Reduced models in the medium frequency range for general dissipative structural-dynamics systems

Abstract This paper presents a theoretical approach for constructing a reduced model, in the medium frequency range, in the area of structural dynamics for a general three-dimensional anisotropic and inhomogeneous viscoelastic bounded medium. All the results presented can be used for beams, plates and shells. The boundary value problem in the frequency domain and its variational formulation are presented. For a given medium frequency band, an energy operator which is intrinsic to the dynamic system is introduced and mathematically studied. This energy operator depends on the dissipative part of the dynamical system. It is proved that this operator is a positive-definite symmetric trace operator in a Hilbert space and that its dominant eigensubspace allows a reduced model to be constructed using the Ritz-Galerkin method. A finite dimension approximation of the continuous case is presented (for instance using the finite element method). An effective construction of the dominant subspace using the subspace iteration method is developed. Finally, an example is given to validate the concepts and the algorithms.