Skip to main content
SpringerLink
Log in

Nonlinear Dynamical Responses of Rotary Cylindrical Shells with Internal Resonance

  • Published:
Acta Mechanica Solida Sinica Aims and scope Submit manuscript

Abstract

The nonlinear forced vibration response of a thin, elastic, rotary cylindrical shell to a harmonic excitation is investigated in this study. Nonlinearities due to the large-amplitude shell motion are considered by using Donnell’s nonlinear shallow-shell theory, with consideration of the effect of viscous structural damping. Different from the conventional Donnell’s nonlinear shallow-shell equations, an improved nonlinear model without employing the Airy stress function is utilized to study the nonlinear dynamics of thin shells. The system is discretized using the Galerkin method, while a model involving two degrees of freedom and allowing for the traveling wave response of the shell is adopted. The method of harmonic balance is applied to study the nonlinear dynamic responses of the two-degree-of-freedom system. In addition, the stability of steady-state solutions is analyzed in detail. Finally, results are given for exploring the effects of different parameters on the nonlinear dynamic response with internal resonance.

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. Chu HN. Influence of large amplitudes on flexural vibrations of a thin circular cylindrical shell. J Aerosp Sci. 1961;28:602–9.

    Article  MathSciNet  MATH  Google Scholar 

  2. Amabili M, Pellicano F, PaÏdoussis MP. Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid. J Fluids Struct. 1998;12:883–918.

    Article  Google Scholar 

  3. Moussaoui F, Benamar R. Non-linear vibrations of shell-type structures: a review with bibliography. J Sound Vib. 2002;255:161–84.

    Article  Google Scholar 

  4. Amabili M, Pellicano F, PaÏdoussis MP. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part I: stability. J Sound Vib. 1999;225:655–99.

    Article  Google Scholar 

  5. Amabili M, Pellicano F, PaÏdoussis MP. Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid, part II: large-amplitude vibrations without flow. J Sound Vib. 1999;228:1103–24.

    Article  Google Scholar 

  6. Amabili M, PaÏdoussis MP. Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl Mech Rev. 2003;56:349–81.

    Article  Google Scholar 

  7. Zhang W, Hao YX, Yang J. Nonlinear dynamics of FGM circular cylindrical shell with clamped–clamped edges. Compos Struct. 2012;94:1075–86.

    Article  Google Scholar 

  8. Liu YZ, Hao YX, Zhang W, Chen J, Li SB. Nonlinear dynamics of initially imperfect functionally graded circular cylindrical shell under complex loads. J Sound Vib. 2015;348:294–328.

    Article  Google Scholar 

  9. Hao YX, Chen LH, Zhang W, Lei JG. Nonlinear oscillations, bifurcations and chaos of functionally graded materials plate. J Sound Vib. 2008;312:862–92.

    Article  Google Scholar 

  10. Zhang W, Yang J, Hao YX. Chaotic vibrations of an orthotropic FGM rectangular plate based on third-order shear deformation theory. Nonlinear Dyn. 2010;59:619–60.

    Article  MATH  Google Scholar 

  11. Hao YX, Zhang W, Yang J. Analysis on nonlinear oscillations of a cantilever FGM rectangular plate based on third-order plate theory and asymptotic perturbation method. Compos B: Eng. 2011;42:402–13.

    Article  Google Scholar 

  12. Zhang W, Hao YX, Yang J. Nonlinear dynamics of FGM circular cylindrical shell with clamped-clamped edges. Compos Struct. 2012;94:1075–86.

    Article  Google Scholar 

  13. Hao YX, Zhang W, Yang J. Nonlinear dynamics of cantilever FGM cylindrical shell under 1:2 internal resonance relations. Mech Adv Mater Struct. 2013;20:819–33.

    Article  Google Scholar 

  14. Karagiozis KN, Amabili M, Paidoussis MP, Misra AK. Nonlinear vibrations of fluid-filled clamped circular cylindrical shells. J Fluids Struct. 2005;21:579–95.

