Tolerance modelling of free vibrations of medium thickness functionally graded plates

In many works different averaging methods based on the known Hencky-Bolle-type plate assumptions are applied to describe medium thickness functionally graded plates with a microstructure. The effect of the microstructure size plays a role in various thermomechanical problems of similar plates, cf. Brillouin [1], Baron [2], Jedrysiak [3], Cheng et al. [4]. Unfortunately, governing equations of most of averaged models do not include the effect of the microstructure size on the overall behaviour of these plates. This lack of them is corrected by using the tolerance modelling approach, cf. Wozniak and Wierzbicki [5], Wozniak et al. [6,7]. This method leads to model differential equations with slowly-varying, smooth coefficients. Hence, the new non-asymptotic model of microstructured medium thickness functionally graded plates is proposed. An example formulas of free vibration frequencies will be obtained - the fundamental lower frequency of macrovibrations and the first higher frequency of microvibrations related to the plate microstructure. A certain analysis of them will be shown for various distribution functions of material properties of functionally graded plates.