Comparison of Various Modal Vector Estimation Methods Used in Modal Parameter Estimation

Abstract

Modal vectors can be estimated in a number of different ways in modern modal parameter estimation (MPE) methods. At the very least, when using single frequency response function (FRF), single degree of freedom (SDOF) methods, the modal vectors are estimated from a least squares estimate of the residues of a partial fraction model in the frequency domain, with or without various residuals, or an equivalent model in the time domain. Once the MPE methods involve multiple input, multiple output (MIMO) FRFs, many options exist. These MIMO MPE methods often involve a matrix polynomial equation that is solved using eigenvalue-eigenvector methods. Depending on the spatial dimensionality of the MIMO FRF matrix, these methods estimate an eigenvector that can be used as an estimate of the modal vector directly or an estimate of a portion (subset) of the modal vector. Alternately, these eigenvectors can be used as weighting in a MIMO version of estimating the residues of the partial fraction model using a weighted least squares method, with or without residuals. These weighting vectors can be normalized to remove arbitrary rotations in the eigenvectors and/or real normalized to influence the potential nature of complex-valued modal vectors. This paper reviews all of the approaches using MIMO FRF data on a simple structure where the modal vectors can be expected to be nearly normal modes. Both modal vectors and the associated modal scaling and mean phase correlation (MPC) are evaluated to document the similarities and differences.