In–plane vibration analysis of plates with periodic skeletal truss micro–structures

The present contribution focuses on the dynamic in–plane behaviour of two–dimen- sional lattices consisting of periodic micro–skeletal truss structures. An established homogenization asymptotic technique is applied to derive the partial differential equations describing the in–plane vibration of an equivalent homogeneous plate. A discretized formulation is adopted for studying a planar reticulated medium, whose periodic pattern in this specific case is described by a rectangular reference–cell composed by truss elements. The size of the reference–cell is assumed to be much smaller than the length–scale of the whole structure. The adopted homogenization technique employs the Fourier transform of the dynamic equilibrium equations in both the time and space domains, leading to the definition of the so–called symbol of the periodic structure, which embodies all its elastodynamic properties. Upon Taylor series expansions and inverse transformation, the asymptotic value of the symbol yields continuous dynamic equilibrium equations, whose solution describes the system’s response as the reference–cell characteristic dimension tend to zero. The resulting set of partial differential equations is then analyzed for deriving explicit relationships between reference–cell properties and homogenized continuum properties, i.e. parent material properties. Specific selection criteria are found for geometric and material parameters of the reference–cell in view of getting orthotropic as well as isotropic homogenized plates. Finally, the effects of micro–structural properties on the vibrating behaviour of the homogenized plate are investigated in terms of modal analysis, studying their influence on natural frequencies.