On the modelling of dynamic behavior of periodic lattice structures

The aim of this contribution is to propose and apply a new approach to the formulation of mathematical models for the analysis of dynamic behavior of dense periodic lattice structures (space or plane trusses) of an arbitrary form. The modelling approach is carried out on two levels. First, we formulate a discrete model, represented by the system of finite difference equations with respect to the spatial coordinates. The obtained equations describe both low- and high-frequency wave propagation problems. Second. two continuum models are derived directly from the finite difference equations and represented respectively by the second- and the fourth-order PDEs with constant coefficients. These models have a physical sense provided that the considerations are restricted to the long wave propagation phenomena. The proposed approach is applied to the vibration analysis for a certain plane lattice structure. Special attention is given to the determination of the range of applicability of the continuum models.