Skip to main content
SpringerLink
Log in

Existence of invariant curves for a Fermi-type impact absorber

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

Abstract

In this paper we study an impact absorber which is similar to the Fermi accelerator and can be described as a ball moves in a periodically oscillating ring with a wall and reflects elastically from the wall. First, Poincaré map of the system is established. The existence of invariant curves for the map is proved based on Moser’s twist theorem. Accordingly, the velocities of the ball are always bounded for any initial motion for all time. Moreover, the symmetry of the Poincaré map is discussed. Finally, some numerical simulations are given to demonstrate the theoretical results.

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

Similar content being viewed by others

References

  1. Kolmogorov, A.N.: On the preservation of conditionally periodic motions under a small change in Hamilton’s function. Dokl. Akad. Nauk SSSR 98, 527–530 (1954)

    MathSciNet  MATH  Google Scholar 

  2. Arnold, V.I.: Proof of a theorem of A. N. Kolmogorov on the invariance of quasi-periodic motions under small perturbations of the Hamiltonian. Russ. Math. Surv. 18, 9–36 (1963)

    Article  Google Scholar 

  3. Moser, J.: On invariant curves of area preserving mappings of an annulus. Nachr. Akad. Wiss. Gott., Math. Phys. Kl. 1–20 (1962)

  4. Siegel, C.L., Moser, J.: Lectures on Celestial Mechanics. Springer, Berlin (1971)

    Book  Google Scholar 

  5. Levi, M.: KAM theory for particles in periodic potentials Ergod. Theory Dyn. Syst. 10, 777–785 (1990)

    Article  Google Scholar 

  6. Laederich, S., Levi, M.: Invariant curves and time-dependent potentials. Ergod. Theory Dyn. Syst. 11, 365–378 (1991)

    Article  MathSciNet  Google Scholar 

  7. Zharnitsky, V.: Invariant tori in Hamiltonian systems with impacts. Commun. Math. Phys. 211, 289–302 (2000)

    Article  MathSciNet  Google Scholar 

  8. Ortega, R.: Asymmetric oscillators and twist mappings. J. Lond. Math. Soc. 53, 325–342 (1996)

    Article  MathSciNet  Google Scholar 

  9. Qian, D., Sun, X.: Invariant tori for asymptotically linear impact oscillators. Sci. China Ser. Math. 49, 669–687 (2006)

    Article  MathSciNet  Google Scholar 

  10. Fermi, E.: On the origin of the cosmic radiation. Phys. Rev. 15, 1169–1174 (1949)

    Article  Google Scholar 

  11. Douady, R.: Applications du théorème des tores invariants. Thése de 3éme Cycle, University of Paris VII (1982)

  12. Dolgopyat, D.: Bouncing balls in non-linear potentials. Discrete Contin. Dyn. Syst. 22, 165–182 (2017)

    Article  MathSciNet  Google Scholar 

  13. Pun, D., Liu, Y.B.: On the design of the piecewise linear vibration absorber. Nonlinear Dyn. 22, 393–413 (2000)

    Article  Google Scholar 

  14. Ladeira, D.G., Leonel, E.D.: Dynamical properties of a dissipative hybrid Fermi-Ulam-bouncer model. Chaos 17, 823 (2007)

    Article  MathSciNet  Google Scholar 

  15. de Carvalho, R.E., Sousa, F.C., Leonel, E.D.: Fermi acceleration on the annular billiard: a simplified version. Phys. Rev. E 73, 066229 (2006)

    Article  MathSciNet  Google Scholar 

  16. Ladeira, D.G., Leonel, E.D.: Dynamics of a charged particle in a dissipative Fermi-Ulam model. Commun. Nonlinear Sci. Numer. Simul. 20, 546–558 (2015)

    Article  Google Scholar 

  17. Gelfreich, V., Turaev, D.: Fermi acceleration in non-autonomous billiards. J. Phys. A: Math. Theor. 41, 212003 (2008)

    Article  MathSciNet  Google Scholar 

  18. Felix, J.L.P., José Balthazar, M.: Comments on a nonlinear and nonideal electromechanical damping vibration absorber, Sommerfeld effect and energy transfer. Nonlinear Dyn. 55, 1–11 (2009)

    Article  Google Scholar 

  19. Kruger, T., Pustyl’nikov, L.D., Troubetzkoy, S.E.: Acceleration of bouncing balls in external fields. Nonlinearity 8, 397–410 (1994)

    Article  MathSciNet  Google Scholar 

  20. Medeiros, E.S., Souza, S.L.T.D., Medrano, T.R.O., et al.: Periodic window arising in the parameter space of an impact oscillator. Phys. Lett. A 374, 2628–2635 (2010)

    Article  Google Scholar 

  21. Luo, A.C.J.: Period-doubling induced chaotic motion in the LR model of a horizontal impact oscillator. Chaos Solitons Fractals 19, 823–839 (2004)

    Article  MathSciNet  Google Scholar 

  22. Pustyl’nikov, L.D.: Existence of invariant curves for maps close to degenerate maps, and a solution of the Fermi-Ulam problem. Russ. Acad. Sci. Sb. Math. 82, 113–124 (1995)

    Google Scholar 

  23. Rüssmann, H., Kleine, Nenner. I.: Über invariante kurven differenzierbarer abbildungen eines kreisrings, Nachr. Acad. Wiss., Göttingen, Math. Phys. K1. II, 67–105 (1970)

  24. Yue, Y., Xie, J.H.: Symmetry and bifurcations of a two-degree-of-freedom vibro-impact system. J. Sound Vib. 314, 228–245 (2008)

    Article  Google Scholar 

  25. Arnold, V.I.: Mathematical Methods of Classical Mechanics, 2nd edn. Springer, New York (1989)

    Book  Google Scholar 

  26. Ortega, R.: Asymmetric oscillators and twist mappings. J. Lond. Math. Soc. 53, 325–342 (1996)

    Article  MathSciNet  Google Scholar 

  27. Herman, M.: Sur les courbes invariantes par les diffémorphismes de l’anneau. Astérisque 103, 1–221 (1983)

    Google Scholar 

  28. Herman, M.: Sur les courbes invariantes par les diffémorphismes de l’anneau. Astérisque 104, 1–243 (1983)

    Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundations of China (11732014, 11672249). The authors express their gratitude to Dr. Hebai Chen for helpful suggestions and to the reviewers for fruitful comments and suggestions.

Author information

Authors and Affiliations

  1. School of Mechanics and Engineering, Southwest Jiaotong University, Chengdu, 610031, China

    Zhenbang Cao, Xiaoming Zhang & Jianhua Xie

  2. School of Mathematics and Statistics, Hexi University, Zhangye, 734000, China

    Denghui Li

  3. State Key Laboratory of Advanced Design and Manufacture for Vehicle Body, Hunan University, Changsha, 410082, China

    Shan Yin

Authors
  1. Zhenbang Cao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiaoming Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Denghui Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Shan Yin
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Jianhua Xie
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Denghui Li.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Cao, Z., Zhang, X., Li, D. et al. Existence of invariant curves for a Fermi-type impact absorber. Nonlinear Dyn 99, 2647–2656 (2020). https://doi.org/10.1007/s11071-019-05437-0

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-019-05437-0

Keywords

Search

Navigation