A complete methodology for estimating a set of reduced full-rank mass, damping and stiffness matrices of a structurefrom dynamic testing to engineering applications

Abstract

This thesis is devoted to developing and testing an integrated methodology to obtain a linear physical model that represents the dynamic behaviour of a real structure. It is conceived to be applied in an autonomous way, so, if required, it could belong to a larger SHM system receiving data from a data logging device and processing it in order to periodically assess the state of the structure. The methodology is split into three stages: first, the recorded time signals, measured at a certain number of points n and containing both inputs and outputs, are transformed into a set of frequency response functions; then, a set of m modes are autonomously identified from the estimated FRFs by sequentially applying different techniques; and, finally, a set of real-valued and symmetric mass, damping and stiffness matrices of dimension n x n are estimated from the identified modal model, which can be complete (m = n) or incomplete (m < n). The whole methodology is conceived so it can be applied to proportionally and non-proportionally damped viscous models. In previous works, other authors have shown that an inverse procedure can be applied to complete or incomplete proportionally damped models in order to obtain real-valued symmetric matrices that accurately represent the same measured dynamic behaviour, being the obtained physical matrices rank-deficient if the modal model is incomplete. Other authors have established conditions under which a complete non-proportionally damped model can provide similar, yet meaningful results. This work is aimed at including them all, as well as incomplete non-proportionally damped models, in a robust methodology that, in general, does not provide rank-deficient matrices. It is demonstrated that, for the case in which an incomplete non-proportionally damped modal model is considered, a certain condition linking n and m must be met in order to obtain symmetric and real-valued physical matrices. The estimated physical matrices contain the same dynamic behaviour as the monitored structure on a subset of degrees of freedom, so the methodology can also be seen as a way of condensing the continuous structure into the $n$ monitored degrees of freedom. The terms of those matrices do not have any direct physical meaning; however, since they contain the relevant dynamic behaviour, they prove to be useful for some applications such as structural modifications or the addition of substructures. Some examples are provided to demonstrate that, as long as the values of the modifications remain within a reasonable range, the estimated physical models accurately reproduce the same effects and simulation results as the original structures subjected to the same modifications.