High-frequency dynamics of heterogeneous slender structures

This paper gives an overview of the theoretical modeling of high-frequency linear dynamics of built-up structures including the influence of uncertainties by a probabilistic approach. Its analytical developments are enlightened by a preliminary discussion on the vibrational responses of such systems as observed from some experiments conducted in a broad frequency range of excitation. The paper first reviews the main engineering approaches used so far to address the higher frequency domain, namely the statistical energy analysis and the vibrational conductivity analogy. Both methods establish heuristic steady diffusion equations to describe the spatial distribution of the vibrational energy. It is then argued that several limitations and assumptions which restrict their range of validity may be released if a wave transport model is invoked. The latter describes the multiple reflections of high-frequency elastic waves in heterogeneous (possibly random) media adopting a kinetic point of view pertaining to the associated energy density. Transient transport equations evolve into unsteady diffusion equations after long times, supporting in this respect the engineering approaches. Thus the second part of the paper is devoted to a generic presentation of some recent works on kinetic transport models for application to structural dynamics. This objective requires the extension of the existing results of that theory to include dissipation and boundary effects. The proposed models are illustrated by a numerical example showing their consistency with an SEA computation, and the concurrence of a time domain simulation with a frequency domain result.

► Review of engineering approaches for high-frequency vibrations of built-up systems.
► Influence of uncertainties by a parametric probabilistic approach.
► We develop kinetic models for transient dynamics in terms of energetic observables.
► Damping and boundary conditions for transport equations.
► Diffusion limits with equipartition rules within substructures are exhibited.