A Nonsymmetric State-Variable Decomposition for Modal Analysis1

A modal decomposition strategy based on state-variable ensembles is formulated. A nonsymmetric, generalized eigenvalue problem is constructed. The data-based eigenvalue problem is related to the generalized eigenvalue problem associated with free-vibration solutions of the state-variable formulation of linear multi-degree-of-freedom systems. For linear free-response data, the inverse-transpose of the eigenvector matrix converges to the state-variable modal eigenvectors, and the eigenvalues of the nonsymmetric eigenvalue problem approximate those of the state-variable model. As such, the eigenvalues lead to estimates of frequencies and modal damping. The interpretation holds for linear systems with multi-modal free responses, whether damping is large or small, modal or nonmodal, and without the need of input data.