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Shock and Vibration
Volume 2016, Article ID 4097123, 30 pages
http://dx.doi.org/10.1155/2016/4097123
Research Article

A Unified Spectro-Geometric-Ritz Method for Vibration Analysis of Open and Closed Shells with Arbitrary Boundary Conditions

1College of Mechanical and Electrical Engineering, Harbin Engineering University, Harbin 150001, China
2College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China

Received 18 September 2015; Accepted 16 December 2015

Academic Editor: Tai Thai

Copyright © 2016 Dongyan Shi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. E. Carrera, “Theories and finite elements for multilayered, anisotropic, composite plates and shells,” Archives of Computational Methods in Engineering, vol. 9, no. 2, pp. 87–140, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  2. K. M. Liew, C. W. Lim, and S. Kitipornchai, “Vibration of shallow shells: a review with bibliography,” Applied Mechanics Reviews, vol. 50, no. 8, pp. 431–444, 1997. View at Publisher · View at Google Scholar · View at Scopus
  3. M. S. Qatu, R. W. Sullivan, and W. Wang, “Recent research advances on the dynamic analysis of composite shells: 2000–2009,” Composite Structures, vol. 93, no. 1, pp. 14–31, 2010. View at Publisher · View at Google Scholar · View at Scopus
  4. A. W. Leissa, Vibration of Shells, vol. 288, Scientific and Technical Information Office, National Aeronautics and Space Administration, Washington, DC, USA, 1973.
  5. J. N. Reddy, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, 2004.
  6. G. B. Warburton, “Vibration of thin cylindrical shells,” Journal of Mechanical Engineering Science, vol. 7, no. 4, pp. 399–407, 1965. View at Publisher · View at Google Scholar
  7. C. W. Bert and M. Malik, “Free vibration analysis of thin cylindrical shells by the differential quadrature method,” Journal of Pressure Vessel Technology, vol. 118, no. 1, pp. 1–12, 1996. View at Publisher · View at Google Scholar · View at Scopus
  8. K. Y. Lam and C. T. Loy, “Effects of boundary conditions on frequencies of a multi-layered cylindrical shell,” Journal of Sound and Vibration, vol. 188, no. 3, pp. 363–384, 1995. View at Publisher · View at Google Scholar · View at Scopus
  9. C. T. Loy and K. Y. Lam, “Vibration of thick cylindrical shells on the basis of three-dimensional theory of elasticity,” Journal of Sound and Vibration, vol. 226, no. 4, pp. 719–737, 1999. View at Publisher · View at Google Scholar · View at Scopus
  10. M. R. Mofakhami, H. H. Toudeshky, and S. H. Hashemi, “Finite cylinder vibrations with different end boundary conditions,” Journal of Sound and Vibration, vol. 297, no. 1-2, pp. 293–314, 2006. View at Publisher · View at Google Scholar · View at Scopus
  11. L. Dai, T. Yang, J. Du, W. L. Li, and M. J. Brennan, “An exact series solution for the vibration analysis of cylindrical shells with arbitrary boundary conditions,” Applied Acoustics, vol. 74, no. 3, pp. 440–449, 2013. View at Publisher · View at Google Scholar · View at Scopus
  12. Y. Chen, G. Jin, and Z. Liu, “Free vibration analysis of circular cylindrical shell with non-uniform elastic boundary constraints,” International Journal of Mechanical Sciences, vol. 74, pp. 120–132, 2013. View at Publisher · View at Google Scholar · View at Scopus
  13. S. M. R. Khalili, A. Davar, and K. Malekzadeh Fard, “Free vibration analysis of homogeneous isotropic circular cylindrical shells based on a new three-dimensional refined higher-order theory,” International Journal of Mechanical Sciences, vol. 56, no. 1, pp. 1–25, 2012. View at Publisher · View at Google Scholar · View at Scopus
