Free vibrations of thin periodic plates interacting with an elastic periodic foundation

A dynamic behaviour of thin periodic plates interacting with an elastic periodic foundation is analysed. The main aim is to investigate free vibrations and calculate frequencies for these plates. The presented modelling approach to the linear-elastic plates, having periodic structure in planes parallel to the midplane, is adopted from the tolerance averaging method developed by Woźniak and Wierzbicki (Averaging techniques in thermomechanics of Composite Solids, Wydawnictwo Politechniki Częstochowskiej, Częstochowa, 2000) for periodic composites. The obtained plate model describes the effect of the periodicity cell size on the overall plate behaviour. In this paper this effect is analysed in the free vibrations problem. Two special cases are taken into account: a square plate with a periodic thickness interacting with a homogeneous foundation and a square plate with a constant thickness interacting with a periodic foundation. A certain justification of the model is also presented.