Numerical Method Based On Initial Value-Finite Differences for Free Vibration of Stepped Thickness Plates

The main objective of the present paper is to derive an easy numerical technique for the analysis of the free vibration through the stepped regions of plates. Based on the utilities of the step by step integration initial values IV and Finite differences FD methods, the present improved Initial Value Finite Differences (IVFD) technique is achieved. The first initial conditions are formulated in convenient forms for the step by step integrations while the upper and lower edge conditions are expressed in finite difference modes. Also compatibility conditions are created due to the sudden variation of plate thickness. The present method (IVFD) is applied to solve the fourth order partial differential equation of motion for stepped plate across two different panels under the sudden step compatibility in addition to different types of end conditions. The obtained results are examined and the validity of the present method is proved showing excellent efficiency and rapid convergence.

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

• Vibrations
• Step by Step Integration
• Stepped plate
• Boundary.

[1] A. M. Farag "Closed form solution for vibrating surfaces of partially
restrained and clamped double-panel plates" European Journal of
scientific Research. Vol 29 N0.3 pp.320-333, 2009.
[2] Y. Xiang, G.W. Wei "Exact solutions for buckling and vibration of
stepped rectangular mindlin plates" International journal of solids and
structures, vol. 41, pp 279-294, 2004.
[3] S. Kukla ,M. Szewczyk " free vibration of annular plates of stepped
thickness resting on winkler elastic foundation" scientific research of
institute of mathematics and computer science, Poland, 2008.
[4] M. El-sayad, A. M. Farag "numerical solution of vibrating double and
triple-panel stepped thickness plates" Applied & Computational
Mathematics.vol 1, issue 3, 2012
[5] S.J. Guo, A. J. Keane, M. Moshrefi "vibration analysis of stepped
thickness plates" journal of sound and vibration, vol. 204(4), pp 645-
657, 1997.
[6] 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.
[7] A. M. Farag, "Mathematical analysis of free and forced vibration of
rectangular plate", Ph.D Thesis, Faculty of engineering, Alexandria
university, 1994.
[8] 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.
[9] 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.
[10] 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.
[11] W. Yu, R. Mittra, T. Su, Y Liu, X. Yang, "Parallel finite-difference
time-domain method." ARTECH HOUSE, 2006.
[12] D. J. Duffy, "Finite Difference Methods in Financial Engineering A
Partial Differential Equation Approach", [email protected], 2006.
[13] H. Al-Khaiat., H. H. West., "Analysis of plates by the initial value
method". Computer & structure vol.24 No.3, pp, 475-483, 1986.
[14] H. Al-Khaiat., "Free vibration analysis of orthotropic plates by the initial
value method". Computer & structure vol.33 No.6, pp, 1431-1435, 1989.
[15] S. Timoshenko, S. Woiowesky-krieger, "Theory of plates and shells",
McGRAW-HILL, 1959.
[16] Y. F. Xing, B.Liu, "New exact solutions for free vibrations of thin
orthotropic rectangular plates" , Elsevier, Composite Structure, 89, pp,
567-574, 2009.
[17] 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.
[18] 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.
[19] I. Chern "Finite difference methods for solving differential equations",
Department of Mathematics, National Taiwan University, 2009.
[20] J. Awrejcewicz ,"Numerical Analysis - Theory and Application",
Published by InTech, JanezaTrdine 9, 51000 Rijeka, Croatia, 2011.
[21] 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.
[22] N. Baddour ,"Recent Advances in Vibrations Analysis", Published by
InTech, JanezaTrdine 9, 51000 Rijeka, Croatia, 2011.
[23] M. A. El-Sayadand 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.
[24] H. Yunshan "Dsc-Ritz method for the free vibration analysis of mindlin
plate" Msc department of computer science, national university of
Singapore, 2003.
[25] H. Khov, W. L. Li b, R. F. Gibson An accurate solution method for the
static and dynamic deflections of orthotropic plates with general
boundary conditions Composite Structures 90, 474-481, (2009).
[26] 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.
[27] G. M. Oosterhout, P. J. Van Derhoogt and R. M. Spiering. "Accurate
calculation 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.