  14. G. Jin, T. Ye, Y. Chen, Z. Su, and Y. Yan, “An exact solution for the free vibration analysis of laminated composite cylindrical shells with general elastic boundary conditions,” Composite Structures, vol. 106, pp. 114–127, 2013. View at Publisher · View at Google Scholar · View at Scopus
  15. E. Viola and F. Tornabene, “Vibration analysis of conical shell structures using GDQ method,” Far East Journal of Applied Mathematics, vol. 25, no. 1, pp. 23–39, 2006. View at Google Scholar · View at MathSciNet
  16. K. M. Liew, T. Y. Ng, and X. Zhao, “Free vibration analysis of conical shells via the element-free kp-Ritz method,” Journal of Sound and Vibration, vol. 281, no. 3–5, pp. 627–645, 2005. View at Publisher · View at Google Scholar · View at Scopus
  17. X. Zhao and K. M. Liew, “Free vibration analysis of functionally graded conical shell panels by a meshless method,” Composite Structures, vol. 93, no. 2, pp. 649–664, 2011. View at Publisher · View at Google Scholar · View at Scopus
  18. C. Shu, “An efficient approach for free vibration analysis of conical shells,” International Journal of Mechanical Sciences, vol. 38, no. 8-9, pp. 935–949, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  19. F. Tornabene, “Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution,” Computer Methods in Applied Mechanics and Engineering, vol. 198, no. 37, pp. 2911–2935, 2009. View at Publisher · View at Google Scholar · View at Scopus
  20. F. Tornabene, E. Viola, and D. J. Inman, “2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures,” Journal of Sound and Vibration, vol. 328, no. 3, pp. 259–290, 2009. View at Publisher · View at Google Scholar · View at Scopus
  21. C.-P. Wu and C.-Y. Lee, “Differential quadrature solution for the free vibration analysis of laminated conical shells with variable stiffness,” International Journal of Mechanical Sciences, vol. 43, no. 8, pp. 1853–1869, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  22. F. Tornabene and E. Viola, “Vibration analysis of spherical structural elements using the GDQ method,” Computers & Mathematics with Applications, vol. 53, no. 10, pp. 1538–1560, 2007. View at Publisher · View at Google Scholar · View at Scopus
  23. J. Lee, “Free vibration analysis of spherical caps by the pseudospectral method,” Journal of Mechanical Science and Technology, vol. 23, no. 1, pp. 221–228, 2009. View at Publisher · View at Google Scholar · View at Scopus
  24. B. P. Gautham and N. Ganesan, “Free vibration characteristics of isotropic and laminated orthotropic spherical caps,” Journal of Sound and Vibration, vol. 204, no. 1, pp. 17–40, 1997. View at Publisher · View at Google Scholar · View at Scopus
  25. J.-H. Kang and A. W. Leissa, “Three-dimensional vibrations of thick spherical shell segments with variable thickness,” International Journal of Solids and Structures, vol. 37, no. 35, pp. 4811–4823, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  26. Y. Qu, X. Long, S. Wu, and G. Meng, “A unified formulation for vibration analysis of composite laminated shells of revolution including shear deformation and rotary inertia,” Composite Structures, vol. 98, pp. 169–191, 2013. View at Publisher · View at Google Scholar · View at Scopus
  27. Y. Qu, X. Long, G. Yuan, and G. Meng, “A unified formulation for vibration analysis of functionally graded shells of revolution with arbitrary boundary conditions,” Composites Part B: Engineering, vol. 50, pp. 381–402, 2013. View at Publisher · View at Google Scholar · View at Scopus
