Archive of Applied Mechanics

, Volume 87, Issue 6, pp 961–988 | Cite as

A unified spectro-geometric-Ritz solution for free vibration analysis of conical–cylindrical–spherical shell combination with arbitrary boundary conditions

Original

Abstract

In the present article, a spectro-geometric-Ritz solution for free vibration analysis of conical–cylindrical–spherical shell combinations with arbitrary boundary conditions is presented. The classical theory of small displacements of thin shells is employed to formulate the theoretical model. The admissible functions of each shell components are described as a combination of a two-dimensional (2-D) Fourier cosine series and auxiliary functions. As an innovative point of this work, the auxiliary functions are introduced to accelerate the convergence of the series representations and eliminate all the relevant discontinuities with the displacement and its derivatives at the boundaries and the junction between the shell components. The artificial spring technique is adopted here to model the boundary condition and coupling condition, respectively. All the expansion coefficients are considered as the generalized coordinates and determined by Ritz procedure. Convergence and comparison studies for both open and closed conical–cylindrical–spherical shells with arbitrary boundary conditions are carried out to verify the reliability and accuracy of the present solutions. Some selected mode shapes are illustrated to enhance the understanding of the research topic. It is found the present method exhibits stable and rapid convergence characteristics, and the present results, including the natural frequencies and the mode shapes, agree closely with those solutions obtained from the finite element analyses and results in the literature. The effects of the geometrical dimensions of the shell combinations on the natural frequencies are also investigated.

Keywords

Vibration analysis Conical–cylindrical–spherical shell A spectro-geometric-Ritz method Arbitrary boundary conditions 

Notes

Acknowledgements

The authors gratefully acknowledge the financial supports from the National Natural Science Foundation of China (Nos. 51505096 and U1430236) and the Natural Science Foundation of Heilongjiang Province of China (No. E2016024).

Compliance with ethical standards

Conflicts of interest

The authors declare that they have no conflict of interest.

