In this paper, we provide a comprehensive definition and classification of various sources of uncertainty within the fields of structural dynamics, system identification, and structural health monitoring (SHM), with a primary focus on the latter. Utilizing the classical input-output system representation as a main contextual framework, we present a taxonomy of uncertainties, intended for consistent classification of uncertainties in SHM applications: (i) input uncertainty; (ii) model form uncertainty; (iii) model parameter/variable uncertainty; (iv) measurement uncertainty; and (v) inherent variability. We then critically review methods and algorithms that address these uncertainties in the context of key SHM tasks: system identification and model inference, model updating, accounting for environmental and operational variability (EOV), virtual sensing, damage identification, and prognostic health management. A benchmark shear frame model with hysteretic links is employed as a running example to illustrate the application of selected methods and algorithmic tools. Finally, we discuss open challenges and future research directions in uncertainty quantification for SHM.

A method is introduced for the identification of the nonlinear governing equations of dynamical systems in the presence of discontinuous and nonsmooth nonlinear forces, such as the ones generated by frictional contacts, based on noisy measurements. The so-called Physics Encoded Sparse Identification of Nonlinear Dynamics (PhI-SINDy) builds upon the existing RK4-SINDy identification scheme, incorporating known physics and domain knowledge in three different ways (biases). In this way, it addresses the discontinuous behavior of frictional systems when stick–slip phenomena are observed, which can not be captured by existing state-of-the-art approaches. The potential of PhI-SINDy is highlighted through a plethora of case studies, starting from a simple yet representative Single Degree of Freedom (SDOF) oscillator with a Coulomb friction contact under harmonic load, using both synthetic and experimental noisy measurements. An alternative friction law, namely the Dieterich-Ruina one, is also considered as well as a more realistic excitation time series, which was generated based on the Jonswap spectrum. Lastly, a Multi Degree of Freedom system with single and multiple friction contacts is used as a testbed, showcasing the applicability of PhI-SINDy to more complicated systems and/or multiple sources of discontinuous nonlinearities.

Manual inspection and assessment of structures on a large scale is labour intensive and often infeasible, while data-driven machine learning techniques can fail to identify relevant failure mechanisms and suffer from poor generalization to previously unseen conditions, particularly when limited information is available. We propose a physics-informed variational autoencoder formulation for disentangled representation learning of confounding sources in the measurements with the aim of computing the posterior distribution of latent parameters of a physics-based model and predicting the response of a structure when limited measurements are available. The latent space of the autoencoder is augmented with a set of physics-based latent variables that are interpretable and allow for domain knowledge in the form of prior distributions and physics-based models to be included in the autoencoder formulation. To prevent the data-driven components of the model from overriding the known physics, a regularization term is included in the training objective that imposes constraints on the latent space and the generative model prediction. The feasibility of the proposed approach is evaluated on a synthetic case study.

The application of vibration-based Structural Health Monitoring (SHM) for damage detection is characterised by three fundamental aspects: the features extracted as representative of the structural condition that can be directly linked to some form of damage, the metrics selected as novelty or damage index, and the statistical model or classifier built to identify underlying patterns indicative of changes in the structure's state. Focusing on the first step to improve the performance of vibration-based SHM strategies, the extracted features should be robust to noise, sensitive to the presence of a specific type of damage. Further, damage should induce patterns that are distinguishable from environmental and operational variabilities and other forms of damage such as ageing phenomena. In this paper, the problem is framed as an outlier detection problem and the the use of different modal parameters as Damage Sensitive Features (DSFs) is investigated, evaluating them based on the detection performance of an unsupervised One-Class Support Vector Machine (OCSVM) classifier. In particular, an artificial dataset is generated from the calibrated numerical model of a three-storey steel frame structure in different damage scenarios. The methodology is validated against available experimental data. For the case investigated the natural frequencies were sensitive to damage and robust to noise.

