Best linear approximation of nonlinear and nonstationary systems using Operational Modal Analysis

Operational Modal Analysis (OMA) is a widely used class of tools to identify the modal properties of existing civil engineering structures and mechanical systems under ambient vibrations or normal operating conditions by only employing measured responses. The OMA techniques, however, are confined to estimate modal properties of linear and time-invariant systems, which do not always comply with real life applications. Additionally, even though structures are found to be nonlinear and nonstationary (i.e., time-varying), it is often engineering practice to apply linear models due to their straightforward insight into the dynamic behaviour of the considered system and mathematical properties when compared to more time consuming consideration of nonlinearities and nonstationarities. With that point of view, the concept of evaluating a best linear approximation of a nonlinear response is well established within Experimental Modal Analysis, i.e., deterministic modal analysis where both the system excitation and the corresponding response are measured. Along these lines, the paper takes the first step towards extending the concept of a best linear approximation to correlation-driven OMA in terms of estimated natural frequencies, damping ratios, and mode shapes. This paper proves that the correlation function matrix calculated from the measured response of a nonlinear and nonstationary system can be approximated by a set of free decays of the best linear approximation. In other words, the paper derives and shows through appropriately designed numerical simulations that the linear modal model, estimated using the output-only framework of conventional correlation-driven (i.e., covariance-driven) OMA, leads to suboptimal minimisation of the difference between the true response of a nonlinear and/or nonstationary system and the response of the linear modal model in a least squares sense. Moreover, the paper demonstrates, using an additional numerical simulation case study, that the output-only framework causes an underestimation of the error that is inevitably involved when estimating linear modal properties from a nonlinear and nonstationary response.