The variational theory of complex rays: a predictive tool for medium-frequency vibrations

Abstract

Today, the main numerical modeling techniques for the analysis of medium-frequency vibrations are all based on finite element or boundary element approaches. In order to represent small-wavelength phenomena in complex structures such as car chassis, satellites or ships, these techniques require a huge number of degrees of freedom (at least seven elements per wavelength are required to represent oscillating solutions). In addition, the solution obtained is highly sensitive to material properties and boundary conditions. The use of high-frequency approaches such as the statistical energy analysis (SEA) or any of its improvements does not appear to be suitable for medium-frequency vibrations: the vibrational behavior becomes too smooth and, in general, the coupling loss factor cannot be calculated in a predictive way. In this paper, a new and multiscale approach to the calculation of the vibrations of elastic structures in the medium-frequency range, called the “variational theory of complex rays”, is presented. The feasibility and effectiveness of the method are demonstrated for structures made of plate assemblies. Several 3D examples are analyzed and compared to the results from a classical finite element approach and from a SEA code. This comparison shows that our method is able to predict the effective quantities at a very low numerical cost.