Addressing challenges in nonlinear structural dynamics: Bayesian methods for input, system, and output identification

The field of structural dynamics is essential for predicting how systems respond to external excitations. This is crucial for optimizing performance, mitigating damage, and guiding maintenance and operational decisions. As engineering projects grow increasingly ambitious, the complexity of these systems intensifies, resulting in more pronounced nonlinear and non-stationary behaviours, as well as greater uncertainty in the physical understanding of these structures. This compounding of factors necessitates the integration of data-driven, uncertainty-aware approaches with physical models to effectively capture the dynamic behaviour. This thesis seeks to addresses these challenges by developing new Bayesian methodologies towards system identification, prediction, and input estimation in nonlinear dynamic systems.

Input identification plays a critical role in structural dynamics, involving the recovery of external forces from output-only measurements. This process is essential for characterising latent forces, providing insight into operational loads, and aiding prognosis through direct fatigue load analysis. The Gaussian process latent force model (GPLFM) has emerged as a powerful tool for input identification, allowing for the recovery of distributions over temporal latent forcing functions in the presence of measurement uncertainty. However, the application of GPLFM faces several significant challenges.

Existing approaches for joint input and state identification using GPLFM have been effective when system nonlinearities are static; however, they have not yet been applied to systems with dynamic nonlinearities. These systems not only introduce complex, time-varying nonlinearities into the system's transition function but also necessitate additional hidden states, which are not simply derivatives or integrals of other states. To address these challenges, this thesis investigates the used of the GPLFM for joint input-state identification in systems with hysteretic nonlinearities.

Furthermore, a limitation of the standard GPLFM is the assumption that the estimated force can be modelled a priori as a stationary process. This assumption can become inadequate when dealing with non-stationary forces, such as those encountered in operational loads such as wind, wave, traffic, or seismic loads. To address this, a new hierarchical state-space formulation of the GPLFM is developed, designed to effectively capture smooth, non-stationary forcing functions.

The identification of nonlinearities in dynamic systems remains a challenge across engineering disciplines, serving to uncover governing physical phenomena and predict system responses to new inputs. To meet these objectives, the GPLFM can also be applied to the identification of latent restoring forces. Latent restoring force identification provides a pathway to infer critical, often nonlinear, internal mechanisms that restore equilibrium. Assuming input-output measurements are available, existing methods for latent restoring force identification have been shown to be effective. However, for many structural applications, access to input-output data is not feasible. Therefore, this work presents a new methodology to perform latent restoring force identification using output-only measurements. By combining the GPLFM latent restoring force framework with novel post-processing techniques, it is demonstrated that it is possible to jointly recover the temporal functions of the latent states, the latent restoring force, and the latent input force, as well as a GP representation of the underlying restoring force surface.

In addition to latent force identification, this thesis explores the under-exploited interface of probabilistic numerics and Bayesian (parametric) system identification. In engineering, accurately modelling nonlinear dynamic systems from data contaminated by noise is both essential and complex. Established Sequential Monte Carlo (SMC) methods, used for the Bayesian identification of these systems, facilitate the quantification of measurement uncertainty in the parameter identification process. However, these methods require numerical integration of nonlinear continuous-time ordinary differential equations (ODEs) to align theoretical models with discretely sampled data. For most nonlinear systems, the absence of closed-form solutions necessitates numerical approximations for this step, which introduces numerical uncertainty into the parameter evaluation process. This thesis develops a new methodology to efficiently identify latent states and system parameters from noisy measurements while simultaneously incorporating probabilistic solutions to ODEs into the identification process.

By addressing these challenges, this thesis bridges critical gaps in Bayesian system identification, prediction, and input estimation, advancing both the theoretical foundations and practical applications of uncertainty-aware tools for the structural dynamicist.