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ISSN:2164-6376 (print)
ISSN:2164-6414 (online)
Discontinuity, Nonlinearity, and Complexity

Dimitry Volchenkov (editor), Dumitru Baleanu (editor)

Dimitry Volchenkov(editor)

Mathematics & Statistics, Texas Tech University, 1108 Memorial Circle, Lubbock, TX 79409, USA

Email: dr.volchenkov@gmail.com

Dumitru Baleanu (editor)

Cankaya University, Ankara, Turkey; Institute of Space Sciences, Magurele-Bucharest, Romania

Email: dumitru.baleanu@gmail.com

Period-3 Motions to Chaos in an Inverted Pendulum with a Periodic Base Movement

Discontinuity, Nonlinearity, and Complexity 10(4) (2021) 663--680 | DOI:10.5890/DNC.2021.12.007

Chuan Guo, Albert C.J. Luo

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

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Abstract

A bifurcation tree of period-3 motions to chaos in an inverted pendulum with a periodic base movement is presented through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are achieved semi-analytically, and the corresponding stability and bifurcation of the periodic motions are also carried out. Frequency-amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Numerical simulations of periodic motions on the bifurcation are completed. The numerical and analytical results are presented for comparison. Except for period-1 motion to chaos studied before, this study focuses on other periodic motions to chaos existing in the inverted pendulum with periodic base movement. In the earthquake testing, one tests structures from 1hz to 33 hz. However, during such a frequency range, one not only can observe the period-1 motion to chaos, but one can observe period-3 motion to chaos in such an inverted pendulum with a periodic base movement. Thus, in the building design, period-3 motions to chaos should be considered, which have different dynamical behaviors from the period-1 motions to chaos.