    Article  Google Scholar 

  15. Pellicano F. Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads. Commun Nonlinear Sci Numer Simul. 2009;14:3449–62.

    Article  MATH  Google Scholar 

  16. Wang YQ, Guo XH, Li YG, Li J. Nonlinear traveling wave vibration of a circular cylindrical shell subjected to a moving concentrated harmonic force. J Sound Vib. 2010;329:338–52.

    Article  Google Scholar 

  17. Wang Y, Liang L, Guo X, Li J, Liu J, Liu P. Nonlinear vibration response and bifurcation of circular cylindrical shells under traveling concentrated harmonic excitation. Acta Mechanica Solida Sinica. 2013;26:277–91.

    Article  Google Scholar 

  18. Kurylov Y, Amabili M. Polynomial versus trigonometric expansions for nonlinear vibrations of circular cylindrical shells with different boundary conditions. J Sound Vib. 2010;329:1435–49.

    Article  Google Scholar 

  19. Amabili M. Internal resonances in non-linear vibrations of a laminated circular cylindrical shell. Nonlinear Dyn. 2012;69:755–70.

    Article  MathSciNet  MATH  Google Scholar 

  20. Zhang W, Yang SW, Mao JJ. Nonlinear radial breathing vibrations of CFRP laminated cylindrical shell with non-normal boundary conditions subjected to axial pressure and radial line load at two ends. Compos Struct. 2018;190:52–78.

    Article  Google Scholar 

  21. Wang YQ, Liang L, Guo XH. Internal resonance of axially moving laminated circular cylindrical shells. J Sound Vib. 2013;332:6434–50.

    Article  Google Scholar 

  22. Zhang YF, Zhang W, Yao ZG. Analysis on nonlinear vibrations near internal resonances of a composite laminated piezoelectric rectangular plate. Eng Struct. 2018;173:89–106.

    Article  Google Scholar 

  23. Bryan GH. On the beats in the vibrations of a revolving cylinder or bell. In: Proceedings of the Cambridge Philosophical Society. 1890;101–11.

  24. DiTaranto RA, Lessen M. Coriolis acceleration effect on the vibration of a rotating thin-walled circular cylinder. ASME J Appl Mech. 1964;31:700–1.

    Article  Google Scholar 

  25. Srinivasan AV, Lauterbach GF. Traveling waves in rotating cylindrical shells. J Eng Ind. 1971;93:1229–32.

    Article  Google Scholar 

  26. Huang SC, Soedel W. Effects of Coriolis acceleration on the forced vibration of rotating cylindrical shells. ASME J Appl Mech. 1988;55:231–3.

    Article  Google Scholar 

  27. Ng TY, Lam KY, Reddy JN. Parametric resonance of a rotating cylindrical shell subjected to periodic axial loads. J Sound Vib. 1998;214:513–29.

    Article  Google Scholar 

  28. Hua L, Lam KY. Frequency characteristics of a thin rotating cylindrical shell using the generalized differential quadrature method. Int J Mech Sci. 1998;40:443–59.

    Article  MATH  Google Scholar 

  29. Liew KM, Ng TY, Zhao X, Reddy JN. Harmonic reproducing kernel particle method for free vibration analysis of rotating cylindrical shells. Comput Methods Appl Mech Eng. 2002;191:4141–57.

    Article  MATH  Google Scholar 

  30. Liew KM, Hu YG, Ng TY, Zhao X. Dynamic stability of rotating cylindrical shells subjected to periodic axial loads. Int J Solids Struct. 2006;43:7553–70.

    Article  MATH  Google Scholar 

  31. Zhang XM. Parametric analysis of frequency of rotating laminated composite cylindrical shells with the wave propagation approach. Comput Methods Appl Mech Eng. 2002;191:2029–43.

    Google Scholar 

  32. Sun SP, Chu SM, Cao DQ. Vibration characteristics of thin rotating cylindrical shells with various boundary conditions. J Sound Vib. 2012;331:4170–86.

