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Beam lattice materials are characterized by a periodic microstructurerealizing a geometrically regular pattern of elementary cells. In thesemicrostructured materials, the dispersion properties governing the free dynamicpropagation of elastic waves can be studied by formulating parametriclagrangian models and applying the Floquet-Bloch theory. Within this framework,governing the wave propagation by means of spectral design techniques and/orenergy dissipation mechanisms is a major issue of theoretical and appliedinterest. Specifically, the wave propagation can be inhibited by purposelydesigning the microstructural parameters to open band gaps in the materialspectrum at target center frequencies. Based on these motivations, a generaldynamic formulation for determining the dispersion properties of beam latticemetamaterials, equipped with local resonators is presented. The mechanism oflocal resonance is realized by tuning periodic auxiliary masses,viscoelastically coupled with the beam lattice microstructure. As peculiaraspect, the viscoelastic coupling is derived by a mechanical formulation basedon the Boltzmann superposition integral, whose kernel is approximated by aProny series. Consequently, the free propagation of damped waves is governed bya linear homogeneous system of integral-differential equations of motion.Therefore, differential equations of motion with frequency-dependentcoefficients are obtained by applying the bilateral Laplace transform. Thecorresponding complex-valued branches characterizing the dispersion spectrumare determined and parametrically analyzed. Particularly, the spectracorresponding to Taylor series approximations of the equation coefficients areinvestigated.

Free and forced wave propagation in beam lattice metamaterials with viscoelastic resonators