  28. A. Selmane and A. A. Lakis, “Dynamic analysis of anisotropic open cylindrical shells,” Computers & Structures, vol. 62, no. 1, pp. 1–12, 1997. View at Publisher · View at Google Scholar · View at Scopus
  29. S. Kandasamy and A. V. Singh, “Free vibration analysis of skewed open circular cylindrical shells,” Journal of Sound and Vibration, vol. 290, no. 3, pp. 1100–1118, 2006. View at Publisher · View at Google Scholar · View at Scopus
  30. X. M. Zhang, G. R. Liu, and K. Y. Lam, “Frequency analysis of cylindrical panels using a wave propagation approach,” Applied Acoustics, vol. 62, no. 5, pp. 527–543, 2001. View at Publisher · View at Google Scholar · View at Scopus
  31. S. D. Yu, W. L. Cleghorn, and R. G. Fenton, “On the accurate analysis of free vibration of open circular cylindrical shells,” Journal of Sound and Vibration, vol. 188, no. 3, pp. 315–336, 1995. View at Publisher · View at Google Scholar · View at Scopus
  32. N. S. Bardell, J. M. Dunsdon, and R. S. Langley, “On the free vibration of completely free, open, cylindrically curved isotropic shell panels,” Journal of Sound and Vibration, vol. 207, no. 5, pp. 647–669, 1997. View at Publisher · View at Google Scholar · View at Scopus
  33. N. S. Bardell, J. M. Dunsdon, and R. S. Langley, “Free vibration of thin, isotropic, open, conical panels,” Journal of Sound and Vibration, vol. 217, no. 2, pp. 297–320, 1998. View at Publisher · View at Google Scholar · View at Scopus
  34. C. W. Lim and S. Kitipornchai, “Effects of subtended and vertex angles on the free vibration of open conical shell panels: a conical co-ordinate approach,” Journal of Sound and Vibration, vol. 219, no. 5, pp. 813–835, 1999. View at Publisher · View at Google Scholar · View at Scopus
  35. X. Zhao, Q. Li, K. M. Liew, and T. Y. Ng, “The element-free kp-Ritz method for free vibration analysis of conical shell panels,” Journal of Sound and Vibration, vol. 295, no. 3–5, pp. 906–922, 2006. View at Publisher · View at Google Scholar · View at Scopus
  36. K. M. Liew, L. X. Peng, and T. Y. Ng, “Three-dimensional vibration analysis of spherical shell panels subjected to different boundary conditions,” International Journal of Mechanical Sciences, vol. 44, no. 10, pp. 2103–2117, 2002. View at Publisher · View at Google Scholar · View at Scopus
  37. T. Ye, G. Jin, Y. Chen, and S. Shi, “A unified formulation for vibration analysis of open shells with arbitrary boundary conditions,” International Journal of Mechanical Sciences, vol. 81, pp. 42–59, 2014. View at Publisher · View at Google Scholar · View at Scopus
  38. M. S. Qatu and E. Asadi, “Vibration of doubly curved shallow shells with arbitrary boundaries,” Applied Acoustics, vol. 73, no. 1, pp. 21–27, 2012. View at Publisher · View at Google Scholar · View at Scopus
  39. S. Jiang, T. Yang, W. L. Li, and J. Du, “Vibration analysis of doubly curved shallow shells with elastic edge restraints,” Journal of Vibration and Acoustics, vol. 135, no. 3, Article ID 034502, 2013. View at Publisher · View at Google Scholar · View at Scopus
  40. F. Tornabene, E. Viola, and N. Fantuzzi, “General higher-order equivalent single layer theory for free vibrations of doubly-curved laminated composite shells and panels,” Composite Structures, vol. 104, pp. 94–117, 2013. View at Publisher · View at Google Scholar · View at Scopus
  41. F. A. Fazzolari and E. Carrera, “Advances in the Ritz formulation for free vibration response of doubly-curved anisotropic laminated composite shallow and deep shells,” Composite Structures, vol. 101, pp. 111–128, 2013. View at Publisher · View at Google Scholar · View at Scopus
