A Semi-Analytical Solution for Free Vibration and Modal Stress Analyses of Circular Plates Resting on Two-Parameter Elastic Foundations

Document Type: Research Paper


Faculty of Mechanical Engineering, K.N. Toosi University of Technology


In the present research, free vibration and modal stress analyses of thin circular plates with arbitrary edge conditions, resting on two-parameter elastic foundations are investigated.  Both Pasternak and Winkler parameters are adopted to model the elastic foundation. The differential transform method (DTM) is used to solve the eigenvalue equation yielding the natural frequencies and mode shapes of the circular plates. Accuracy of obtained results is evaluated by comparing the results with those available in the well-known references. Furthermore, effects of the foundation stiffness parameters and the edge conditions on the natural frequencies, mode shapes, and distribution of the maximum in-plane modal stresses are investigated.


[1] Leissa A.W., 1969, Vibration of Plates, NASA SP-160, Office of Technology Utilization, NASA, Washington DC

[2] Leissa A.W., 1977, Recent research in plate vibrations, 1973-1979: classical theory, The Shock and Vibration Digest 9(10): 13-24.

[3] Leissa A.W., 1981, Plate vibration research, 1976-1980: classical theory, The Shock and Vibration Digest 13(9): 11-22.

[4] Leissa A.W., 1987, Recent studies in plate vibrations, 1981-1985, part I: classical theory, The Shock and Vibration Digest 19(2): 11-18.

[5] Airey J., 1911, The vibration of circular plates and their relation to Bessel functions, Proceeding of the Physical Society of London 23: 225-232.

[6] Irie T., Yamada G., Aomura S., 1980, Natural frequencies of Mindline circular plates, Journal of Applied Mechanics 47: 652-655.

[7] Ahmadian M.T., Mojahedi M., Moeenfard H., 2009, Free Vibration Analysis of a Nonlinear Beam Using Homotopy and Modified Lindstedt-Poincare Methods, Journal of Solid Mechanics 1(1): 29-36.

[8] Ganji D.D., Alipour M.M., Fereidoon A.H., Rostamiyan Y., 2010, Analytic approach to investigation of fluctuation and frequency of the oscillators with odd and even nonlinearities, International Journal of Engineering, in Press.

[9] Shaban M., Ganji D.D., Alipour M.M., 2010, Nonlinear fluctuation, frequency and stability analyses in free vibration of circular sector oscillation systems, Current Applied Physics, doi: 10.1016/j.cap.2010.03.005.

[10] Liew K.M., Han J.B., Xiao Z.M., 1997, Vibration analysis of circular Mindlin plates using differential quadrature method, Journal of Sound and Vibration 205(5): 617-30.

[11] Wu T.Y., Wang Y.Y., Liu G.R., 2002, Free vibration analysis of circular plates using generalized differential quadrature rule, Computer Methods in Applied Mechanics and Engineering 191: 5365-5380.

[12] Rokni Damavandi Taher H., Omidi M., Zadpoor A.A., Nikooyan A.A., 2006, Free vibration of circular and annular plates with variable thickness and different combinations of boundary conditions, Journal of Sound and Vibration 296: 1084-1092.

[13] Liew K.M., Yang B., 2000, Elasticity solutions for free vibrations of annular plates from three-dimensional analysis, International Journal of Solids and Structure 37: 7689-7702.

[14] Zhou D., Au F.T.K., Cheung Y.K., Lo S.H., 2003, Three-dimensional vibration analysis of circular and annular plates via the Chebyshev-Ritz method, International Journal of Solids and Structures 40: 3089-3105.

[15] Chen W., Shen Z.J., Shen L.J., Yuan, G.W., 2005, General solutions and fundamental solutions of varied orders to the vibrational thin, the Berger, and the Winkler plates, Engineering Analysis with Boundary Elements 29: 699-702.

[16] Gupta U.S., Ansari A.H., Sharma S., 2006, Buckling and vibration of polar orthotropic circular plate resting on Winkler foundation, Journal of Sound and Vibration, 297: 457-476.

[17] Gupta U.S., Lal R., Sharma S., 2006, Vibration analysis of non-homogeneous circular plate of nonlinear thickness variation by differential quadrature method, Journal of Sound and Vibration, 298: 892-906.

[18] Pasternak P.L., 1954, On a new method of analysis of an elastic foundation by means of two foundation constants, Gosudarstvennoe Izdatelstvo Literaturi po Stroitelstvu I Arkhitekture, Moscow, USSR (in Russian).

[19] Daloğlu A., Doğangün A., Ayvaz Y., 1999, Dynamic analysis of foundation plates using a consistent Vlasov model, Journal of Sound and Vibration, 224(5): 941-951.

[20] Celep Z., Güler K., 2007, Axisymmetric forced vibrations of an elastic free circular plate on a tensionless two parameter foundation, Journal of Sound and Vibration, 301: 495-509.

[21] Hosseini Hashemi, Sh., Rokni Damavandi Taher, H., Omidi M., 2008, 3-D free vibration analysis of annular plates on Pasternak elastic foundation via p-Ritz method, Journal of Sound and Vibration, 311: 1114-1140.

[22] Zhou D., Lo S.H., Au F.T.K., Cheung Y.K., 2006, Three-dimensional free vibration of thick circular plates on Pasternak foundation, Journal of Sound and Vibration, 292: 726-741.

[23] Abdel-Halim Hassan I.H., 2004, Differential transformation technique for solving higher-order initial value problems, Applied Mathematics and Computation, 154: 299-311.

[24] Momani S, Noor M.A., 2007, Numerical comparison of methods for solving a special fourth-order boundary value problem, Applied Mathematics and Computation, 191: 218-224.

[25] Ertürk V.S., Momani S., Odibat Z., 2008, Application of generalized differential transform method to multi-order fractional differential equation, Communications in Nonlinear Scince Numercal Simulation, 13: 1642-1654.

[26] Reddy J.N., 2007, Theory and Analysis of Elastic Plates and Shells, CRC / Taylor & Francis, second edition.