In order to maximize the properties of viscoelastic materials, they are often used as an interlayer between two elastic layers inducing shearing in the core layer and thus passive damping. This paper focuses on the design, characterization and modelling of sandwich structures with periodic and viscoelastic core. The viscoelastic polymers used are blends of isotactic polypropylene grafted maleic anhydride (iPP-g-MAH), selected for its adhesion properties to metal surfaces, with other polymers in order to adapt the mechanical properties. The periodicity of the sandwich structure, is obtained by alternating two materials with specific viscoelastic properties into the core, this allows the resonance frequencies to be located in bundles creating frequency ranges without any resonance peaks. The latter, called "Band Gaps", are simulated using finite element modelling. Parametric studies are then conducted to study the impact of variations in geometric and mechanical properties on the width and position of these Band Gaps. Samples of periodic sandwich composites developed in the laboratory are tested on a vibrating pot in order to experimentally determine the damping properties of the sandwich. The results obtained are then compared with those obtained numerically in order to validate the parameters used in the modelling.