An important consideration when using sequential Bayesian filters for the estimation of unknown states and inputs, is the potential impact of modelling errors on the accuracy and precision of the estimation. When the modelling errors are explicitly accounted for in the estimation, one can speak of bias-aware filtering. One approach that has been suggested to achieve bias-aware filtering is the use of Gaussian process latent force models. Using this approach, the biases are represented as Gaussian processes and identified from the vibration data in conjunction with the additional unknown quantities. In this contribution, we focus on modelling biases due to miscalibrated dynamic properties in linear system models. We analyze the treatment of such biases using Gaussian process latent force models, and explore the robustness of the approach to changes in sensor configurations using simulated data from a cantilever beam.

To reduce uncertainties associated with its structural reassessment, the Zeeland Bridge in the Netherlands is currently the subject of a field lab, which will run for 2 years. In this contribution, numerical investigations to study the dependencies between variables associated with uncertain structural properties of the bridge and various response/measurement quantities are presented. Initial focus is on load testing of the bridge to obtain insight into the possibly varying response in different spans of the bridge. Parametric studies to expose input-output parameter dependencies are performed on a representative subsystem of the bridge, and the results are used to assist in the design of a measurement campaign and the development of a robust model updating strategy for the bridge.

In applied structural dynamics, input and state estimation techniques are used to identify unmeasurable response quantities and unknown forces from limited sensor data. These estimators depend on mechanical state-space models, which can be inaccurate due to uncertainties in the systems' mass, damping, and stiffness (or equivalently, the modal parameters of a linear system). This work examines the impact of such model errors and their effects are characterized by a theoretical framework for error propagation to the estimated quantities. The results are exemplified in a numerical example of a mass-spring system. The characterization is a first step towards more focused attempts to develop algorithms in the realm of input and state estimation that are able to deal more robustly with numerical models subject to uncertainty.

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.

Wind loading is an essential aspect in the design and assessment of long-span bridges, but it is often not well-known and cannot be measured directly. Most structural health monitoring systems can easily measure structural responses at discrete locations using accelerometers. This data can be combined with reduced-order modal models in Kalman filter-based algorithms for an inverse estimation of wind loads and system states. As a further development, this work investigates the incorporation of Gaussian process latent force models (GP-LFMs), which can characterize the evolution of the wind loading. The Hardanger Bridge, a 1310 m long suspension bridge instrumented with a monitoring system for wind and vibrations, is used as a case study. It is shown how the LFMs can be enriched with physical information about the stochastic wind loads using monitoring anemometer data and aerodynamic coefficients from wind tunnel tests. It is found that the estimates of the modal wind loads and modal states obtained from a Kalman filter and Rauch–Tung–Striebel smoother are stable for acceleration output only, thus avoiding the accumulation of errors. The proposed approach demonstrates how physical or environmental data can be injected as valuable information for global monitoring strategies and virtual sensing in bridges.

Fatigue assessment in offshore wind turbine support structures requires the monitoring of strains below the mudline, where the highest bending moments occur. However, direct measurement of these strains is generally impractical. This paper presents the validation of a virtual sensing technique based on the Gaussian process latent force model for dynamic strain monitoring. The dataset, taken from an operating near-shore turbine in the Westermeerwind Park in the Netherlands, provides a unique opportunity for validation of strain estimates at locations below the mudline using strain gauges embedded within the monopile foundation.

This paper presents a general framework for estimating the state and unknown inputs at the level of a system subdomain using a limited number of output measurements, enabling thus the component-based vibration monitoring or control and providing a novel approach to model updating and hybrid testing applications. Under the premise that the system subdomain dynamics are driven by the unknown (i) externally applied inputs and (ii) interface forces, with the latter representing the unmodeled system components, the problem of output-only response prediction at the substructure level can be tailored to a Bayesian input-state estimation context. As such, the solution is recursively obtained by fusing a Reduced Order Model (ROM) of the structural subdomain of interest with the available response measurements via a Bayesian filter. The proposed framework is without loss of generality established on the basis of fixed- and free-interface domain decomposition methods and verified by means of three simulated Wind Turbine (WT) structure applications of increasing complexity. The performance is assessed in terms of the achieved accuracy on the estimated unknown quantities.

