The Galerkin method and RungeKutta method are used for dynamical analysis of shells to give expressions of natural frequencies and non-linear dynamic responses. Numerical results show the essential influence of characteristics of functionally graded materials and dimension ratios on the dynamical behaviors of shells. | Vietnam Journal of Mechanics, VAST, Vol. 32, No. 4 (2010), pp. 199 – 210 NON - LINEAR VIBRATION OF FUNCTIONALLY GRADED SHALLOW SPHERICAL SHELLS Dao Huy Bich1 and Le Kha Hoa2 1 Vietnam National University, Hanoi 2 Military Logistic Academy Abstract. The present paper deals with the non-linear vibration of functionally graded shallow spherical shells. The properties of shell material are graded in the thickness direction according to the power law distribution in terms of volume fractions of the material constituents. In the derived governing equations geometric non-linearity in all strain-displacement relations of the shell is considered. From the deformation compatibility equation and the motion equation a system of partial differential equations for stress function and deflection of shell is obtained. The Galerkin method and RungeKutta method are used for dynamical analysis of shells to give expressions of natural frequencies and non-linear dynamic responses. Numerical results show the essential influence of characteristics of functionally graded materials and dimension ratios on the dynamical behaviors of shells. 1. INTRODUCTION In recent years, functionally graded materials (FGMs) have gained considerable attention in the engineering applications. FGMs are microscopically inhomogeneous, in which the material properties vary smoothly and continuously from one surface to the other. Studies on the stability and vibration of functionally graded plates, cylindrical and shallow shells have been carried out. About vibration of FGM plates Vel . and Batra . [1] gave three dimensional exact solution for the vibration of FGM rectangular plates; Ferreira . et al. [2] received natural frequencies of FGM plates by meshless method. Natural frequencies and buckling stresses of FGM plates and shallow shells were analyzed by Hiroyuki Matsunaga [3,4] using 2-D higher-order deformation theory. Pradyumna S. and Bandyspadhyay . received natural frequencies of FGM .
