Computational Initial Value Method for Vibration Analysis of Symmetrically Laminated Composite Plate

Authors: Ahmed M. Farag, Wael F. Mohamed, Atef A. Ata, Burhamy M. Burhamy

Abstract:

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.

Keywords: Free Vibrations, Initial Value, Finite Differences, Laminated plates.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1060511

References:

[1] J. N. Reddy "Mechanics of Laminated Composite plates and shells Theory and Analysis" , library of congress,1997.
[2] Y.Y. Wang, K.Y. Lam, G.R. Liu "A strip element method for the transient analysis of symmetric laminated plates", pergamon, International Journal of Solids and Structures vol. 38, pp, 241- 259,(2001).
[3] A.A., Khdeir, J.N., Reddy. "Exact solution for the transient response of symmetric cross-ply laminates using a higher-order plate theory". Composite Science and Technology 34, 205-224, 1989.
[4] H.Thai, S.Kim,"Levy-type solution for free vibration analysis of orthotropic plates based on two variable refined plate theory", Elsevier, Applied Mathematical Modeling 36, pp, 3870-3882, 2012.
[5] T. Kant, K. Swaminathan " Analytical solutions for the static analysis of laminated composite and sandwich plates based on a higher order refined theory" Elsevier, Composite Structures. vol. 56, pp. 329-344, 2002.
[6] H. Yunshan "Dsc-Ritz method for the free vibration analysis of mindlin plate" Msc, department of computer science, national university of singhapore, 2003.
[7] S.Guenfoud, S.V.Bosakov, F.Debra" A Ritz-s method based solution for the contact problem of a deformable rectangular plate on an elastic quarter-space" International Journal of Solids & Structures, Elsevier, 47 (14-15): 1822- 1829, 2010.
[8] M. A. El-Sayad, S.A. Ghazy "Rayleigh-Ritz Method for Free Vibration of Mindlin Trapezoidal Plates" International Journal of Emerging Technology and Advanced Engineering, Volume 2, Issue 5, May 2012)
[9] G. M. Oosterhout, P. J. Van Derhoogt and R. M. Spiering. " "Accuratecalculation methods for natural frequencies of plates with special attention of the higher modes" Journal of Sound and Vibration" vol. 183(1), pp. 33-47, 1995.
[10] L.R. Chung and T.Y. Chung "Vibration Analysis of Symmetrically Laminated Composite Rectangular Plates" Proceedings of the Third (1993) International Offshore and Polar Engineering Conference, Vol. (IV), Singapore, 6-11 June, 1993.
[11] R. Kolar "Dynamics of Shear Deformable Laminated Composites Using Raleigh Ritz Method" NASA Dryden Flight Research Center, Dryden, California. Department of Aeronautics & Astronautics, Naval Postgraduate School, 699 Dyer Road, ldg 234, Rm 245, Monterey, CA 93943, U.S.A,2002.
[12] S. O. Eruslu and M. A.Gdu "Free vibration analysis of short fiber reinforced laminated plates with first shear deformation theory" Turkish J. Eng. Env. Sci. 36, 95 - 107, (2012).
[13] A.Ergun, N.Kunbasar, "A new approach of improved finite difference scheme on plate bending analysis", Scientific research and essays vol.6(1),pp, 6-17, 2011.
[14] C.B.Dolicanin, V.B. Nikolic, D.C. Dolicanin, "Application of finite difference method to study of the phenomenon in the theory of thin plates", Appl. Math. Inform. And Mech. Vol.2 ,1 , pp, 29-43, 2010.
[15] R. J. LeVeque, "Finite difference methods for ordinary and partial differential equations: steady-state and time-dependent problems", the Society for Industrial and Applied Mathematics, 2007.
[16] R. Pedro. "Geometricallynon-linear oscillations ofcomposite laminated plates by the hierarchical finite element method" M.sc in computational methods in sciences and engineering , university of Coimbra, 2004.
[17] A. K. Noor , M. D. Mnthers "Shear-fleixble finite element models of laminated plates and shells" National aeronautics and space administration" Lnrzgley Reseccrch Ceizter Hampton, v a. 23665Washington D.C. December, 1975.
[18] A.M.Farag, "Mathematical analysis of free and forced vibration of rectangular plate" , Ph.D Thesis, Faculty of engineering, Alexandria university, 1994.
[19] H. Al-Khaiat., H.H.West., "Analysis of plates by the initial value method". Computer & structure vol.24 No.3, pp, 475-483, 1986.
[20] H. Al-Khaiat., "free vibration analysis of orthotropic plates by the initial value method". Computer & structure vol.33 No.6, pp,1431-1435, 1989.
[21] A.d.Reis, E.L.Albuquerque, F.L.Torsani "Computation of moments and stresses in laminated composite plates by the boundary element method", Elsevier, Engineering and analysis with boundary elements, 35, pp, 105- 113, 2011.
[22] E.L.Albuquerque, P.Sollero, W.S. Venturini, M.H.Aliabadi, "Boundary element analysis of anisotropic Kirchhoff plates", Elsevier, International journal of solids and structures 43,pp,4029-4046, 2006.
[23] W. Portilho de Paiva, P. Sollero and E. L. Albuquerque ,"Treatment of hyper singularities in boundary element anisotropic plate bending problems" Latin American Journal of Solids and Structures 1,pp , 49-73, (2003).
