Dynamics of One-Directional Functionally Graded Plates with Different Sizes of Microstructure: Theoretical Tolerance...

The dynamics of thin elastic one-directional non-periodic plates are considered in this paper. The structure of these plates is, at a macro level, functionally graded along the x₁-axis, but at the micro level it is non-periodic (tolerance-periodic). In the plates, the effect of a microstructure size on their behaviour can play a crucial role. The tolerance modelling method allows for this effect to be taken into account. This paper mainly proposes that tolerance modelling leads to model equations of two different tolerance models for one-directional functionally graded plates with two kinds of tolerance-periodic microstructures, i.e., (a) those having a microstructure size that is an order of the plate thickness, d~l, and (b) those having the plate thickness that is smaller than a microstructure size, d << l. Derived model equations are characterised by slowly varying coefficients. A subset of these coefficients is contingent on the microstructure size. The models presented herein determine formulas for both fundamental lower-order vibration frequencies and higher-order vibration frequencies, which are related to the microstructure. These models of such plates are implemented in a rudimentary example of free vibrations. Using the Ritz method, formulas of frequencies are obtained.