The traditional wind load assessment for long-span bridges relies on assumed models for the wind field and aerodynamic coefficients from wind tunnel tests, which usually introduce some uncertainties. Recent studies have shown that large deviations can exist between the predicted and observed wind-induced dynamic response of suspension bridges. In studies of the dynamical behavior of bridges, inverse force identification methods can therefore be an interesting tool in the assessment of possible uncertainties involved in the modeling of wind loads. This paper presents a novel case study of the identification of the dynamic wind loads on the 1310 ​m long Hardanger bridge, a suspension bridge equipped with a monitoring system for wind and vibrations. The modal wind loads are identified from acceleration data using an algorithm for model-based joint input and state estimation. Several data sets with different wind conditions are presented. The wind loads are studied in the time and frequency domains and are compared to the mean velocity and turbulence characteristics of the wind.

A vessel's response to waves is dependent on a large number of parameters, some of which are both frequency and direction dependent. To predict vessel response, these parameters are used to construct response amplitude operators (RAOs) that act as transfer functions between the directional wave spectra and the motion spectra of the vessel. In particular situations, however, vessel motions predicted using RAOs calculated with general-purpose radiation-diffraction codes and measured wave spectra are found to deviate from measured vessel responses. To address this problem, a methodology for calibrating RAOs based on measurements of the directional wave spectra and vessel motions is proposed. Use is made of a vector fitting method through which the frequency dependent hydrodynamic properties of the vessel can be approximated by a ratio of two polynomials, thus greatly reducing the number of parameters that need to be calibrated. The reduced set of parameters is subsequently related to previously identified causes of RAO inaccuracy in order to arrive at optimization algorithms for identifying more accurate RAOs from the measurements. It is shown that the RAOs can be improved with accuracy in situations where the discrepancies are caused by imprecise estimates for the vessel's radii of gyration, center of gravity, or viscous damping. When the discrepancies in the RAOs are related to the potential mass, damping and wave forces, however, the problem becomes highly non-convex and it is not possible to find a unique RAO that satisfies the data.

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.

The traditional wind load assessment for long-span bridges rely on assumed models for the wind field and aerodynamic coefficients from wind tunnel tests, which usually introduces some uncertainties. It is therefore desired to develop tools that can utilize full-scale vibration response data from existing bridges in order to study the wind loading in detail for in-situ conditions. This paper presents a novel case study of inverse identification of dynamic wind loads on the 1310 m long Hardanger bridge, a suspension bridge equipped with a network of accelerometers. The identification method used is an extented Kalman-type filter for joint input, state, and parameter estimation. A system model considering the still-air modes in addition to a quasi-steady submodel for the self-excited forces of the bridge is presente. The coefficients for self-excited lift and pitching moment are considered unknown and are jointly estimated with the buffeting forces.

The dynamic behaviour of long-span bridges is governed by stochastic loads from typically ambient excitation sources. In real life, these loads cannot be measured directly at full scale. However, inverse methods can be utilised to identify these unknown forces using response measurements together with a numerical model of the relevant structure. This paper presents a case study of full-scale identification of the wave forces on the Bgsøysund bridge, a long-span pontoon bridge that has been monitored since 2013. First, a numerical model of the structure is formed, resulting in a reduced-order state-space model that takes into account the frequency-dependent hydrodynamic mass and damping from the fluid, based on fitting of rational transfer functions. Using acceleration data of the structure measured during several events of moderate and strong seas, the wave forces are identified using stochastic-deterministic methods for combined state and input estimation. In addition, a separate frequency-domain assessment of the wave forces is performed, in which the spectral density of the first-order wave forces is constructed from an estimated directional wave field model driven by wave elevation data. When compared in the frequency domain, the force estimates from the two approaches are of comparable magnitude. However, uncertainties in the assumptions and models behind the force estimates from the two approaches still play a significant role.

Numerical predictions of the dynamic response of complex structures are often uncertain due to uncertainties inherited from the assumed load effects. Inverse methods can estimate the true dynamic response of a structure through system inversion, combining measured acceleration data with a system model. This article presents a case study of full-field dynamic response estimation of a long-span floating bridge: the Bergøysund Bridge in Norway. This bridge is instrumented with a network of 14 triaxial accelerometers. The system model consists of 27 vibration modes with natural frequencies below 2 Hz, obtained from a tuned finite element model that takes the fluid-structure interaction with the surrounding water into account. Two methods, a joint input-state estimation algorithm and a dual Kalman filter, are applied to estimate the full-field response of the bridge. The results demonstrate that the displacements and the accelerations can be estimated at unmeasured locations with reasonable accuracy when the wave loads are the dominant source of excitation.