Model Validation and Uncertainty Quantification, Volume 3

Time-domain coupled loads analysis (CLA) is used to determine the response of a launch vehicle and payload system to transient forces, such as liftoff, engine ignitions and shutdowns, jettison events, and atmospheric flight loads, such as buffet. CLA, using Hurty/Craig-Bampton (HCB) component models, is the accepted method for the establishment of design-level loads for launch systems. However, uncertainty in the component models flows into uncertainty in predicted system results. Uncertainty in the structural responses during launch is a significant concern because small variations in launch vehicle and payload mode shapes and their interactions can result in significant variations in system loads. Uncertainty quantification (UQ) is used to determine statistical bounds on prediction accuracy based on model uncertainty. In this paper uncertainty is treated at the HCB component-model level. In an effort to account for model uncertainties and statistically bound their effect on CLA predictions, this work combines CLA with UQ in a process termed variational coupled loads analysis (VCLA). The modeling of uncertainty using a parametric approach, in which input parameters are represented by random variables, is common, but its major drawback is the resulting uncertainty is limited to the form of the nominal model. Uncertainty in model form is one of the biggest contributors to uncertainty in complex built-up structures. Model-form uncertainty can be represented using a nonparametric approach based on random matrix theory (RMT). In this work, UQ is performed using the hybrid parametric variation (HPV) method, which combines parametric with nonparametric uncertainty at the HCB component model level. The HPV method requires the selection of dispersion values for the HCB fixed-interface (FI) eigenvalues, and the HCB mass and stiffness matrices. The dispersions are based upon component test-analysis modal correlation results. During VCLA, random component models are assembled into an ensemble of random systems using a Monte Carlo (MC) approach. CLA is applied to each of the ensemble members to produce an ensemble of system-level responses for statistical analysis. The proposed methodology is demonstrated through its application to a buffet loads analysis of NASA’s Space Launch System (SLS) during the transonic regime 50 s after liftoff. Core stage (CS) section shears and moments are recovered, and statistics are computed.

The engineer starts with paper and pencil. From the basic idea to the axiomatic model, this is all he or she needs. If the axiomatic model does not reproduce the results of the experiment due to too many simplifications, the axiomatic model shall be extended. The aim is to guarantee the desired functionality at an early design stage and thus ensure a safe design. Mathematical models of a new vibration absorber technology of different complexity are utilized in order to predict its dynamic response under different operation conditions. Such a prediction of the dynamic response is subject to model uncertainty.The focus of this paper is on model’s uncertainty resulting from model’s complexity. The model’s complexity is designed to be as simple as possible for an efficient optimisation approach. Since development is domain-specific, system and boundaries must first be defined. For this purpose, the modules are cut out of the overall system. The system boundaries, of the Fluid Dynamic Vibration Absorber have first to be found for a mathematical model but also for an experimental setup to get reliable empiric data. After this the proposed dynamic response of mathematical model is investigated. At a hydraulic transmission in an oscillating system, there are several approaches to modelling the oscillating flow and damping. Damping plays a decisive role in vibration absorbers. The occurring uncertainty of prediction of the dynamic response of the different models has to be quantified, especially if it represents a risk for vehicle occupants. Therefore, a Bayesian interval hypothesis-based method is used to quantify this uncertainty. It turns out that the choice of model boundary is a crucial one for model confidence.

The field of structural health monitoring (SHM) applies damage detection techniques to provide timely in-situ system condition assessment. Previously, researchers have suggested a fundamental axiom for SHM that states, “damage will increase the complexity of a system.” One way this increased complexity can manifest itself is in the increased complexity of sensor data recorded from the structure when damage occurs. The question then becomes how to best quantify the increase in complexity of those data. Information complexity is one such approach and within this framework various information entropy quantities have been proposed as measures of complexity. The literature has shown that there are multiple information entropy measures, including; Shannon Entropy, Rényi Entropy, Permutation Entropy, Sample Entropy, Approximate Entropy, and Spectral Entropy. With multiple measures proposed to quantify information entropy; a study to compare the relative effectiveness of these entropy measures in the context of SHM is needed. Therefore, the objective of this paper is to compare the effectiveness of entropy-based methods in distinguishing between “Healthy” and “Unhealthy” labeled datasets. The labeled datasets considered in this study were obtained from a 4DOF impact oscillator, a rotating machine with a damaged bearing, and an impact oscillator excited by a rotating machine. Furthermore, two methods were used in this study to classify the results from the different entropy measures; Naïve-Bayes classification, and K-means clustering. Effectiveness of a given entropy measure is determined by the number of misclassifications produced when compared to the true labels. The analysis showed that entropy measures obtained from data corresponding to sensors closer to the damage source had fewer misclassifications for the datasets tested. For the datasets considered in this study, the researchers concluded that each dataset had a different most effective entropy measure. The study would need to be expanded to include other classification methods and other datasets to define more precisely which entropy measure is the most effective in identifying the increase in complexity associated with damage and, hence, distinguishing between healthy and damaged data.

Axial and lateral loads of lightweight beam truss structures e.g. used in automotive engineering may lead to undesired structural vibration that can be reduced near a structural resonance frequency via resonant piezoelectric shunt-damping. In order to tune the electrical circuits to the desired structural resonance frequency within a model-based approach, an adequate mathematical model of the beam truss structure is required. Piezo-elastic truss supports with integrated piezoelectric stack transducers can transfer the axial and lateral forces and may be used for vibration attenuation of single beams or whole beam truss structures. For usage in a single beam test setup, the piezo-elastic support’s casing is clamped rigidly and is connected to the beam via a membrane-like spring element that allows for rotation as well as axial and lateral displacements of the beam. In this contribution, the piezo-elastic support is integrated into a two-dimensional beam truss structure comprising seven beams, where its casing is no longer clamped rigidly but is subject to axial, lateral and rotational displacements. Based on the previously verified and validated model of the single beam test setup, two different complex mathematical models of the piezo-elastic support integrated in the two-dimensional beam truss structure are derived in this contribution. The two mathematical models differ in their number of degrees of freedom for the piezo-elastic support as well as in the assumption of rigid or compliant casing. By comparing numerically and experimentally determined structural resonance frequencies and vibration amplitudes, the model that more adequately predicts the truss structure’s vibration behavior is selected on basis of the normalized root mean squared error. For future works, the more adequate model will be used to tune electrical circuits for resonant piezoelectric shunt-damping in a three-dimensional truss structure.

Various applications in structural dynamics may require the real-time estimation of unknown input. A recently developed joint input-state estimator for linear systems treats the unknown input as white Gaussian noise with finite covariance. The performance of this finite input covariance (FIC) estimator is validated using simulated data from a finite element model of the Air Force Research Laboratory’s experimental testbed called the DROPBEAR (Dynamic Reproduction of Projectiles in Ballistic Environments for Advanced Research). The estimator performance is compared with a few well-known estimators, including the augmented Kalman filter (AKF) and the weighted least squares (WLS) estimators. The results show that the FIC estimator is capable of accurately estimating an impact load applied to the beam when acceleration is measured at a small number of locations. Additionally, the results show that the FIC estimator eliminates the low-frequency drift error that other well-known estimators are susceptible to.

