DOI QR코드

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The G. D. Q. method for the harmonic dynamic analysis of rotational shell structural elements

  • Viola, Erasmo (D.I.S.T.A.R.T. - Scienza delle Costruzioni, University of Bologna) ;
  • Artioli, Edoardo (D.I.S.T.A.R.T. - Scienza delle Costruzioni, University of Bologna)
  • 투고 : 2003.08.04
  • 심사 : 2004.01.13
  • 발행 : 2004.06.25

초록

This paper deals with the modal analysis of rotational shell structures by means of the numerical solution technique known as the Generalized Differential Quadrature (G. D. Q.) method. The treatment is conducted within the Reissner first order shear deformation theory (F. S. D. T.) for linearly elastic isotropic shells. Starting from a non-linear formulation, the compatibility equations via Principle of Virtual Works are obtained, for the general shell structure, given the internal equilibrium equations in terms of stress resultants and couples. These equations are subsequently linearized and specialized for the rotational geometry, expanding all problem variables in a partial Fourier series, with respect to the longitudinal coordinate. The procedure leads to the fundamental system of dynamic equilibrium equations in terms of the reference surface kinematic harmonic components. Finally, a one-dimensional problem, by means of a set of five ordinary differential equations, in which the only spatial coordinate appearing is the one along meridians, is obtained. This can be conveniently solved using an appropriate G. D. Q. method in meridional direction, yielding accurate results with an extremely low computational cost and not using the so-called "delta-point" technique.

키워드

참고문헌

  1. Artioli, E. and Viola, E. (2003), "On the harmonic elastic analysis of straight-meridian shells of revolution, by means of a G.D.Q. solution technique", Technical Report No. 100, DISTART, University of Bologna, Italy.
  2. Bellman, R. and Casti, J. (1971), "Differential quadrature and long-term integration", Journal Mathematics Analytic Applications, 34, 235-238. https://doi.org/10.1016/0022-247X(71)90110-7
  3. Bellman, R. and Casti, J. (1972), "Differential quadrature: a technique for the rapid solution of nonlinear partial differential equations", J. Comp. Ph., 10(1), 40-52. https://doi.org/10.1016/0021-9991(72)90089-7
  4. Bert, C.W. and Malik, M. (1996), "Differential quadrature method in computational mechanics: a review", Appl. Mech. Rev., 49, 1-27. https://doi.org/10.1115/1.3101882
  5. Bert, C.W. and Malik, M. (1996), "Free vibration analysis of thin cylindrical shells by the differential quadrature method", Journal of Pressure Vessel Technology, 118, 1-12. https://doi.org/10.1115/1.2842156
  6. Gould, P.L. (1999), Analysis of Shells and Plates, Upper Saddle River, Prentice Hall.
  7. Gould, P.L. (1985), Finite Element Analysis of Shells of Revolution, Pitman Advanced Publishing Program.
  8. Jiang, W. and Redekop, D. (2002), "Polar axisymmetric vibration of a hollow toroid using the differential quadrature method", J. Sound Vib., 251(4), 761-765. https://doi.org/10.1006/jsvi.2001.3865
  9. Kim, J.G. (1998), "A higher-order harmonic element for shells of revolution based on the modified mixed formulation", Ph.D. Thesis, Dept. of Mechanical Design and Production Engineering, Seoul National University.
  10. Kunieda, H. (1984), "Flexural axisymmetric free vibrations of a spherical dome: exact results and approximate solutions", J. Sound Vib., 92(1), 1-10. https://doi.org/10.1016/0022-460X(84)90368-7
  11. Lam, K.Y., Li, H. and Hua, L. (1997), "Vibration analysis of a rotating truncated circular conical shell", Int. J. Solids Struct., 34, 2183-2197. https://doi.org/10.1016/S0020-7683(96)00100-X
  12. Lam, K.Y., Li, H. and Hua, L. (2000), "Generalized differential quadrature for frequency of rotating multilayered conical shell", J. Eng. Mech., 126, 1156-1162. https://doi.org/10.1061/(ASCE)0733-9399(2000)126:11(1156)
  13. Li, H. and Lam, K.Y. (2001), "Orthotropic influence on frequency characteristics of a rotating composite laminated conical shell by the generalized differential quadrature method", Int. J. Solids Struct., 38, 3995- 4015. https://doi.org/10.1016/S0020-7683(00)00272-9
  14. Luah, M.H. and Fan, S.C. (1989), "General free vibration analysis of shells of revolution using the spline finite element method", Comput. Struct., 33(5), 1153-1162. https://doi.org/10.1016/0045-7949(89)90454-9
  15. Ng, T.Y., Hua, L. and Lam, K.Y. (2003), "Generalized differential quadrature for free vibration of rotating composite laminated conical shell with various boundary conditions", Int. J. Mech. Sci., 45, 567-587. https://doi.org/10.1016/S0020-7403(03)00042-0
  16. Ng, T.Y., Li, H., Lam, K.Y. and Chua, C.F. (2003), "Frequency analysis of rotating conical panels: a generalized differential quadrature approach", J. Appl. Mech., 70, 601-605. https://doi.org/10.1115/1.1577600
  17. Reddy, J.N. (1984), Energy and Variational Methods in Applied Mechanics, John Wiley & Sons.
  18. Redekop, D. and Xu, B. (1999), "Vibration analysis of toroidal panels using the differential quadrature method", Thin Walled Structures, 34, 217-231. https://doi.org/10.1016/S0263-8231(99)00010-5
  19. Reissner, E. and Wan, F.Y.M. (1967), "On stress strain relations and strain displacement relations of the linear theory of shells", The Folke-Odqvist Volume, 487-500.
  20. Reissner, E. (1969), "On the equations of non-linear shallow shell theory", Studies Appl. Math., 48, 171-175. https://doi.org/10.1002/sapm1969482171
  21. Sen, S.K. and Gould, P.L. (1974), "Free vibration of shells of revolution using FEM", J. the Eng. Mech. Div., ASCE, 100, 283-303.
  22. Shu, C. and Richards, B.E. (1992), "Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations", Int. J. Num. Meth. Fl., 15(3), 791-798. https://doi.org/10.1002/fld.1650150704
  23. Wu, T.Y. and Liu, G.R. (2000), "Axisymmetric bending solution of shells of revolution by the generalized differential quadrature rule", Int. J. Pressure Vessel and Piping, 77, 149-157. https://doi.org/10.1016/S0308-0161(00)00006-5

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  14. NURBS-Based Collocation Methods for the Structural Analysis of Shells of Revolution vol.6, pp.12, 2016, https://doi.org/10.3390/met6030068
  15. Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery vol.112, 2014, https://doi.org/10.1016/j.compstruct.2014.01.039
  16. 2-D GDQ solution for free vibrations of anisotropic doubly-curved shells and panels of revolution vol.93, pp.7, 2011, https://doi.org/10.1016/j.compstruct.2011.02.006
  17. FGM and laminated doubly curved shells and panels of revolution with a free-form meridian: A 2-D GDQ solution for free vibrations vol.53, pp.6, 2011, https://doi.org/10.1016/j.ijmecsci.2011.03.007