Large Vibration Amplitudes of Circular Functionally Graded Thin Plates Resting on Winkler Elastic Foundations

Authors: El Kaak, Rachid, El Bikri, Khalid, Benamar, Rhali

Abstract:

This paper describes a study of geometrically nonlinear free vibration of thin circular functionally graded (CFGP) plates resting on Winkler elastic foundations. The material properties of the functionally graded composites examined here are assumed to be graded smoothly and continuously through the direction of the plate thickness according to a power law and are estimated using the rule of mixture. The theoretical model is based on the classical Plate theory and the Von-Kármán geometrical nonlinearity assumptions. An homogenization procedure (HP) is developed to reduce the problem considered here to that of isotropic homogeneous circular plates resting on Winkler foundation. Hamilton-s principle is applied and a multimode approach is derived to calculate the fundamental nonlinear frequency parameters which are found to be in a good agreement with the published results. On the other hand, the influence of the foundation parameters on the nonlinear fundamental frequency has also been analysed.

Keywords: Functionally graded materials, nonlinear vibrations, Winkler foundation.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1056707

References:

[1] M. Koizumi The concept of FGM. Ceram Trans Func Grad Mater 1993; 34:3-10.W.-K. Chen, Linear Networks and Systems (Book style).Belmont, CA: Wadsworth, 1993, pp. 123-135.
[2] M. Koizumi FGM activities in Japan. Composite B 1997;28:1-4.
[3] S .Suresh, Mortensen A. Fundamentals of Functionally Graded Materials: Processing and Thermomechanical Behavior of Graded Metals and Metal-Ceramic Composites.London, UK: IOM Communications Ltd, 1998.
[4] Y .Miyamoto, Kaysser W A, Rabin B H, et al. Functionally Graded Materials: Design, Processing and Applications.Boston, UK: Kluwer Academic Publishers, 1999.
[5] A. Allahverdizadeh, M.H. Naei, M. Nikkhah Bahrami, Nonlinear free and forced vibration analysis of thin circular functionally graded plates, Journal of Sound and Vibration 310 (2008) 966-984.
[6] A.Zerkane,K.El Bikri,R.Benamar," A homogenization procedure for nonlinear free vibration analysis of functionally graded beams resting on nonlinear elastic foundations"
[7] H.Shen Shen,Functionally graded materials : nonlinear analysis of plates and shells. Taylor & Francis Group, LLC. 2009
[8] R. Benamar, M.M.K. Bennouna, R.G. White, The effects of large vibration amplitudes on the mode shapes and natural frequencies of thin elastic structures, part II: fully clamped rectangular isotropic plates, Journal of Sound and Vibration 164 (1991) 399-424.
[9] M. El Kadiri, R. Benamar, R.G. White, The non-linear free vibration of fully clamped rectangular plates: second non-linear mode for various plate aspect ratios, Journal of Sound and Vibration 228 (2) (1999) 333- 358.