Free Vibration of Annular Plates by Discrete Singular Convolution and Differential Quadrature Methods

Document Type : Research Paper


1 Akdeniz University Civil ENG.DEPT.

2 Akdeniz University Mechanical Engineering Dept.

3 Civil Engineering Dept.


Plates and shells are significant structural components in many engineering and industrial applications. In this study, the free vibration analysis of annular plates is investigated. To this aim, two different numerical methods including the differential quadrature and the discrete singular convolution methods are performedfor numerical simulations. Moreover, the Frequency values are obtained via these two methods and finally, the performance of these methods is investigated.


Main Subjects

[1]     Qatu, M., Vibration of Laminated Shells and Plates, Academic Press, U.K., 2004.
[2]     Soedel, W., Vibrations of shells and plates, Third Edition, CRC Press, 2004.
[3]     Leissa, A.W., Vibration of shells, Acoustical Society of America, 1993.
[4]     Reddy, J.N., Mechanics of laminated composite plates and shells: theory and analysis. (2nd ed.) New York: CRC Press, 2003.
[5]     Tornabene, F., Fantuzzi, N., “Mechanics of laminated composite doubly-curved shell structures, the generalized differential quadrature method and the strong formulation finite element method”, Società Editrice Esculapio, 2014.
[6]     Leissa AW., Vibration analysis of plates. NASA, SP-160, 1969.
[7]     Blevins R.D., “Formulas for natural frequency and mode shapes.” Malabur-Florida: Robert E. Krieger, 1984.
[8]     Vogel, S.M. and Skinner, D.W., “Natural frequencies of transversely vibrating uniform annular plates”, J. Appl. Meh., Vol. 32, pp. 926-931, 1965.
[9]     Civalek, Ö., “Finite Element analyses of plates and shells”. Elazığ: Fırat University, (in Turkish) 1998.
[10] Wei, G.W., 2001, “Vibration analysis by discrete singular convolution, Journal of Sound and Vibration”, Vol. 244, pp. 535-553, 2001.
[11] Wei, G.W., “Discrete singular convolution for beam analysis, Engineering Structures”, Vol. 23, pp.1045-1053, 2001.
[12] Wei, G.W., Zhou Y.C., Xiang, Y., “Discrete singular convolution and its application to the analysis of plates with internal supports. Part 1: Theory and algorithm”, Int J Numer Methods Eng., Vol. 55, pp.913-946, 2002.
[13] Wei, G.W., Zhou Y.C., Xiang, Y., “The determination of natural frequencies of rectangular plates with mixed boundary conditions by discrete singular convolution”, Int J Mechanical Sciences, Vol. 43, pp.1731-1746, 2001.
[14] Wei, G.W., “A new algorithm for solving some mechanical problems”, Comput. Methods Appl. Mech. Eng., Vol. 190, pp. 2017-2030, 2001.
[15] Lim, C.W., Li Z.R., and Wei, G.W.,  “DSC-Ritz method for high-mode frequency analysis of thick shallow shells”, International Journal for Numerical Methods in Engineering, Vol. 62, pp.205-232, 2005.
[16] Civalek, Ö., “An efficient method for free vibration analysis of rotating truncated conical shells”, Int. J. Pressure Vessels and Piping, Vol. 83, pp. 1-12, 2006.
[17] Civalek, Ö., Gürses, M. “Free vibration analysis of rotating cylindrical shells using discrete singular convolution technique”. Int J Press Vessel Pip, Vol. 86, No. 10, pp. 677-683, 2009.
[18] Civalek, Ö., “Vibration analysis of laminated composite conical shells by the method of discrete singular convolution based on the shear deformation theory”, Compos Part B Eng, Vol. 45, No. 1, pp. 1001-1009, 2013.
[19] Civalek, Ö., “Free vibration analysis of composite conical shells using the discrete singular convolution algorithm”, Steel Compos Struct, Vol. 6, No. 4, pp.353-366, 2006.
