Closed Form Solutions to the Equation of Motion Avoiding Mass Matrix Inversion and Eigenvectors Decomposition

Solutions of the vibrations equations are generally obtained, in the linear case, by methods involving either matrix exponential computation or matrix eigendecomposition. However, these methods lead to a loss of symmetry because of the necessary inversion of the mass matrix. In doing so, one can introduce ill-conditioned matrices and, thus, compute eigenvectors with poor accuracy. In order to avoid these inconveniences, our method, which is an extension of Le Verrier-Souriau algorithm, provides solutions to the vibrations equations without inverting the mass matrix or computing eigenvectors. Moreover, we can solve damping systems even when the damping matrix has no specific properties such as the Basile property.