The decreasing cost and improved sensor and monitoring system technology (e.g., fiber optics and strain gauges) have led to more measurements in close proximity to each other. When using such spatially dense measurement data in Bayesian system identification strategies, the correlation in the model prediction error can become significant. The widely adopted assumption of uncorrelated Gaussian error may lead to inaccurate parameter estimation and overconfident predictions, which may lead to suboptimal decisions. This article addresses the challenges of performing Bayesian system identification for structures when large datasets are used, considering both spatial and temporal dependencies in the model uncertainty. We present an approach to efficiently evaluate the log-likelihood function, and we utilize nested sampling to compute the evidence for Bayesian model selection. The approach is first demonstrated on a synthetic case and then applied to a (measured) real-world steel bridge. The results show that the assumption of dependence in the model prediction uncertainties is decisively supported by the data. The proposed developments enable the use of large datasets and accounting for the dependency when performing Bayesian system identification, even when a relatively large number of uncertain parameters is inferred.

Multi-modal distributions of some physics-based model parameters are often encountered in engineering due to different situations such as a change in some environmental conditions, and the presence of some types of damage and non-linearity. In statistical model updating, for locally identifiable parameters, it can be anticipated that multi-modal posterior distributions would be found. The full characterization of these multi-modal distributions is important as methodologies for structural condition monitoring in structures are frequently based in the comparison of the damaged and healthy models of the structure. The characterization of posterior multi-modal distributions using state-of-the-art sampling techniques would require a large number of simulations of expensive-to-run physics-based models. Therefore, when a limited number of simulations can be run, as it often occurs in engineering, the traditional sampling techniques would not be able to capture accurately the multi-modal distributions. This could potentially lead to large numerical errors when assessing the performance of an engineering structure under uncertainty. Therefore, an approach is proposed for drastically reducing the number of models runs while yielding accurate estimates of highly multi-modal posterior distributions. This approach introduces a cyclical annealing schedule into the Variational Bayes Monte Carlo (VBMC) method to improve the algorithm's phase of exploration and the finding of high probability areas in the multi-modal posteriors throughout the different cycles. Three numerical and one experimental investigations are used to compare the proposed cyclical VBMC with the standard VBMC algorithm, the monotonic VBMC and the Transitional Ensemble Markov Chain Monte Carlo (TEMCMC). It is shown that the standard VBMC fails in capturing multi-modal posteriors as it is unable to escape already found regions of high posterior density. In the presence of highly multi-modal posteriors, the proposed cyclical VBMC algorithm outperforms all the other approaches in terms of accuracy of the resulting posterior, and number of model runs required.

Often overlooked, vibration transmission through the entire body of an animal is an important factor in understanding vibration sensing in animals. To investigate the role of dynamic properties and vibration transmission through the body, we used a modal test and lumped parameter modelling for a spider. The modal test used laser vibrometry data on a tarantula, and revealed five modes of the spider in the frequency range of 20-200 Hz. Our developed and calibrated model took into account the bounce, pitch and roll of the spider body and bounce of all the eight legs. We then performed a parametric study using this calibrated model, varying factors such as mass, inertia, leg stiffness, damping, angle and span to study what effect they had on vibration transmission. The results support that some biomechanical parameters can act as physical constraints on vibration sensing. But also, that the spider may actively control some biomechanical parameters to change the signal intensity it can sense. Furthermore, our analysis shows that the parameter changes in front and back legs have a greater influence on whole system dynamics, so may be of particular importance for active control mechanisms to facilitate biological sensing functions.

A self-supervised classification algorithm is proposed for detecting and isolating sensor faults of health monitoring devices. This is achieved by automatically extracting information from failure investigations. This approach uses (i) failure reports for extracting comprehensive failure labels; (ii) recorded data of a faulty monitoring device and the information of the failure type for selecting fault-sensitive features. The features-label pairs are then used to train a classification algorithm, so that when a new set of measurements becomes available, the algorithm is capable of identifying with a high accuracy one of the possible failure types included in the training data set. The proposed approach is successfully applied to the failure investigations conducted on a low-cost wearable device, displaying similar challenges encountered in SHM.