In structural health monitoring (SHM), model updating is concerned with identifying and updating system parameters (e.g. stiffness and mass) based on the measured response data of the monitored structure. With the increasing number of SHM systems deployed on modern structures in recent years, real-time model updating has become possible. This allows a digital twin of the monitored structure to be built, such that the structural behaviour can be monitored and predicted simultaneously throughout its life-cycle. In real applications, the structural response data are normally measured under operational conditions where the environment and loading condition cannot be directly controlled, which leads to significant identification uncertainty. The system model can also be complex, meaning that identifying system parameters directly from measured response data is challenging and time consuming. Focusing on the above concern, a real-time updating strategy for a SHM digital twin is proposed in this work. An intermediate model is used for environmental condition estimation and divergence analysis in order to increase the updating efficiency. A Bayesian system identification approach is adopted so that the identification uncertainty can be fully accounted for. Synthetic and laboratory examples are presented to illustrate the proposed updating strategy.

When designing complex engineering machines, such as an aero-engine, there are many factors that need to be kept in consideration. However in essence they boil down to two important elements: Safety and operational efficiency. Essential though these two factors are, they are also at odds with each other in terms of their demands on eventual designs. The need for operational efficiency drives engineering solution towards ever-so lighter structures with optimised complex designs often requiring the utilisation of novel materials. All important safety on the other hand requires that a deeper understanding of underlying physical behaviour is acquired to ensure that the mechanical integrity is never at risk. The job of engineers operating in this field is to make sure that they utilise all available tools to map the solution space for an optimal balance of these two conflicting demands.

Real-time hybrid simulation is a method to obtain the response of a system subjected to dynamic excitation by combining loading-rate-sensitive numerical and physical substructures. The interfaces between physical and numerical substructures are usually implemented using closed-loop-controlled actuation systems. In current practice, the parameters that characterize the hybrid model are deterministic. However, the effect of uncertainties may be significant.Stochastic hybrid simulation is an extension of the deterministic hybrid simulation where the parameters of the system are treated as random variables with known probability distributions. The results are probability distributions of the structural response quantities of interest. The arising question is to what extent does the actuation control system at the interface between physical and numerical substructures affect the outcomes of stochastic hybrid simulations. This question is most acute for real-time hybrid simulations.The response of a benchmark stochastic prototype to random excitation will be computed. Then, a part of the prototype will be replaced by a hybrid model whose substructure interfaces are actuated in closed-loop control. A controller that guarantees robustness and stability of the interfaces will be designed. The parameters of this hybrid model will be treated as random variables in repeated real-time hybrid response simulations to the same random excitation. The difference between the prototype and hybrid model responses will be used to determine if the controller design has an effect on the simulation outcomes, to predict such effects, and to propose guidelines for real-time controller design such that it has a predictable effect on the hybrid simulation.Additional criteria based on peak and root mean square tracking errors, as well as energy errors, are addressed in order to assess the overall system performance. Based on simulation data, surrogate models will be developed. Multiple additional runs of the surrogate models will give insight into the robustness and performance of the control system under uncertainties. Global sensitivity analysis of the overall system response will also be performed, identifying the most sensitive stochastic input variables. Cross-check validation of the results will take place using different meta-modeling techniques.

A novel approach to deal with nonlinear system identification of civil structures subjected to unmeasured excitations is presented. Using only sparse global dynamic structural response, mechanics-based nonlinear finite element (FE) model parameters and unmeasured inputs are estimated. Unmeasured inputs are represented by a time-varying autoregressive (TAR) model. Unknown FE model parameters and TAR model parameters are jointly estimated using an unscented Kalman filter. The proposed method is validated using numerically simulated data from a 3D steel frame subjected to seismic base excitation. Six material parameters and one component of the base excitation are considered as unknowns. Excellent input and model parameter estimations are obtained, even for low order TAR models.

Critical to railway infrastructure assessment is the tracing of interaction between railway vehicle and track.This is a non-trivial task characterized by non-stationary dynamics appearing due to changing operational conditions. To ensure reliable dynamic characterization of rail infrastructure, we propose a modeling methodology of locally non-stationary parametric time-series models to accommodate long-term variability. In the proposed approach, the non-stationary vibration, as measured from axle-box accelerometers, is modelled by means of a parametric time-series model with explicit dependence on the vehicle velocity and variations due to the infrastructure type. The postulated time-series model is demonstrated through real-world data stemming from the on-board measurement system of a Swiss Railway diagnostic vehicle, where the main drivers of uncertainty are the vehicle speed, the track type and its condition. The model will be exploited to build real-time indicators for railway infrastructure condition assessment.

Environmental tests are typically conducted in order to reproduce the operational response of a system. Non-realistic excitation mechanisms and mismatches between operational and test boundary conditions, represent relevant limitations of the currently adopted testing procedures. The Boundary Condition Challenge (BCC) addresses the assessment of a testing practice that allows to reproduce structural responses, which better represent the operational environment, thereby allowing for a more precise prediction of potential failure mechanisms. In this framework, Virtual Sensing (VS) can be used to estimate the complete response field of the component in operation, and compare this to the one delivered via testing. If the full-field strain is estimated, stress fields can be derived and component failure can be predicted. This work focuses on employing VS for augmenting the information acquired from physical sensors during environmental tests on the Box Assembly with Removable Component (BARC) benchmark. In order to apply VS techniques such as Kalman-type filters, a Reduced Order Model (ROM) of the system has been built taking into account the boundary conditions employed during testing. Moreover, an existing Optimal Sensor Placement (OSP) strategy has been adopted for configuring the positioning of sensors to be used during environmental tests.

The ground-steering system is a part of the nose landing gear, which is fundamental to an aircraft’s safety. A sensing mechanism estimates the wheel direction, which is then fed back to the controller in order to calculate the error between the desired steering angle and the actual steering angle. As in many safety-critical control systems, the sensing mechanism for the nose wheel direction requires the use of multiple redundant sensors to estimate the same controlled signal. A virtual sensing technique is commonly employed, which estimates the steering angle using the measurements of multiple remote displacement sensors. The wheel position is then calculated on the basis of the nonlinear alignment of these sensors.In practice, however, each sensor is subject to uncertainty, minor and major faults and there is also ambiguity associated with the estimate of the steering angle because of the geometric nonlinearity. The redundant sensor outputs are thus different from each other, and it is important to reliably estimate the controlled signal under these conditions.This paper presents the development of a digital twin of the ground-steering system, in which the effect of uncertainties and faults can be systematically analysed. A number of state estimation algorithms are investigated under several scenarios of uncertainty and sensor faults. Two of these algorithms are based on a least squares estimation approach, the other algorithm, instead, calculates the steering angle estimate using a soft-computing approach. It is shown that the soft-computing estimation algorithm is more robust than the least squares based methods in the presence of uncertainties and sensor faults. The propagation of an uncertainty interval from the sensor outputs to the steering angle estimate is also investigated, in order to calculate the error bounds on the estimated controlled signal. The optimal arrangement of the sensors is also investigated using a parametric study of the uncertainty propagation, in which the optimal model parameters are the ones that generates the smallest uncertainty interval for the estimate.