References

  1. [1]  Guo, C. and Luo, A.C.J. (2020), Periodic motions on bifurcation trees in an inverted pendulum with a periodically moving base, International Journal of Dynamics and Control, {DOI}:10.1007/s40435-020-00647-6.
  2. [2]  Stephenson, A. (1908), On induced stability, Philosophical Magazine, 15, 233--237.
  3. [3]  Landau, L.D. and Lifshitz, E.M. (1960), Mechanics, Oxford: Pergamon Press.
  4. [4]  Hsu, C.S. (1975), Limit cycle oscillations of parametrically excited second-order nonlinear systems, ASME Journal of Applied Mechanics, 42, 176--182.
  5. [5]  Clifford, M.J. and Bishop, S.R. (1994), Approximating the escape zone for the parametrically excited pendulum Journal of Sound and Vibration, 174, 572--576.
  6. [6]  Capecchi, D. and Bishop, S.R. (1994), Periodic oscillations and attracting basins for a parametrically excited pendulum, International Journal for the Dynamics and Stability of Systems, 9, 123--143.
  7. [7]  Broer, H.W., Hoveijin, I., and van Noort, M. (1998), A reversible bifurcation analysis of the inverted pendulum, Physica D: Nonlinear Phenomena, 112, 50-63.
  8. [8]  Lenci, S. and Rega, G. (2000), Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation, Chaos, Solutions {$\&$ Fractals}, 11(15), 2453-2472.
  9. [9]  Lagrange, J.L. (1788), Mecanique Analytique, 2, (edition Albert Balnchard: Paris, 1965).
  10. [10]  Poincare, H. (1899), Methodes Nouvelled de la Mecanique Celeste, 3, Gauthier-Villars, Paris.
  11. [11]  van der Pol, B. (1920) A theory of the amplitude of free and forced triode vibrations, Radio Review, 1, 701-710, 754-762.
  12. [12]  Krylov, N.M. and Bogolyubov, N.N. (1935), Methodes approchees de la mecanique non-lineaire dans leurs application a lAeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance sy rapportant, Kiev: Academie des Sciences dUkraine (in French).
  13. [13]  Hayashi, C. (1964), Nonlinear oscillations in Physical Systems, McGraw-Hill Book Company, New York.
  14. [14]  Barkham, P.G.D. and Soudack, A.C. (1969), An extension to the method of Krylov and Bogoliubov, International Journal of Control, 10, 377-392.
  15. [15]  Garcia-Margallo, J. and Bejarano, J.D. (1987), A generalization of the method of harmonic balance, Journal of Sound and Vibration, 116, 591-595.
  16. [16]  Coppola, V.T. and Rand, R.H. (1990), Averaging using elliptic functions: Approximation of limit cycle, Acta Mechanica, 81, 125-142.
  17. [17]  Maaita, J.O. (2016), A theorem on the bifurcations of the slow invariant manifold of a system of two linear oscillators coupled to a k-order nonlinear oscillator, Journal of Applied Non-linear Dynamics, 5(2), 193--197.
  18. [18]  Yamgou\{e}, S.B., Nana, B., and Pelap, F.B. (2017), Approximate analytical solutions of a nonlinear oscil-lator equation modeling a constrained mechanical system, Journal of Applied Nonlinear Dynamics, 6(1), 17--26.
  19. [19]  Shayak, B. and Vyas. P. (2017), Krylov Bogoliubov type analysis of variants of the Mathieu equation, Journal of Applied Nonlinear Dynamics, 6(1), 57--77.
  20. [20]  Rajamani, S. and Rajasekar. S. (2017), Variation of response amplitude in parametrically driven sin-gle Duffing oscillator and unidirectionally coupled Duffing oscillators, Journal of Applied Nonlinear Dynamics, 6(1), 121--129.
  21. [21]  Luo, A.C.J. (2012), Continuous Dynamical Systems, HEP/L&H Scientific, Beijing/Glen Carbon.
  22. [22]  Luo, A.C.J. (2014), On analytical routes to chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 24 (4), 1430013 (28 pages).
  23. [23]  Luo, A.C.J. (2014), Toward Analytical Chaos in Nonlinear Dynamical Systems, Wiley.
  24. [24]  Luo, A.C.J. (2014), Analytical Routes to Chaos in Nonlinear Engineering. Wiley.
  25. [25]  Luo, A.C.J. and Huang, J.Z. (2012), Analytical dynamics of period-$m$ flows and chaos in nonlinear systems, International Journal of Bifurcation and Chaos, 22, 1-29.
  26. [26]  Luo, A.C.J. and Yu, B. (2013), Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator, Journal of Vibration and Control, 21(5), 907-918.
  27. [27]  Luo, A.C.J. and Yu, B. (2015), Bifurcation trees of period-1 motions to chaos in a two-degree-of-freedom, nonlinear oscillator, International Journal of Bifurcation and Chaos, 25(13), Article No:1550179 (40 pages).
  28. [28]  Yu, B. and Luo, A.C.J. (2017), Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring, International Journal of Dynamics and Control, 5(3), 436--453.
  29. [29]  Luo, A.C.J. (2013), Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay, International Journal of Dynamics and Control, 1, 330-359.
  30. [30]  Luo, A.C.J. and Jin, H.X. (2014), Bifurcation trees of period-m motion to chaos in a Time-delayed, quadratic nonlinear oscillator under a periodic excitation, Discontinuity, Nonlinearity, and Complexity, 3, 87-107.
  31. [31]  Wang, Y.F. and Liu, Z.W. (2015), A matrix-based computational scheme of generalized harmonic balance method for periodic solutions of nonlinear vibratory systems, Journal of Applied Nonlinear Dynamics, 4(4), 379-389.
  32. [32]  Luo, A.C.J. (2015), Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems, International Journal of Bifurcation and Chaos, 25(3), Article No.1550044.
  33. [33]  Guo, Y. and Luo, A.C.J. (2016), Routes of periodic motions to chaos in a periodically forced pendulum, International Journal of Dynamics and Control, 5(3), 551-569.
  34. [34]  Luo, A.C.J. and Xing, S.Y. (2017), Time-delay effects on periodic motions in a periodically forced, time-delayed, hardening duffing oscillator, Journal of Vibration Testing and Systems Dynamics, 1(1), 73-91.
  35. [35]  Xing, S.Y. and Luo, A.C.J. (2017), Towards infinite bifurcation trees of period-1 motions to chaos in a time-delayed, twin-well Duffing oscillator, Journal of Vibration Testing and System Dynamics, 1(4), 353-392.
  36. [36]  Xing, S.Y. and Luo, A.C.J. (2018), On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator, International Journal of Dynamics and Control, 6(4), 1429--1464.
  37. [37]  Xu, Y.Y. and Luo, A.C.J. (2019), A series of symmetric period-1 motions to chaos in a two-degree-of-freedom van der Pol-Duffing oscillator, Journal of Vibration Testing and System Dynamics, 2(2), 119-153.
  38. [38]  Luo, A.C.J. and Guo, C. (2019), A period-1 motion to chaos in a periodically forced, damped, double-pendulum, Journal of Vibration Testing and System Dynamics, 3(3), 259-280.

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