A Power Series Solution for Free Vibration of Variable Thickness Mindlin Circular Plates with Two-Directional Material Heterogeneity and Elastic Foundations

Document Type : Research Paper


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


In the present paper, a semi-analytical solution is presented for free vibration analysis of circular plates with complex combinations of the geometric parameters, edge-conditions, material heterogeneity, and elastic foundation coefficients. The presented solution covers many engineering applications. The plate is assumed to have a variable thickness and made of a heterogeneous material whose properties vary in both radial and transverse directions. While the edge is simply-supported, clamped, or free; the bottom surface of the plate is resting on a two-parameter (Winkler-Pasternak) elastic foundation. A comprehensive sensitivity analysis including evaluating effects of various parameters is carries out. Mindlin theory is employed for derivation of the governing equations whereas the differential transform method is used to solve the resulted equations. In this regard, both the in-plane and rotary inertia are considered. Results show that degradations caused by a group of the factors (e.g., the geometric parameters) in the global behavior of the structure may be compensated by another group of factors of different nature (e.g, the material heterogeneity parameters). Moreover, employing the elastic foundation leads to higher natural frequencies and postponing the resonances.    


[1] 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.
[2] 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.
[3] Ramaiah G.K., Vijayakumar K., 1973, Natural frequencies of polar orthotropic annular plates, Journal of Sound and Vibration 26: 517-531.
[4] Narita Y., 1984, Natural frequencies of completely free annular and circular plates having polar orthotropy, Journal of Sound and Vibration 92: 33-38.
[5] Lin C.C., Tseng C.S., 1998, Free vibration of polar orthotropic laminated circular and annular plates, Journal of Sound and Vibration 209: 797-810.
[6] Prakash T., Ganapathi M., 2006, Asymmetric flexural vibration and thermoelastic stability of FGM circular plates using finite element method, Composites Part B 37: 642–649.
[7] Efraim E., Eisenberger M., 2007, Exact vibration analysis of variable thickness thick annular isotropic and FGM plates, Journal of Sound and Vibration 299: 720-738.
[8] Nie G.J., Zhong Z., 2007, Semi-analytical solution for three-dimensional vibration of functionally graded circular plates, Computer Methods in Applied Mechanics and Engineering 196: 4901-4910.
[9] Dong C.Y., 2008, Three-dimensional free vibration analysis of functionally graded annular plates using the Chebyshev-Ritz method, Materials and Design 29: 1518-1525.
[10] Malekzadeh P., 2009, Three-dimensional free vibration analysis of thick functionally graded plates on elastic foundations, Composite Structures 89: 367-373.
[11] Wang Y., Xu R.Q., Ding H.J., 2009, Free axisymmetric vibration of FGM circular plates, Applied Mathematics and Mechanics 30: 1077-1082.
[12] Nie G.J., Zhong Z., 2010, Dynamic analysis of multi-directional functionally graded annular plates, Applied Mathematical Modelling 34: 608-616.
[13] Arikoglu A., Ozkol I., 2005, Solution of boundary value problems for integro-differential equations by using differential transform method, Applied Mathematicsand Computation 168: 1145-1158.
[14] Chen C.K., Ho S.H., 1998, Application of differential transformation to eigenvalue problems, Applied Mathematicsand Computation 79: 173-188.
[15] Malik M., Dang H.H., 1998, Vibration analysis of continuous systems by differential transformation, Applied Mathematicsand Computation 96: 17-26.
[16] Yeh Y.L., Jang M.J., Wang C.C., 2006, Analyzing the free vibrations of a plate using finite difference and differential transformation method, Applied Mathematicsand Computation 178: 493-501.
[17] Yeh Y.-L., Wang C.C., Jang M.-J., 2007, Using finite difference and differential transformation method to analyze of large deflections of orthotropic rectangular plate problem, Applied Mathematicsand Computation 190: 1146-1156.
[18] Yalcin H.S., Arikoglu A., Ozkol I., 2009, Free vibration analysis of circular plates by differential transformation method, Applied Mathematicsand Computation 212: 377-386.
[19] Shariyat M., Alipour MM., 2011, Differential transform vibration and modal stress analyses of circular plates made of two-directional functionally graded materials, resting on elastic foundations, Archives of Applied Mechanics 81: 1289-1306.
[20] Alipour M.M., Shariyat M., Shaban M., 2010, A semi-analytical solution for free vibration of variable thickness two-directional-functionally graded plates on elastic foundations, International Journal of Mechanics and Materials in Design 6: 293-304.
[21] Reddy J.N., 2007, Theory and Analysis of Elastic Plates and Shells, CRC/Taylor & Francis, Second edition.
[22] Reddy J.N., 2004, Mechanics of Laminated Composite Plates and Shells: Theory and Analysis, CRC Press, Second edition.
[23] Hosseini-Hashemi Sh., Fadaee M., Es’haghi M., 2010, A novel approach for in-plane, out-of-plane frequency analysis of functionally graded circular/annular plates, International Journal of Mechanical Sciences 52: 1025-1035.
[24] Irie T., Yamada G., Takagi K., 1980, Natural Frequencies of Circular Plates, Journal of Applied Mechanics 47: 652-655.
[25] Gupta U.S., Lal R., Sharma S., 2007, Vibration of non-homogeneous circular Mindlin plates with variable thickness, Journal of Sound and Vibration 302: 1-17.