Today, in aerospace and automotive industries, structural components are more and more designed up to their functional limits, pursuing weight minimization without compromising the mechanical integrity. Especially in the aforementioned domains, the fracture behavior is of utmost importance in this respect. Yet, due to the complexity of the underlying physical phenomena and corresponding models, taking fracture into account in a virtual design process still proves to be highly challenging. Nevertheless, in recent years, there has been a strong development in various numerical simulation techniques that can simulate crack initiation and growth, i.e., techniques such as the Extended Finite Element Method (XFEM), meshless methods as the Element Free Galerkin Method (EFG) or the Phase Field Fracture (PFF) modeling approach. In this paper, the objective is to examine to what extend these approaches are applicable in a design context by assessing their relative value regarding validity and uncertainty. Also, some numerical aspects regarding efficiency and stability are discussed. The work is based on the study of a specific case. First, standard compact tension experiments are performed. These results are used to identify the necessary material properties and associated parametric uncertainty levels for each of the considered fracture modelling approaches. Next, a validation study is performed in which stereo Digital Image Correlation (DIC) is applied during the actual crack propagation phase in the same compact tension test. The model form uncertainty is assessed by comparing the set of DIC full field displacement and strain measurements with the range of the numerical predictions including uncertainty of the different modelling strategies.

Reliability of sensor information in today’s highly automated systems is crucial. Neglected and not quantifiable uncertainties lead to lack of knowledge which results in erroneous interpretation of sensor data. Physical redundancy is an often-used approach to reduce the impact of lack of knowledge but in many cases is infeasible and gives no absolute certainty about which sensors and models to trust. However, structural models can link spatially distributed sensors to create analytical redundancy. By using existing sensor data and models, analytical redundancy comes with the benefits of unchanged structural behavior and cost efficiency. The detection of conflicting data using analytical redundancy reveals lack of knowledge, e.g. in sensors or models, and supports the inference from conflict to cause.We present an approach to enforce analytical redundancy by using an information model of the technical system formalizing sensors, physical models and the corresponding uncertainty in a unified framework. This allows for continuous validation of models and the verification of sensor data. This approach is applied to a structural dynamic system with various sensors based on an aircraft landing gear system.

Wind turbines are complex mechatronic systems designed to withstand high dynamic loads over a considerable period due to environmental influences. Following the guidelines from DNVGL or IEC, the design process of modern wind turbines is governed by a rather rough classification of annual mean wind speed and turbulence intensity resulting in very conservative design loads compared to the actual load conditions at the specific erection site of the wind turbine. Thus, structural reserves are very likely at the end of the turbine’s approved lifetime. Driven by this, a research collaboration between different institutions from wind industry and research facilities was initiated with the aim to exploit these structural reserves by means of continuous, model-based fatigue monitoring of individual wind turbine structures. Within this contribution, the authors present the fundamental process chain, arising challenges and first solutions employing the broad expertise of the individual project partners. Generally, the following key requirements are set for the monitoring concept: Precise estimation of the endured fatigue loads at critical spots by means of a digital twin; low sensor installation effort on the operating turbine comprising a minimal set of long-term reliable measurement hardware at accessible positions; online data processing on the turbine control system unit (SCADA). The research project comprises the design of state estimators for tower and blade structure, the definition of an optimal sensor network, the development of damage models for materials of blades and tower, concept studies on a scaled testbed of the turbine and the tests on the real wind turbine.

Structural Health Monitoring (SHM) Techniques have recently started to draw significant attention in engineering applications due to the need of maintenance cost reductions and catastrophic failures prevention. Most of the current research on SHM focuses on developing either purely experimental models or stays on purely numerical data without real application validation. The potential of SHM methods however could be unlocked, having accurate enough numerical models and classifiers that not only recognize but also locate or quantify the structural damage. The present study focuses on the implementation of a methodology to bridge the gap between SHM models with numerically generated data and correspondence with measurements from the real structure to provide reliable damage predictions. The methodology is applied in a composite carbon fiber tube truss structure which is constructed, using aluminum elements and steel bolts for the connections. The composite cylindrical parts are produced on a spinning axis by winded carbon fibers, cascaded on specified number of plies, in various angles and directions. 3D FE models of the examined cylindrical parts are developed in robust finite element analysis software simulating each carbon fiber ply and resin matrix and analyzed against static and dynamic loading to investigate their linear and nonlinear response. In addition, experimental tests on composite cylindrical parts are conducted based on the corresponding analysis tests. The potential damage to the structure is set as loose bolts defining a multiclass damage identification problem. The SHM procedure starts with optimal modeling of the structure using an updated Finite Element (FE) model scheme, for the extraction of the most accurate geometrical and physical numerical model. To develop a high-fidelity FE model for reliable damage prediction, modal residuals and mode shapes are combined with response residuals and time-histories of strains and accelerations by using the appropriate updating algorithm. Next, the potential multiclass damage is simulated with the optimal model through a series of stochastic FE load cases for different excitation characteristics. The acceleration time series obtained through a network of optimally placed sensors are defined as the feature vectors of each load case, which are to be fed in a supervised Neural Network (NN) classifier. The necessary data processing, feature learning and limitations of the NN are discussed. Finally, the learned NN is tested against the real structure for different damage cases identification.

In this paper, a scheme to predict stress-stiffening effect on structures life assessment is proposed. The adopted methodology combines modal analysis and polynomial chaos expansion-based meta-model to evaluate stress distribution. It manages pre-stresses sources caused by manufacturing processes and quality control uncertainties.Experimental modal analysis is performed on a set of components with nominal values affected by manufacturing and process uncertainties to prove component and assembly variability. Meta-models are developed upon a small sample of experimental observations and stress stiffening effects are measured. Numerical and experimental investigations of stress-stiffening effects on the dynamic behaviour of an assembled automotive steel wheel are, then, reported.The pre-stress distribution is successfully used to estimate fatigue life of the structure in different load cases. The effectiveness of modal analysis, when combined with numerical/statistical approaches in industrial environments is pointed out.

