# ANALYSIS OF A FINITE STAGGERED PERIODIC BEAM DYNAMICS UNDER DIFFERENT BOUNDARY CONDITIONS

## Palavras-chave:

metamaterials, attenuation band, computational efficiency## Resumo

Passive solutions for the vibrations control via metamaterials have gained relevance due to the possi-

bility of developing a considerable vibration attenuation on lightweight and compact structures, with competitive

performance in certain frequency bands. In this context, the modelling of these periodic systems has central impor-

tance in the design of such structures, allowing for sensitivity analyses, design refinement, and even, optimisation.

Two families of modelling procedures can be highlighted: those that require the full system to be modelled, in-

cluding the full structural domain, boundary conditions and external loads (if needed), and those that are based on

a single unit-cell model under periodic boundary conditions, more compact than the latter but with some important

limitations, e.g. the representation of global boundary conditions and external loads. Typical solutions for the

modelling of dynamic systems make use of numerical methods, such as the Finite Element Method (FEM); how-

ever, this may become an unlikely alternative in systems that have a very large number of degrees of freedom due

to the high computational costs involved. Regarding the modelling of periodic systems, the Wave Finite Element

(WFE) approach has been presented as an interesting alternative due to the possibility of reducing the computation

effort, as it is based on an FEM model of a single unit cell, with the force/displacement relationships between 2

adjacent unit cells determined through the transfer matrix. The WFE requires the resolution of an eigenvalue prob-

lem, which may present ill-conditioning depending on the properties of the system. In this study, the dynamics of

staggered periodic beams are evaluated using the WFE approach to calculate the forced response in the frequency

domain of such systems subjected to a bending load. The influence of different boundary conditions imposed on

an increasing number of unit cells (the so-called metastructure) is evaluated against vibration attenuation perfor-

mance. In addition, the results obtained via FE and WFE are compared and contrasted both, in terms of accuracy,

and in terms of computational effort.