Abstract
This work takes up the challenge of deriving the ‘uncertainty law’ for close modes, i.e., closed form analytical expressions for the remaining uncertainty of modal parameters identified using (output-only) ambient vibration data. In principle the uncertainty law can be obtained from the inverse of the Fisher information matrix of modal parameters. The key mathematical challenges stem from analytical treatment of entangled stochastic dynamics with a large number of modal parameters of different nature and the quest for closed form expressions for identification uncertainty, whose possibility is questionable. Fortunately the problem still admits insightful closed form solution under long data, high signal-to-noise ratio and wide resonance band for identification. Up to modelling assumptions and the use of probability, the uncertainty law dictates the achievable precision of modal properties regardless of the identification algorithm used. A companion paper discusses the insights, verification, scientific implications and recommendation for ambient test planning.
Highlights
Operational modal analysis (OMA) shows great promise as a feasible and economically viable means for obtaining in-situ modal properties of structures [1,2,3]
Closed form analytical asymptotic expressions have been derived for the posterior c.o.v.s of natural frequencies, damping ratios and mode shapes for two close modes following classically damped stochastic linear dynamics identified with long ambient data, high s/n ratio and wide resonance band; see (31), (34), (36) and (44)
Under wide band situation the problem still admits remarkably simple expressions and they have been discovered in this work
Summary
Operational modal analysis (OMA) shows great promise as a feasible and economically viable means for obtaining in-situ modal properties of structures [1,2,3]. Managing ID uncertainty requires another level of advance in knowledge beyond the ability to calculate, involving insights on how it depends on various factors that may affect the quality of ID results It is to discover the fundamental law of how the information about modal properties is extracted from ambient data following stochastic dynamics assumptions. Such discovery need not be possible, but recent research in Bayesian OMA shows that for wellseparated modes with long and high signal-to-noise (s/n) ratio data it is possible to obtain closed form analytical expressions for the ID uncertainty in terms of test configuration and environment [20]. Semi-empirical correction factors have been developed to account for the effect of finite bandwidth in the companion paper
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