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Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance

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Abstract

This study is devoted to the experimental validation of a theoretical model of large amplitude vibrations of thin spherical shells described in a previous study by the same authors. A modal analysis of the structure is first detailed. Then, a specific mode coupling due to a 1:1:2 internal resonance between an axisymmetric mode and two companion asymmetric modes is especially addressed. The structure is forced with a simple-harmonic signal of frequency close to the natural frequency of the axisymmetric mode. The experimental setup, which allows precise measurements of the vibration amplitudes of the three involved modes, is presented. Experimental frequency response curves showing the amplitude of the modes as functions of the driving frequency are compared to the theoretical ones. A good qualitative agreement is obtained with the predictions given by in the model. Some quantitative discrepancies are observed and discussed, and improvements of the model are proposed.

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References

  1. Nayfeh, A.H.: Nonlinear Interactions. Wiley, New York (2000)

    MATH  Google Scholar 

  2. Touzé, C., Chaigne, A.: Lyapunov exponents from experimental time series: Application to cymbal vibrations. Acustica 86, 557–567 (2000)

    Google Scholar 

  3. Chaigne, A., Touzé, C., Thomas, O.: Nonlinear vibrations and chaos in gongs and cymbals. Acoust. Sci. Technol. 26(5), 403–409 (2005)

    Article  Google Scholar 

  4. Qatu, M.S.: Recent research advances in the dynamic behavior of shells: 1989–2000: Part 2. Homogeneous shells. ASME Appl. Mech. Rev. 55(5), 415–434 (2002)

    Google Scholar 

  5. 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. ASME Appl. Mech. Rev. 56(4), 349–381 (2003)

    Article  Google Scholar 

  6. Alhazza, K.A., Alhazza, A.A.: A review of the vibrations of plates and shells. Shock Vib. Dig. 36(5), 377–395 (2004)

    Article  Google Scholar 

  7. Thomas, O., Touzé, C., Chaigne, A.: Non-linear vibrations of free-edge thin spherical shells: Modal interaction rules and 1:1:2 internal resonance. Int. J. Sol. Struct. 42(11–12), 3339–3373 (2005)

    Google Scholar 

  8. Hoppmann, W.H., II, Baronet, C.N.: A study of the vibrations of shallow spherical shells. J. Appl. Mech. 30(3), 329–334 (1963)

    Google Scholar 

  9. Amabili, M., Pellegrini, M., Tommesani, M.: Experiments on largeamplitude vibrations of a circular cylindrical panel. J. Sound Vib. 260(3), 537–547 (2003)

    Article  Google Scholar 

  10. Amabili, M.: Theory and experiments for large-amplitude vibrations of empty and fluid-filled circular cylindrical shells with imperfections. J. Sound Vib. 262(4), 921–975 (2003)

    Article  Google Scholar 

  11. Evensen, H.A., Evan-Iwanowsky, R.M.: Dynamic response and stability of shallow spherical shells subject to time-dependant loading. AIAA J. 5(5), 969–976 (1967)

    Article  MATH  Google Scholar 

  12. Yasuda, K., Kushida, G.: Nonlinear forced oscillations of a shallow spherical shell. Bull. JSME 27(232), 2233–2240 (1984)

    MathSciNet  Google Scholar 

  13. Touzé, C., Thomas, O., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part 1: Theory. J. Sound Vib. 258(4), 649–676 (2002)

    Article  Google Scholar 

  14. Thomas, O., Touzé, C., Chaigne, A.: Asymmetric non-linear forced vibrations of free-edge circular plates. Part 2: Experiments. J. Sound Vib. 265(5), 1075–1101 (2003)

    Article  Google Scholar 

  15. Chu, H.-N., Herrmann, G.: Influence of large amplitudes on free flexural vibrations of rectangular elastic plates. J. Appl. Mech. 23, 532–540 (1956)

    MATH  MathSciNet  Google Scholar 

  16. Efstathiades, G.J.: A new approach to the large-deflection vibrations of imperfect circular disks using Galerkin’s procedure. J. Sound Vib. 16(2), 231–253 (1971)

    Article  MATH  Google Scholar 

  17. Nayfeh, A.H., Mook, D.T.: Nonlinear Oscillations. Wiley, New York (1979)

    MATH  Google Scholar 

  18. Touzé, C., Thomas, O.: Non-linear behaviour of free-edge shallow spherical shells: Effect of the geometry. Int. J. NonLinear Mech. (in press)

  19. Touzé, C., Thomas, O., Chaigne, A.: Hardening/softening behaviour in non-linear oscillations of structural systems using non-linear normal modes. J. Sound Vib. 273(1–2), 77–101 (2004)

    Google Scholar 

  20. Touzé, C., Thomas, O.: Type of non-linearity of shallow spherical shells using non-linear normal modes. In: Fifth Euromech Non-linear Dynamics Conference (ENOC 2005), Eindhoven, August 2005

  21. Touzé, C., Thomas, O., Huberdeau, A.: Asymptotic non-linear normal modes for large amplitude vibrations of continuous structures. Comput. Struct. 82(31–32), 2671–2682 (2004)

    Google Scholar 

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Authors and Affiliations

  1. Structural Mechanics and Coupled Systems Laboratory, CNAM, 2 rue Conté, 75003, Paris, France

    O. Thomas

  2. ENSTA-UME, Chemin de la Hunière, 91761, Palaiseau Cedex, France

    C. Touzé

Authors
  1. O. Thomas
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  2. C. Touzé
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  3. É. Luminais
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Correspondence to O. Thomas.

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Thomas, O., Touzé, C. & Luminais, É. Non-linear vibrations of free-edge thin spherical shells: Experiments on a 1:1:2 internal resonance. Nonlinear Dyn 49, 259–284 (2007). https://doi.org/10.1007/s11071-006-9132-y

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  • DOI: https://doi.org/10.1007/s11071-006-9132-y

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