Modal Analysis of Elastic Vibrations of Incompressible Materials Based on a Variational Multiscale Finite Element Method

In this study, we extend the standard modal analysis technique that is used to approximate vibration problems of elastic materials to incompressible elasticity. The second order time derivative of the displacements in the inertia term is utilized, and the problem is transformed into an eigenvalue problem in which the eigenfunctions are precisely the amplitudes, and the eigenvalues are the squares of the frequencies. The finite element formulation that is based on the variational multiscale concept preserves the linearity of the eigenproblem, and accommodates arbitrary interpolations. Several eigenvalues and eigenfunctions are computed, and then the time approximation to the continuous solution is obtained taking a few modes of the whole set, those with higher energy. We present an example of the vibration of a linear incompressible elastic material showing how our approach is able to approximate the problem. It is shown how the energy of the modes associated to higher frequencies rapidly decreases, allowing one to get good approximate solution with only a few modes.