In the present paper, an improved initial value numerical technique is presented to analyze the free vibration of symmetrically laminated rectangular plate. A combination of the initial value method (IV) and the finite differences (FD) devices is utilized to develop the present (IVFD) technique. The achieved technique is applied to the equation of motion of vibrating laminated rectangular plate under various types of boundary conditions. Three common types of laminated symmetrically cross-ply, orthotropic and isotropic plates are analyzed here. The convergence and accuracy of the presented Initial Value-Finite Differences (IVFD) technique have been examined. Also, the merits and validity of improved technique are satisfied via comparing the obtained results with those available in literature indicating good agreements., {"references":["J. N. Reddy \"Mechanics of Laminated Composite plates and shells\nTheory and Analysis\" , library of congress,1997.","Y.Y. Wang, K.Y. Lam, G.R. Liu \"A strip element method for the\ntransient analysis of symmetric laminated plates\", pergamon,\nInternational Journal of Solids and Structures vol. 38, pp, 241-\n259,(2001).","A.A., Khdeir, J.N., Reddy. \"Exact solution for the transient response of\nsymmetric cross-ply laminates using a higher-order plate theory\".\nComposite Science and Technology 34, 205-224, 1989.","H.Thai, S.Kim,\"Levy-type solution for free vibration analysis of\northotropic plates based on two variable refined plate theory\", Elsevier,\nApplied Mathematical Modeling 36, pp, 3870-3882, 2012.","T. Kant, K. Swaminathan \" Analytical solutions for the static analysis of\nlaminated composite and sandwich plates based on a higher order\nrefined theory\" Elsevier, Composite Structures. vol. 56, pp. 329-344,\n2002.","H. Yunshan \"Dsc-Ritz method for the free vibration analysis of mindlin\nplate\" Msc, department of computer science, national university of\nsinghapore, 2003.","S.Guenfoud, S.V.Bosakov, F.Debra\" A Ritz-s method based solution for\nthe contact problem of a deformable rectangular plate on an elastic\nquarter-space\" International Journal of Solids & Structures, Elsevier, 47\n(14-15): 1822- 1829, 2010.","M. A. El-Sayad, S.A. Ghazy \"Rayleigh-Ritz Method for Free Vibration\nof Mindlin Trapezoidal Plates\" International Journal of Emerging\nTechnology and Advanced Engineering, Volume 2, Issue 5, May 2012)","G. M. Oosterhout, P. J. Van Derhoogt and R. M. Spiering. \"\n\"Accuratecalculation methods for natural frequencies of plates with\nspecial attention of the higher modes\" Journal of Sound and Vibration\"\nvol. 183(1), pp. 33-47, 1995.\n[10] L.R. Chung and T.Y. Chung \"Vibration Analysis of Symmetrically\nLaminated Composite Rectangular Plates\" Proceedings of the Third\n(1993) International Offshore and Polar Engineering Conference, Vol.\n(IV), Singapore, 6-11 June, 1993.\n[11] R. Kolar \"Dynamics of Shear Deformable Laminated Composites Using\nRaleigh Ritz Method\" NASA Dryden Flight Research Center, Dryden,\nCalifornia. Department of Aeronautics & Astronautics, Naval\nPostgraduate School, 699 Dyer Road, ldg 234, Rm 245, Monterey, CA\n93943, U.S.A,2002.\n[12] S. O. Eruslu and M. A.Gdu \"Free vibration analysis of short fiber\nreinforced laminated plates with first shear deformation theory\" Turkish\nJ. Eng. Env. Sci. 36, 95 - 107, (2012).\n[13] A.Ergun, N.Kunbasar, \"A new approach of improved finite difference\nscheme on plate bending analysis\", Scientific research and essays\nvol.6(1),pp, 6-17, 2011.