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Exact Solutions for Free Vibration of Cylindrical Shells by a Symplectic Approach

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Abstract

Purpose

In engineering applications, the vibration analysis of cylindrical shells is highly significant for the design of engineering structures subjected to external dynamic excitations. In practice, the explicit expression would make a better and more quick design comparing to the numerical results. However, the studies on analytical method were concentrated on the semi-inverse method. To overcome the limitation, we introduce a new analytical treatment for the free vibration of cylindrical shells.

Methods

Under the framework of the Hamiltonian system, an analytical symplectic approach is introduced into the free vibration of cylindrical shells. In the symplectic space, exact solutions are obtained in a rational and systematic way without any trial functions.

Results

Analytical frequency equation for arbitrary boundary conditions is derived. Highly accurate natural frequencies and analytical vibration mode functions are obtained simultaneously.

Conclusion

In this paper, the governing equations for Donnell’s and Reissner’s shell theories can be successfully transformed into the Hamiltonian form by introducing a full state vector. The free vibration of cylindrical shells is reduced into a symplectic eigenproblem. Numerical results are compared with those previously published in literature and good agreement is observed. In addition, the proposed method can be extended to micro/nanoscale cylindrical shells.

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References

  1. Farshidianfar A, Oliazadeh P (2012) Free vibration analysis of circular cylindrical shells: comparison of different shell theories. Int J Mech Appl 25:74–80

    Google Scholar 

  2. Qatu MS (2002) Recent research advances in the dynamic behavior of shells: 1989-2000, Part 2: homogeneous shells. Appl Mech Rev 55:415–434

    Article  Google Scholar 

  3. El-Mously M (2003) Fundamental natural frequencies of thin cylindrical shells: a comparative study. J Sound Vib 264:1167–1186

    Article  Google Scholar 

  4. Qatu MS, Sullivan RW, Wang W (2010) Recent research advances on the dynamic analysis of composite shells: 2000–2009. Compos Struct 93:14–31

    Article  Google Scholar 

  5. Yang TLW, Dai L (2012) Vibrations of cylindrical shells. InTech, Croatia

    Book  Google Scholar 

  6. Leissa A (1993) Vibration of Shells. Acoustical society of America, New York

    Google Scholar 

  7. Forsberg K (1964) Influence of boundary conditions on the modal characteristics of thin cylindrical shells. AIAA J 2:2150–2157

    Article  Google Scholar 

  8. Haft EE, Smith BL (1968) Natural frequencies of clamped cylindrical shells. AIAA J 6:720–721

    Article  Google Scholar 

  9. Smith BL, Vronay DF (1970) Free vibration of circular cylindrical shells of finite length. AIAA J 8:601–603

    Article  Google Scholar 

  10. Zhang XM, Liu GR, Lam KY (2001) Vibration analysis of thin cylindrical shells using wave propagation approach. J Sound Vib 239:397–403

    Article  Google Scholar 

  11. Matsunaga H (2008) Free vibration and stability of functionally graded shallow shells according to a 2D higher-order deformation theory. Compos Struct 84:132–146

    Article  Google Scholar 

  12. Matsunaga H (2009) Free vibration and stability of functionally graded circular cylindrical shells according to a 2D higher-order deformation theory. Compos Struct 88:519–531

    Article  Google Scholar 

  13. Vel SS (2010) Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells. Compos Struct 92:2712–2727

    Article  Google Scholar 

  14. Xing Y, Liu B, Xu T (2013) Exact solutions for free vibration of circular cylindrical shells with classical boundary conditions. Int J Mech Sci 75:178–188

    Article  Google Scholar 

  15. Ye T, Jin G, Chen Y, Ma X, Su Z (2013) Free vibration analysis of laminated composite shallow shells with general elastic boundaries. Compos Struct 106:470–490

    Article  Google Scholar 

  16. Malekzadeh P, Farid M, Zahedinejad P (2008) A three-dimensional layerwise-differential quadrature free vibration analysis of laminated cylindrical shells. Int J Press Vessels Pip 85:450–458

    Article  Google Scholar 

  17. Tornabene F, Viola E (2008) Free vibration analysis of functionally graded panels and shells of revolution. Meccanica 44:255–281

    Article  MATH  Google Scholar 

  18. Tornabene F (2009) Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution. Comput Methods Appl Mech Eng 198:2911–2935

    Article  MATH  Google Scholar 

  19. Tornabene F, Viola E (2009) Free vibrations of four-parameter functionally graded parabolic panels and shells of revolution. Eur J Mech A Solids 28:991–1013

    Article  MATH  Google Scholar 

  20. Tornabene F (2011) 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution. Compos Struct 93:1854–1876

    Article  Google Scholar 

  21. Tornabene F (2011) Free vibrations of anisotropic doubly-curved shells and panels of revolution with a free-form meridian resting on Winkler-Pasternak elastic foundations. Compos Struct 94:186–206

    Article  Google Scholar 

  22. Tornabene F, Liverani A, Caligiana G (2011) FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: a 2-D GDQ solution for free vibrations. Int J Mech Sci 53:446–470

    Article  Google Scholar 

  23. Viola E, Tornabene F, Fantuzzi N (2013) General higher-order shear deformation theories for the free vibration analysis of completely doubly-curved laminated shells and panels. Compos Struct 95:639–666

