Asymptotic analysis of high-frequency modulation in periodic systems. Analytical study of discrete and continuous structures

Abstract

This paper addresses the modeling of large scale modulations of high-frequency mechanical waves propagating in discrete and continuum periodic reticulated structures. By means of an asymptotic approach implemented using a multi-cells approach, we derive (i) the equations governing the large scale modulations, (ii) the specific features of the modulation propagation, and (iii) the domain of validity of the description. This approach provides specific information on the physics of large scale modulations that is not directly accessible from the Bloch wave decomposition nor from a multi-scattering approach. The theoretical formulation enables (i) the identification of the frequency range for which evolutions of the rapidly oscillating wave motions arise over long distances, and (ii) simple calculation of the high-frequency wave field based on a two-step procedure that separate the (multi-) cell and the large modulation scales.The asymptotic method is introduced by studying the academic case of a 1D spring-mass chain. This allows us to establish the correspondence between the modulation phenomena and the classical approaches of homogenization and Floquet–Bloch waves.High-frequency modulations in reticulated beams are then investigated. The dispersion equation for longitudinal and transversal vibrations are first established. Then, from the asymptotic analysis, the equations governing the structure behavior in the standard homogenization regime and in the modulation regimes are derived. These results are illustrated and supported by numerical simulations. General comments on modulation and potential application are developed in the conclusions.