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An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells

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Abstract

Using Donnell non-linear shallow shell equations in terms of the displacements and the potential flow theory, this work presents a qualitatively accurate low dimensional model to study the non-linear dynamic behavior and stability of a fluid-filled cylindrical shell under lateral pressure and axial loading. First, the reduced order model is derived taking into account the influence of the driven and companion modes. For this, a modal solution is obtained by a perturbation technique which satisfies exactly the in-plane equilibrium equations and all boundary, continuity, and symmetry conditions. Finally, the equation of motion in the transversal direction is discretized by the Galerkin method. The importance of each mode in the proposed modal expansion is studied using the proper orthogonal decomposition. The quality of the proposed model is corroborated by studying the convergence of frequency–amplitude relations, resonance curves, bifurcation diagrams, and time responses. The parametric analysis clarifies the influence of the lateral and axial loads on the non-linear vibrations and stability of the liquid-filled shell. Finally, the global response of the system is investigated in order to quantify the degree of safety of the shell in the presence of external perturbations through the use of bifurcation diagrams and basins of attraction. This allows one to evaluate the safety and dynamic integrity of the cylindrical shell in a dynamic environment.

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References

  1. Evensen, D.A.: Some observations on the nonlinear vibration of thin cylindrical shells. AIAA J. 1, 2857–2858 (1963)

    Article  Google Scholar 

  2. Chen, J.C., Babcock, C.D.: Nonlinear vibration of cylindrical shells. AIAA J. 13, 868–876 (1975)

    Article  MATH  Google Scholar 

  3. 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, 655–699 (2003)

    Article  Google Scholar 

  4. Kubenko, V.D., Koval’chuk, P.S.: Nonlinear problems of the vibration of thin shells (review). Int. Appl. Mech. 34, 703–728 (1998)

    Article  MathSciNet  Google Scholar 

  5. Amabili, M.: Nonlinear Vibrations and Stability of Shell and Plates. Cambridge University Press, New York (2008)

    Book  Google Scholar 

  6. 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)

    Article  MATH  Google Scholar 

  7. Shaw, S.W., Pierre, C.: Normal modes for nonlinear vibratory systems. J. Sound Vib. 164, 85–124 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  8. Vakakis, A.F.: Nonlinear normal modes (NNMs) and their applications in vibration theory: an overview. Mech. Syst. Signal Process. 11, 3–22 (1997)

    Article  Google Scholar 

  9. Steindl, A., Troger, H.: Methods for dimension reduction and their applications in nonlinear dynamics. Int. J. Solids Struct. 38, 2131–2147 (2001)

    Article  MATH  Google Scholar 

  10. Rega, G., Troger, H.: Dimension reduction of dynamical systems: methods, models, applications. Nonlinear Dyn. 41, 1–15 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  11. Amabili, M., Sarkar, A., Païdoussis, M.P.: Reduced-order models for nonlinear vibrations of cylindrical shells via the proper orthogonal decomposition method. J. Fluids Struct. 18, 227–250 (2003)

    Article  Google Scholar 

  12. Amabili, M., Sarkar, A., Païdoussis, M.P.: Chaotic vibrations of circular cylindrical shells: Galerkin versus reduced-order models via the proper orthogonal decomposition method. J. Sound Vib. 290, 736–762 (2006)

    Article  Google Scholar 

  13. Amabili, M., Touzé, C.: Reduced-order models for nonlinear vibrations of fluid-filled circular cylindrical shells: comparison of POD and asymptotic nonlinear normal modes methods. J. Fluids Struct. 23, 885–903 (2007)

    Article  Google Scholar 

  14. Touzé, C., Amabili, M.: Nonlinear normal modes for damped geometrically nonlinear systems: application to reduced-order modeling of harmonically forced structures. J. Sound Vib. 298, 958–981 (2006)

    Article  Google Scholar 

  15. Gonçalves, P.B., Batista, R.C.: Nonlinear vibration analysis of fluid-filled cylindrical shells. J. Sound Vib. 127, 133–143 (1988)

    Article  Google Scholar 

  16. Silva, F.M.A.: Low dimensional models for nonlinear vibration and stability analysis of cylindrical shells. D.Sc. thesis, Catholic University of Rio de Janeiro, PUC-Rio, Rio de Janeiro, Brazil (2008)

  17. 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, 641–663 (2008)

    Article  Google Scholar 

  18. Mallon, N.J., Fey, R.H.B., Nijmeijer, H.: Dynamic stability of a thin cylindrical shell with top mass subjected to harmonic base-acceleration. Int. J. Solids Struct. 45, 1587–1613 (2008)

    Article  MATH  Google Scholar 

  19. 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, 3149–3170 (2010)

    Article  Google Scholar 

  20. Thompson, J.M.T.: Chaotic behavior triggering the escape from a potential well. Proc. R. Soc. Lond. Ser. A 421, 195–225 (1989)

    Article  MATH  Google Scholar 

  21. Soliman, M.S., Thompson, J.M.T.: Integrity measures quantifying the erosion of smooth and fractal basins of attraction. J. Sound Vib. 135, 453–475 (1989)

