Uncertainty quantification of the Modal Assurance Criterion in operational modal analysis

The Modal Assurance Criterion (MAC) is a modal indicator designed to decide whether the mode shapes used in its computation are corresponding to the same mode. During structural monitoring, it can be applied to evaluate changes in the mode shapes. When the mode shapes are estimated from measurement data, the MAC inherits their statistical properties, thus is afflicted with statistical uncertainty. The evaluation of this uncertainty is particularly relevant when the MAC estimate is close to 1, where 1 indicates equal mode shapes. In structural monitoring, it can be used to assess changes in mode shapes after early damage. While the framework for uncertainty quantification of modal parameters is well-known and developed in the context of subspace-based system identification methods, uncertainty quantification for the MAC has not been developed yet. A particular challenge for its statistical characterization is its boundedness in the interval between 0 and 1. In this paper it is shown that this boundedness yields two different distributions of the MAC estimates. The MAC computed between estimates of different mode shapes is inside the interval (0,1), and a Gaussian approximation of its distribution is obtained. When the MAC is computed between estimates of equal mode shapes, the resultant MAC estimate is close to 1, and the classical Gaussian approximation is inadequate. In this case it is shown that the MAC estimate is linked to a quadratic form of the mode shapes, whose distribution can be approximated by a scaled and shifted χ2 distribution. For both cases, uncertainty bounds related to the MAC estimates are established. The proposed frameworks are validated by extensive Monte Carlo simulations and then applied to evaluate mode shape changes due to damage during monitoring of the S101 Bridge.