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

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.

Authors:

• Ahmed M. Farag
• Wael F. Mohamed
• Atef A. Ata
• Burhamy M. Burhamy

Keywords:

• Free Vibrations
• Initial Value
• Finite Differences
• Laminated plates.

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