Skip to main content
SpringerLink
Log in

Experiments on the dynamics and stability of cantilevered circular cylindrical shells conveying airflow

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

Abstract

The present experimental study investigates the dynamics and stability of soft cantilevered circular cylindrical shells of various length-to-radius (L/R) ratios subjected to subsonic internal airflow. The results indicate that the system loses stability by flutter through a strongly subcritical bifurcation. Longer shells lose stability at lower flow velocities and the oscillations become more irregular as the flow velocity is increased beyond the critical value, suggesting the presence of a chaotic component in the oscillations. The chaotic component is shown to be dependent on the L/R. The results also show that for high enough flow velocities, flutter can no longer be excited. Finally, the effect of the outflow jet on the stability of the system is examined by mounting a flat horizontal plate at different distances over the free end of the shell. It is found that changing the outflow jet direction from axial to radial not only has a stabilizing effect, but also weakens the chaotic behaviour of the oscillations.

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
Fig. 14
Fig. 15
Fig. 16
Fig. 17
Fig. 18
Fig. 19

Similar content being viewed by others

References

  1. Amabili, M.: Nonlinear Mechanics of Shells and Plates in Composite, Soft and Biological Materials. Cambridge University Press, New York (2018)

    MATH  Google Scholar 

  2. Kaneko, S., et al.: Flow-Induced Vibrations. Classifications and Lessons from Practical Experiences, 2nd edn. Academic Press, London (2014)

    Google Scholar 

  3. Païdoussis, M.P.: Fluid–Structure Interactions: Slender Structures and Axial Flow, vol. 1, 2nd edn. Academic Press, Oxford (2014)

    Google Scholar 

  4. Dowell, E.H.: Aeroelasticity of Plates and Shells. Noordhoff International Publishing, Leyden (1975)

    MATH  Google Scholar 

  5. Dowell, E.H.: A Modern Course in Aeroelasticity, Fifth Revised and Enlarged Edition. Springer, Dordrecht (2015)

    Google Scholar 

  6. Païdoussis, M.P., Denise, J.-P.: Flutter of cylindrical shells conveying fluid. J. Sound Vib. 16(3), 459–461 (1971)

    MATH  Google Scholar 

  7. Païdoussis, M.P., Denise, J.-P.: Flutter of thin cylindrical shells conveying fluid. J. Sound Vib. 20(1), 9–26 (1972)

    MATH  Google Scholar 

  8. Shayo, L.K., Ellen, C.H.: Theoretical studies of internal flow-induced instabilities of cantilever pipes. J. Sound Vib. 56(4), 463–474 (1978)

    MATH  Google Scholar 

  9. El Chebair, A., Païdoussis, M.P., Misra, A.K.: Experimental study of annular-flow-induced instabilities of cylindrical shells. J. Fluids Struct. 3(4), 349–364 (1989)

    Google Scholar 

  10. Païdoussis, M.P., Nguyen, V.B., Misra, A.K., Païdoussis, M.P., Misra, A.K.: A theoretical study of the stability of cantilevered coaxial cylindrical shells conveying fluid. J. Fluids Struct. 5(8), 913–930 (1991)

    MATH  Google Scholar 

  11. Païdoussis, M.P., Misra, A.K., Nguyen, V.B.: Internal- and annular-flow-induced instabilities of a clamped-clamped or cantilevered cylindrical shell in a coaxial conduit: the effects of system parameters. J. Sound Vib. 159(2), 193–205 (1992)

    MATH  Google Scholar 

  12. Nguyen, V.B., Païdoussis, M.P., Misra, A.K.: An experimental study of the stability of cantilevered coaxial cylindrical shells conveying fluid. J. Fluids Struct. 7(8), 913–930 (1993)

    Google Scholar 

  13. Nguyen, V.B., Païdoussis, M.P., Misra, A.K.: A CFD-based model for the study of the stability of cantilevered coaxial cylindrical shells conveying viscous fluid. J. Sound Vib. 176(1), 105–125 (1994)

    MATH  Google Scholar 

  14. Lakis, A.A., Laveau, A.: Non-linear dynamic analysis of anisotropic cylindrical shells containing a flowing fluid. Int. J. Solids Struct. 28(9), 1079–1094 (1991)

    MATH  Google Scholar 

  15. Evensen, D.A.: Nonlinear Flexural Vibrations of Thin-Walled Circular Cylinders. In: NASA TN D-4090 (1967)

  16. Evensen, D.A.: Nonlinear vibrations of an infinitely long cylindrical shell. AIAA J. 6(7), 1401–1403 (1968)

    MATH  Google Scholar 

  17. Amabili, M., Pellicano, F., Païdoussis, M.P.: Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part I: stability. J. Sound Vib. 225(4), 655–699 (1999)

