Skip to main content
SpringerLink
Log in

Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads

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

Abstract

The global bifurcations and multi-pulse orbits of an aero-thermo-elastic functionally graded material (FGM) truncated conical shell under complex loads are investigated with the case of 1:2 internal resonance and primary parametric resonance. The method of multiple scales is utilized to obtain the averaged equations. Based on the averaged equations obtained, the normal form theory is employed to find the explicit expressions of normal form associated with a double zero and a pair of pure imaginary eigenvalues. The energy-phase method developed by Haller and Wiggins is used to analyze the multi-pulse homoclinic bifurcations and chaotic dynamics of the FGM truncated conical shell. The analytical results obtained here indicate that there exist the multi-pulse Shilnikov-type homoclinic orbits for the resonant case which may result in chaos in the system. Homoclinic trees which describe the repeated bifurcations of multi-pulse solutions are found. The diagrams show a gradual breakup of the homoclinic tree in the system as the dissipation factor is increased. Numerical simulations are presented to illustrate that for the FGM truncated conical shell, the multi-pulse Shilnikov-type chaotic motions can occur. The influence of the structural-damping, the aerodynamic-damping, and the in-plane and transverse excitations on the system dynamic behaviors is also discussed by numerical simulations. The results obtained here mean the existence of chaos in the sense of the Smale horseshoes for the FGM truncated conical shell.

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

Similar content being viewed by others

References

  1. Leissa, A.W.: Vibration of Shells. Acoustical Society of America, New York (1993)

    Google Scholar 

  2. Shu, C.: An efficient approach for free vibration analysis of conical shells. Int. J. Mech. Sci. 38, 935–949 (1996)

    Article  MATH  Google Scholar 

  3. Liew, K.M., Ng, T.Y., Zhao, X.: Free vibration analysis of conical shells via the element-free kp–Ritz method. J. Sound Vib. 281, 627–645 (2005)

    Article  Google Scholar 

  4. Jin, G.Y., Su, Z., Ye, T.G., Jia, X.Z.: Three-dimensional vibration analysis of isotropic and orthotropic conical shells with elastic boundary restraints. Int. J. Mech. Sci. 89, 207–221 (2014)

    Article  Google Scholar 

  5. Sofiyev, A.H.: The stability of functionally graded truncated conical shells subjected to aperiodic impulsive loading. Int. J. Solids Struct. 41, 3411–3424 (2004)

    Article  MATH  Google Scholar 

  6. Sofiyev, A.H.: Thermoelastic stability of functionally graded truncated conical shells. Compos. Struct. 77, 56–65 (2007)

    Article  Google Scholar 

  7. Sofiyev, A.H.: The buckling of FGM truncated conical shells subjected to combined axial tension and hydrostatic pressure. Compos. Struct. 92, 488–498 (2010)

    Article  Google Scholar 

  8. Sofiyev, A.H.: Influence of the initial imperfection on the non-linear buckling response of FGM truncated conical shells. Int. J. Mech. Sci. 53, 753–761 (2011)

    Article  Google Scholar 

  9. Sofiyev, A.H.: The vibration and stability behavior of freely supported FGM conical shells subjected to external pressure. Compos. Struct. 89, 356–366 (2009)

    Article  Google Scholar 

  10. Sofiyev, A.H.: On the vibration and stability of clamped FGM conical shells under external loads. J. Compos. Mater. 45, 771–788 (2011)

    Article  Google Scholar 

  11. Najafov, A.M., Sofiyev, A.H.: The non-linear dynamics of FGM truncated conical shells surrounded by an elastic medium. Int. J. Mech. Sci. 66, 33–44 (2013)

    Article  Google Scholar 

  12. Bhangale, R.K., Ganesan, K., Padmanabhan, C.: Linear thermoelastic buckling and free vibration behavior of functionally graded truncated conical shells. J. Sound Vib. 292, 341–371 (2006)

    Article  Google Scholar 

  13. Tornabene, F., Viola, E., Inman, D.J.: 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures. J. Sound Vib. 328, 259–290 (2009)

    Article  Google Scholar 

  14. Setoodeh, A.R., Tahani, M., Selahi, E.: Transient dynamic and free vibration analysis of functionally graded truncated conical shells with non-uniform thickness subjected to mechanical shock loading. Compos. Part B 43, 2161–2171 (2012)

    Article  Google Scholar 

  15. Malekzadeh, P., Fiouz, A.R., Sobhrouyan, M.: Three-dimensional free vibration of functionally graded truncated conical shells subjected to thermal environment. Int. J. Press. Vessel. Pip. 89, 210–221 (2012)

    Article  Google Scholar 

  16. Jooybar, N., Malekzadeh, P., Fiouz, A.R., Vaghefi, M.: Thermal effect on free vibration of functionally graded truncated conical shell panels. Thin Walled Struct. 103, 45–61 (2016)

    Article  Google Scholar 

  17. Malekzadeh, P., Daraie, M.: Dynamic analysis of functionally graded truncated conical shells subjected to asymmetric moving loads. Thin Walled Struct. 84, 1–13 (2014)

    Article  Google Scholar 

  18. Heydarpour, Y., Aghdam, M.M., Malekzadeh, P.: Free vibration analysis of rotating functionally graded carbon nanotube-reinforced composite truncated conical shells. Compos. Struct. 117, 187–200 (2014)

