Exact 3D solutions and finite element 2D models for free vibration analysis of plates and cylinders

References

[1] A.W. Leissa, Vibration of Plates, NASA SP-160, Washington, 1969. Search in Google Scholar

[2] A.W. Leissa, Vibration of Shells, NASA SP-288, Washington, 1973. Search in Google Scholar

[3] S. Werner, Vibrations of Shells and Plates, 3rd edition: revised and expanded, CRC Press, New York: Marcel Dekker Inc., 2004. Search in Google Scholar

[4] S. Brischetto and E. Carrera, Importance of higher order modes and refined theories in free vibration analysis of composite plates, Journal of Applied Mechanics, 77, 1-14, 2010. 10.1115/1.3173605Search in Google Scholar

[5] S. Brischetto, Exact elasticity solution for natural frequencies of functionally graded simply-supported structures, CMES: Computer Modeling in Engineering & Sciences, 95, 391-430, 2013. Search in Google Scholar

[6] S. Brischetto, A continuum elastic three-dimensional model for natural frequencies of single-walled carbon nanotubes, Composites Part B: Engineering, 61, 222-228, 2014. 10.1016/j.compositesb.2014.01.046Search in Google Scholar

[7] S. Brischetto, Three-dimensional exact free vibration analysis of spherical, cylindrical, and flat one-layered panels, Shock and Vibration, 2014, 1-29, 2014. 10.1155/2014/479738Search in Google Scholar

[8] MSC Nastran, Products (http://www.mscsoftware.com/ product/msc-nastran, accessed on 4th November 2014). Search in Google Scholar

[9] S. Aimmanee and R.C. Batra, Analytical solution for vibration of an incompressible isotropic linear elastic rectangular plate, and frequencies missed in previous solutions, Journal of Sound and Vibration, 302, 613-620, 2007. 10.1016/j.jsv.2006.11.029Search in Google Scholar

[10] R.C. Batra and S. Aimmanee, Letter to the Editor: Missing frequencies in previous exact solutions of free vibrations of simply supported rectangular plates, Journal of Sound and Vibration, 265, 887-896, 2003. 10.1016/S0022-460X(02)01568-7Search in Google Scholar

[11] S. Srinivas, C.V. Joga Rao and A.K. Rao, An exact analysis for vibration of simply-supported homogeneous and laminated thick rectangular plates, Journal of Sound and Vibration, 12, 187-199, 1970. 10.1016/0022-460X(70)90089-1Search in Google Scholar

[12] S. Srinivas, A.K. Rao and C.V.J. Rao, Flexure of simply supported thick homogeneous and laminated rectangular plates, Zeitschrift für Angewandte Mathematik und Mechanik, 49, 449- 458, 1969. 10.1002/zamm.19690490802Search in Google Scholar

[13] R.C. Batra, S. Vidoli and F. Vestroni, Plane wave solutions and modal analysis in higher order shear and normal deformable plate theories, Journal of Sound and Vibration, 257, 63-88, 2002. 10.1006/jsvi.2002.5029Search in Google Scholar

[14] J.Q. Ye, A three-dimensional free vibration analysis of cross-ply laminated rectangular plates with clamped edges, Computer Methods in Applied Mechanics and Engineering, 140, 383-392, 1997. 10.1016/S0045-7825(96)01112-7Search in Google Scholar

[15] A. Messina, Three dimensional free vibration analysis of crossply laminated plates through 2D and exact models, 3rd International Conference on Integrity, Reliability and Failure, Porto (Portugal), 20-24 July 2009. Search in Google Scholar

[16] Y.K. Cheung and D. Zhou, Three-dimensional vibration analysis of cantilevered and completely free isosceles triangular plates, International Journal of Solids and Structures, 39, 673- 687, 2002. 10.1016/S0020-7683(01)00196-2Search in Google Scholar

[17] K.M. Liew and B. Yang, Three-dimensional elasticity solutions for free vibrations of circular plates: a polynomials-Ritz analysis, Computer Methods in Applied Mechanics and Engineering, 175, 189-201, 1999. 10.1016/S0045-7825(98)00368-5Search in Google Scholar

[18] Y.B. Zhao, G.W. Wei and Y. Xiang, Discrete singular convolution for the prediction of high frequency vibration of plates, International Journal of Solids and Structures, 39, 65-88, 2002. 10.1016/S0020-7683(01)00183-4Search in Google Scholar

[19] G.W. Wei, Y.B. Zhao and Y. Xiang, A novel approach for the analysis of high-frequency vibrations, Journal of Sound and Vibration, 257, 207-246, 2002. 10.1006/jsvi.2002.5055Search in Google Scholar

[20] H. Rokni Damavandi Taher, M. Omidi, A.A. Zadpoor and A.A. Nikooyan, Short Communication: Free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions, Journal of Sound and Vibration, 296, 1084-1092, 2006. 10.1016/j.jsv.2006.03.022Search in Google Scholar

[21] Y. Xing and B. Liu, New exact solutions for free vibrations of rectangular thin plates by symplectic dual method, Acta Mechanica Sinica, 25, 265-270, 2009. 10.1007/s10409-008-0208-4Search in Google Scholar