References

  1. 1.
    Leissa, A.W.: Vibration of Shells, vol. 288. Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington (1973)MATHGoogle Scholar
  2. 2.
    Liew, K., Lim, C., Kitipornchai, S.: Vibration of shallow shells: a review with bibliography. Appl. Mech. Rev. 50(8), 431–444 (1997)CrossRefGoogle Scholar
  3. 3.
    Qatu, M.S.: Recent research advances in the dynamic behavior of shells: 1989–2000, part 1: laminated composite shells. Appl. Mech. Rev. 55(4), 325–350 (2002)CrossRefGoogle Scholar
  4. 4.
    Qinkai, H., Fulei, C.: Effect of rotation on frequency characteristics of a truncated circular conical shell. Arch. Appl. Mech. 83(12), 1789–1800 (2013)CrossRefMATHGoogle Scholar
  5. 5.
    Civalek, Ö., Gürses, M.: Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique. Int. J. Press. Ves. Pip. 86(10), 677–683 (2009)CrossRefMATHGoogle Scholar
  6. 6.
    Talebitooti, M., Ghayour, M., Ziaei-Rad, S., Talebitooti, R.: Free vibrations of rotating composite conical shells with stringer and ring stiffeners. Arch. Appl. Mech. 80(3), 201–215 (2010)CrossRefMATHGoogle Scholar
  7. 7.
    Civalek, Ö.: The determination of frequencies of laminated conical shells via the discrete singular convolution method. J. Mech. Mater. Struct. 1(1), 163–182 (2006)CrossRefGoogle Scholar
  8. 8.
    Tornabene, F., Fantuzzi, N., Viola, E., Ferreira, A.: Radial basis function method applied to doubly-curved laminated composite shells and panels with a general higher-order equivalent single layer formulation. Compos. Pt. B-Eng. 55, 642–659 (2013)CrossRefGoogle Scholar
  9. 9.
    El Damatty, A., Saafan, M., Sweedan, A.: Dynamic characteristics of combined conical-cylindrical shells. Thin Wall Struct. 43(9), 1380–1397 (2005)CrossRefGoogle Scholar
  10. 10.
    Kalnins, A.: Free vibration of rotationally symmetric shells. J. Acoust. Soc. Am. 36(7), 1355–1365 (1964)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Rose, J., Mortimer, R., Blum, A.: Elastic-wave propagation in a joined cylindrical-conical-cylindrical shell. Exp. Mech. 13(4), 150–156 (1973)CrossRefGoogle Scholar
  12. 12.
    Hu, W.C., Raney, J.: Experimental and analytical study of vibrations of joined shells. AIAA J. 5(5), 976–980 (1967)CrossRefGoogle Scholar
  13. 13.
    Irie, T., Yamada, G., Muramoto, Y.: Free vibration of joined conical-cylindrical shells. J. Sound Vib. 95(1), 31–39 (1984)CrossRefGoogle Scholar
  14. 14.
    Galletly, G., Mistry, J.: The free vibrations of cylindrical shells with various end closures. Nucl. Eng. Des. 30(2), 249–268 (1974)CrossRefGoogle Scholar
  15. 15.
    Benjeddou, A.: Vibrations of complex shells of revolution using B-spline finite elements. Comput. Struct. 74(4), 429–440 (2000)CrossRefGoogle Scholar
  16. 16.
    Efraim, E., Eisenberger, M.: Exact vibration frequencies of segmented axisymmetric shells. Thin Wall Struct. 44(3), 281–289 (2006)CrossRefGoogle Scholar
  17. 17.
    Caresta, M., Kessissoglou, N.J.: Free vibrational characteristics of isotropic coupled cylindrical-conical shells. J. Sound Vib. 329(6), 733–751 (2010)CrossRefGoogle Scholar
  18. 18.
    Ma, X., Jin, G., Xiong, Y., Liu, Z.: Free and forced vibration analysis of coupled conical-cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 88, 122–137 (2014)CrossRefGoogle Scholar
  19. 19.
    Chen, M., Xie, K., Jia, W., Xu, K.: Free and forced vibration of ring-stiffened conical-cylindrical shells with arbitrary boundary conditions. Ocean Eng. 108, 241–256 (2015)CrossRefGoogle Scholar
  20. 20.
    Cheng, L., Nicolas, J.: Free vibration analysis of a cylindrical shell–circular plate system with general coupling and various boundary conditions. J. Sound Vib. 155(2), 231–247 (1992)CrossRefMATHGoogle Scholar
  21. 21.
    Lee, Y.-S., Choi, M.-H.: Free vibrations of circular cylindrical shells with an interior plate using the receptance method. J. Sound Vib. 248(3), 477–497 (2001)CrossRefGoogle Scholar
  22. 22.
    Liang, S., Chen, H.: The natural vibration of a conical shell with an annular end plate. J. Sound Vib. 294(4), 927–943 (2006)CrossRefGoogle Scholar
  23. 23.
    Redekop, D.: Vibration analysis of a torus-cylinder shell assembly. J. Sound Vib. 277(4), 919–930 (2004)CrossRefGoogle Scholar
  24. 24.
    Ma, X., Jin, G., Shi, S., Ye, T., Liu, Z.: An analytical method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions. J. Vib. Control 23(2), 305–328 (2015). doi: 10.1177/1077546315576301 MathSciNetCrossRefGoogle Scholar
  25. 25.
    Lee, Y.-S., Yang, M.-S., Kim, H.-S., Kim, J.-H.: A study on the free vibration of the joined cylindrical-spherical shell structures. Comput. Struct. 80(27), 2405–2414 (2002)CrossRefGoogle Scholar
  26. 26.
    Lee, J.: Free vibration analysis of a hermetic capsule by pseudospectral method. J. Mech. Sci. Technol. 26(4), 1011–1015 (2012)CrossRefGoogle Scholar
