Autonomous Modal Parameter Estimation: Statisical Considerations R.J. Allemang, A.W. Phillips, D.L. Brown Structural Dynamics Research Laboratory School of Dynamic Systems College of Engineering and Applied Science University of Cincinnati Cincinnati, OH 45221-0072 USA Email: Randall.Allemang@UC.EDU ABSTRACT 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. σr = Singular value r from cluster. λr = S domain polynomial root. λr = Complex modal frequency (rad/sec). zr = Z domain polynomial root. {ψr} = Base vector (modal vector). { φr} = 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. 1. Introduction The desire to estimate modal parameters automatically, once a set or multiple sets of test data are acquired, has been a subject of great interest for more than 40 years [1-24] . In the 1960s, even when modal testing was limited to analog test methods, several researchers were exploring the idea of an automated test procedure for determining modal parameters [1-3] . Today, with the increased memory and compute power of current computers used to process test data, an automated or autonomous, modal parameter estimation procedure is entirely possible and is being attempted by numerous researchers. During the development of a new autonomous modal parameter estimation procedure, it became obvious that, since a large number of possible solutions were being evaluated, that this development was a natural way to introduce statistical evaluations into the modal analysis estimation process. This paper reviews some of the statistical estimates that can aid any user in evaluating possible modal parameter estimation solutions. The larger question concerning autonomous modal parameter estimation is the intended user. Is the autonomous modal parameter estimation procedure expected to give results sufficiently robust for the novice user? This implies that the user could have no experience with modal analysis and, therefore, have no experiential judgement to use in assessing the quality of the results. The use of the term wizard implies that this is the desired situation. In contrast, the user could be very knowledgeable in the theory and experienced. For this case, the autonomous modal parameter estimation procedure is simply an efficient mechanism for sorting a very large number of solutions into a final set of solutions that satisfies a set of criteria T. Proulx (ed.), Modal Analysis Topics, Volume 3, Conference Proceedings of the Society for Experimental Mechanics Series 6, 385 DOI 10.1007/978-1-4419-9299-4_33, © The Society for Experimental Mechanics, Inc. 2011
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