[22] S.S. Vel and R.C. Batra, Three-dimensional exact solution for the vibration of functionally graded rectangular plates, Journal of Sound and Vibration, 272, 703-730, 2004. 10.1016/S0022-460X(03)00412-7Search in Google Scholar

[23] W.-Q. Chen, H.-J. Ding and R.-Q. Xu, On exact analysis of free vibrations of embedded transversely isotropic cylindrical shells, International Journal of Pressure Vessels and Piping, 75, 961- 966, 1998. 10.1016/S0308-0161(98)00060-XSearch in Google Scholar

[24] B. Gasemzadeh, R. Azarafza, Y. Sahebi and A. Motallebi, Analysis of free vibration of cylindrical shells on the basis of three dimensional exact elasticity theory, Indian Journal of Science and Technology, 5, 3260-3262, 2012. 10.17485/ijst/2012/v5i9.10Search in Google Scholar

[25] N.N. Huang, Exact analysis for three-dimensional free vibrations of cross-ply cylindrical and doubly-curved laminates, Acta Mechanica, 108, 23-34, 1995. 10.1007/BF01177325Search in Google Scholar

[26] J.N. Sharma, D.K. Sharma and S.S. Dhaliwal, Three-dimensional free vibration analysis of a viscothermoelastic hollow sphere, Open Journal of Acoustics, 2, 12-24, 2012. 10.4236/oja.2012.21002Search in Google Scholar

[27] J.N. Sharma and N. Sharma, Three-dimensional free vibration analysis of a homogeneous transradially isotropic thermoelastic sphere, Journal of Applied Mechanics, 77, 1-9, 2010. 10.1115/1.3172141Search in Google Scholar

[28] K.P. Soldatos and J. Ye, Axisymmetric static and dynamic analysis of laminated hollow cylinders composed of monoclinic elastic layers, Journal of Sound and Vibration, 184, 245-259, 1995. 10.1006/jsvi.1995.0315Search in Google Scholar

[29] A.E. Armenakas, D.C. Gazis and G. Herrmann, Free Vibrations of Circular Cylindrical Shells, Pergamon Press, Oxford, 1969. Search in Google Scholar

[30] A. Bhimaraddi, A higher order theory for free vibration analysis of circular cylindrical shells, International Journal of Solids and Structures, 20, 623-630, 1984. 10.1016/0020-7683(84)90019-2Search in Google Scholar

[31] H. Zhou, W. Li, B. Lv and W.L. Li, Free vibrations of cylindrical shellswith elastic-support boundary conditions, Applied Acoustics, 73, 751-756, 2012. 10.1016/j.apacoust.2012.02.008Search in Google Scholar

[32] S.M.R. Khalili, A. Davar and K.M. 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, 56, 1-25, 2012. 10.1016/j.ijmecsci.2011.11.002Search in Google Scholar

[33] S.S. Vel, Exact elasticity solution for the vibration of functionally graded anisotropic cylindrical shells, Composite Structures, 92, 2712-2727, 2010. 10.1016/j.compstruct.2010.03.012Search in Google Scholar

[34] 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, 226, 719-737, 1999. 10.1006/jsvi.1999.2310Search in Google Scholar

[35] Y.Wang, R. Xu, H. Ding and J. Chen, Three-dimensional exact solutions for free vibrations of simply supportedmagneto-electroelastic cylindrical panels, International Journal of Engineering Science, 48, 1778-1796, 2010. 10.1016/j.ijengsci.2010.09.022Search in Google Scholar

[36] E. Efraim and M. Eisenberger, Exact vibration frequencies of segmented axisymmetric shells, Thin-Walled Structures, 44, 281-289, 2006. 10.1016/j.tws.2006.03.006Search in Google Scholar

[37] J.-H. Kanga and A.W. Leissa, Three-dimensional vibrations of thick spherical shell segments with variable thickness, International Journal of Solids and Structures, 37, 4811-4823, 2000. 10.1016/S0020-7683(99)00182-1Search in Google Scholar

[38] 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, 44, 2103-2117, 2002. 10.1016/S0020-7403(02)00167-4Search in Google Scholar

[39] F.B. Hildebrand, E. Reissner and G.B. Thomas, Notes on the Foundations of the Theory of Small Displacements of Orthotropic Shells, NACA Technical Note No. 1833, Washington, 1949. Search in Google Scholar

[40] F. Tornabene, Meccanica delle Strutture a Guscio in Materiale Composito, Societá Editrice Esculapio, Bologna (Italy), 2012. Search in Google Scholar

[41] W. Soedel, Vibration of Shells and Plates, Marcel Dekker, Inc., New York, 2004. 10.4324/9780203026304Search in Google Scholar

[42] G.B. Gustafson, Systems of Differential Equations, free available on http://www.math.utah.edu/gustafso/, accessed on 16th September 2014. Search in Google Scholar

[43] W.E. Boyce and R.C. DiPrima, Elementary Differential Equations and Boundary Value Problems, John Wiley & Sons, Ltd., New York, 2001. Search in Google Scholar

[44] D. Zwillinger, Handbook of Differential Equations, Academic Press, New York, 1997. Search in Google Scholar

[45] C. Molery and C. Van Loan, Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later, SIAM Review, 45, 1-46, 2003. 10.1137/S00361445024180Search in Google Scholar