A digital twin is a powerful new concept in computational modelling that aims to produces a one-to-one mapping of a physical structure, operating in a specific context, into the digital domain. This technology could therefore provide improved and robust decision making for asset management. Although the applications of digital twins vary, this paper focuses on digital twins for structural dynamic systems. A key consideration in developing a digital twin is in the construction of a workflow that defines decisions and interactions within the modelling framework. This process will generally be bespoke to specific applications, however key principles will apply. Furthermore, a workflow will provide a methodology for identifying poor predictive performance and systematically improving predictions via optimal decision making. In this paper a three storey building structure is introduced as a case study in order to motivate the challenges and technologies required of a digital twin. The context of this case study is to develop a digital twin of the building structure that consistently predicts the acceleration response of the three floors given an unknown structural state, caused by a contact nonlinearity between two floors. This reflects realistic challenges for a digital twin in that the physical twin will degrade with age, and its response may change under various loading scenarios, unforeseen in the initial model development phase. Key elements within a potential workflow for this application are discussed. These include indicating when model updating schemes become problematic and how augmenting physics-based models with a data-based component can provide information about poor predictive performance. These techniques are linked to hybrid testing, as a potential method for improving model development based on the physical structure in a controlled offline manner. Finally, the impact of these procedures are discussed for model based control methods in terms of vibration attenuation performance, but also robustness against model uncertainties and external disturbances. The workflow and key technologies investigated in this specific case study are expected to outline the general processes that apply to digital twins more broadly, and provide a clearer understanding of how a digital twin should be implemented.

The knowledge of the dynamic behavior of a mechanical system in a certain operating scenario is essential in many industrial applications. In particular, nowadays, the accurate and concurrent identification of the response fields and external loads represents a challenging target. Several experimental techniques, some exploiting a coupling with simulated solutions based on predictive methodologies, have recently been proposed and are in current use. However, in practice there is a common issue in the selection of the optimal types of sensors and their measurement location selection in order to reconstruct the desired quantities (e.g., loads, displacement or acceleration field) for a desired accuracy and dynamic range. This paper focuses on a Kalman filter approach for multiple input/state estimation, combining operational measurement and numerical model data. In the presented framework, an existing Optimal Sensor Placement (OSP) strategy for load identification is discussed and an improvement of this sensor selection is proposed. The reference OSP approach, previously proposed by the authors, is mainly focused on system observability, which is only a minimum requirement to obtain a stable estimator. For this reason it does not necessarily lead to the most accurate estimator or the highest dynamic range. In this work, we propose two alternative metrics based respectively on estimator covariance convergence and closed-loop estimator bandwidth with respect to the available set of measurements. The existing OSP is compared with the proposed metrics for multiple input/state estimation, showing improved accuracy of estimated quantities when these new metrics are accounted for in the sensor selection.

Population-based structural health monitoring (PBSHM) involves utilising knowledge from one set of structures in a population and applying it to a different set, such that predictions about the health states of each member in the population can be performed and improved. Central ideas behind PBSHM are those of knowledge transfer and mapping. In the context of PBSHM, knowledge transfer involves using information from a structure, defined as a source domain, where labels are known for a given feature, and mapping these onto the unlabelled feature space of a different, target domain structure. If the mapping is successful, a machine learning classifier trained on the transformed source domain data will generalise to the unlabelled target domain data; i.e. a classifier built on one structure will generalise to another, making Structural Heath Monitoring (SHM) cost-effective and applicable to a wide range of challenging industrial scenarios. This process of mapping features and labels across source and target domains is defined as domain adaptation, a subcategory of transfer learning. However, a key assumption in conventional domain adaptation methods is that there is consistency between the feature and label spaces. This means that the features measured from one structure must be the same dimension as the other (i.e. the same number of spectral lines of a transmissibility), and that labels associated with damage locations, classification and assessment, exist on both structures. These consistency constraints can be restrictive, limiting to which types of population domain adaptation can be applied. This paper, therefore, provides a mathematical underpinning for when domain adaptation is possible in a structural dynamics context, with reference to topology of a graphical representation of structures. By defining when conventional domain adaptation is applicable in a structural dynamics setting, approaches are discussed that could overcome these consistency restrictions. This approach provides a general means for performing transfer learning within a PBSHM context for structural dynamics-based features.

The development of low-cost measurement devices for dynamic identification of structures has gained considerable interest in recent times. The high cost of traditional systems has led several authors to evaluate the feasibility of using affordable acquisition systems. This article aims to quantify the uncertainties of the modal parameters of a flexible structure, instrumented with traditional and low-cost measurement devices. The evaluated low-cost acquisition systems were smartphones and Raspberry Pi microprocessors, while the conventional system consisted of highly sensitive piezoelectric accelerometers. The instrumented structure was a steel pedestrian bridge located in the city of Barranquilla. Bayesian methodologies were used to identify the modal parameters and their associated uncertainties in terms of probability distributions. The results obtained show the feasibility of using low-cost devices to determine the dynamic properties of a flexible structure.

Population-based structural health monitoring is the process of utilising information from a group of structures in order to perform and improve inferences that generalise to the complete population. A significant challenge in inferring a general representation for structures is that feature spaces will be inconsistent for a wide variety of populations and datasets. This scenario, where the dimensions of the feature spaces for each structure are different, occurs for a variety of reasons. Firstly, the group of structures themselves may be a heterogeneous population, where differences occur due to topology, leading to inconsistency in modal-based features. Secondly, feature spaces may be inconsistent across the population due to differences in the raw data (i.e. different sample frequencies etc.) and feature extraction processing. In this context, where feature spaces are inconsistent between different structure in a population, a general model that describes their behaviours becomes challenging to infer. This issue is because dimensionality reduction must be performed such that each domain’s feature set projects to a consistent shared latent space where a model can be inferred. This paper introduces a technique, kernelised Bayesian transfer learning, that seeks to learn a projection matrix and kernel embedding that map to a latent space where a discriminative classifier can be inferred in a Bayesian manner, using variational inference. This algorithm allows a general discriminative classifier to be inferred across a population where the feature spaces for each structure are inconsistent. A numerical case study is presented, demonstrating the effectiveness of this approach and for providing a discussion of its implications for population-based structural health monitoring.

One can estimate unmeasured acceleration spectral density responses of a structure utilizing measured responses from a relatively small number of accelerometers and the active mode shapes provided from a finite element model. The objective in this paper is to demonstrate a similar concept, but purely based on information from a laboratory pre-test. Response predictions can only be calculated at degrees of freedom that have been instrumented in the experimental pre-test, but greater accuracy may be possible than with a finite element-based expansion. A multi-reference set of frequency response functions is gathered in the laboratory pre-test of the field hardware. Two response instrumentation sets are included in the pre-test. One set corresponds to the measurements that will be taken in the field environment. The second set is the field responses that are of great interest but will not be measured in the field environment due to logistical constraints. For example, the second set would provide definition of the component field environment. A set of basis vectors is extracted from the pre-test experimental data in each of multiple frequency bands. Then the field environment is applied to the hardware and the data gathered from the field accelerometers. The basis vectors are then used to expand the response from the field accelerations to the other locations of interest. The proof of concept is provided with an acoustic test environment on the Modal Analysis Test Vehicle. Predicted acceleration spectral density simulations at 14 degrees of freedom (known as “truth responses”) are compared against truth acceleration measurements collected for this work from the acoustic environment. Due to the segregated bandwidth analysis, the required number of field accelerometers to provide the simulation is much smaller than the number of modes in the entire frequency bandwidth.