[20] Civalek, Ö., “The determination of frequencies of laminated conical shells via the discrete singular convolution method”, J Mech Mater Struct, Vol. 1, No. 1, pp. 163-182, 2006.
[21] Civalek, Ö., “A four-node discrete singular convolution for geometric transformation and its application to numerical solution of vibration problem of arbitrary straight-sided quadrilateral plates”, Appl Math Model, Vol. 33, pp. 300–314, 2009.
[22] Civalek, Ö., “Three-dimensional vibration, buckling and bending analyses of thick rectangular plates based on discrete singular convolution method”, Int J Mech Sci, Vol. 49, No.6, pp.752-765, 2007.
[23] Demir, Ç., Mercan, K., Civalek, Ö., “Determination of critical buckling loads of isotropic, FGM and laminated truncated conical panel”, Compos Part B: Eng Vol. 94, pp. 1-10, 2016.
[24] Civalek, Ö., “Fundamental frequency of isotropic and orthotropic rectangular plates with linearly varying thickness by discrete singular convolution method”, Appl Math Model, Vol. 33, No. 10, pp. 3825-3835, 2009.
[25] Civalek, Ö., “Free vibration analysis of symmetrically laminated composite plates with first-order shear deformation theory (FSDT) by discrete singular convolution method”, Finite Elem Anal Des Vol. 44, pp. 725-731, 2008.
[26] Civalek, Ö., “Vibration analysis of conical panels using the method of discrete singular convolution”, Commun Numer Methods Eng, Vol. 24, pp. 169-181, 2008.
[27] Civalek, Ö., Korkmaz, A, Demir, Ç., “Discrete singular convolution approach for buckling analysis of rectangular Kirchhoff plates subjected to compressive loads on two opposite edges”. Adv Eng Softw, Vol. 41, pp. 557-560, 2010.
[28] Civalek, Ö., “Analysis of thick rectangular plates with symmetric cross-ply laminates based on first-order shear deformation theory”, J Compos Mater, Vol. 42, pp. 2853–2867, 2008.
[29] Seçkin, A., Sarıgül, A.S., “Free vibration analysis of symmetrically laminated thin composite plates by using discrete singular convolution (DSC) approach: algorithm and verification”, J Sound Vib, Vol. 315, pp. 197-211, 2008.
[30] Seçkin, A., “Modal and response bound predictions of uncertain rectangular composite plates based on an extreme value model”, J Sound Vib, Vol. 332, pp.1306-1323, 2013.
[31] Xin, L., Hu, Z., “Free vibration of layered magneto-electro-elastic beams by SSDSC approach”, Compos Struct, Vol. 125, pp. 96-103, 2015.
[32] Xin. L., Hu, Z., “Free vibration of simply supported and multilayered magnetoelectro-elastic plates”, Comp Struct, Vol. 121, pp. 344-350, 2015.
[33] Wang, X., Xu, S., “Free vibration analysis of beams and rectangular plates with free edges by the discrete singular convolution”, J Sound Vib, Vol. 329, pp. 1780-1792, 2010.
[34] Wang, X., Wang, Y., Xu, S., “DSC analysis of a simply supported anisotropic rectangular plate”, Compos Struct, Vol. 94, pp. 2576-2584, 2012.
[35] Duan, G., Wang, X., Jin, C., “Free vibration analysis of circular thin plates with stepped thickness by the DSC element method”, Thin Walled Struct, Vol. 85, pp. 25-33, 2014.
[36] Baltacıoğlu, A.K., Civalek, Ö., Akgöz, B., Demir, F., “Large deflection analysis of laminated composite plates resting on nonlinear elastic foundations by the method of discrete singular convolution”, Int J Pres Vessel Pip, Vol. 88, pp. 290-300, 2011.
[37] Civalek, Ö., Akgöz, B., “Vibration analysis of micro-scaled sector shaped graphene surrounded by an elastic matrix”, Comp. Mater. Sci., Vol. 77, pp. 295-303, 2013.