    Google Scholar 

  18. Amabili, M., Pellicano, F., Païdoussis, M.P.: Non-linear dynamics and stability of circular cylindrical shells containing flowing fluid. Part IV: large-amplitude vibrations with flow. J. Sound Vib. 237(4), 641–666 (2000)

    Google Scholar 

  19. Amabili, M., Pellicano, F., Païdoussis, M.P., Païdoussis, M.P.: Non-linear dynamics and stability of circular cylindrical shells conveying flowing fluid. Comput. Struct. 80(9–10), 899–906 (2002)

    MATH  Google Scholar 

  20. Amabili, M., Pellicano, F., Païdoussis, M.P.: Nonlinear stability of circular cylindrical shells in annular and unbounded axial flow. J. Appl. Mech. 68(6), 827–834 (2001)

    MATH  Google Scholar 

  21. Amabili, M., Pellicano, F.: Nonlinear supersonic flutter of circular cylindrical shells. AIAA J. 39(4), 564–573 (2001)

    MATH  Google Scholar 

  22. Amabili, M., Pellicano, F.: Multimode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells. J. Appl. Mech. 69(2), 117–129 (2002)

    MATH  Google Scholar 

  23. Olson, M.D., Fung, Y.C.: Supersonic flutter of circular cylindrical shells subjected to internal pressure and axial compression. AIAA J. 4(5), 858–864 (1966)

    Google Scholar 

  24. Olson, M.D., Fung, Y.C.: Comparing theory and experiment for the supersonic flutter of circular cylindrical shells. AIAA J. 5(10), 1849–1856 (1967)

    Google Scholar 

  25. Paak, M., Païdoussis, M.P., Misra, A.K.: Nonlinear dynamics and stability of cantilevered circular cylindrical shells conveying fluid. J. Sound Vib. 332(14), 3474–3498 (2013)

    Google Scholar 

  26. Paak, M., Païdoussis, M.P., Misra, A.K.: Influence of steady viscous forces on the non-linear behaviour of cantilevered circular cylindrical shells conveying fluid. Int. J. Nonlinear Mech. 58, 167–183 (2014)

    Google Scholar 

  27. Chiba, M.: Non-linear hydroelastic vibration of a cantilever cylindrical tank—II. Experiment (liquid-filled case). Int. J. Non. Linear. Mech. 28(5), 601–612 (1993)

    Google Scholar 

  28. Chiba, M.: Non-linear hydroelastic vibration of a cantilever cylindrical tank—I. Experiment (empty case). Int. J. Nonlinear Mech. 28(5), 591–599 (1993)

    Google Scholar 

  29. Chiba, M.: Experimental studies on a nonlinear hydroelastic vibration of a clamped cylindrical tank partially filled with liquid. J. Press. Vessel Technol. 115(4), 381–388 (1993)

    Google Scholar 

  30. Mallon, N.J., Fey, R.H.B., Nijmeijer, H.: Dynamic stability of a base-excited thin orthotropic cylindrical shell with top mass: simulations and experiments. J. Sound Vib. 329(15), 3149–3170 (2010)

    Google Scholar 

  31. Pellicano, F., Barbieri, M., Zippo, A., Strozzi, M.: Experiments on shells under base excitation. J. Sound Vib. 369, 209–227 (2016)

    Google Scholar 

  32. Pellicano, F.: Vibrations of circular cylindrical shells: theory and experiments. J. Sound Vib. 303(1–2), 154–170 (2007)

    Google Scholar 

  33. Pellicano, F.: Dynamic stability and sensitivity to geometric imperfections of strongly compressed circular cylindrical shells under dynamic axial loads. Commun. Nonlinear Sci. Numer. Simul. 14(8), 3449–3462 (2009)

    MATH  Google Scholar 

  34. Pellicano, F., Avramov, K.V.: Linear and nonlinear dynamics of a circular cylindrical shell connected to a rigid disk. Commun. Nonlinear Sci. Numer. Simul. 12(4), 496–518 (2007)

    MATH  Google Scholar 

  35. Pellicano, F.: Dynamic instability of a circular cylindrical shell carrying a top mass under base excitation: experiments and theory. Int. J. Solids Struct. 48(3–4), 408–427 (2011)

    MATH  Google Scholar 

  36. Popov, A.A.: Parametric resonance in cylindrical shells: a case study in the nonlinear vibration of structural shells. Eng. Struct. 25(6), 789–799 (2003)

