Devaux C., Joly N., Pascal J.

Quadratic variables have proved to be useful to model high frequency dynamics problems. This is illustrated by the development of energy methods, among which Statistical Energy Analysis (SEA) whose resulting information consists in averaged quadratic quantities describing the behavior on a population of modes of the subsystems. In fact quadratic variables can be time- and space-averaged, keeping a strong energy meaning for some of them like enery densities or the structural intensity. With the aim to model power transfers in a middle frequency range for one-dimensional structures and to overcome some limitations of SEA, especially the lack of spatial information of the responses, such averaged quadratic variables are considered here. Because of the wave vector combinations, quadratic variables hold two kinds of components matching with two different scale lengths: the smallest one is the size of the half wavelength and results of interferences of propagative waves, whereas the largest one accounts for dissipative phenomena and global energy transfers at a large scale compared to the wavelength. When space-averaged, quadratic variables for plane waves are found to be proportionnal to their large scale components. This link means space-averaged quadratic variables are evanescent solutions of propagation equations and can be used in a middle frequency range to represent a resonant behavior with a much lower computational cost than element methods. So the main part of this work deals with establishing boundary conditions for the space-averaged structural intensity, either for active or passive junctions. Once the model is complete for this variable, space-averaged energy densities can be derived. Numerical examples are given and when compared to the solution of the displacement formulation, they prove the relevance of this energy method to model global energy transfers along the one-dimensional dissipative structure in a middle frequency range.