Asymptotic identification uncertainty of close modes in Bayesian operational modal analysis

Abstract

Close modes are not typical subjects in operational modal analysis (OMA) but they do occur in structures with modes of similar dynamic properties such as tall buildings and towers. Compared to well-separated modes they are much more challenging to identify and results can have significantly higher uncertainty especially in the mode shapes. There are algorithms for identification (ID) and uncertainty calculation but the value itself does not offer any insight on ID uncertainty, which is necessary for its management in ambient test planning. Following a Bayesian approach, this work investigates analytically the ID uncertainty of close modes under asymptotic conditions of long data and high signal-to-noise ratio, which are nevertheless typical in applications. Asymptotic expressions for the Fisher Information Matrix (FIM), whose inverse gives the asymptotic ‘posterior’ (i.e., given data) covariance matrix of modal parameters, are derived explicitly in terms of governing dynamic properties. By investigating analytically the eigenvalue properties of FIM, we show that mode shape uncertainty occurs in two characteristic types of mutually uncorrelated principal directions, one perpendicular (Type 1) and one within the ‘mode shape subspace’ spanned by the mode shapes (Type 2). Uncertainty of Type 1 was found previously in well-separated modes. It is uncorrelated from other modal parameters (e.g., frequency and damping), diminishes with increased data quality and is negligible in applications. Uncertainty of Type 2 is a new discovery unique to close modes. It is potentially correlated with all modal parameters and does not vanish even for noiseless data. It reveals the intrinsic complexity and governs the achievable precision limit of OMA with close modes. Theoretical findings are verified numerically and applied with field data. This work has not reached the ultimate goal of ‘uncertainty law’, i.e., explicitly relating ID uncertainty to test configuration for understanding and test planning, but the analytical expressions of FIM and understanding about its eigenvalue properties shed light on possibility and provide the pathway to it.