    Google Scholar 

  37. Gonçalves, P.B., del Prado, Z.G.: Effect of non-linear modal interaction on the dynamic instability of axially excited cylindrical shells. Comput. Struct. 82(31), 2621–2634 (2004)

    Google Scholar 

  38. Gonçalves, P.B., Silva, F.M.A., Del Prado, Z.J.G.N.: Global stability analysis of parametrically excited cylindrical shells through the evolution of basin boundaries. Nonlinear Dyn. 50(1), 121–145 (2007)

    MathSciNet  MATH  Google Scholar 

  39. Gonçalves, P.B., Silva, F.M.A., Del Prado, Z.J.G.N.: Transient and steady state stability of cylindrical shells under harmonic axial loads. Int. J. Nonlinear Mech. 42(1), 58–70 (2007)

    MATH  Google Scholar 

  40. Kumar, A., Patel, B.P.: Experimental study on nonlinear vibration of cantilever cylindrical shell. Int. J. Struct. Stab. Dyn. 15(06), 1450085 (2015)

    Google Scholar 

  41. Popov, A.A.: Auto-parametric resonance in thin cylindrical shells using the slow fluctuation method. Thin Wall. Struct. 42(3), 475–495 (2004)

    Google Scholar 

  42. Popov, A.A.: The application of Hamiltonian dynamics and averaging to nonlinear shell vibration. Comput. Struct. 82(31), 2659–2670 (2004)

    Google Scholar 

  43. Avramov, K.V.: Nonlinear forced vibrations of a cylindrical shell with two internal resonances. Int. Appl. Mech. 42(2), 169–175 (2006)

    MathSciNet  Google Scholar 

  44. Avramov, K.V., Mikhlin, Y.V., Kurilov, E.: Asymptotic analysis of nonlinear dynamics of simply supported cylindrical shells. Nonlinear Dyn. 47(4), 331–352 (2007)

    MathSciNet  MATH  Google Scholar 

  45. Gonçalves, P.B., Silva, F.M.A., Del Prado, Z.J.G.N.: Low-dimensional models for the nonlinear vibration analysis of cylindrical shells based on a perturbation procedure and proper orthogonal decomposition. J. Sound Vib. 315(3), 641–663 (2008)

    Google Scholar 

  46. Rougui, M., Moussaoui, F., Benamar, R.: Geometrically non-linear free and forced vibrations of simply supported circular cylindrical shells: a semi-analytical approach. Int. J. Nonlinear Mech. 42(9), 1102–1115 (2007)

    MATH  Google Scholar 

  47. Avramov, K.V.: Multidimensional models of traveling waves and nonlinear modes in cylindrical shells. Int. Appl. Mech. 47(1), 70–77 (2011)

    MathSciNet  MATH  Google Scholar 

  48. Païdoussis, M.P.: Fluid-Structure Interactions: Slender Structures and Axial Flow: 2, 2nd edn. Academic Press, Oxford (2016)

    Google Scholar 

  49. Amabili, M., Païdoussis, M.P.: Review of studies on geometrically nonlinear vibrations and dynamics of circular cylindrical shells and panels, with and without fluid-structure interaction. Appl. Mech. Rev. 56(4), 349–356 (2003)

    Google Scholar 

  50. Amabili, M.: Nonlinear Vibrations and Stability of Shells and Plates. Cambridge University Press, Cambridge (2008)

    MATH  Google Scholar 

  51. Alijani, F., Amabili, M.: Non-linear vibrations of shells: a literature review from 2003 to 2013. Int. J. Nonlinear Mech. 58, 233–257 (2014)

    Google Scholar 

  52. Gholami, I., Amabili, M., Païdoussis, M.P.: Dynamic divergence of circular cylindrical shells conveying airflow. Mech. Syst. Signal Process. 166, 108496 (2022)

    Google Scholar 

  53. Gholami, I., Amabili, M., Païdoussis, M.P.: Experimental parametric study on dynamic divergence instability and chaos of circular cylindrical shells conveying airflow. Mech. Syst. Signal Process. 169, 108755 (2022)

    Google Scholar 

  54. Mallat, S.: A Wavelet Tour of Signal Processing: The Sparse Way, 2nd edn. Academic Press, Burlington (2009)

    MATH  Google Scholar 

  55. Malraison, B., Atten, P., Bergé, P., Dubois, M.: Dimension of strange attractors: an experimental determination for the chaotic regime of two convective systems. J. Phys. Lett. 44(22), 897–902 (1983)

    Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

Thanks go to Prof. Jovan Nedić of McGill University for lending us the high-speed camera. The authors are also grateful to Dr. Giovanni Ferrari for helping with conducting experimental modal analysis.