Truncation error is ubiquitous in computational sciences, yet, rarely quantified and often ignored altogether. By definition, it is the difference between the exact-but-unknown solution of continuous equations that one wishes to solve, such as conservation laws, and what the computational software (finite elements, finite volumes, etc.) calculates. We contend that the commonly-accepted representation of truncation error as a single-term power-law (i.e., ε(∆x) = β · ∆x p where ∆x is the level of mesh resolution in the calculation and p is the accuracy of the numerical method) is inadequate and can lead to an erroneous quantification. The remedy proposed is to model this error as a series expansion of integer-valued powers (i.e., ε(∆x) = β 1 · ∆x + β 2 · ∆x 2 + … + β N · ∆xN where the expansion order N is unknown and potentially infinite). This representation is consistent with the theoretical form of truncation error derived from Modified Equation Analysis. Because N and the regression coefficients β k are not known, we further propose to use an info-gap model to numerically derive bounds of truncation error. These bounds, ‖y Exact − y(∆x)‖ ≤ U(∆x), would express the worst-case error between what is calculated at resolution ∆x and what is exact but unknown. Reporting such bounds is essential to assess the quality of a numerical simulation, much like an experimental uncertainty should accompany a measurement. The discussion proposed is, for the most part, conceptual and future efforts will focus on the numerical implementation of these ideas.

The digital twin paradigm that aims to integrate the information obtained from sensor data, physics models, operational data and inspection/maintenance/repair history of the system or component of interest, can potentially be used to optimize operational parameters that achieve a desired performance or reliability goal. In this paper, we discuss such a methodology for intelligent operation planning in mechanical systems. The proposed approach discusses two key components of the problem: damage diagnosis and damage prognosis. We consider the problem of diagnosis and prognosis of fatigue crack growth in a metal component, as an example. We discuss a probabilistic, Lamb-wave-scattering-based crack diagnosis framework that incorporates both aleatory and epistemic uncertainties in the diagnosis process. We build a Bayesian network for the Lamb-wave pitch-catch NDE using a low-fidelity physics-based model of the same. We perform global sensitivity analysis to quantify the contribution of various parameters to the variance of the damage-sensitive output signal feature(s) using this model. We retain the parameters with higher contribution, and build a medium-fidelity, one-way coupled, multi-physics model to simulate the piezoelectric effect and Lamb wave propagation. We perform Bayesian diagnosis of crack growth using the medium-fidelity model, considering data corrupted by measurement noise, and fuse the information from multiple sensors. We build a finite-element-based high-fidelity model for crack growth under uniaxial cyclic loading, and calibrate a phenomenological (low-fidelity) fatigue crack growth model using the high-fidelity model output. We use the resulting multi-fidelity model in a probabilistic crack growth prognosis framework; thus achieving both accuracy and computational efficiency. We integrate the probabilistic diagnosis and prognosis engines to estimate the damage state using both sensor data as well as model prediction.

For widespread adoption of Structural Health Monitoring (SHM) strategies, a key challenge is to produce tools which automate the creation and learning of effective algorithms with minimal input from expert practitioners. Classification of damage in a structure is one of the key steps in constructing a useful SHM system. The artificial neural network has been shown, for a number of years, to be a powerful tool for building such a classifier. While the learning of the parameters in the network has been widely addressed in the literature, the hyperparameters of the model (i.e. the topology) remain a challenge. This paper investigates the use of an evolutionary algorithm tailored to this learning problem, namely, Neuro-Evolution of Augmenting Topologies. The effectiveness of this approach is considered with regard to classification accuracy, user input, and generalisation. A benchmark structure a representative three-storey building is used to demonstrate the use of this methodology.

Characterisation of structural dynamic systems remains a key feature of modern engineering analysis. Commonly, methods for performing this identification will be based on deterministic solutions such as a least-squares approach. However, quantifying uncertainty in the parameters of a given model can add significant value. This quantification can be propagated forward into further analyses such as model updating, fatigue damage assessment, and activities related to structural health monitoring. The state-space representation of structural dynamic systems is a flexible and powerful framework for performing identification tasks due to its flexibility, natural dynamic structure, and computational efficiency. The current work is concerned with recovering Bayesian solutions to these systems, in terms of the posterior distribution over the states and the parameters. The Bayesian solution offers a number of advantages; these include the ability to formally incorporate prior knowledge (for example from existing models of the system such as from finite element models) and to recover the posterior distributions over the parameters of the system and the internal states (displacements and velocities). Two algorithms are compared here, the first is a traditional Markov Chain Monte Carlo approach based on the Metropolis-Hasting algorithm, the second, a Sequential Monte Carlo approach called Iterated Batch Importance Sampling.

Model form uncertainty often arises in structural engineering problems when simplifications and assumptions in the mathematical modelling process admit multiple possible models. It is well known that all models incorporate a model error that is captured by a discrepancy due to missing or incomplete physics in the mathematical model. As an example, this discrepancy can be modelled as a function based upon Gaussian processes and its confidence bounds can be seen as a measure of adequacy for the respective model. Assessment of model form uncertainty can be conducted by comparing the confidence bounds of competing discrepancy functions. In this paper, a modular active spring-damper system is considered that was designed to resemble a suspension strut as part of an aircraft landing gear and is excited by dynamic drop tests. In previous research about the suspension strut, different mathematical system models with respect to different linear and non-linear assumptions for damping and stiffness properties to describe the dynamic system behaviour of the suspension strut were compared by means of the confidence intervals of their discrepancy functions. The results indicated that the initial conditions used for exciting the system model were inadequate. The initial conditions themselves constitute a mathematical model, so that model form uncertainty inherent to the initial condition model can effect the system model. The propagation of model form uncertainty within the model will be analysed in this paper by considering two cases: In the first case, the system model is excited with an inadequate initial condition model, while in the second case, experimentally measured initial conditions will be employed that represent the true value except for measurement errors. The comparison of both shows how model form uncertainty propagates through the model chain from the initial condition model to the system model.