The design of complex engineering systems is an often long and articulated process that highly relies on engineers’ expertise and professional judgment. As such, the typical pitfalls of activities involving the human factor often manifest themselves in terms of lack of completeness or exhaustiveness of the analysis, inconsistencies across design choices or documentation, as well as an implicit degree of subjectivity. An approach is proposed to assist systems engineers in the automatic generation of systems diagrams from unstructured natural language text. Natural Language Processing (NLP) techniques are used to extract entities and their relationships from textual resources (e.g., specifications, manuals, technical reports, maintenance reports) available within an organisation, and convert them into Systems Modelling Language (SysML) diagrams, with particular focus on structure and requirement diagrams. The intention is to provide the users with a more standardised, comprehensive and automated starting point onto which subsequently refine and adapt the diagrams according to their needs. The proposed approach is flexible and open-domain. It consists of six steps which leverage open-access tools, and it leads to an automatic generation of SysML diagrams without intermediate modelling requirement, but through the specification of a set of parameters by the user. The applicability and benefits of the proposed approach are shown through six case studies having different textual sources as inputs, and benchmarked against manually defined diagram elements.

The friction force at joints of engineering structures is usually unknown and not directly identifiable. This contribution explores a procedure for obtaining the governing equation of motion and correctly identifying the unknown Coulomb friction force of a mass-springdashpot system. In particular, a single degree-of-freedom system is investigated both numerically and experimentally. The proposed procedure extends the state-of-the-art datadriven sparse identification of nonlinear dynamics (SINDy) algorithm by developing a methodology that explicitly imposes constraints encoding knowledge of the nonsmooth dynamics experienced during stick-slip phenomena. The proposed algorithm consists of three steps: (i) data segregation of mass-motion from mass-sticking during stick-slip response; (ii) application of SINDy on the mass-motion dataset to obtain the functional form of the governing equation; and (iii) applying sticking and slipping conditions to identify the unknown system parameters. It is shown that the proposed approach yields an improved estimate of the uncertain system parameters such as stiffness, viscous damping, and magnitude of friction force (all mass normalized) for various signal-to-noise ratios compared to SINDy.

Virtual sensing techniques have gained traction in applications to the structural health monitoring of monopile-based offshore wind turbines, as the strain response below the mudline, which is a primary indicator of fatigue damage accumulation, is impractical to measure directly with physical instrumentation. The Gaussian process latent force model (GPLFM) is a generalized Bayesian virtual sensing technique which combines a physics-driven model of the structure with a data-driven model of latent variables of the system to extrapolate unmeasured strain states. In the GPLFM, unknown sources of excitation are modeled as a Gaussian process (GP) and endowed with a structured covariance relationship with response states, using properties of the GP covariance kernel as well as correlation information supplied by the mechanical model. It is shown that posterior inference of the latent inputs and states is performed by Gaussian process regression of measured accelerations, computed efficiently using Kalman filtering and Rauch–Tung–Striebel smoothing in an augmented state-space model. While the GPLFM has been previously demonstrated in numerical studies to improve upon other virtual sensing techniques in terms of accuracy, robustness, and numerical stability, this work provides one of the first cases of in-situ validation of the GPLFM. The predicted strain response by the GPLFM is compared to subsoil strain data collected from an operating offshore wind turbine in the Westermeerwind Park in the Netherlands. A number of test cases are conducted, where the performance of the GPLFM is evaluated for its sensitivity to varying operational and environmental conditions, to the instrumentation scheme of the turbine, and to the fidelity of the mechanical model. In particular, this paper discusses the capacity of the GPLFM to achieve relatively robust strain predictions under high model uncertainty in the soil-foundation system of the offshore wind turbine by attributing sources of model error to the estimated stochastic input.