    Article  Google Scholar 

  24. Toorani MH, Lakis AA (2000) General equations of anisotropic plates and shells including transverse shear deformations, rotary inertia and initial curvature effects. J Sound Vib 237:561–615

    Article  Google Scholar 

  25. Patel BP, Gupta SS, Loknath MS, Kadu CP (2005) Free vibration analysis of functionally graded elliptical cylindrical shells using higher-order theory. Compos Struct 69:259–270

    Article  Google Scholar 

  26. Kadoli R, Ganesan N (2006) Buckling and free vibration analysis of functionally graded cylindrical shells subjected to a temperature-specified boundary condition. J Sound Vib 289:450–480

    Article  Google Scholar 

  27. Zhao X, Lee YY, Liew KM (2009) Thermoelastic and vibration analysis of functionally graded cylindrical shells. Int J Mech Sci 51:694–707

    Article  Google Scholar 

  28. Lee H, Kwak MK (2015) Free vibration analysis of a circular cylindrical shell using the Rayleigh-Ritz method and comparison of different shell theories. J Sound Vib 353:344–377

    Article  Google Scholar 

  29. Cammalleri M, Costanza A (2016) A closed-form solution for natural frequencies of thin-walled cylinders with clamped edges. Int J Mech Sci 110:116–126

    Article  Google Scholar 

  30. Qu Y, Hua H, Meng G (2013) A domain decomposition approach for vibration analysis of isotropic and composite cylindrical shells with arbitrary boundaries. Compos Struct 95:307–321

    Article  Google Scholar 

  31. Qu Y, Long X, Yuan G, Meng G (2013) A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions. Compos B Eng 50:381–402

    Article  Google Scholar 

  32. Viswanathan KK, Kim KS, Lee JH, Koh HS, Lee JB (2009) Free vibration of multi-layered circular cylindrical shell with cross-ply walls, including shear deformation by using spline function method. J Mech Sci Technol 22:2062–2075

    Article  Google Scholar 

  33. Yao WA, Zhong WX, Lim CW (2009) Symplectic elasticity. World Scientific Publishing Co.Pte.Ltd, Singapore

    Book  MATH  Google Scholar 

  34. Lim CW, Xu XS (2010) Symplectic elasticity: theory and applications. Appl Mech Rev 63:050802

    Article  Google Scholar 

  35. Lim CW, Lü C, Xiang Y, Yao W (2009) On new symplectic elasticity approach for exact free vibration solutions of rectangular Kirchhoff plates. Int J Eng Sci 47:131–140

    Article  MathSciNet  MATH  Google Scholar 

  36. Zhou ZH, Wong KW, Xu XS, Leung AYT (2011) Natural vibration of circular and annular thin plates by Hamiltonian approach. J Sound Vib 330:1005–1017

    Article  Google Scholar 

  37. Lim CW (2010) Symplectic elasticity approach for free vibration of rectangular plates. Adv Vib Eng 9:159–163

    Google Scholar 

  38. Qiu WB, Zhou ZH, Xu XS (2016) The dynamic behavior of circular plates under impact loads. J Vib Eng Technol 4:111–116

    Google Scholar 

  39. Loy C, Lam K, Shu C (1997) Analysis of cylindrical shells using generalized differential quadrature. Shock Vib 4:193–198

    Article  Google Scholar 

  40. Eringen AC (1983) On differential equations of nonlocal elasticity and solutions of screw dislocation and surface waves. J Appl Phys 54:4703–4710

    Article  Google Scholar 

  41. Eringen AC (2002) Nonlocal continuum field theories. Springer-Verlag, New York

    MATH  Google Scholar 

  42. Li C (2016) On vibration responses of axially travelling carbon nanotubes considering nonlocal weakening effect. J Vib Eng Technol 4:175–181

    Google Scholar 

  43. Li C, Yu YM, Fan XL, Li S (2015) Dynamical characteristics of axially accelerating weak visco-elastic nanoscale beams based on a modified nonlocal continuum theory. J Vib Eng Technol 3:565–574

    MathSciNet  Google Scholar 

  44. Duan WH, Wang CM, Zhang YY (2007) Calibration of nonlocal scaling effect parameter for free vibration of carbon nanotubes by molecular dynamics. J Appl Phys 101:024305

    Article  Google Scholar 

  45. Wang CY, Zhang J, Fei YQ, Murmu T (2012) Circumferential nonlocal effect on vibrating nanotubules. Int J Mech Sci 58:86–90

    Article  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China [11672054, 11372070]; and the National Basic Research Program of China (973 Program) [2014CB046803].

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Authors and Affiliations

  1. State Key Laboratory of Structure Analysis of Industrial Equipment and Department of Engineering Mechanics, Dalian University of Technology, Dalian, 116024, People’s Republic of China

    Zhen Zhen Tong, Yi Wen Ni, Zhen Huan Zhou, Xin Sheng Xu, Sheng Bo Zhu & Xu Yuan Miao

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  1. Zhen Zhen Tong
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  2. Yi Wen Ni
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  3. Zhen Huan Zhou
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  4. Xin Sheng Xu
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Correspondence to Xin Sheng Xu.

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Tong, Z.Z., Ni, Y.W., Zhou, Z.H. et al. Exact Solutions for Free Vibration of Cylindrical Shells by a Symplectic Approach. J. Vib. Eng. Technol. 6, 107–115 (2018). https://doi.org/10.1007/s42417-018-0021-8

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  • DOI: https://doi.org/10.1007/s42417-018-0021-8

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