    Article  MathSciNet  Google Scholar 

  22. Rega, G., Lenci, S.: Identifying, evaluating and controlling dynamical integrity measures in non-linear mechanical oscillators. Nonlinear Anal. 63, 902–914 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  23. Gonçalves, P.B., Silva, F.M.A., Rega, G., Lenci, S.: Global dynamics and integrity of a two-dof model of a parametrically excited cylindrical shell. Nonlinear Dyn. 63, 61–82 (2011)

    Article  MATH  Google Scholar 

  24. Gonçalves, P.B., Del Prado, Z.J.G.N.: Low-dimensional Galerkin models for nonlinear vibration and instability analysis of cylindrical shells. Nonlinear Dyn. 41, 129–145 (2005)

    Article  MATH  Google Scholar 

  25. ABAQUS. Standard user’s manual. Version 6.2, Hibbit, Karlsson, & Sorensen, Inc. USA (2001)

  26. Gonçalves, P.B., Del Prado, Z.J.G.N.: Effect of nonlinear modal interaction on the dynamic instability of axially excited cylindrical shells. Comput. Struct. 82, 2621–2634 (2004)

    Article  Google Scholar 

  27. Amabili, M., Pellicano, F., Païdoussis, M.P.: Nonlinear vibrations of simply supported, circular cylindrical shells, coupled to quiescent fluid. J. Fluids Struct. 12, 883–918 (1998)

    Article  Google Scholar 

  28. Kim, Y.W., Lee, Y.S., Ko, S.H.: Coupled vibration of partially fluid-filled cylindrical shells with ring stiffeners. J. Sound Vib. 276, 869–897 (2004)

    Article  Google Scholar 

  29. Li, D., Xu, J.: A new method to determine the periodic orbit of nonlinear dynamic system and its period. Eng. Comput. 20, 316–322 (2005)

    Article  Google Scholar 

  30. Wolter, C.: An introduction to model reduction based on Karhunen-Loève expansions, M.Sc. dissertation, Catholic University of Rio de Janeiro, PUC-Rio, Rio de Janeiro, Brazil (2001)

  31. Allgower, E., Georg, K.: Numerical Continuation Methods. Springer, Berlin (1990)

    Book  MATH  Google Scholar 

  32. Pellicano, F., Amabili, M.: Stability and vibration of empty and fluid-filled circular cylindrical shells under static and periodic axial loads. Int. J. Solids Struct. 40, 3229–3251 (2003)

    Article  MATH  Google Scholar 

  33. Lansbury, A.N., Thompson, J.M.T., Stewart, H.B.: Basin erosion in the twin-well Duffing oscillator: two distinct bifurcation scenarios. Int. J. Bifurc. Chaos 2, 505–532 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  34. Soliman, M.S., Thompson, J.M.T.: Global dynamics underlying sharp basin erosion in nonlinear driven oscillators. Phys. Rev. A 45, 3425–3431 (1992)

    Article  Google Scholar 

  35. Batista, R.C., Gonçalves, P.B.: Nonlinear lower bounds for shell buckling design. J. Constr. Steel Res. 29, 101–120 (1994)

    Article  Google Scholar 

  36. Catellani, G., Pellicano, F., Dall’Asta, D., Amabili, M.: Parametric instability of a circular cylindrical shell with geometric imperfections. Compos. Struct. 82, 2635–2645 (2004)

    Article  Google Scholar 

  37. Amabili, M., Pellicano, F.: Multi-mode approach to nonlinear supersonic flutter of imperfect circular cylindrical shells. J. Appl. Mech. 69, 117–129 (2002)

    Article  MATH  Google Scholar 

  38. Pellicano, F., Amabili, M.: Dynamic instability and chaos of empty and fluid-filled circular cylindrical shells under periodic axial loads. J. Sound Vib. 293, 227–252 (2006)

    Article  Google Scholar 

  39. Del Prado, Z.J.G.N., Gonçalves, P.B., Païdoussis, M.P.: Nonlinear vibrations and imperfection sensitivity of a cylindrical shell containing axial fluid flow. J. Sound Vib. 327, 211–230 (2009)

    Article  Google Scholar 

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Authors and Affiliations

  1. Civil Engineering School, Federal University of Goiás, UFG, 74605-220, Goiânia, GO, Brazil

    Frederico M. A. Silva & Zenón J. G. N. del Prado

  2. Civil Engineering Department, Pontifical Catholic University of Rio de Janeiro, PUC-Rio, 22451-900, Rio de Janeiro, RJ, Brazil

    Paulo B. Gonçalves

Authors
  1. Frederico M. A. Silva
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  2. Paulo B. Gonçalves
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  3. Zenón J. G. N. del Prado
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Correspondence to Paulo B. Gonçalves.

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Silva, F.M.A., Gonçalves, P.B. & del Prado, Z.J.G.N. An alternative procedure for the non-linear vibration analysis of fluid-filled cylindrical shells. Nonlinear Dyn 66, 303–333 (2011). https://doi.org/10.1007/s11071-011-0037-z

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  • DOI: https://doi.org/10.1007/s11071-011-0037-z

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