Full-field response monitoring in structural systems driven by a set of identified equivalent forces

Kalman-type filters for coupled input-state estimation can be used to estimate the full-field dynamic response of structures from only a limited set of vibration measurements. The use of these coupled estimators allows for response prediction to be performed in the absence of any knowledge of both the dynamic evolution and spatial distribution of the excitation forces, where often a set of response-driving equivalent forces is identified from the measurements. In this contribution, a rigorous analysis of the concept of equivalent force based response monitoring is performed, with the aim to clearly establish its limitations and ranges of applicability. It is shown that, unlike commonly assumed, the success of this type of response monitoring cannot be related solely to whether the chosen set of equivalent forces satisfy the controllability requirements, but will depend on the bandwith of the excitation forces in combination with the extent/characteristics of the sensor network. Arguments are instantiated using simple numerical examples where a comparison is made between the theoretical assumptions used to derive the filters and the physical situation. Included in the analyses are situations where (a) the applied and equivalent loads are concentrated and collocated, (b) the applied and equivalent loads are concentrated and non-collocated, (c) modal equivalent loads are used to represent concentrated non-moving forces, and (d) modal equivalent loads are used to represent concentrated moving forces. Results are applicable to any Kalman-type coupled input-state estimator derived using the principles of minimum-variance unbiased estimation.