Higher order vibrations of thin periodic plates

A problem of higher order vibrations of thin periodic plates related to a plate periodicity is discussed. The applied model, based on the Kirchhoff plate theory assumptions and additional hypothesis of tolerance averaging [Woźniak Cz, Wierzbicki E. Averaging techniques in thermomechanics of composite solids. Tolerance averaging versus homogenization. Poland, Częstochowa: Częstochowa University of Technology; 2000], makes it possible to take into account the effect of the periodicity cell size on the overall plate behaviour. Assuming more functions describing oscillations of the periodicity cell, we can investigate within the model not only fundamental vibrations but also higher order vibrations related to a periodic plate structure. These functions have to be properly chosen approximations of solutions to eigenvalue problem for natural vibrations of separated cell with periodic boundary conditions, [Jędrysiak J. On vibrations of thin plates with one-dimensional periodic structure. Int J Eng Sci 2000;38:2023-2043]. Using the known two-dimensional plate models—the orthotropic plate model and finite element method as well as the discrete model, certain justifications of the proposed model are made.