  42. F. Tornabene, N. Fantuzzi, M. Bacciocchi, and E. Viola, “Higher-order theories for the free vibrations of doubly-curved laminated panels with curvilinear reinforcing fibers by means of a local version of the GDQ method,” Composites Part B: Engineering, vol. 81, pp. 196–230, 2015. View at Publisher · View at Google Scholar
  43. F. A. Fazzolari and E. Carrera, “Refined hierarchical kinematics quasi-3D Ritz models for free vibration analysis of doubly curved FGM shells and sandwich shells with FGM core,” Journal of Sound and Vibration, vol. 333, no. 5, pp. 1485–1508, 2014. View at Publisher · View at Google Scholar · View at Scopus
  44. T. Ye, G. Jin, Y. Chen, X. Ma, and Z. Su, “Free vibration analysis of laminated composite shallow shells with general elastic boundaries,” Composite Structures, vol. 106, pp. 470–490, 2013. View at Publisher · View at Google Scholar · View at Scopus
  45. N. M. Price, M. Liu, R. Eatock Taylor, and A. J. Keane, “Vibrations of cylindrical pipes and open shells,” Journal of Sound and Vibration, vol. 218, no. 3, pp. 361–387, 1998. View at Publisher · View at Google Scholar · View at Scopus
  46. C. W. Lim and K. M. Liew, “Vibratory behaviour of shallow conical shells by a global Ritz formulation,” Engineering Structures, vol. 17, no. 1, pp. 63–70, 1995. View at Publisher · View at Google Scholar · View at Scopus
  47. Y. K. Cheung, W. Y. Li, and L. G. Tham, “Free vibration analysis of singly curved shell by spline finite strip method,” Journal of Sound and Vibration, vol. 128, no. 3, pp. 411–422, 1989. View at Publisher · View at Google Scholar · View at Scopus
  48. Z. Su, G. Jin, S. Shi, T. Ye, and X. Jia, “A unified solution for vibration analysis of functionally graded cylindrical, conical shells and annular plates with general boundary conditions,” International Journal of Mechanical Sciences, vol. 80, pp. 62–80, 2014. View at Publisher · View at Google Scholar · View at Scopus
  49. D. Shi, Q. Wang, X. Shi, and F. Pang, “A series solution for the in-plane vibration analysis of orthotropic rectangular plates with non-uniform elastic boundary constraints and internal line supports,” Archive of Applied Mechanics, vol. 85, no. 1, pp. 51–73, 2015. View at Publisher · View at Google Scholar · View at Scopus
  50. D. Shi, Q. Wang, X. Shi, and F. Pang, “Free vibration analysis of moderately thick rectangular plates with variable thickness and arbitrary boundary conditions,” Shock and Vibration, vol. 2014, Article ID 572395, 25 pages, 2014. View at Publisher · View at Google Scholar
  51. D. Shi, Q. Wang, X. Shi, and F. Pang, “An accurate solution method for the vibration analysis of Timoshenko beams with general elastic supports,” Proceedings of the Institution of Mechanical Engineers Part C: Journal of Mechanical Engineering Science, 2014. View at Publisher · View at Google Scholar
  52. X. Shi, D. Shi, W. L. Li, and Q. Wang, “A unified method for free vibration analysis of circular, annular and sector plates with arbitrary boundary conditions,” Journal of Vibration and Control, 2014. View at Publisher · View at Google Scholar
  53. T. Ye, G. Jin, and Y. Zhang, “Vibrations of composite laminated doubly-curved shells of revolution with elastic restraints including shear deformation, rotary inertia and initial curvature,” Composite Structures, vol. 133, pp. 202–225, 2015. View at Publisher · View at Google Scholar
  54. X. Xie, H. Zheng, and G. Jin, “Free vibration of four-parameter functionally graded spherical and parabolic shells of revolution with arbitrary boundary conditions,” Composites Part B: Engineering, vol. 77, pp. 59–73, 2015. View at Publisher · View at Google Scholar