  27. 27.
    Wu, S., Qu, Y., Hua, H.: Vibrations characteristics of joined cylindrical-spherical shell with elastic-support boundary conditions. J. Mech. Sci. Technol. 27(5), 1265–1272 (2013)CrossRefGoogle Scholar
  28. 28.
    Tornabene, F., Brischetto, S., Fantuzzi, N., Viola, E.: Numerical and exact models for free vibration analysis of cylindrical and spherical shell panels. Compos. Pt. B-Eng. 81, 231–250 (2015)CrossRefGoogle Scholar
  29. 29.
    Qu, Y., Chen, Y., Long, X., Hua, H., Meng, G.: A modified variational approach for vibration analysis of ring-stiffened conical-cylindrical shell combinations. Eur. J. Mech. A-Solids 37, 200–215 (2013)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Qu, Y., Wu, S., Chen, Y., Hua, H.: Vibration analysis of ring-stiffened conical-cylindrical-spherical shells based on a modified variational approach. Int. J. Mech. Sci. 69, 72–84 (2013)CrossRefGoogle Scholar
  31. 31.
    Qu, Y., Hua, H., Meng, G.: Vibro-acoustic analysis of coupled spherical-cylindrical-spherical shells stiffened by ring and stringer reinforcements. J. Sound Vib. 355, 345–359 (2015)CrossRefGoogle Scholar
  32. 32.
    Bardell, N., Dunsdon, J., Langley, R.: Free vibration of thin, isotropic, open, conical panels. J. Sound Vib. 217(2), 297–320 (1998)CrossRefGoogle Scholar
  33. 33.
    Selmane, A., Lakis, A.A.: Dynamic analysis of anisotropic open cylindrical shells. Comput. Struct. 62(1), 1–12 (1997)CrossRefMATHGoogle Scholar
  34. 34.
    Lim, C., Kitipornchai, S.: Effects of subtended and vertex angles on the free vibration of open conical shell panels: a conical co-ordinate approach. J. Sound Vib. 219(5), 813–835 (1999)CrossRefGoogle Scholar
  35. 35.
    Fan, S., Luah, M.: Free vibration analysis of arbitrary thin shell structures by using spline finite element. J. Sound Vib. 179(5), 763–776 (1995)CrossRefGoogle Scholar
  36. 36.
    Kandasamy, S., Singh, A.V.: Free vibration analysis of skewed open circular cylindrical shells. J. Sound Vib. 290(3), 1100–1118 (2006)CrossRefGoogle Scholar
  37. 37.
    Zhang, X., Liu, G., Lam, K.: Frequency analysis of cylindrical panels using a wave propagation approach. Appl. Acoust. 62(5), 527–543 (2001)CrossRefGoogle Scholar
  38. 38.
    Yu, S., Cleghorn, W., Fenton, R.: On the accurate analysis of free vibration of open circular cylindrical shells. J. Sound Vib. 188(3), 315–336 (1995)CrossRefMATHGoogle Scholar
  39. 39.
    Ye, T., Jin, G., Chen, Y., Shi, S.: A unified formulation for vibration analysis of open shells with arbitrary boundary conditions. Int. J. Mech. Sci. 81, 42–59 (2014)CrossRefGoogle Scholar
  40. 40.
    Ye, T., Jin, G., Su, Z., Jia, X.: A unified Chebyshev-Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions. Arch. Appl. Mech. 84(4), 441–471 (2014)CrossRefMATHGoogle Scholar
  41. 41.
    Su, Z., Jin, G., Ye, T.: Free vibration analysis of moderately thick functionally graded open shells with general boundary conditions. Compos. Struct. 117, 169–186 (2014)CrossRefGoogle Scholar
  42. 42.
    Shi, D., Zhao, Y., Wang, Q., Teng, X., Pang, F.: A unified spectro-geometric-Ritz method for vibration analysis of open and closed shells with arbitrary boundary conditions. Shock Vib. 2016 (2016). doi: 10.1155/2016/4097123
  43. 43.
    Zhao, Y.-K., Shi, D.-Y., Wang, Q.-S., Meng, H.: Free vibration analysis of coupled open conical-cylindrical shells with arbitrary boundary conditions. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, vol. 6 (2015). Institute of Noise Control Engineering, pp. 839–850Google Scholar
  44. 44.
    Li, W.L.: Free vibrations of beams with general boundary conditions. J. Sound Vib. 237(4), 709–725 (2000)CrossRefGoogle Scholar
  45. 45.
    Li, W.L.: Comparison of Fourier sine and cosine series expansions for beams with arbitrary boundary conditions. J. Sound Vib. 255(1), 185–194 (2002). doi: 10.1006/jsvi.2001.4108 CrossRefGoogle Scholar
  46. 46.
    Li, W.L.: Vibration analysis of rectangular plates with general elastic boundary supports. J. Sound Vib. 273(3), 619–635 (2004)CrossRefGoogle Scholar
  47. 47.
    Du, J., Li, W.L., Jin, G., Yang, T., Liu, Z.: An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges. J. Sound Vib. 306(3–5), 908–927 (2007). doi: 10.1016/j.jsv.2007.06.011 CrossRefGoogle Scholar
  48. 48.
    Du, J.T., Liu, Z.G., Li, W.L., Zhang, X.F., Li, W.Y.: Free in-plane vibration analysis of rectangular plates with elastically point-supported edges. J. Vib. Acoust. Trans-ASME (2010). doi: 10.1115/1.4000777
  49. 49.
    Shi, D., Wang, Q., Shi, X., Pang, F.: A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Arch. Appl. Mech. 85(1), 51–73 (2015)CrossRefGoogle Scholar
  50. 50.
    Shi, X., Shi, D., Li, W.L., Wang, Q.: A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions. J. Vib. Control 22(2), 442–456 (2014). doi: 10.1177/1077546314533580 CrossRefGoogle Scholar
