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Nonlinear vibrations of truncated conical shells considering multiple internal resonances

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Abstract

The geometrically nonlinear vibration response of truncated thin conical shells is studied for the first time considering the one-to-one internal resonance, a phenomenon typically observed in symmetric structures such as conical shells. The Novozhilov nonlinear shell theory, retaining all nonlinear terms in the in-plane strain–displacement relationships of the three mid-surface displacements, is applied to study nonlinear vibrations of truncated conical shells. In-plane inertia is also taken into account, and a relatively large number of generalized coordinates, associated with the global discretization of the shell, is considered. This gives very accurate numerical solutions for simply supported, truncated thin conical shells. The effect of an exact one-to-one internal resonance, due to the axial symmetry of conical shells, is fully considered and the results are presented for different excitation levels. The numerical results show that also an almost exact one-to-one internal resonance with a mode presenting a different number of circumferential waves can also arise, which further complicates the nonlinear vibrations and leads to 1:1:1:1 internal resonance. The numerical model was augmented with additional generalized coordinates to capture this phenomenon. Pitchfork, Neimark–Sacker and period-doubling bifurcations of the forced vibration responses arising from internal resonances are detected, followed and presented, showing complex nonlinear dynamics.

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References

  1. Leissa, A.: Vibration of shells (NASA SP-288). US Government Printing Office, Washington, DC (1973). Reprinted by the Acoustical Society of America (1993)

  2. Lindholm, U.S., Hu, W.C.: Non-symmetric transverse vibrations of truncated conical shells. Int. J. Mech. Sci. 8(9), 561–579 (1966)

    Article  Google Scholar 

  3. Dong, S.B.: Free vibration of laminated orthotropic cylindrical shells. J. Acoust. Soc. Am. 44(6), 1628–1635 (1968)

    Article  Google Scholar 

  4. Irie, T., Yamada, G., Kaneko, Y.: Free vibration of a conical shell with variable thickness. J. Sound Vib. 82(1), 83–94 (1982)

    Article  Google Scholar 

  5. Irie, T., Yamada, G., Tanaka, K.: Natural frequencies of truncated conical shells. J. Sound Vib. 92, 447–453 (1984)

    Article  Google Scholar 

  6. Leissa, A.W., So, J.: Three-dimensional vibrations of truncated hollow cones. J. Vib. Control 1(2), 145–158 (1995). https://doi.org/10.1177/107754639500100202

    Article  Google Scholar 

  7. Liew, K., Ng, T., Zhao, X.: Free vibration analysis of conical shells via the element-free kp-Ritz method. J. Sound Vib. 281(3–5), 627–645 (2005)

    Article  Google Scholar 

  8. Firouz-Abadi, R., Rahmanian, M., Amabili, M.: Free vibration of moderately thick conical shells using a higher order shear deformable theory. J. Vib. Acoust. 136(5), 051001 (2014)

    Article  Google Scholar 

  9. Kerboua, Y., Lakis, A., Hmila, M.: Vibration analysis of truncated conical shells subjected to flowing fluid. Appl. Math. Model. 34(3), 791–809 (2010)

    Article  MathSciNet  Google Scholar 

  10. Sofiyev, A.H.: Review of research on the vibration and buckling of the FGM conical shells. Compos. Struct. 211, 301–317 (2019). https://doi.org/10.1016/j.compstruct.2018.12.047

    Article  Google Scholar 

  11. Sun, C.L., Lu, S.Y.: Nonlinear dynamic behavior of heated conical and cylindrical shells. Nucl. Eng. Des. 7(2), 113–122 (1968). https://doi.org/10.1016/0029-5493(68)90053-8

    Article  Google Scholar 

  12. Ueda, T.: Non-linear free vibrations of conical shells. J. Sound Vib. 64(1), 85–95 (1979)

    Article  Google Scholar 

  13. Evensen, D.A.: Nonlinear flexural vibrations of thin-walled circular cylinders. NASA TN D-4090, Washington, DC (1967)

  14. Dumir, P.: Nonlinear axisymmetric response of orthotropic thin truncated conical and spherical caps. Acta Mech. 60(1–2), 121–132 (1986)

    Article  Google Scholar 

  15. Xu, C., Xia, Z., Chia, C.: Non-linear theory and vibration analysis of laminated truncated, thick, conical shells. Int. J. Nonlinear Mech. 31(2), 139–154 (1996)

    Article  Google Scholar 

  16. Yonggang, W., Xinzhi, W., Huifang, S.: Nonlinear free vibration of orthotropic shallow shells of revolution under the static loads. Appl. Math. Mech. 18(6), 585–591 (1997)

    Article  Google Scholar 

  17. Fu, Y., Chen, C.: Non-linear vibration of elastic truncated conical moderately thick shells in large overall motion. Int. J. Nonlinear Mech. 36(5), 763–771 (2001)

    Article  Google Scholar 

  18. Krysko, V., Awrejcewicz, J., Shchekaturova, T.: Chaotic vibrations of spherical and conical axially symmetric shells. Arch. Appl. Mech. 74(5–6), 338–358 (2005)