Aircraft engines are one of the most heavily instrumented parts of an aircraft, and the data from various types of instrumentation across these engines are continuously monitored both offline and online for potential anomalies. Different measurements (temperatures, vibration, etc.) are influenced by various flight parameters (e.g. throttle position) and environmental conditions (outside temperature, pressure, humidity, etc.). Identification of the mutual interactions and causation underpins the understanding of emerging structures in such a complex system in which key parameters might be nonlinearly dependent. A simple cross-correlation analysis among the different sensors would fall short in an effort to paint a complete picture as the system involved is multifractal and evolves in multiple time scales with strong non-stationary signals. In the present case of aero-engine responses, dynamics among the different parts of the engine are particularly complex and understanding the cross-correlation among different parameters would enable the development of a data-driven model for quantities of interest.In the current work, firstly, to identify the mutual interaction of different sensor parameters to vibration, multi-scale de-trended partial cross-correlation analysis (MS-DPCCA) coefficient method is applied to various parameters on the aero-engine gas path (e.g. temperature, rotor speed, etc). Secondly, a new approach to model building is carried out using the state of the art input-output Dynamic Mode Decomposition (ioDMD) combined with a multi-resolution approach to estimate the state coefficients. The model is validated using a completely different engine test data.

Aircraft engines are one of the most heavily instrumented parts of an aircraft, and the data from various types of instrumentation across these engines are continuously monitored both offline and online for potential anomalies. The different measurements (temperatures, vibration, etc.) are influenced by the flight parameters (e.g., throttle position) and environmental conditions (outside temperature, pressure, humidity, etc.). Because of all the interdependencies among the various parameters measured, aero-engine vibration responses vary widely depending on the operating and prevailing environmental conditions. Variations in the measured values can, therefore, result from variations in the flight parameters and conditions rather than being due to abnormal behavior. The majority of vibration assessment is done by monitoring engine vibration levels to fundamental shaft rotational orders. However, focus on shaft orders in isolation may not expose the full picture when a range of other factors are also known to be in effect during operation of a complex machine such as an aero engine.In the present work, a data-driven approach for modeling the vibration response of a complex aero-engine system using the data collected from various sensors, as inputs, is presented. With the new age of sensor fusion, where data is plentiful, the present work opens a new window of opportunity to model a quantity of interest using all the other sensor data in a more holistic sense. A dynamical data-driven approach inspired from the recently developed dynamic mode decomposition (DMD) method is presented and the efficacy is studied by applying it to aircraft engine data.

This paper studies the performance of recursive and batch Bayesian methods for nonlinear model updating. Unscented Kalman filter (UKF) is selected to represent the recursive Bayesian method, and two UKF approaches are investigated and compared, i.e., non-adaptive UKF and adaptive UKF. The proposed new adaptive filter, forgetting factor adaptive UKF, estimates the model parameters and measurement noise covariance in an online manner. The forgetting factor adaptive UKF is based on the principle of matching the covariance of residuals to its theoretical values by updating the measurement noise covariance. The performance of non-adaptive UKF, adaptive UKF and batch Bayesian method are investigated when applied to a numerical nonlinear 3-story 3-bay steel frame structure for parameter estimation of material properties. Different types of modeling errors are considered in the 21 updating models to study the effects of modeling errors on model updating. It is found that adaptive UKF approach provides the most accurate parameter estimations, while batch Bayesian approach gives the smallest errors on response predictions.

Data-driven models in Structural Health Monitoring (SHM) generally require comprehensive datasets, recorded from systems in operation, which are rarely available. One potential solution to this problem, considers that information might be transferred, in some sense, between similar systems. As a result, a population-based approach to SHM suggests methods to both model and transfer this valuable information, by considering different groups of structures as populations. Specifically, in this work, a method is proposed to model a population of nominally-identical systems, where (complete) datasets are only available from a subset of members. The framework attempts to build a general model, referred to as the population form, which can be used to make predictions across a group of homogeneous systems. First, the form is demonstrated through applications to a simulated population – with a single experimental (test-rig) member; secondly, the form is applied to data recorded from a group of operational wind turbines.

In structural engineering, different assumptions and simplifications during the mathematical modeling process may lead to a set of competing mathematical models with different complexity and functional relationships. The quantification of the resulting model form uncertainty for competing mathematical models may be used to select the model that predicts the experimental measurements of a system most adequately for given requirements, e.g. highest possible accuracy or low computational costs. In this paper, a review and application of four selected approaches to detect and quantify model form uncertainty using experimental measurements and simulation data are conducted: detection of model form uncertainty by (1) parameter estimation with optimal design of experiments as well as quantification of model form uncertainty by (2) the area validation metric for comparing the cumulative density function of a numerical simulation with measurements, (3) a non-parametric regression approach to describe the model error and (4) a Gaussian process based quantification of model form uncertainty. As an exemplary system, an experimental load-carrying truss structure is considered with its system output being the maximum axial tensions in selected truss members due to static and dynamic loads. In two competing mathematical models, the truss structure is either assembled with rigidly connected beams or pin jointed rods. The proposed approaches (1) to (4) detect and quantify model form uncertainty in the two competing models of the load-carrying truss structure, which are subsequently compared and evaluated in terms of their simulation accuracy. Depending on the model requirements, the adequate truss structure model with quantified model form uncertainty may then be selected for further investigations.

This paper presents the application of a recursive nonlinear system identification approach in the context of a structural system exhibiting geometric and material nonlinearities. The structure of interest consists of a building frame structure equipped with an adaptive negative stiffness device (ANSD) for vibration suppression. To reduce the dynamic vibration response the ANSD exerts a nonlinear displacement-dependent lateral force to the structure resulting in a geometric-type nonlinearity. In addition, when the modified structure is subjected to severe loading the primary structure shows yielding in its structural elements resulting on material nonlinearity. The unscented Kalman filter is employed herein to recursively estimate the parameters that define a mathematical model of the modified structure that accounts for both sources of nonlinearity. An illustrative example demonstrates the effectiveness of the approach and its ability to accurately estimate the parameters that define the mathematical model when both sources of nonlinearity are non-negligible.

Active buckling control of compressively loaded beam-columns provides a possibility to increase the maximum bearable axial load compared to passive beam-columns. Reliable mathematical beam-column models that adequately describe the lateral dynamic behavior are required for the model-based controller synthesis in order to avoid controller instability for real testing and application. This paper presents an adequate mathematical beam-column model for the active buckling control in a tetrahedron truss structure. Furthermore, it discusses model form uncertainty arising from model simplification of the global tetrahedron model to three local beam-column models. An experimental tetrahedron truss structure that comprises three passive beams and three active beam-columns with piezo-elastic supports for active buckling control is investigated. The tetrahedron is clamped at the three base nodes and free at the top node. In the two piezo-elastic supports of each active beam-column, integrated piezoelectric stack actuators compensate lateral deflections due to increasing axial compressive loads and may, thus, prevent buckling. In previous works, active buckling control was investigated for a single beam-column that was clamped rigidly in an experimental test setup. A verified and validated single beam-column model with compliant boundary conditions was used to represent the piezo-elastic supports for active buckling control. The mathematical model of the active beam-columns is calibrated with experimental data from all three nominally identical active beam-columns to account for uncertainty in manufacturing, assembly or mounting. Subsequently, they are compared with respect to the transfer functions and the first eigenfrequencies. It is shown that the boundary conditions of the single beam-column model may be calibrated to adequately describe the boundary conditions within the tetrahedron truss structure. Thus, it will be used for the model-based controller synthesis in future investigations on the active buckling control of the tetrahedron truss structure.