    Article  MATH  Google Scholar 

  19. Malekzadeh, P., Heydarpour, Y.: Free vibration analysis of rotating functionally graded truncated conical shells. Compos. Struct. 97, 176–188 (2013)

    Article  Google Scholar 

  20. Yang, S.W., Hao, Y.X., Zhang, W., Li, S.B.: Nonlinear dynamic behavior of functionally graded truncated conical shell under complex loads. Int. J. Bifurc. Chaos 25, 1550025 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  21. Haller, G., Wiggins, S.: N-pulse homoclinic orbits in perturbations of resonant Hamiltonian systems. Arch. Ration. Mech. Anal. 130, 25–101 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  22. Wiggins, S.: Global Bifurcations and Chaos: Analytical Methods. Springer, New York (1988)

    Book  MATH  Google Scholar 

  23. Kovacic, G., Wiggins, S.: Orbits homoclinic to resonances, with an application to chaos in a model of the forced and damped Sine–Gordon equation. Phys. D 57, 185–225 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  24. Kovacic, G., Wettergren, T.A.: Homoclinic orbits in the dynamics of resonantly driven coupled pendula. Z. Angew. Math. Phys. 47, 221–264 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Camassa, R., Kovacic, G., Tin, S.K.: A Melnikov method for homoclinic orbits with many pulses. Arch. Ration. Mech. Anal. 143, 105–193 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  26. Haller, G., Wiggins, S.: Orbits homoclinic to resonances: the Hamiltonian case. Phys. D 66, 298–346 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  27. Haller, G., Wiggins, S.: Multi-pulse jumping orbits and homoclinic trees in a modal truncation of the damped-forced nonlinear Schrödinger equation. Phys. D 85, 311–347 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  28. Malhotra, N., Namachchivaya, N.S., McDonald, R.J.: Multipulse orbits in the motion of flexible spinning discs. J. Nonlinear Sci. 12, 1–26 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  29. McDonald, R.J., Namachchivaya, N.S.: Pipes conveying pulsating fluid near a 0:1 resonance: global bifurcations. J. Fluids Struct. 21, 665–687 (2005)

    Article  Google Scholar 

  30. Yao, M.H., Zhang, W.: Multipulse Shilnikov orbits and chaotic dynamics for nonlinear nonplanar motion of a cantilever beam. Int. J. Bifurc. Chaos 15, 3923–3952 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  31. Zhang, W., Yao, M.H.: Multi-pulse orbits and chaotic dynamics in motion of parametrically excited viscoelastic moving belt. Chaos Solitons Fractals 28, 42–66 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  32. Li, S.B., Zhang, W., Yao, M.H.: Using energy-phase method to study global bifurcations and Shilnikov type multipulse chaotic dynamics for a nonlinear vibration absorber. Int. J. Bifurc. Chaos 22, 1250001 (2012)

    Article  MATH  Google Scholar 

  33. Malhotra, N., Namachchivaya, N.S.: Global dynamics of parametrically excited nonlinear reversible systems with nonsemisimple 1:1 resonance. Phys. D 89, 43–70 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Zhang, W., Liu, Z.M., Yu, P.: Global dynamics of a parametrically and externally excited thin plate. Nonlinear Dyn. 24, 245–268 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  35. Zhang, W., Yao, M.H., Zhang, J.H.: Using the extended Melnikov method to study the multi-pulse global bifurcations and chaos of a cantilever beam. J. Sound Vib. 319, 541–569 (2009)

  36. Malhotra, N., Namachchivaya, N.S.: Chaotic motion of shallow arch structures under 1:1 internal resonance. J. Eng. Mech. 123, 620–627 (1997)

  37. Yu, W.Q., Chen, F.Q.: Global bifurcations of a simply supported rectangular metallic plate subjected to a transverse harmonic excitation. Nonlinear Dyn. 59, 129–141 (2010)

  38. Zhang, W., Hao, W.L.: Multi-pulse chaotic dynamics of six-dimensional non-autonomous nonlinear system for a composite laminated piezoelectric rectangular plate. Nonlinear Dyn. 73, 1005–1033 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  39. Zhang, W., Wang, F.X., Zu, J.W.: Computation of normal forms for high dimensional non-linear systems and application to non-planar non-linear oscillations of a cantilever beam. J. Sound Vib. 278, 949–974 (2004)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fengxian An

  2. Faculty of Mathematics and Physics, Huaiyin Institute of Technology, Huaian, 223003, People’s Republic of China

    Fengxian An

  3. Department of Mathematics, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, People’s Republic of China

    Fangqi Chen

Authors
  1. Fengxian An
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Fangqi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding authors

Correspondence to Fengxian An or Fangqi Chen.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

This research was supported by the National Natural Science Foundation of China (11572148), and the National Research Foundation for the Doctoral Program of Higher Education of China (20133218110025).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

An, F., Chen, F. Multi-pulse chaotic motions of functionally graded truncated conical shell under complex loads. Nonlinear Dyn 89, 1753–1778 (2017). https://doi.org/10.1007/s11071-017-3550-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-017-3550-x

Keywords

Search

Navigation