[24] G. D. Hatzigeorgiou, D. E. Beskos, "Static and dynamic analysis of inelastic solids and structures by the BEM", Journal of the Serbian Society for Computational Mechanics / Vol. 2 / No. 1, 2008 / pp. 1-27.
[25] W. Portilho de Paiva, P. Sollero and E. L. Albuquerque ,"Treatment of hyper singularities in boundary element anisotropic plate bending problems" Latin American Journal of Solids and Structures 1,pp , 49-73, (2003).
[26] W. Han, M. petyt "Lineear vibration Analysis of Laminated Rectangular Plates Using The Hierarchical Finite Element MethodÔÇöI. Free Vibration Analysis." Pergamon , computer & structures vol. 61,No. 4,pp. 705-712, 1996.
[27] G. Davi, A. Millazzo "A meshfree method for transeverse vibration of anisotropic plates" pergamon, International Journal of solid & structures vol. 40,pp. 5229-5249, 2003.
[28] J.N., Reddy, E.J. Barebero, "A plate bending element based on a generalized laminated plate theory", International journal of numerical methods in engineering, vol. 28, pp, 2275-2292, 1989.
[29] C. Wanji, W. Zhen, "A Selective Review on Recent Development of Displacement-Based Laminated Plate Theories", Recent Patents on Mechanical Engineering,vol 1, pp29-44, 2008 .
[30] L.G.Nallim, F.J. Bellomo, R.D. Quinteros, and S. Oller, "Dynamical Analysis of long fiber-reinforced laminated plates with elastically restrained edges", Hindawi publishing corporation, Advanced in acouastic and vibration, vol. 2012.
[31] M. Haddad, Y. Gourinat, M. Charlotte, "Equivalence Theory Applied to Anisotropic Thin Plates", scientific research engineering,vol 3,pp 669- 67, 2011.
[32] S.T. chow, K.M. Liew, K.Y. Lam "Transverse Vibration of Symmetrically Laminated Rectangular Composite Plates", computer & structures vol. 20,pp213-226, 1992.
[33] O. Civalek, O. Kiracioglu "discrete singular convolution for free vibration analysis of anisotropic plates", Mathematical and computational applications. vol. 12, No. 3, pp. 151-160, 2007.
[34] L. Demasi "Quasi-3D analysis of free vibration of anisotropic plates", Elsevier, Composite Structures. vol. 74, pp. 449-457, 2006.
[35] R.C. batra, J. Jin "Natural frequencies of functionally graded anisotropic rectangular plate", Elsevier, Journal of sound and vibration. vol. 282, pp. 509-56, 2005.
[36] W.Yu, R. Mittra, T. Su, Y Liu, X. Yang, "Parallel finite-difference timedomain method" ARTECH HOUSE, 2006.
[37] D. J. Duffy, "Finite Difference Methods in Financial Engineering A Partial Differential Equation Approach", [email protected],John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO198SQ, England, 2006.
[38] I. Chern "Finite difference methods for solving differential equations", Department of Mathematics, book, National Taiwan University, 2009.
[39] J. Awrejcewicz ,"Numerical Analysis - Theory and Application", Published by In. Tech, Janeza Trdine 9, 51000 Rijeka, Croatia, 2011.
[40] A.A. Kuleshov, "Finite difference method for the model of small transverse vibrations in thin elastic plates" Proceeding of the 4th WSEAS international conference of finite differences, pp, 19-22, 2010.
[41] Y.F.Xing, B.Liu, "New exact solutions for free vibrations of thin orthotropic rectangular plates", Elsivier, Composite Structure, 89, pp, 567-574, 2009.
[42] A.K.Gupta, N.Agarwal, H.Kaur, "Free vibration analysis of nonhomogeneous orthotropic visco-elastic elliptic plate of non-uniform thickness", Int. J. of Appl. Math and Mech. 7(6): pp, 1-18, 2011.
[43] A.M. Farag, and A.S. Ashour "Free vibration of orthotropic skew plates", Journal of vibration and acoustics, ASMF, vol. 122, pp, 313- 317, 2000.
[44] R.P.Shimpi, H.G.Patel, "A two variable refined plate theory for orthotropic plate analysis." Aerospace Engineering, Indian Institute of Technology Bombay, pp, 6783-6799, 2003.
[45] H. Khov a, W. L. Lib, Ronald F. Gibson An accurate solution method for the static and dynamic deflections of orthotropic plates with general boundary conditions Composite Structures 90 (2009) 474-481
[46] H. Thai, S. Kim "Levy-type solution for free vibration analysis of orthotropic plates based on refined plate theory" Elsevier, Applied mathematical modeling. vol. 36, pp. 3870-3882, 2012.
[47] N. Baddour ,"Recent Advances in Vibrations Analysis", Published by InTech, Janeza, Trdine 9, 51000 Rijeka, Croatia, 2011.
[48] S. Timoshenko, S. Woiowesky-krieger, "Theory of plates and shells", McGRAW-HILL, 1959.
[49] A.A. khdeir, L. Librescu" Analysis of symmetric cross-ply laminated elastic plates using a higher-order theory: Part IIÔÇöBuckling and free vibration" Composite Structures, Volume 9, Issue 4, Pages 259-277, 1988.
[50] L. G. Nallim, R. O. Grossi. "Vibration of angle-ply symmetric laminated composite plates with edges elastically restrained", Composite structures, vol.81, pp88-83, 2007