In this study, a hierarchical Bayesian model inversion method is developed for site characterization and site response using surface wave measurements and H/V spectral ratios. The hierarchical Bayesian method estimates a velocity profile with uncertainty assessment using both surface wave derived dispersion curves and ambient noise derived H/V spectral ratios. In the proposed hierarchical inversion framework, prior distributions are assumed for the soil parameters and shear wave velocity distribution at each soil layer is estimated. The proposed inversion process is evaluated experimentally when applied to field measurements at a site in the greater Boston area. In this application, soil stratigraphy is assumed from a boring log a-priori and prior Gaussian distributions are assumed for the soil parameters using SPT blowcounts and/or local shear wave velocity data. The prior probability distributions of shear wave velocities are updated to their posterior distributions through the hierarchical Bayesian inference where the mean and covariance of model parameters are estimated as hyperparameters. The Metropolis-Hastings algorithm and Gibbs sampler is used for estimating the updating parameters and hyperparameters. The hierarchical Bayesian approach provides a mechanism to use both the dispersion curve and the H/V spectral ratio in the shear wave velocity inversion. The estimated uncertainty bounds derived from the hierarchical Bayesian approach using dispersion curves and H/V spectral ratios are more informative than those using classical Bayesian inference schemes and dispersion curves alone.

Load-bearing structures with kinematic functions like a suspension of a vehicle and an aircraft landing gear enable and disable degrees of freedom and are part of many mechanical engineering applications. In most cases, the load path going through the load-bearing structure is predetermined in the design phase. However, if parts of the load-bearing structure become weak or suffer damage, e.g. due to deterioration or overload, the load capacity may become lower than designed. In that case, load redistribution can be an option to adjust the load path and, thus, reduce the effects of damage or prevent further damage. For an adequate numerical prediction of the load redistribution capability, an adequate mathematical model with calibrated model parameters is needed. Therefore, the adequacy of an exemplary load-bearing structure’s mathematical model is evaluated and its predictability is increased by model parameter uncertainty quantification and reduction. The mathematical model consists of a mechanical part, a friction model and the electromagnetic actuator to achieve load redistribution, whereby the mechanical part is chosen for calibration in this paper. Conventionally, optimization algorithms are used to calibrate the model parameters deterministically. In this paper, the model parameter calibration is formulated to achieve a model prediction that is statistically consistent with the data gained from an experimental test setup of the exemplary load-bearing structure. Using the R2 sensitivity analysis, the most influential parameters for the model prediction of interest, i.e. the load path going through the load-bearing structure represented by the support reaction forces, are identified for calibration. Subsequently, BAYESIAN inference based calibration procedure using the experimental data and the selected model parameters is performed. Thus, the mathematical model is adjusted to the actual operating conditions of the experimental load-bearing structure via the model parameters and the model prediction accuracy is increased. Uncertainty represented by originally large model parameter ranges can be reduced and quantified.

Higher accuracy in estimation of stress distribution, especially in the critical locations with stress variation, is important for a more reliable prediction of possible damage or prognosis of the structure. A structure can have several such locations and it is not economically feasible to monitor all of them with traditional sensing methods. Moreover, these methods often provide response measurements at a few localized points and demand deployment in large numbers to get a distributed response. Hybrid submodeling is a data-driven physics-based submodeling method to achieve a refined estimate of distributed structural response in and around the critical locations or other locations of interest. This method uses measured response on the pre-meditated boundaries around the location of interest to drive the corresponding submodel of the structural component or connection to achieve a more spatially refined response. The method accumulates and propagates uncertainty in different stages of response estimation such as uncertainty in sensor location, measurement noise, uncertainty in submodel geometry, material properties, and boundary condition uncertainty at submodel boundaries. This article demonstrates uncertainty propagation in the hybrid-submodeling method, considering uncertainty in the submodel boundary conditions as the only source of uncertainty. This is demonstrated by using DIC digital image correlation (DIC) to measurements to drive the submodel region around a critical location on a structural component. Monte-Carlo simulation (MCS) is used to study the uncertainty propagated to the response in the hybrid-submodeling method. The study observed lower variability in the distribution of response in each location obtained through hybrid submodeling compared to its corresponding distribution from DIC measurements on the plate under the same loading conditions.

The optimality of recursive Bayesian estimators which have been extensively studied and implemented, for problems of state and parameter estimation, as well as for state estimation of systems with unknown inputs, is closely related to the quality of prior information about the process and measurement noise terms. These are typically treated as tuning parameters and therefore adjusted in an ad hoc and rather heuristic manner. Such an approach might be adequate for systems under stable environmental and operational conditions, but is proven insufficient for systems operating in a dynamic environment, where adaptive schemes are required. In this work, a new leave-one-out (LOO) metric is proposed for innovation-based adaptation of noise covariance matrices with the aim of robustly quantifying the actual model errors and properly describing the measurement-related uncertainties.

Accurate and rapid assessment of infrastructure conditions is vital for post-earthquake disaster management and decision making. The authors propose a data-based approach that utilizes limited information from buildings after the earthquake to predict the damage intensity for other buildings in the community. The data-based approach is formulated with Gaussian process regression (GPR). General engineering demand parameters (EDP) that are available after the earthquake on instrumented buildings are assumed to be indicators of damage. The 2011 Tohoku earthquake scenario is considered for demonstrating the applicability of the proposed method to buildings across a large region. A wide variety of steel structure models representing the general buildings in municipal areas are created to populate the dataset. Based on the time history analysis, EDPs are calculated and used as damage labels. Judicious variables describing the earthquake input characteristics and the building attributes are used in the feature space. All the utilized variables are selected based on the accessibility after the earthquake. A set of covariance functions are considered for Gaussian process model and the predictive performances are compared. A spectral analysis is also performed in order to compare with prediction results from GPR algorithm. Finally, it is shown that with a set of handy features and inspecting as low as 5% of the affected buildings in a region, damage estimations are improved significantly compared to the spectral analysis.