    Article  Google Scholar 

  33. Lam KY, Loy CT. Influence of boundary conditions for a thin laminated rotating cylindrical shell. Compos Struct. 1998;41:215–28.

    Article  Google Scholar 

  34. Lee YS, Kim YW. Nonlinear free vibration analysis of rotating hybrid cylindrical shells. Comput Struct. 1999;70:161–8.

    Article  MATH  Google Scholar 

  35. Wang YQ, Guo XH, Chang HH, Li HY. Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape–Part I: Numerical solution. Int J Mech Sci. 2010;52:1217–24.

    Article  Google Scholar 

  36. Wang YQ, Guo XH, Chang HH, Li HY. Nonlinear dynamic response of rotating circular cylindrical shells with precession of vibrating shape–Part II: Approximate analytical solution. Int J Mech Sci. 2010;52:1208–16.

    Article  Google Scholar 

  37. Liu YQ, Chu FL. Nonlinear vibrations of rotating thin circular cylindrical shell. Nonlinear Dyn. 2012;67:1467–79.

    Article  MathSciNet  MATH  Google Scholar 

  38. Han Q, Qin Z, Zhao J, Chu F. Parametric instability of cylindrical thin shell with periodic rotating speeds. Int J Non-Linear Mech. 2013;57:201–7.

    Article  Google Scholar 

  39. Amabili M. Nonlinear vibrations and stability of shells and plates. New York: Cambridge University Press; 2008.

    Book  MATH  Google Scholar 

  40. Wang YQ, Huang XB, Li J. Hydroelastic dynamic analysis of axially moving plates in continuous hot-dip galvanizing process. Int J Mech Sci. 2016;110:201–16.

    Article  Google Scholar 

  41. Wang Y, Ye C, Zu JW. Identifying the temperature effect on the vibrations of functionally graded cylindrical shells with porosities. Appl Math Mech. 2018;39:1587–604.

    Article  MathSciNet  Google Scholar 

  42. Wang YQ, Zu JW. Nonlinear steady-state responses of longitudinally traveling functionally graded material plates in contact with liquid. Compos Struct. 2017;164:130–44.

    Article  Google Scholar 

  43. Wang YQ. Electro-mechanical vibration analysis of functionally graded piezoelectric porous plates in the translation state. Acta Astronautica. 2018;143:263–71.

    Article  Google Scholar 

  44. Wang YQ, Zu JW. Vibration behaviors of functionally graded rectangular plates with porosities and moving in thermal environment. Aerosp Sci Technol. 2017;69:550–62.

    Article  Google Scholar 

  45. Wang YQ, Zu JW. Instability of viscoelastic plates with longitudinally variable speed and immersed in ideal liquid. Int J Appl Mech. 2017;9:1750005.

    Article  Google Scholar 

  46. Wolfram S. The mathematica book. Cambridge: Cambridge University Press; 1999.

    MATH  Google Scholar 

  47. Peng GL. Fortran 95 program. Beijing: China Electric Power Press; 2002 (in Chinese).

    Google Scholar 

Download references

Acknowledgements

This research was supported by the National Natural Science Foundation of China (Project No. 11672188).

Author information

Authors and Affiliations

  1. School of Aerospace Engineering, Shenyang Aerospace University, Shenyang, 110136, China

    Yufei Zhang & Jintang Liu

  2. School of Mechanical Engineering and Automation, Northeastern University, Shenyang, 110819, China

    Bangchun Wen

Authors
  1. Yufei Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jintang Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bangchun Wen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yufei Zhang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, Y., Liu, J. & Wen, B. Nonlinear Dynamical Responses of Rotary Cylindrical Shells with Internal Resonance. Acta Mech. Solida Sin. 32, 186–200 (2019). https://doi.org/10.1007/s10338-019-00080-z

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10338-019-00080-z

Keywords

Search

Navigation