Uncertainty quantification for the Modal Phase Collinearity of complex mode shapes

In Operational Modal Analysis, the modal parameters, i.e. natural frequencies, damping ratios and mode shapes, are estimated from vibration measurements, where they are related to the eigenstructure of a linear system [1]. Only the measured output data are required, such as accelerations, displacements, velocities or strains, that are recorded on the structure during unknown, unmeasured, ambient excitation conditions. The resulting modal parameter estimates are never equal to the exact parameters of the structure, since they are computed from data of finite length that is moreover afflicted with measurement noise. They are hence impaired with statistical uncertainties. These uncertainties can be quantified or accounted for, which is often crucial in practice when interpreting the outcome from the related system identification algorithms. In this context, explicit expressions for the variance computation of the modal parameter estimates have been developed The Modal Phase Collinearity (MPC) is a modal indicator designed to decide whether the mode shape used in its computation is a real or complex-valued vector. Its estimate inherits the statistical properties of the corresponding mode shape estimate. While the statistical framework for the uncertainty quantification of modal parameters is well-known and developed in the context of subspace-based system identification methods, uncertainty quantification for the MPC estimate has not been carried out yet. In this paper, the uncertainty quantification of the MPC estimates is developed when the corresponding mode shapes are complex-valued vectors. In this case, the theoretical value of the MPC is strictly lower than 1 and it is shown that the distribution of the MPC estimate can be approximated as Gaussian. The computation of its variance and the resulting confidence intervals of the MPC estimate are developed. The proposed framework is validated in Monte Carlo simulations and illustrated on experimental data of an offshore structure. (c) 2020 Elsevier Ltd. All rights reserved.