrejcewicz, J., Kudra, G., Lamarque, C.-H.: Dynamics investigation of three coupled rods with a horizontal barrier. Spec. Issue Meccanica 38(6), 687–698 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Awrejcewicz, J., Kudra, G., Lamarque, C.-H.: Investigation of triple pendulum with impacts using fundamental solution matrices. Int. J. Bif. Chaos 14(12), 4191–4213 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Wolf, A., Swift, J.B., Swinney, H.L., Vastano, J.A.: Determining, Lyapunov exponents from a time series. Physica D16, 285–317 (1985)

    MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Automatics and Biomechanics, Technical University of Łódź, 1/15 Stefanowskiego, 90-924, Łódź, Poland

    Jan Awrejcewicz, Grzegorz Kudra & Grzegorz Wasilewski

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  1. Jan Awrejcewicz
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  2. Grzegorz Kudra
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  3. Grzegorz Wasilewski
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Correspondence to Jan Awrejcewicz.

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Awrejcewicz, J., Kudra, G. & Wasilewski, G. Experimental and numerical investigation of chaotic regions in the triple physical pendulum. Nonlinear Dyn 50, 755–766 (2007). https://doi.org/10.1007/s11071-007-9235-0

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  • DOI: https://doi.org/10.1007/s11071-007-9235-0

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