Funding

The authors gratefully acknowledge the financial support by the Natural Sciences and Engineering Research Council of Canada (NSERC) for CRDPJ 530933–18 grant and Pipeline Research Council International, Canada (PRCI).

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montreal, Quebec, H3A 0C3, Canada

    Iman Gholami, Marco Amabili & Michael P. Païdoussis

Authors
  1. Iman Gholami
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Marco Amabili
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Michael P. Païdoussis
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

IG contributed to data curation, formal analysis, investigation, methodology, software, visualization, and writing—original draft. MA contributed to conceptualization, funding acquisition, methodology, project administration, supervision, and writing—review and editing. MPP contributed to conceptualization, funding acquisition, methodology, project administration, supervision, and writing—review and editing.

Corresponding author

Correspondence to Michael P. Païdoussis.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Availability of data and material

The raw numerical data are freely available upon request.

Additional information

Publisher's Note

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

Appendix

Appendix

Appendix: Detailed results on the unstable oscillations at V > Vrest, i

In this appendix, an extensive analysis of the oscillations at \(V>{V}_{rest,i}\) is presented in Figs. 14 and 15. For this purpose, two data points of the same shell (the same L/R ratio), but different flow velocities (one equal to and the other greater than \({V}_{rest,i}\)), are selected, as listed in Table

Table 5 Selected data points, identified in Fig. 16c, to be analysed in terms of strength of the chaotic component in Figs. 14 and 15
Full size table

5.

Note that in Fig. 20a,

Fig. 20
figure 20

Study of the unstable oscillations at \(V>{V}_{\mathrm{rest},\mathrm{i}}\) (comparison between points C and D in Fig. 16c). Plots on the left side correspond to point C (\({V}_{\mathrm{rest},\mathrm{i}}\)), and on the right side to point D (\(V>{V}_{\mathrm{rest},\mathrm{i}}\)). a Time histories; b PSDs; c phase portraits; d pseudo-phase spaces

Full size image

two lasers are employed to measure the velocity of the shell at two target points; the ‘lower’ and the ‘upper’ measurement points located at \(\frac{1}{5}\) L and \(\frac{1}{3}\) L from the bottom of the shell, respectively (L being the length of the shells). However, the signal from the upper measurement point is used to obtain all the experimental results other than Fig. 20a. As discussed in Sect. 2, the amplitude of oscillations at \(V>{V}_{rest,i}\) undergoes frequent sudden reductions which make the oscillations look like intermittent bursts of oscillations, eventually leading to restabilization of the shell. This behaviour is clearly observed in the time history of oscillations at \(V>{V}_{rest,i}\) in Fig. 20a.

According to Figs. 20 and 21 , for point D, (1) the PSD consists of wide cone-like peaks, (2) the phase portrait displays orbits which tend to fill the phase plane,

Fig. 21
figure 21

Study of the unstable oscillations at \(V>{V}_{\mathrm{rest},\mathrm{i}}\) (comparison between points C and D in Fig. 16c). Plots on the left side correspond to C (\({V}_{\mathrm{rest},\mathrm{i}}\)), and on the right side to D (\(V>{V}_{\mathrm{rest},\mathrm{i}}\)). a PDF; b autocorrelations

Full size image

(3) the pseudo-phase space reconstruction shows a scattered cloud of points, (4) the PDF exhibits a Gaussian distribution, and (5) the autocorrelation dies out rapidly with time.

On the other hand, for point C, (1) the PSD contains relatively sharper spikes, (2) the phase portrait consists of relatively cleaner overlapping orbits, (3) the pseudo-phase space reconstruction tends to form a closed curve, (4) the PDF shows relatively less deviation from the double-masted shape towards Gaussian distribution, and (5) the autocorrelation demonstrates a statistical similarity between delayed versions of oscillations.

In conclusion, the oscillations associated with point C display a weaker chaotic behaviour. This is in agreement with the results of the largest Lyapunov exponent in Fig. 16c, that at point D, \({\lambda }_{1}\) is almost twice as large as for point C, implying less predictability of motions at point D.

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

Gholami, I., Amabili, M. & Païdoussis, M.P. Experiments on the dynamics and stability of cantilevered circular cylindrical shells conveying airflow. Nonlinear Dyn 111, 7005–7026 (2023). https://doi.org/10.1007/s11071-023-08245-9

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-08245-9

Keywords

Search

Navigation