A space-averaged quadratic method for predicting one-dimensional power transfers

A method, which is based on space-averaged quadratic variables and does not assume the restrictive assumptions of the Statistical Energy Analysis (SEA) such as high overlap or damping conditions, is described. The results are spatially more detailed than an SEA response, and of a lower computational cost than the one of the usual displacement formulation. Some of the quadratic variables used, like energy densities or structural intensity, retain a strong energy meaning, even when time- and space-averaged. Their governing equation is derived for a homogeneous and isotropic medium with hysteretic damping. The main originality of this work lies in considering complex structural intensity and establishing appropriate energetic boundary conditions for both active and reactive space-averaged structural intensities by using the usual boundary conditions for the displacement field and the stress tensor. The numerical examples prove that this space-averaged quadratic method is well suited to describe global energy transfers along one-dimensional dissipative structures in a frequency range for which the overlap is too low to obtain a quadratic response from the SEA and solving the wave equation by using the Finite Element Method (FEM) would require more elements. Besides, describing junctions with impedances makes the resonant behaviour of the system still accessible on frequency responses.