A Semi-Analytical Study of Geometrically Nonlinear Free Axisymmetric Vibrations of Thin Circular Functionally Graded Plates Using Iterative and Explicit Analytical Solution

Abstract:

. This paper deals with nonlinear free axisymmetric vibrations of functionally graded thin circular plates (FGCP) whose properties vary through its thickness. The inhomogeneity of the plate is characterized by a power law variation of the Young’s modulus and mass density of the material along the thickness direction, whereas Poisson’s ratio is assumed to be constant. The theoretical model is based on Hamilton’s principle and spectral analysis using a basis of admissible Bessel’s functions to yield the frequencies of the circular plates under clamped boundary conditions on the basis of the classical plate theory. The large vibration amplitudes problem, reduced to a set of non-linear algebraic equations, is solved numerically. The non-linear to linear frequency ratios are presented. Then, explicit analytical solutions are presented, based on the semi-analytical model previously developed by EL Kadiri et al. [1-2] for beams and rectangular plates, which allow direct and easy calculation for the first non-linear axisymmetric mode shape, with their associated non-linear frequencies of FG circular plates and which are expected to be very useful in engineering applications and in further analytical developments. An excellent agreement is found with the results obtained by the iterative method.

• Periodical: Applied Mechanics and Materials (Volume 704)
• Pages: 131-136
• DOI: https://doi.org/10.4028/www.scientific.net/AMM.704.131
• Online since: December 2014
• Authors: El Kaak Rachid, Khalid El Bikri, Benamar Rhali
• Keywords: Circular Plates, Functionally Graded Materials (FGMs), Non-Linear Vibration

References

[1] M. El Kadiri, R. Benamar, R.G. White, Improvement of the semi-analytical method, for determining the geometrically non-linear response of thin straight structures. Part I: application to clamped–clamped and simply supported–clamped beams, Journal of Sound and Vibration 249 (2) (2002).

DOI: 10.1006/jsvi.2001.3808

[2] M. El Kadiri, R. Benamar, Improvement of the semi-analytical method, for determining the geometrically nonlinear response of thin straight structures. Part II: first and second non-linear mode shapes of fully clamped rectangular plates, Journal of Sound and Vibration 257 (2002).

DOI: 10.1006/jsvi.2002.5026

[3] 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.

DOI: 10.1016/j.jsv.2007.08.011

[4] G.N. Praveen, J.N. Reddy, Nonlinear transient thermo elastic analysis of functionally graded ceramic-metal plates, International Journal of Solids and Structures 35 (1998)4457–4476.

DOI: 10.1016/s0020-7683(97)00253-9

[5] J. Yang, H.S. Shen, Dynamic response of initially stressed functionally graded rectangular thin plates, Composite Structures 54 (2001) 497–508.

DOI: 10.1016/s0263-8223(01)00122-2

[6] S. TIMOSHENKO, S. WEINSOWSKY-KRIEGER and R. M. JONES 1975 Mechanics of Composite Materials, 51. International Student Edition, McGraw-Hill Kogakusha, Ltd. Tokyo.