Autonomous Modal Parameter Estimation: Statisical Considerations

Autonomous modal parameter estimations may involve sorting a large number of possible solutions to develop one consistent estimate of the modal parameters (frequency, damping, mode shape, and modal scaling). Once the final, consistent estimate of modal parameters is established, this estimate can be compared to related solutions from the larger set of solutions to develop statistical attributes for the final, consistent set of modal parameters. These attributes will include sample size, standard deviation and other familiar variance estimates. New variance estimates are introduced to categorize the modal vector solution. These modal vector statistics are based upon the residual contributions in a set of correlated modal vectors that are used to estimate a single modal vector. Examples of this statistical information is included for a number of realistic data cases. Nomenclature N = Number of vectors in cluster. σ r = Singular value r from cluster. λ r = S domain polynomial root. λ r = Complex modal frequency (rad/sec). zr = Z domain polynomial root. {ψ r} = Base vector (modal vector). {φ r} = Pole weighted base vector (state vector). Std. Dev. = Standard deviation. NMVR1 = Normalized modal vector residual 1. NMVR2 = Normalized modal vector residual 2. NSVR1 = Normalized state vector residual 1. NSVR2 = Normalized state vector residual 2.