[38] Gürses, M., Civalek, Ö., Korkmaz, A., Ersoy, H., “Free vibration analysis of symmetric laminated skew plates by discrete singular convolution technique based on first-order shear deformation theory”, Int J Numer Methods Eng, Vol.79, pp. 290-313, 2009.
[39] Baltacıoglu, A.K., Akgöz, B., Civalek, Ö., “Nonlinear static response of laminated composite plates by discrete singular convolution method”, Compos Struct, Vol. 93, pp. 153-161, 2010.
[40] Gürses, M., Akgöz, B., Civalek, Ö., “Mathematical modeling of vibration problem of nano-sized annular sector plates using the nonlocal continuum theory via eight-node discrete singular convolution transformation”, Appl Math Comput, Vol. 219, pp. 3226–3240, 2012.
[41] Mercan, K., Civalek, Ö., “DSC method for buckling analysis of boron nitride nanotube (BNNT) surrounded by an elastic matrix”, Comp Struct, Vol. 143, pp. 300-309, 2016.
[42] Akgöz, B., Civalek, O., “A new trigonometric beam model for buckling of strain gradient microbeams”, Int J Mech Sci Vol. 81, pp. 88-94, 2014.
[43] Civalek, Ö., Akgöz, B., “Static analysis of single walled carbon nanotubes (SWCNT) based on Eringen’s nonlocal elasticity theory”, Int J Eng Appl Sci, Vol. 1, pp. 47-56, 2009.
[44] Demir, Ç., Civalek, Ö., “Torsional and longitudinal frequency and wave response of microtubules based on the nonlocal continuum and nonlocal discrete models”, Appl Mathl Model, Vol. 37, pp. 9355-9367, 2013.
[45] Akgöz, B., Civalek, Ö., “Shear deformation beam models for functionally graded microbeams with new shear correction factors”, Comp Struct, Vol. 112, pp. 214-225, 2014.
[46] Xin, L., Hu, Z., “Free vibration analysis of laminated cylindrical panels using discrete singular convolution”, Comp Struct, Vol. 149, pp. 362-368, 2016.
[47] Tornabene, F., Fantuzzi, N., Viola, E., Ferreira, A.J.M., “Radial basis function method applied to doubly-curved laminated composite shells and panels with a General Higher-order Equivalent Single Layer formulation”, Compos Part B: Eng, Vol. 55, pp. 642–659, 2013.
[48] Tornabene, F., Viola, E., “Inman DJ. 2-D differential quadrature solution for vibration analysis of functionally graded conical, cylindrical shell and annular plate structures”, J Sound Vib, Vol. 328, pp. 259–290, 2009.
[49] Civalek, Ö., “Application of Differential Quadrature (DQ) and Harmonic Differential Quadrature (HDQ) For Buckling Analysis of Thin Isotropic Plates and Elastic Columns”, Eng Struct, Vol. 26, No. 2, pp. 171-186,2004.
[50] Civalek, Ö., “Geometrically non-linear static and dynamic analysis of plates and shells resting on elastic foundation by the method of polynomial differential quadrature (PDQ)”, PhD. Thesis, Fırat University, (in Turkish), Elazığ, 2004.
[51] Liew, K.M., Han, J-B, Xiao, Z.M., and Du, H., “Differentiel Quadrature Method for Mindlin plates on Winkler foundations”, Int. J. Mech. Sci., Vol. 38, No. 4, pp. 405-421, 1996.
[52] Striz, A.G., Wang, X., Bert, C.W., “Harmonic differential quadrature method and applications to analysis of structural components”, Acta Mechanica, Vol.  111, pp. 85-94, 1995.
[53] Bert, C.W., Wang, Z., Striz, A.G., “Static and free vibrational analysis of beams and plates by differential quadrature method”, Acta Mechanica, Vol. 102, pp. 11-24, 1994.
[54] Du, H., Lim, M.K., Lin, R.M., “Application of generalized differential quadrature method to vibration analysis”, J Sound Vib,Vol. 181, No. 2, 279-93, 1995.
[55] Mercan, K, Demir, Ç., Civalek, Ö., "Vibration analysis of FG cylindrical shells with power-law index using discrete singular convolution technique." Curv Layer Struct, Vol. 3, No.1, pp. 82-90, 2016.