    Article  Google Scholar 

  19. Sofiyev, A.: The non-linear vibration of FGM truncated conical shells. Compos. Struct. 94(7), 2237–2245 (2012)

    Article  MathSciNet  Google Scholar 

  20. Sofiyev, A.: Large-amplitude vibration of non-homogeneous orthotropic composite truncated conical shell. Compos. B Eng. 61, 365–374 (2014)

    Article  Google Scholar 

  21. Soıyev, A., Kuruoglu, N.: Large-amplitude vibration of the geometrically imperfect FGM truncated conical shell. J. Vib. Control 21(1), 142–156 (2015)

    Article  Google Scholar 

  22. Chan, D.Q., Quan, T.Q., Kim, S.-E., Duc, N.D.: Nonlinear dynamic response and vibration of shear deformable piezoelectric functionally graded truncated conical panel in thermal environments. Eur. J. Mech. A/Solids 77, 103795 (2019)

    Article  MathSciNet  Google Scholar 

  23. Ansari, R., Hasrati, E., Torabi, J.: Nonlinear vibration response of higher-order shear deformable FG-CNTRC conical shells. Compos. Struct. 222, 110906 (2019)

    Article  Google Scholar 

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

    Book  Google Scholar 

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

    Book  Google Scholar 

  26. Amabili, M.: A comparison of shell theories for large-amplitude vibrations of circular cylindrical shells: Lagrangian approach. J. Sound Vib. 264(5), 1091–1125 (2003)

    Article  Google Scholar 

  27. Amabili, M., Balasubramanian, P., Ferrari, G.: Travelling wave and non-stationary response in nonlinear vibrations of water-filled circular cylindrical shells: experiments and simulations. J. Sound Vib. 381, 220–245 (2016)

    Article  Google Scholar 

  28. Amabili, M.: Nonlinear vibrations of angle-ply laminated circular cylindrical shells: skewed modes. Compos. Struct. 94(12), 3697–3709 (2012)

    Article  Google Scholar 

  29. 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)

    Article  Google Scholar 

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

    Article  Google Scholar 

  31. Novozhilov, V.V.: Foundations of the Nonlinear Theory of Elasticity. Graylock Press, Rochester (1953)

    MATH  Google Scholar 

  32. Amabili, M.: Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections. J. Sound Vib. 262, 921–975 (2003)

    Article  Google Scholar 

  33. Amabili, M.: Non-linearities in rotation and thickness deformation in a new third-order thickness deformation theory for static and dynamic analysis of isotropic and laminated doubly curved shells. Int. J. Nonlinear Mech. 69, 109–128 (2015)

    Article  Google Scholar 

  34. Amabili, M.: Nonlinear damping in large-amplitude vibrations: modelling and experiments. Nonlinear Dyn. 93, 1–14 (2018)

    Article  Google Scholar 

  35. Amabili, M.: Nonlinear damping in nonlinear vibrations of rectangular plates: derivation from viscoelasticity and experimental validation. J. Mech. Phys. Solids 118, 275–292 (2018)

    Article  MathSciNet  Google Scholar 

  36. Balasubramanian, P., Ferrari, G., Amabili, M.: Identification of the viscoelastic response and nonlinear damping of a rubber plate in nonlinear vibration regime. Mech. Syst. Signal Process. 111, 376–398 (2018)

    Article  Google Scholar 

  37. Alijani, F., Amabili, M., Balasubramanian, P., Carra, S., Ferrari, G., Garziera, R.: Damping for large-amplitude vibrations of plates and curved panels, Part 1: modeling and experiments. Int. J. Nonlinear Mech. 85, 23–40 (2016)

    Article  Google Scholar 

  38. Doedel, E.J., Champneys, A.R., Fairgrieve, T.F., Kuznetsov, Y.A., Sandstede, B., Wang, X.: Continuation and bifurcation software for ordinary differential equations (with HomCont). AUTO97, Concordia University, Canada (1997)

  39. Amabili, M.: Internal resonances in non-linear vibrations of a laminated circular cylindrical shell. Nonlinear Dyn. 69, 755–770 (2012)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The authors acknowledge the financial support of the NSERC Discovery Grant and the Canada Research Chairs.

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Authors and Affiliations

  1. Canada Research Chair (Tier 1), Department of Mechanical Engineering, McGill University, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, PQ, H3A 0C3, Canada

    Marco Amabili

  2. Department of Mechanical Engineering, McGill University, Macdonald Engineering Building, 817 Sherbrooke Street West, Montreal, PQ, H3A 0C3, Canada

    Prabakaran Balasubramanian

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  1. Marco Amabili
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Correspondence to Marco Amabili.

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Amabili, M., Balasubramanian, P. Nonlinear vibrations of truncated conical shells considering multiple internal resonances. Nonlinear Dyn 100, 77–93 (2020). https://doi.org/10.1007/s11071-020-05507-8

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