Skip to main content
SpringerLink
Log in

Periodic motions on bifurcation trees in an inverted pendulum with a periodically moving base

  • Published:
International Journal of Dynamics and Control Aims and scope Submit manuscript

Abstract

This paper studies periodic motions of an inverted pendulum with a periodically moving base. A bifurcation tree of period-1 to period-8 motions is predicted through a discrete implicit mapping method. The stable and unstable periodic motions on the bifurcation tree are obtained semi-analytically, and the corresponding stability and bifurcation of the periodic motions are determined by eigenvalue analysis. Numerical simulations are completed for illustration of motion complexity in the inverted pendulum with a periodically moving base. The numerical and analytical results are presented for comparison. Frequency–amplitude characteristics of the bifurcation tree are presented through the finite Fourier series analysis. Through such results, one can better understand the building vibration during earthquake. In addition, stabilization of the inverted pendulum can be based on the periodically moving base.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  1. Stephenson A (1908) On induced stability. Philos Mag 15:233–237

    Article  Google Scholar 

  2. Landau LD, Lifshitz EM (1960) Mechanics. Pergamon Press, Oxford

    MATH  Google Scholar 

  3. Hsu CS (1975) Limit cycle oscillations of parametrically excited second-order nonlinear systems. ASME J Appl Mech 42:176–182

    Article  Google Scholar 

  4. Clifford MJ, Bishop SR (1994) Approximating the escape zone for the parametrically excited pendulum. J Sound Vib 174:572–576

    Article  MathSciNet  Google Scholar 

  5. Capecchi D, Bishop SR (1994) Periodic oscillations and attracting basins for a parametrically excited pendulum. Int J Dyn Stab Syst 9:123–143

    MathSciNet  MATH  Google Scholar 

  6. Broer HW, Hoveijin I, van Noort M (1998) A reversible bifurcation analysis of the inverted pendulum. Phys D 112:50–63

    Article  MathSciNet  Google Scholar 

  7. Lenci S, Rega G (2000) Periodic solutions and bifurcations in an impact inverted pendulum under impulsive excitation. Chaos, Solut Fractals 11(15):2453–2472

    Article  MathSciNet  Google Scholar 

  8. Lagrange JL (1788) Mecanique analytique. In: Albert B (ed), Paris, 1965, vol 2

  9. Poincare H (1899) Methodes Nouvelled de la Mecanique Celeste, vol 3. Gauthier-Villars, Paris

    Google Scholar 

  10. van der Pol B (1920) A theory of the amplitude of free and forced triode vibrations. Radio Review 1(701–710):754–762

    Google Scholar 

  11. Krylov NM, Bogolyubov NN (1935) Methodes approchees de la mecanique non-lineaire dans leurs application a l’Aeetude de la perturbation des mouvements periodiques de divers phenomenes de resonance s’y rapportant. Academie des Sciences d’Ukraine, Kiev (in French)

    Google Scholar 

  12. Hayashi C (1964) Nonlinear oscillations in physical systems. McGraw-Hill Book Company, New York

    MATH  Google Scholar 

  13. Barkham PGD, Soudack AC (1969) An extension to the method of Krylov and Bogoliubov. Int J Control 10:377–392

    Article  Google Scholar 

  14. Garcia-Margallo J, Bejarano JD (1987) A generalization of the method of harmonic balance. J Sound Vib 116:591–595

    Article  MathSciNet  Google Scholar 

  15. Coppola VT, Rand RH (1990) Averaging using elliptic functions: approximation of limit cycle. Acta Mech 81:125–142

    Article  MathSciNet  Google Scholar 

  16. Luo ACJ (2012) Continuous dynamical systems. HEP/L&H Scientific, Beijing/Glen Carbon

    MATH  Google Scholar 

  17. Luo ACJ (2014) On analytical routes to chaos in nonlinear systems. Int J Bifurc Chaos 24(4):1430013

    Article  MathSciNet  Google Scholar 

  18. Luo ACJ (2014) Toward analytical chaos in nonlinear dynamical systems. Wiley, Hoboken

    Book  Google Scholar 

  19. Luo ACJ (2014) Analytical routes to chaos in nonlinear engineering. Wiley, Hoboken

    Book  Google Scholar 

  20. Luo ACJ, Huang JZ (2012) Analytical dynamics of period-m flows and chaos in nonlinear systems. Int J Bifurc Chaos 22, 1250093

    Article  MathSciNet  Google Scholar 

  21. Luo ACJ, Yu B (2013) Complex period-1 motions in a periodically forced, quadratic nonlinear oscillator. J Vib Control 21(5):907–918

    MathSciNet  Google Scholar 

  22. Luo ACJ, Yu B (2015) Bifurcation trees of period-1 motions to chaos in a two-degree-of-freedom, nonlinear oscillator. Int J Bifurc Chaos 25(13), 1550179

    Article  MathSciNet  Google Scholar 

  23. Yu B, Luo ACJ (2017) Analytical period-1 motions to chaos in a two-degree-of-freedom oscillator with a hardening nonlinear spring. Int J Dyn Control 5(3):436–453

    Article  MathSciNet  Google Scholar 

  24. Luo ACJ (2013) Analytical solutions for periodic motions to chaos in nonlinear systems with/without time-delay. Int J Dyn Control 1:330–359

    Article  Google Scholar 

  25. Luo ACJ, Jin HX (2014) Bifurcation trees of period-m motion to chaos in a Time-delayed, quadratic nonlinear oscillator under a periodic excitation. Discontin Nonlinearity Complex 3:87–107

    Article  Google Scholar 

  26. Luo ACJ (2015) Periodic flows to chaos based on discrete implicit mappings of continuous nonlinear systems. Int J Bifurc Chaos 25(3), 1550044

    Article  MathSciNet  Google Scholar 

  27. Guo Y, Luo ACJ (2016) Routes of periodic motions to chaos in a periodically forced pendulum. Int J Dyn Control 5(3):551–569

    Article  MathSciNet  Google Scholar 

  28. Luo ACJ, Xing SY (2016) Time-delay effects on periodic motions in a Duffing oscillator. In: Mark E, Macau EEN, Sanjuan MAF (eds) Chaotic, fractional, and complex dynamics: new insights and perspectives. Understanding complex systems. Springer, Cham

    Google Scholar 

  29. Xing SY, Luo ACJ (2017) Towards infinite bifurcation trees of period-1 motions to chaos in a time-delayed, twin-well Duffing oscillator. J Vib Test Syst Dyn 1(4):353–392

    Google Scholar 

  30. Xing SY, Luo ACJ (2018) On possible infinite bifurcation trees of period-3 motions to chaos in a time-delayed, twin-well Duffing oscillator. Int J Dyn Control 6(4):1429–1464

    Article  MathSciNet  Google Scholar 

  31. Xu XY, Luo ACJ (2019) A series of symmetric period-1 motions to chaos in a two-degree-of-freedom van der Pol-Duffing oscillator. J Vib Test Syst Dyn 2(2):119–153

    Google Scholar 

  32. Luo ACJ, Guo C (2019) A period-1 motion to chaos in a periodically forced, damped, double-pendulum. J Vib Test Syst Dyn 3(3):259–280

    Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Chuan Guo & Albert C. J. Luo

Authors
  1. Chuan Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Albert C. J. Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Albert C. J. Luo.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Guo, C., Luo, A.C.J. Periodic motions on bifurcation trees in an inverted pendulum with a periodically moving base. Int. J. Dynam. Control 9, 410–423 (2021). https://doi.org/10.1007/s40435-020-00647-6

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s40435-020-00647-6

Keywords

Search

Navigation