Free Vibration Analysis of Thin Functionally Graded Plate Bands with Microstructure as a Function of Material Inhomogeneity Distribution and Boundary Conditions

An analysis of free vibrations for thin functionally graded plate bands is presented in this work. On the microlevel these plate bands have a tolerance-periodic microstructure in planes parallel to the mid-plane. Partial differential equations with tolerance-periodic, highly oscillating, non-continuous coefficients describe the vibrations of such plates. Here, the influence of microstructure inhomogeneity is shown on free vibration frequencies of these plate bands with different boundary conditions. This analysis was carried out within the framework of two models of these plates. The models are represented by equations with smooth, slowly varying coefficients. One of these models, called the tolerance model, takes into account the effect of the microstructure size. Hence, it leads not only to formulas of fundamental lower-order vibration frequencies, but also to formulas of higher-order vibration frequencies, which are related to the microstructure. The analyses of free vibration frequencies for thin functionally graded plate bands with different boundary conditions are presented. The formulas of frequencies are obtained using the Ritz method. A comparison of some calculated results to the results obtained by the FEM is also shown.