On assessing the posterior mode shape uncertainty in ambient modal identification

Ambient vibration tests have gained increasing popularity in practice as they provide an economical means of identifying the actual modal properties of a structure in its working state without the need for artificial loading. Since the signal-to-noise ratio cannot be directly controlled, the uncertainty associated with the identified modal parameters is often a primary concern. In a Bayesian context, the posterior most probable value of the mode shape (given the measured data) is the point that minimizes the log-likelihood function; the posterior covariance matrix is equal to the inverse of the Hessian evaluated at the most probable value. This article discusses two technical issues associated with assessing the posterior uncertainty of the mode shape. We discuss how to impose a Euclidean norm constraint on the mode shapes when calculating the Hessian of the log-likelihood function. Second, we investigate the modal assurance criterion (MAC) of the most probable mode shape with a random mode shape consistent with the posterior distribution. The expectation of this MAC is found to provide a global measure of the posterior uncertainty. Ambient modal identifications of a footbridge and a super-tall building are used to illustrate the theoretical findings.