Skip to main content
SpringerLink
Log in

Motion analysis of magnetic spring pendulum

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In order to analyze the motion characteristics of the spring pendulum under the action of magnetic field force, the motion of the spring pendulum will be studied by applying a uniform magnetic field in the vertical direction. Firstly, a first-order approximate solution is given by studying the micro-vibration around its equilibrium point. And an approximate solution similar to the Foucault pendulum is also presented in the case of a soft spring with strong ductility. Then, according to the resonance conditions of mechanical vibration, the internal resonance phenomenon of magnetic spring pendulum is discovered, and then the conclusion that the energy of the system is cyclically transmitted between the three modes of breathing, oscillating and deflection is presented subsequently. Finally, the influence of magnetic field strength on the motion stability of the spring pendulum is explored, and not only the bifurcation phenomenon at its equilibrium point is found, but also the complex dynamic behavior including chaotic motion occurs.

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
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

Data Availability

All data generated or analyzed during this study are included in this published article.

References

  1. Zhang, P., Li, H.N., Tian, L., Zhang, Z.Q.: Seismic vibration control of transmission tower with a spring pendulum. WEE 32(01), 210–218 (2016)

    Google Scholar 

  2. Wang, Q., Li, H.N., Zhang, P.: Calculation model of impact vibration reducing system of spring pendulum. J. Shenyang Jianzhu U: Nat. Sci. Ed. 34(02), 222–228 (2018)

    Google Scholar 

  3. Vitt, A., Gorelik, G.: Oscillations of an elastic pendulum as an example of the oscillations of two parametrically coupled linear systems. Tech Phys. 3(2–3), 294–307 (1933)

    Google Scholar 

  4. Broucke, R., Baxa, P.A.: Periodic solutions of a spring-pendulum system. Celest. Mech. Dyn. Astr. 8(2), 261–267 (1973)

    Article  MATH  Google Scholar 

  5. Aldoshin, G.T., Yakovlev, S.P.: On swinging spring chaotic oscillations. In: AIP Conference Proceedings, vol. 1, p. 030001. AIP Publishing LLC, Melville (2018)

    Google Scholar 

  6. Awrejcewicz, J., Starosta, R., Sypniewska-Kamiska, G.: Stationary and transient resonant response of a spring pendulum. Procedia IUTAM 19, 201–208 (2016)

    Article  Google Scholar 

  7. Olsson, M.G.: Why does a mass on a spring sometimes misbehave. Am. J. Phys. 44(12), 1211–1212 (1976)

    Article  Google Scholar 

  8. Amer, T.S., Bek, M.A., Hamada, I.S.: On the motion of harmonically excited spring pendulum in elliptic path near resonances. Adv. Math. Phys. 2016, 1–15 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gonzalez-Buelga, A., Kyrychko, Y., Wagg, D.J.: Bifurcations on a spring-pendulum oscillator. PAMM 7(1), 1030303–1030304 (2007)

    Article  Google Scholar 

  10. Boeck, T., Sanjari, S.L., Becker, T.: Dynamics of a magnetic pendulum in the presence of an oscillating conducting plate. PAMM 20(1), e202000083 (2021)

    Article  Google Scholar 

  11. Kitio Kwuimy, C.A., Nataraj, C., Belhaq, M.: Chaos in a magnetic pendulum subjected to tilted excitation and parametric damping. Math. Probl. Eng. 10, 1239–1257 (2012)

    MathSciNet  MATH  Google Scholar 

  12. Mann, B.P.: Energy criterion for potential well escapes in a bistable magnetic pendulum. J. Sound Vib. 323(3–5), 864–876 (2009)

    Article  Google Scholar 

  13. Qin, B., Shang, H.L., Jiang, H.M.: Global dynamic behavior analysis of typical magnetic pendulum. Acta Phys. Sin. 70(18), 180501 (2021)

    Article  Google Scholar 

  14. Pili, U.B.: Modeling damped oscillations of a simple pendulum due to magnetic braking. Phys. Educ. 55(3), 035025 (2020)

    Article  Google Scholar 

  15. Goldstein, H., Poole, C., Safko, J.: Classical Mechanics, 3rd edn. Pearson, Upper Saddle River (2002)

    MATH  Google Scholar 

  16. Greiner, W.: Classical Mechanics: Systems of Particles and Hamiltonian Dynamics, 2nd edn., pp. 26–27. Springer, New York (2003)

    Google Scholar 

  17. Frenkel, D., Portugal, R.: Algebraic methods to compute Mathieu functions. J. Phys. A-Math. Gen. 34(17), 3541 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gutiérrez-Vega, J.C., Rodríguez-Dagnino, R.M., Meneses-Nava, M.A., Chávez-Cerda, S.: Mathieu functions, a visual approach. Am. J. Phys. 71(3), 233–242 (2003)

    Article  Google Scholar 

  19. Acar, G., Feeny, B.F.: Floquet-based analysis of general responses of the Mathieu equation. J. Vib. Acoust. 138(4), 041017.1-041017.9 (2016)

    Article  Google Scholar 

  20. Lawrence, R.: Applications of the Mathieu equation. Am. J. Phys. 64(1), 39–44 (1996)

    Article  MathSciNet  Google Scholar 

  21. Ramakrishnan, V., Feeny, B.F.: Resonances of a forced Mathieu equation with reference to wind turbine blades. J. Vib. Acoust. 134(6), 064501.1-064501.5 (2012)

    Article  Google Scholar 

  22. Schulz-DuBois, E.O.: Foucault pendulum experiment by kamerlingh onnes and degenerate perturbation theory. Am. J. Phys. 38(2), 173–188 (1970)

    Article  MathSciNet  Google Scholar 

  23. Opat, G.I.: The precession of a Foucault pendulum viewed as a beat phenomenon of a conical pendulum subject to a Coriolis force. Am. J. Phys. 59(9), 822–823 (1991)

  24. Pan, L.Y.: On rose curve and its application. C.A.S. 10, 236–238 (2008)

    Google Scholar 

  25. Xia, H.Y., Dai, J.P., Zhao, C.: The trajectory property analysis of inner cycloid. M.E.E.T. 46(01), 25–30 (2017)

    Google Scholar 

  26. Wolf, A., Swift, J.B., Swinney, H.L., et al.: Determining Lyapunov exponents from a time series. Phys. D. 16(3), 285–317 (1985)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Physical Science and Technology, Ningbo University, Ningbo, 315211, China

    Yong Meng

Authors
  1. Yong Meng
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yong Meng.

Ethics declarations

Conflict of interest

The author declares that he has no conflicts of interest to report regarding the present study.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Meng, Y. Motion analysis of magnetic spring pendulum. Nonlinear Dyn 111, 6111–6128 (2023). https://doi.org/10.1007/s11071-022-08171-2

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-022-08171-2

Keywords

Mathematics Subject Classification

Search

Navigation