On the hierarchical Bayesian modelling of frequency response functions

Structural health monitoring (SHM) strategies seek to evaluate, predict, and maintain structural integrity, to improve the safety and design service life of structures in operation. Many of these strategies involve monitoring changes in structural dynamics, as damage can affect modal properties and present as changes in the characteristics of the resonance peaks of the frequency response function (FRF). While recent advances have improved the safety and reliability of structures, a number of challenges remain, impeding the practical implementation and generalisation of these systems. Like damage, benign variations, such as those caused by changes in temperature or other environmental fluctuations, can affect dynamic properties, making it difficult to distinguish between damage and normal operating conditions. In addition, newly-deployed structures can have insufficient data to describe the normal operating conditions (i.e., data scarcity), which can impair the development of data-based prediction models. Another common challenge is data loss (i.e., data sparsity), which may result from transmission issues, sensor failure, a sample-rate mismatch between sensors, and other causes. Missing data in the time domain will result in decreased resolution in the frequency domain, which can impair dynamic characterisation. For situations that may benefit from information sharing among datasets, e.g., population-based SHM of similar structures, the hierarchical Bayesian approach provides a useful modelling structure. Hierarchical Bayesian models learn statistical distributions at the population (or parent) and the domain levels simultaneously, to bolster statistical strength among the parameters. As a result, variance is reduced among the parameter estimates, particularly when data are limited. In this paper, a combined probabilistic FRF model is developed for a small population of nominally-identical helicopter blades, using a hierarchical Bayesian structure, to support information transfer in the context of sparse data. The modelling approach is also demonstrated in a traditional SHM context, for a single helicopter blade exposed to varying temperatures, to show how the inclusion of physics-based knowledge can improve generalisation beyond the training data, in the context of scarce data. These models address critical challenges in SHM, by accommodating benign variations that present as differences in the underlying dynamics, while also considering (and utilising), the similarities among the domains.