Uncertainty law in operational modal analysis

Ambient modal identification, or 'operational modal analysis', allows one to identify the modal properties of a structure based on (output-only) measured response (e.g., acceleration) without artificial loading or operation interruption. Since no information about the loading is used in the identification process, the uncertainty associated with the identified modal parameters is often significantly higher than those based on tests with known (forced vibration) or no excitation (free vibration). From a scientific point of view, it is of interest to know what factors the uncertainty depends on and what the scaling relationship is. For planning or specification purposes, it is desirable to have an assessment of the test configuration required to achieve a specified accuracy in the modal parameters. E.g., how much data is needed to achieve a 30% coefficient of variation (c.o.v.) in the damping ratio? For well-separated modes, small damping and sufficiently long data, this paper presents close-form analytical formulas for the accuracy of the modal parameters (natural frequency, damping ratio, mode shape) given the ambient vibration data. These 'uncertainty laws' are derived based on an asymptotic analysis of the posterior (i.e., given data) covariance matrix formulated using a Bayesian approach. Among other results, it is shown rigorously that the posterior c.o.v. of the natural frequency and damping ratio are asymptotically equal to (ζ/2πNcBf )1/2 and (2πζNcBζ )-1/2, respectively; where ζ is the damping ratio; Nc is the data length as a multiple of the natural period; Bf and Bζ are data length factors that depend only on the bandwidth utilized for identification, forwhich explicit expressions have been derived.As the Bayesian approach follows fundamentally from probability logic and random vibration assumptions, they dictate the lower limits that can be achieved by any other method (Bayesian or non-Bayesian) based on the same set of assumptions and data.