On the hierarchical Bayesian modelling of frequency response functions

TL;DR

The paper tackles vibration-based SHM under data scarcity and benign structural variations by developing probabilistic FRF models within a hierarchical Bayesian framework. It uses partial pooling to share information across nominally-identical structures and introduces physics-informed functional relationships to describe how modal parameters change with temperature, enabling extrapolation to unseen conditions. Two case studies—population FRFs for four blades at ambient temperature and temperature-variant FRFs for a single blade—demonstrate variance reduction and accurate extrapolation (NMSE < 5%) while maintaining interpretability through modal parameters and residues. The work advances PBSHM by enabling model generalisation to new domains and conditions through population-level learning and physics-based relationships, with practical implications for robust SHM in data-limited scenarios.

Abstract

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.

Figures (21)

• Figure 1: Comparison of data-pooling approaches, using a simple linear regression example.
• Figure 2: DGM of linear regression model, (a) independent (no pooling) and (b) partial pooling.
• Figure 3: Helicopter blade in a substantiated wall mount.
• Figure 4: Sensor locations on the helicopter blades.
• Figure 5: Real components of the helicopter blades FRFs, (a) full bandwidth and (b) bandwidth of interest, at the drive-point location, from vibration data collected at ambient laboratory temperature.