Abstract
Close modes are much more difficult to identify than well-separated modes and their identification (ID) results often have significantly larger uncertainty or variability. The situation becomes even more challenging in operational modal analysis (OMA), which is currently the most economically viable means for obtaining in-situ dynamic properties of large civil structures and where ID uncertainty management is most needed. To understand ID uncertainty and manage it in field test planning, this work develops 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. The expressions reveal a fundamental definition that quantifies ‘how close is close’ and demystify the roles of various governing factors. The results are verified with synthetic, laboratory and field data. Statistics of governing factors from field data reveal OMA challenges in different situations, now accountable within a coherent probabilistic framework. Recommendations are made for planning ambient vibration tests taking close modes into account. Up to modelling assumptions and the use of probability, the uncertainty law dictates the achievable precision of modal properties regardless of the ID algorithm used. The mathematical theory behind the results in this paper is presented in a companion paper.
Highlights
The modal properties of a structure include primarily the natural frequencies, mode shapes and damping ratios
We show that the c.o.v. of natural frequencies δ fi and damping ratios δζ i are given by ζi 2 π N c i
Being a theoretical ensemble average over long data in hypothetical repeated experiments distributed according to the same likelihood function of Bayesian Operational Modal Analysis (BAYOMA), the exact Fisher Information Matrix (FIM) value does not depend on the particular data set used but rather the ‘true’ modal properties
Summary
The modal properties of a structure include primarily the natural frequencies, mode shapes and damping ratios. It comes with no surprise that close modes are much more difficult to identify than well-separated modes Since their frequencies are close, their detection requires as many measured DOFs (as the number of modes) along directions spanning the MSS so that the data PSD matrix has sufficient rank to show multiple significant lines in the SV spectrum. Consider two classically damped modes ( i = 1,2 ) with natural frequencies fi (Hz), damping ratios ζ i and mode shapes φi (real-valued, confined to measured DOFs and normalised with unit sum of squares), subjected to ambient excitations whose modal forces are assumed to be stochastic stationary with constant PSDs Sii ( g2 / Hz ) and coherence χ = S21 / S11S22 (complex-valued) within the resonance band covering the two modes (so only band-limited white). The effect of ‘leakage’, i.e., smearing of energy over neighbouring frequencies in FFT, is neglected in the scope of uncertainty law because it is asymptotically small for long data
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