Multi-mode Problem

Abstract

This chapter develops the computational algorithm for determining the most probable value (MPV) and covariance matrix of modal parameters in Bayesian operational modal analysis with data in a single setup and for the general case of multiple (possibly close) modes. The partial mode shapes, i.e., confined to the measured degrees of freedom, are not necessarily orthogonal and this presents difficulty in their efficient identification. Representing them via an orthogonal basis of a ‘mode shape subspace’ allows efficient determination in terms of such basis and the associated coordinates. An iterative algorithm is presented for determining the MPV, where the MPVs of different groups of parameters are updated until convergence. The asymptotic behavior of the MPV for modes with high signal-to-noise ratio is analyzed. Analytical formulas are derived for systematically computing the covariance matrix of parameters given the measured data. Examples with synthetic data and field data are presented to illustrate the algorithms and their applications.