Uncertainty law in ambient modal identification-Part I: Theory

Ambient vibration test has gained increasing popularity in practice as it provides an economical means for modal identification without artificial loading. Since the signal-to-noise ratio cannot be directly controlled, the uncertainty associated with the identified modal parameters is a primary concern. From a scientific point of view, it is of interest to know on what factors the uncertainty depends and what the 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. For example, what is the minimum data duration to achieve a 30% coefficient of variation (c.o.v.) in the damping ratio? To address these questions, this work investigates the leading order behavior of the 'posterior uncertainties' (i.e., given data) of the modal parameters in a Bayesian identification framework. In the context of well-separated modes, small damping and sufficient data, it is shown rigorously that, among other results, the posterior c.o.v. of the natural frequency and damping ratio are asymptotically equal to (ζ/2πN_cB_f)^1/2 and 1/(2πζN_cB_ζ)^1/2, respectively; where ζ is the damping ratio; N_c is the data length as a multiple of the natural period; B_f and B_ζ are data length factors that depend only on the bandwidth utilized for identification, for which explicit expressions have been derived. As the Bayesian approach allows full use of information contained in the data, the results are fundamental characteristics of the ambient modal identification problem. This paper develops the main theory. The companion paper investigates the implication of the results and verification with field test data.