MESO-SCALE MODELS OF MULTI-STRUCTURES WITH NEGATIVE INERTIA

Abstract The paper presents applications of meso-scale asymptotic models to the analysis of multi-structures with negative inertia. The focus is on the low-frequency regimes of flexural systems, which contain multiple resonators. Such systems do not have to be periodic, and resonators themselves may represent multi-scale structures. The challenge is to analyse low-frequency vibrations of flexural multi-structures, containing large clusters of resonators in the special regimes of negative inertia. The Floquet-Bloch analysis is considered for the elastic waves propagating through a flexural beam, containing a periodic array of spring-mass resonators. This is followed by the solution of the Wiener-Hopf functional equation, corresponding to the transmission/reflection problem. The surprising observation is in the increase of the wavelength of the transmitted wave packet at frequencies that correspond to the negative inertia of resonators. This effect is explained in the asymptotic model, based on the meso-scale approximations of clusters of resonators within the flexural beam.