A two-stage fast Bayesian spectral density approach for ambient modal analysis. Part I: Posterior most probable value and uncertainty

Abstract

A two-stage fast Bayesian spectral density approach is proposed in this study to extract modal properties for the cases of separated modes and closely spaced modes. A novel technique for variable separation is developed so that the interaction between spectrum variables (i.e., frequency, damping ratio as well as the spectral density of modal excitation and prediction error) and spatial variables (i.e., mode shape components) can be decoupled completely for both cases. In a first stage, the spectrum variables can be identified through a so-called ‘fast Bayesian spectral trace approach’ (FBSTA) by employing the statistical properties of the sum of auto-spectral density, while the spatial variables can be estimated in a follow up second stage through ‘fast Bayesian spectral density approach’ (FBSDA) by using the statistical information of the entire spectral density matrix. This study also reveals the intrinsic relationship between FBSDA and ‘fast Bayesian FFT approach’ formulated recently when multiple sets of measurements are available. The newly developed two-stage Bayesian approach allows for a fast computation of the most probable values and covariance matrix of modal properties. The companion paper is devoted to assembling the local mode shape components corresponding to different setups to form the overall mode shapes using a Bayesian statistical framework and verifying the proposed algorithms through simulated and field testing data.