\n[14] C.B.Dolicanin, V.B. Nikolic, D.C. Dolicanin, \"Application of finite\ndifference method to study of the phenomenon in the theory of thin\nplates\", Appl. Math. Inform. And Mech. Vol.2 ,1 , pp, 29-43, 2010.\n[15] R. J. LeVeque, \"Finite difference methods for ordinary and partial\ndifferential equations: steady-state and time-dependent problems\", the\nSociety for Industrial and Applied Mathematics, 2007.\n[16] R. Pedro. \"Geometricallynon-linear oscillations ofcomposite laminated\nplates by the hierarchical finite element method\" M.sc in computational\nmethods in sciences and engineering , university of Coimbra, 2004.\n[17] A. K. Noor , M. D. Mnthers \"Shear-fleixble finite element models of\nlaminated plates and shells\" National aeronautics and space\nadministration\" Lnrzgley Reseccrch Ceizter Hampton, v a.\n23665Washington D.C. December, 1975.\n[18] A.M.Farag, \"Mathematical analysis of free and forced vibration of\nrectangular plate\" , Ph.D Thesis, Faculty of engineering, Alexandria\nuniversity, 1994.\n[19] H. Al-Khaiat., H.H.West., \"Analysis of plates by the initial value\nmethod\". Computer & structure vol.24 No.3, pp, 475-483, 1986.\n[20] H. Al-Khaiat., \"free vibration analysis of orthotropic plates by the initial\nvalue method\". Computer & structure vol.33 No.6, pp,1431-1435, 1989.\n[21] A.d.Reis, E.L.Albuquerque, F.L.Torsani \"Computation of moments and\nstresses in laminated composite plates by the boundary element method\",\nElsevier, Engineering and analysis with boundary elements, 35, pp, 105-\n113, 2011.\n[22] E.L.Albuquerque, P.Sollero, W.S. Venturini, M.H.Aliabadi, \"Boundary\nelement analysis of anisotropic Kirchhoff plates\", Elsevier, International\njournal of solids and structures 43,pp,4029-4046, 2006.\n[23] W. Portilho de Paiva, P. Sollero and E. L. Albuquerque ,\"Treatment of\nhyper singularities in boundary element anisotropic plate bending\nproblems\" Latin American Journal of Solids and Structures 1,pp , 49-73,\n(2003).\n[24] G. D. Hatzigeorgiou, D. E. Beskos, \"Static and dynamic analysis of\ninelastic solids and structures by the BEM\", Journal of the Serbian\nSociety for Computational Mechanics / Vol. 2 / No. 1, 2008 / pp. 1-27.\n[25] W. Portilho de Paiva, P. Sollero and E. L. Albuquerque ,\"Treatment of\nhyper singularities in boundary element anisotropic plate bending\nproblems\" Latin American Journal of Solids and Structures 1,pp , 49-73,\n(2003).\n[26] W. Han, M. petyt \"Lineear vibration Analysis of Laminated Rectangular\nPlates Using The Hierarchical Finite Element MethodÔÇöI. Free Vibration\nAnalysis.\" Pergamon , computer & structures vol. 61,No. 4,pp. 705-712,\n1996.\n[27] G. Davi, A. Millazzo \"A meshfree method for transeverse vibration of\nanisotropic plates\" pergamon, International Journal of solid & structures\nvol. 40,pp. 5229-5249, 2003.\n[28] J.N., Reddy, E.J. Barebero, \"A plate bending element based on a\ngeneralized laminated plate theory\", International journal of numerical\nmethods in engineering, vol. 28, pp, 2275-2292, 1989.\n[29] C. Wanji, W. Zhen, \"A Selective Review on Recent Development of\nDisplacement-Based Laminated Plate Theories\", Recent Patents on\nMechanical Engineering,vol 1, pp29-44, 2008 .\n[30] L.G.Nallim, F.J. Bellomo, R.D. Quinteros, and S. Oller, \"Dynamical\nAnalysis of long fiber-reinforced laminated plates with elastically\nrestrained edges\", Hindawi publishing corporation, Advanced in\nacouastic and vibration, vol. 2012.