Two non-intrusive uncertainty propagation approaches are proposed for the performance analysis of engineering systems described by expensive-to-evaluate deterministic computer models with parameters defined as interval variables. These approaches employ a machine learning based optimization strategy, the so-called Bayesian optimization, for evaluating the upper and lower bounds of a generic response variable over the set of possible responses obtained when each interval variable varies independently over its range. The lack of knowledge caused by not evaluating the response function for all the possible combinations of the interval variables is accounted for by developing a probabilistic description of the response variable itself by using a Gaussian Process regression model. An iterative procedure is developed for selecting a small number of simulations to be evaluated for updating this statistical model by using well-established acquisition functions and to assess the response bounds. In both approaches, an initial training dataset is defined. While one approach builds iteratively two distinct training datasets for evaluating separately the upper and lower bounds of the response variable, the other one builds iteratively a single training dataset. Consequently, the two approaches will produce different bound estimates at each iteration. The upper and lower response bounds are expressed as point estimates obtained from the mean function of the posterior distribution. Moreover, a confidence interval on each estimate is provided for effectively communicating to engineers when these estimates are obtained at a combination of the interval variables for which no deterministic simulation has been run. Finally, two metrics are proposed to define conditions for assessing if the predicted bound estimates can be considered satisfactory. The applicability of these two approaches is illustrated with two numerical applications, one focusing on vibration and the other on vibro-acoustics.

This study aims at assessing the predictive performance of the Amontons–Coulomb law to reliably predict the cyclic response, inclusive of stick–slip, of a single degree of freedom system in contact with the ground through two versions (steady-state and rate-and-state) of a regularized Dieterich–Ruina law. The assessment is carried out by defining a cost function and a physics-based constraint that enable the identification of the corresponding optimal coefficients of the Amontons–Coulomb law through a multi-start constrained non-linear optimization. The comparative study starts with a sensitivity analysis, aimed at first identifying the most meaningful model parameters for the Dieterich–Ruina law. Subsequently, the cyclic dynamic responses provided by both friction laws are analysed for varying model parameters, and characteristic features are observed within the dynamic forcing–displacement graph and the friction force–velocity plot, that could be directly linked to one friction model or the other. The sensitivity analysis led to the definition of a cost function expressed in terms of the displacement and velocity response differences and a constraint based on the phase difference. The optimization study identified areas of the Dieterich–Ruina's parameter space for which the Amontons–Coulomb law can reliably be used to predict a cyclic stick–slip response. The relevance of these results with respect to problems of modelling and identification of friction are discussed.

Bayesian inference is a popular approach towards parameter identification in engineering problems. Such technique would involve iterative sampling methods which are often robust. However, these sampling methods often require significant computational resources and also the tuning of a large number of parameters. This motivates the development of a sampler called the Transitional Ensemble Markov Chain Monte Carlo. The proposed approach implements the Affine-invariant Ensemble sampler in place of the classical Metropolis–Hastings sampler as the Markov chain Monte Carlo move kernel. In doing so, it allows for the sampling of badly-scaled and highly-anisotropic distributions without requiring extra computational costs. This makes the proposed sampler computationally efficient as a result of having less auxiliary parameters to compute per iteration compared to the standard single particle Transitional Markov Chain Monte Carlo. In addition to such change, an adaptive tuning algorithm is also proposed within the new sampler. This algorithm allows for automatic tuning of the step-size of the Affine-invariant Ensemble sampler. Hence, such proposals not only ensure that the new sampler is “tune-free” for the users, but also improves its robustness by ensuring that the acceptance rate of samples is well-controlled within acceptable bounds. As a result, this approach could be significantly faster compared to standard Transitional Markov Chain Monte Carlo methods on badly scaled and highly skewed distributions, which can be encountered when dealing with complex engineering problems. The proposed sampler will be implemented on 2 benchmark numerical examples of varying complexities to demonstrate its strengths and advantages. In addition, the sampler is validated by investigating its parameter identification capability on an Aluminium Frame using experimental data.