An optimal method for periodic structures design

Periodic structures provide filtering behavior for vibrations, as a result of the repetition in space of unit blocks, or unit cells. In general, they are characterized by an internal mechanical impedance mismatch, so that waves are reflected and transmitted every time a discontinuity is present. The global behavior given by waves superposition is their cancellation, only for specific frequency ranges, generally called stop-bands or band-gaps. The variation of non-dimensional parameters shows how these attenuation regions move in the frequency domain: the correspondent diagrams are the main tools for the design problem and are known as band-maps. The selection of the geometrical, physical and elastic properties of the unit cell is therefore dependent on the designer experience and nothing can be said about the optimality of the proposed solution. Numerical methods are used for the selection of the best cell geometry, in order to get optimal attenuation. Generally, this is a time consuming approach. In this paper, an new method is presented, based on how the waves are reflected and transmitted at cells interface. Both beam and rod case studies are investigated. The algorithm allows matching between band-gap central frequency and the desired value, while the designed attenuation is optimal there, under certain physical and geometrical constraints. Moreover, the design of the bandgap location has been decoupled from the design of the magnitude of attenuation. This approach is purely analytic, therefore the computational efforts required are minimum. In order to validate the analytical model, a passive periodic beam has been manufactured. Its real frequency response is therefore compared to the expected one.