\n[31] M. Haddad, Y. Gourinat, M. Charlotte, \"Equivalence Theory Applied to\nAnisotropic Thin Plates\", scientific research engineering,vol 3,pp 669-\n67, 2011.\n[32] S.T. chow, K.M. Liew, K.Y. Lam \"Transverse Vibration of\nSymmetrically Laminated Rectangular Composite Plates\", computer &\nstructures vol. 20,pp213-226, 1992.\n[33] O. Civalek, O. Kiracioglu \"discrete singular convolution for free\nvibration analysis of anisotropic plates\", Mathematical and\ncomputational applications. vol. 12, No. 3, pp. 151-160, 2007.\n[34] L. Demasi \"Quasi-3D analysis of free vibration of anisotropic plates\",\nElsevier, Composite Structures. vol. 74, pp. 449-457, 2006.\n[35] R.C. batra, J. Jin \"Natural frequencies of functionally graded anisotropic\nrectangular plate\", Elsevier, Journal of sound and vibration. vol. 282, pp.\n509-56, 2005.\n[36] W.Yu, R. Mittra, T. Su, Y Liu, X. Yang, \"Parallel finite-difference timedomain\nmethod\" ARTECH HOUSE, 2006.\n[37] D. J. Duffy, \"Finite Difference Methods in Financial Engineering A\nPartial Differential Equation Approach\", cs-books@wiley.co.uk,John\nWiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex\nPO198SQ, England, 2006.\n[38] I. Chern \"Finite difference methods for solving differential equations\",\nDepartment of Mathematics, book, National Taiwan University, 2009.\n[39] J. Awrejcewicz ,\"Numerical Analysis - Theory and Application\",\nPublished by In. Tech, Janeza Trdine 9, 51000 Rijeka, Croatia, 2011.\n[40] A.A. Kuleshov, \"Finite difference method for the model of small\ntransverse vibrations in thin elastic plates\" Proceeding of the 4th\nWSEAS international conference of finite differences, pp, 19-22, 2010.\n[41] Y.F.Xing, B.Liu, \"New exact solutions for free vibrations of thin\northotropic rectangular plates\", Elsivier, Composite Structure, 89, pp,\n567-574, 2009.\n[42] A.K.Gupta, N.Agarwal, H.Kaur, \"Free vibration analysis of nonhomogeneous\northotropic visco-elastic elliptic plate of non-uniform\nthickness\", Int. J. of Appl. Math and Mech. 7(6): pp, 1-18, 2011.\n[43] A.M. Farag, and A.S. Ashour \"Free vibration of orthotropic skew\nplates\", Journal of vibration and acoustics, ASMF, vol. 122, pp, 313-\n317, 2000.\n[44] R.P.Shimpi, H.G.Patel, \"A two variable refined plate theory for\northotropic plate analysis.\" Aerospace Engineering, Indian Institute of\nTechnology Bombay, pp, 6783-6799, 2003.\n[45] H. Khov a, W. L. Lib, Ronald F. Gibson An accurate solution method\nfor the static and dynamic deflections of orthotropic plates with general\nboundary conditions Composite Structures 90 (2009) 474-481\n[46] H. Thai, S. Kim \"Levy-type solution for free vibration analysis of\northotropic plates based on refined plate theory\" Elsevier, Applied\nmathematical modeling. vol. 36, pp. 3870-3882, 2012.\n[47] N. Baddour ,\"Recent Advances in Vibrations Analysis\", Published by\nInTech, Janeza, Trdine 9, 51000 Rijeka, Croatia, 2011.\n[48] S. Timoshenko, S. Woiowesky-krieger, \"Theory of plates and shells\",\nMcGRAW-HILL, 1959.\n[49] A.A. khdeir, L. Librescu\" Analysis of symmetric cross-ply laminated\nelastic plates using a higher-order theory: Part IIÔÇöBuckling and free\nvibration\" Composite Structures, Volume 9, Issue 4, Pages 259-277,\n1988.\n[50] L. G. Nallim, R. O. Grossi. \"Vibration of angle-ply symmetric laminated\ncomposite plates with edges elastically restrained\", Composite\nstructures, vol.81, pp88-83, 2007"]}