Continuous monitoring of miter gates used in navigation locks is desirable in order to prioritize maintenance and avoid unexpected failures. Substantial economic losses to the marine cargo and associated industries are caused by the closure of these inland waterway structures. Strain gauges are often installed in many of these miter gates for data collection, and various inverse finite element techniques are used to convert the strain gauges data to damage-sensitive features. One of the damage features is the development of a contact-loss “gap” between the components (i.e. quoin blocks) that support the gate laterally, which leads to load re-distribution that can induce overload in some components of the gate. Arguably, a refined finite element model of such structure can be very computationally expensive even when using linear models. An efficient way to solve an inverse problem with time-consuming model evaluations is making use of parallel model evaluations using a Sequential Monte Carlo (SMC) algorithm and parallel solution of the finite element (FE) equations using a commercial FE software. A significant advantage of SMC algorithms is that model evaluations are independent and are able to be run in parallel. In this paper, an expensive high-fidelity model of a miter gate is used to infer the gap extend given a noisy set of strain measurements.

A hierarchical Bayesian modeling (HBM) framework is presented for updating finite element (FE) models. A two stage approach is proposed for which in the first stage the modal data properties (modal frequencies, damping ratios and mode shapes) are estimated using response time histories recorded from multiple independent experiments. In the second stage, the proposed framework provides a reliable approach to account for the uncertainty of the FE model parameters due to the variability in the values of modal data estimated from multiple data sets. This variability arises due to model errors, measurement errors, as well as data processing procedures used to estimate modal data from response time histories. In the proposed framework, the uncertainties are embedded into the FE model parameters by assigning a probability model involving a set of hyper parameters. A formulation is presented for quantifying the uncertainties in the hyper parameter, model parameters and output quantities of interest (QoI) using an efficient asymptotic approximation to process independently the modal data sets. In particular, the posterior distribution of the hyper parameters is analytically formulated as a product of multi-dimensional Gaussian probability distributions. Samples of this distribution are used to estimate model parameter uncertainties as well as uncertainties in response QoI. This combined asymptotic-sampling approach is computationally more efficient than available full sampling approaches. Simulated data from a spring-mass chain model are used to demonstrate that the proposed framework provides reliable and reasonable uncertainty bounds as compared to conventional Bayesian framework that considerably underestimate uncertainties and results in unrealistic predictions of thin uncertainty bounds for response QoI.

Digital twin-based approaches for structural health monitoring (SHM) and damage prognosis (DP) are emerging as a powerful framework for intelligent maintenance of civil structures and infrastructure systems. Model updating of nonlinear mechanics-based Finite Element (FE) models using input and output measurement data with advanced Bayesian inference methods is an effective way of constructing a digital twin. In this regard, the nonlinear FE model updating of a full-scale reinforced-concrete bridge column subjected to seismic excitations applied by a large shake table is considered in this paper. This bridge column, designed according to US seismic design provisions, was tested on the NEES@UCSD Large High-Performance Outdoor Shake Table (LHPOST). The column was subjected to a sequence of ten recorded earthquake ground motions and was densely instrumented with an array of 278 sensors consisting of strain gauges, linear and string potentiometers, accelerometers and Global Positioning System (GPS) based displacement sensors to measure local and global responses during testing. This heterogeneous dataset is used to estimate/update the material and damping parameters of the developed mechanics-based distributed plasticity FE model of the bridge column. The sequential Monte Carlo (SMC) method (set of advanced simulation-based Bayesian inference methods) is used herein for the model updating process. The inherent architecture of SMC methods allows for parallel model evaluations, which is ideal for updating computationally expensive models.

Measured vibration responses are often used to estimate modal (frequency, damping ratio and mode shape) as well as physical (stiffness, damping) parameters of structural systems. These estimates inevitably have some associated uncertainties. It has been shown in past studies that the more informative the measured responses are about the parameters to be identified, the less is the associated uncertainty in the parameter estimates. Hence, when planning an experiment, it becomes necessary to locate sensors such that the measured responses are most informative about the parameters to be identified. In case the experiment consists of planned forced vibration tests, it is further necessary to excite the system at particular degrees of freedom such that the resulting responses are again most informative about the parameters to be identified. With the full set of sensors, such locations of the excitations may be termed as the optimal input locations, in the sense that they lead to minimum uncertainty in the estimated parameters. In this study, a methodology is presented to find such optimal input locations by maximizing an appropriate norm of the associated Fisher information matrix. Here, the element stiffness values are considered as the parameters of interest. Sensitivity of the data to the stiffness of elements are used in the associated Fisher Information Matrix to obtain the optimal input locations. The developed methodology is illustrated using laboratory experiments on a 2-D steel truss excited using an impulse hammer.

In vibration based modal identification of railways bridges, if the mass of the train is not negligible as compared to that of the bridge, the modal parameters of the train-bridge system becomes time-varying, making modal identification challenging. In this study, a method is developed to identify such time-varying modes of a train-bridge system using moving train induced vibration data. The method utilizes the continuous but gradual variation in the modal parameters to perform a window-wise identification of these parameters. In each time window, an auto-regressive model of the system is first identified, which is then used to identify the modes in that window using the Eigensystem Realization Algorithm. A damage index, based on damage locating vectors, to detect damage location using such identified time-varying modes is also discussed. The proposed time-varying modal identification and damage detection methods are illustrated using laboratory experimental data from a two span continuous steel bridge, as well as a damaged version of the bridge, excited using a moving laboratory scale model train.

Many structures in a launch vehicle operate in liquid hydrogen, from the hydrogen fuel tanks through the ducts and valves and into the turbopumps. Calculating the structural dynamic response of these structures is critical for successful qualification, but accurate knowledge of the natural frequencies is based entirely on numerical or analytical predictions since testing in operating conditions is problematic. A comprehensive test/analysis program has therefore been performed at NASA/MSFC to enable accurate prediction of the modal characteristics of the Space Launch System’s (SLS) RS-25 Low Pressure Fuel Turbopump Inducer including the effects of fluid-added mass, mechanical property change at cryogenic temperatures, operation within tight tip clearances, acoustic/structure interaction, and hydroelasticity. The process also has to account for complicated cyclic symmetry mode shapes which can be easily mistuned in test, and geometric, boundary condition, and material modifications between the sub-scale inducer used as a test article and the actual flight component. The first phase of the program, documented previously, focused on testing of a cantilever beam in a number of fluids and temperatures to isolate the effects of fluid-added mass and temperature, while the second phase reported here documents the additional issues associated with the more realistic inducer test article. Preliminary structural dynamic analysis of the flight hardware including variability for the above parameters indicated potential severe resonances, requiring implementation of undesired programmatic constraints, so the improved predictive capability may allow removal of these constraints.

An approach for output-only Bayesian identification of nonlinear systems is presented. The approach is based on a re-parameterization of the posterior distribution of the parameters that define a model class. In the re-parameterization the state predictive distribution is marginalized and recursively estimated in a state estimation step using an unscented Kalman filter, bypassing state augmentation as required by existing recursive methods alleviating the ill-posedness of the inverse problem. The approach circumvents the estimation of the forcing input in applications where only output-measurements are available (output-only identification). Numerical and experimental examples illustrate the effectiveness and advantages of the approach.