  55. X. Ma, G. Jin, S. Shi, T. Ye, and Z. Liu, “An analytical method for vibration analysis of cylindrical shells coupled with annular plate under general elastic boundary and coupling conditions,” Journal of Vibration and Control, 2015. View at Publisher · View at Google Scholar
  56. G. Jin, S. Shi, Z. Su, S. Li, and Z. Liu, “A modified Fourier–Ritz approach for free vibration analysis of laminated functionally graded shallow shells with general boundary conditions,” International Journal of Mechanical Sciences, vol. 93, pp. 256–269, 2015. View at Publisher · View at Google Scholar
  57. G.-Y. Jin, H. Chen, J.-T. Du, T.-J. Yang, and W.-Y. Li, “The influence of edge restraining stiffness on the transverse vibrations of rectangular plate structures,” Journal of Marine Science and Application, vol. 9, no. 4, pp. 393–402, 2010. View at Publisher · View at Google Scholar · View at Scopus
  58. G. Jin, Z. Su, S. Shi, T. Ye, and S. Gao, “Three-dimensional exact solution for the free vibration of arbitrarily thick functionally graded rectangular plates with general boundary conditions,” Composite Structures, vol. 108, no. 1, pp. 565–577, 2014. View at Publisher · View at Google Scholar · View at Scopus
  59. T. Ye, G. Jin, Z. Su, and X. Jia, “A unified Chebyshev-Ritz formulation for vibration analysis of composite laminated deep open shells with arbitrary boundary conditions,” Archive of Applied Mechanics, vol. 84, no. 4, pp. 441–471, 2014. View at Publisher · View at Google Scholar · View at Scopus
  60. G. Jin, T. Ye, X. Ma, Y. Chen, Z. Su, and X. Xie, “A unified approach for the vibration analysis of moderately thick composite laminated cylindrical shells with arbitrary boundary conditions,” International Journal of Mechanical Sciences, vol. 75, pp. 357–376, 2013. View at Publisher · View at Google Scholar · View at Scopus
  61. Z. Su, G. Jin, S. Shi, and T. Ye, “A unified accurate solution for vibration analysis of arbitrary functionally graded spherical shell segments with general end restraints,” Composite Structures, vol. 111, no. 1, pp. 271–284, 2014. View at Publisher · View at Google Scholar · View at Scopus
  62. G. Jin, T. Ye, X. Jia, and S. Gao, “A general Fourier solution for the vibration analysis of composite laminated structure elements of revolution with general elastic restraints,” Composite Structures, vol. 109, no. 1, pp. 150–168, 2014. View at Publisher · View at Google Scholar · View at Scopus
  63. Z. Su, G. Jin, and T. Ye, “Three-dimensional vibration analysis of thick functionally graded conical, cylindrical shell and annular plate structures with arbitrary elastic restraints,” Composite Structures, vol. 118, pp. 432–447, 2014. View at Publisher · View at Google Scholar
  64. T. Ye, G. Jin, Z. Su, and Y. Chen, “A modified Fourier solution for vibration analysis of moderately thick laminated plates with general boundary restraints and internal line supports,” International Journal of Mechanical Sciences, vol. 80, pp. 29–46, 2014. View at Publisher · View at Google Scholar · View at Scopus
  65. W. L. Li, “Free vibrations of beams with general boundary conditions,” Journal of Sound and Vibration, vol. 237, no. 4, pp. 709–725, 2000. View at Publisher · View at Google Scholar · View at Scopus
  66. W. L. Li, “Vibration analysis of rectangular plates with general elastic boundary supports,” Journal of Sound and Vibration, vol. 273, no. 3, pp. 619–635, 2004. View at Publisher · View at Google Scholar · View at Scopus
  67. S. Jiang, W. L. Li, and T. Yang, “A spectro-geometric method for the vibration analysis of built-up structures,” in Proceedings of the INTER-NOISE and NOISE-CON Congress and Conference Proceedings, Institute of Noise Control Engineering, 2013.