  51. 51.
    Jin, G., Ye, T., Chen, Y., Su, Z., Yan, Y.: An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions. Compos. Struct. 106, 114–127 (2013)CrossRefGoogle Scholar
  52. 52.
    Jin, G., Ye, T., Ma, X., Chen, Y., Su, Z., Xie, X.: A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions. Int. J. Mech. Sci. 75, 357–376 (2013)CrossRefGoogle Scholar
  53. 53.
    Chen, Y., Jin, G., Liu, Z.: Flexural and in-plane vibration analysis of elastically restrained thin rectangular plate with cutout using Chebyshev-Lagrangian method. Int. J. Mech. Sci. 89, 264–278 (2014). doi: 10.1016/j.ijmecsci.2014.09.006 CrossRefGoogle Scholar
  54. 54.
    Jin, G., Ma, X., Shi, S., Ye, T., Liu, Z.: A modified Fourier series solution for vibration analysis of truncated conical shells with general boundary conditions. Appl. Acoust. 85, 82–96 (2014). doi: 10.1016/j.apacoust.2014.04.007 CrossRefGoogle Scholar
  55. 55.
    Jin, G., Ye, T., Jia, X., Gao, S.: A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints. Compos. Struct. 109, 150–168 (2014)CrossRefGoogle Scholar
  56. 56.
    Shi, D., Wang, Q., Shi, X., Pang, F.: Free vibration analysis of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions. Shock Vib. 2014 (2014). doi: 10.1155/2014/572395
  57. 57.
    Shi, D., Wang, Q., Shi, X., Pang, F.: A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports. Arch. Appl. Mech. 85(1), 51–73 (2015)CrossRefGoogle Scholar
  58. 58.
    Chen, Y., Jin, G., Zhu, M., Liu, Z., Du, J., Li, W.L.: Vibration behaviors of a box-type structure built up by plates and energy transmission through the structure. J. Sound Vib. 331(4), 849–867 (2012)CrossRefGoogle Scholar
  59. 59.
    Dai, L., Yang, T., Du, J., Li, W., Brennan, M.: An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions. Appl. Acoust. 74(3), 440–449 (2013)CrossRefGoogle Scholar
  60. 60.
    Li, W.L., Bonilha, M.W., Xiao, J.: Vibrations of two beams elastically coupled together at an arbitrary angle. Acta Mech. Solida Sin. 25(1), 61–72 (2012)CrossRefGoogle Scholar
  61. 61.
    Xu, H., Du, J., Li, W.: Vibrations of rectangular plates reinforced by any number of beams of arbitrary lengths and placement angles. J. Sound Vib. 329(18), 3759–3779 (2010)CrossRefGoogle Scholar
  62. 62.
    Shi, D., Wang, Q., Shi, X., Pang, F.: An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports. Proc. Inst. Mech. Eng. C J. Mech. Eng. Sci. 229(13), 2327–2340 (2014). doi: 10.1177/0954406214558675
  63. 63.
    Jin, G., Shi, S., Su, Z., Li, S., Liu, Z.: A modified Fourier-Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions. Int. J. Mech. Sci. 93, 256–269 (2015)CrossRefGoogle Scholar
  64. 64.
    Xie, X., Zheng, H., Jin, G.: Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions. Compos. Pt. B-Eng. 77, 59–73 (2015)CrossRefGoogle Scholar
  65. 65.
    Su, Z., Jin, G., Shi, S., Ye, T., Jia, X.: A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions. Int. J. Mech. Sci. 80, 62–80 (2014)CrossRefGoogle Scholar
  66. 66.
    Li, W., Zhang, X., Du, J., Liu, Z.: An exact series solution for the transverse vibration of rectangular plates with general elastic boundary supports. J. Sound Vib. 321(1), 254–269 (2009)Google Scholar
  67. 67.
    Du, J., Li, W.L., Jin, G., Yang, T., Liu, Z.: An analytical method for the in-plane vibration analysis of rectangular plates with elastically restrained edges. J. Sound Vib. 306(3), 908–927 (2007)CrossRefGoogle Scholar
  68. 68.
    Jiang, S., Yang, T., Li, W., Du, J.: Vibration analysis of doubly curved shallow shells with elastic edge restraints. J. Vib. Acoust. 135(3), 034502 (2013)CrossRefGoogle Scholar
  69. 69.
    Jin, G., Su, Z., Ye, T., Gao, S.: Three-dimensional free vibration analysis of functionally graded annular sector plates with general boundary conditions. Compos. Pt. B-Eng. 83, 352–366 (2015)CrossRefGoogle Scholar
  70. 70.
    Ma, X., Jin, G., Xiong, Y., Liu, Z.: Dynamic analysis of ring stiffened conical-cylindrical shell combinations with general coupling and boundary conditions. Anal. Des. Mar. Struct. V, 365 (2015)Google Scholar
  71. 71.
    Jiang, S., Li, W.L., Yang, T.: A spectro-geometric method for the vibration analysis of built-up structures. In: INTER-NOISE and NOISE-CON Congress and Conference Proceedings, vol 1. Institute of Noise Control Engineering, pp. 948–953 (2013)Google Scholar
  72. 72.
    Shi, D., Shi, X., Li, W.L., Wang, Q.: Free transverse vibrations of orthotropic thin rectangular plates with arbitrary elastic edge supports. J. Vibroeng. 16(1), 389–398 (2014)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.College of Mechanical and Electrical EngineeringHarbin Engineering UniversityHarbinPeople’s Republic of China

Personalised recommendations