Skip to main content

Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions

  • Chapter
  • First Online:
Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members

Part of the book series: Scientific Computation ((SCIENTCOMP))

  • 431 Accesses

Abstract

In the present chapter, novel results of the study of chaotic vibration of flexible circular axially symmetric shallow shells subjected to sinusoidal transverse load are presented for four different boundary conditions are presented. The study is conducted with the use of the finite difference method (FDM), which differentiates the study from the majority of research conducted with the finite element method (FEM).

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Aranda-Iglesias, D., Vadillo, G., Rodriguez-Martinez, J.A.: Oscillatory behaviour of compressible hyperelastic shells subjected to dynamic inflation: a numerical study. Acta Mech. 228(6), 2187–2205 (2017)

    Article  MathSciNet  Google Scholar 

  2. Awrejcewicz, J.: Ordinary Differential Equations and Mechanical Systems. Springer, Berlin (2014)

    Book  Google Scholar 

  3. Awrejcewicz, J., Andrianov, I.V.: Plates and Shells in Nature, Mechanics and Biomechanics. WNT, Fundacja Ksiazka Naukowo-Techniczna, Warsaw (2001) (in Polish)

    Google Scholar 

  4. Awrejcewicz, J., Krysko, V.A., Papkova, I.V.: Dynamics and statics of flexible axially-symmetric shallow shells. Math. Probl. Eng. 2006, ID 35672 (2006)

    Google Scholar 

  5. Banks, J., Brooks, J., Davis, G., Stacey, P.: On Devaney’s definition of chaos. Amer. Math. Monthly 99(4), 332–334 (1992)

    Article  MathSciNet  Google Scholar 

  6. Chen, C., Yuan, J., Mao, Y.: Post-buckling of size-depend micro-plate considering damage effects. Nonlinear Dyn. 90(2), 1301–1314 (2017)

    Article  Google Scholar 

  7. Devaney, R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Reading, MA (1989)

    MATH  Google Scholar 

  8. Feigenbaum, M.J.: The universal metric properties of nonlinear transformations. J. Stat. Phys. 21(6), 669–706 (1979)

    Article  ADS  MathSciNet  Google Scholar 

  9. Gulick, D.: Encounters with Chaos. McGraw-Hill, NewYork (1992)

    MATH  Google Scholar 

  10. Kantz, H.: A robust method to estimate the maximum Lyapunov exponent of a time series. Phys. Lett. A 185, 77–87 (1994)

    Article  ADS  Google Scholar 

  11. Knudsen, C.: Chaos without periodicity. Am. Math. Mon. 101, 563–565 (1994)

    Article  Google Scholar 

  12. Krysko, A.V., Awrejcewicz, J., Zakharova, A.A., Papkova, I.V., Krysko, V.A., Chaotic vibrations of flexible shallow axially symmetric shells. Nonlinear Dyn. 91(4), 2271–2291 (2018)

    Article  Google Scholar 

  13. Lozi, R.: Can we trust in numerical computations of chaotic solutions of dynamical systems? In: World Scientific Series on Nonlinear Science. Topology and Dynamics of Chaos in Celebration of Robert Gilmore’s 70th Birthday, vol. 84, pp. 63–98 (2013)

    Google Scholar 

  14. Mehditabar, A., Rahimi, G.H., Tarahhomi, M.H.: Thermo-elastic analysis of a functionally graded piezoelectric rotating hollow cylindrical shell subjected to dynamic loads. Mech. Adv. Mater. Struct. 0, 1–12 (2017)

    Google Scholar 

  15. Medina, L., Gilat, R., Krylov, S.: Modeling strategies of electrostatically actuated initially curved bistable micro-plates. Int. J. Sol. Struct. 118–119, 1339–1351 (2017)

    Google Scholar 

  16. Reissner, E.: Stress and small displacements of shallow spherical shells. J. Math. Phys. 25, 279–300 (1946)

    Article  Google Scholar 

  17. Rosenstein, M.T., Collins, J.J., De Luca, C.J.: A practical method for calculating largest Lyapunov exponents from small data sets. Phys. D 65, 117–134 (1993)

    Article  MathSciNet  Google Scholar 

  18. Sedov, L.: Similarity and Dimensional Methods in Mechanics. CRC Press, Boca Raton (1993)

    Google Scholar 

  19. Vlasov, V.Z.: General Theory of Shells and Its Application in Engineering. NASA-TT-F-99 (1964)

    Google Scholar 

  20. Vorovich, I.I.: Nonlinear Theory of Shallow Shells. Springer, New York (1998)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2020 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Awrejcewicz, J., Krysko, V.A. (2020). Chaotic Vibrations of Flexible Shallow Axially Symmetric Shells vs. Different Boundary Conditions. In: Elastic and Thermoelastic Problems in Nonlinear Dynamics of Structural Members. Scientific Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-37663-5_14

Download citation

Publish with us

Policies and ethics