On Dynamics of Thin Plates With a Periodic Structure

Abstract

The subject of investigations is elastodynamics of periodic plates like that shown in Fig. l. So far, two general approaches to the modelling of these plates leading to the effective 2D-models (i.e. represented by equations with constant coefficients) have been proposed. The first is based on the asymptotic analysis and multi-scale expansions, cf. [4], and yields what are called homogenized plate equations which have the same form as equations of homogeneous plates, [5,9]. Hence, the main attention in this approach is concentrated on calculations of constant coefficients in a homogenized equation, which are referred to as the effective plate stiffness (cf. [5,9] and [ 10] for an overview of papers on this subject). The second approach applies the tolerance averaging technique as a tool of modelling, [ 14]. We can mention here a series of papers [ 1-3,7,8,11-13] related to dynamics and stability of elastic plates and beams with a periodic structure. The tolerance averaging approach, in contrast to homogenization, makes it possible to describe the local dynamic plate behaviour (e.g. to determine higher order vibration frequencies caused by the plate periodic structure) and takes into account the rotational inertia effect on the plate dynamics. However, until now the effective 2D-models for dynamics of periodic plates, based on the tolerance averaging technique, have been restricted to situations in which the period length is sufficiently large when compared to the plate thickness.