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Über dieses Buch

Model Validation and Uncertainty Quantification, Volume 3: Proceedings of the 37th IMAC, A Conference and Exposition on Structural Dynamics, 2019, the third volume of eight from the Conference brings together contributions to this important area of research and engineering. The collection presents early findings and case studies on fundamental and applied aspects of Model Validation and Uncertainty Quantification, including papers on:

Inverse Problems and Uncertainty Quantification

Controlling Uncertainty

Validation of Models for Operating Environments

Model Validation & Uncertainty Quantification: Decision Making

Uncertainty Quantification in Structural Dynamics

Uncertainty in Early Stage Design

Computational and Uncertainty Quantification Tools

Inhaltsverzeichnis

Frontmatter

Chapter 1. Nondestructive Consolidation Assessment of Historical Camorcanna Ceilings by Scanning Laser Doppler Vibrometry

This paper presents a procedure for the evaluation of the conservation state and the restoration efficiency of nineteenth century camorcanna vaults based on the analysis of objective features extrapolated from nondestructive vibration testing data. As example of application has been chosen the camorcanna vault of the “salone grande” in the nineteenth century Villa Greppi in Monticello Brianza near to Milan, Italy. Non-contact scanning laser Doppler vibrometry has been exploited for the evaluation of the dynamic behavior of the vault before and after rehabilitation. At the first, the structure where frescoes are attached, a cannulated loft spread with mortar, and related aging problem, e.g. painting detachment, are explained. Thus, usual and innovative non- invasive diagnostic techniques are listed, focusing attention on Laser Doppler Vibrometry. Then, Villa Greppi case study is considered, reporting on site equipment and how measurements were taken. Therefore, processed data results are shown, and objective feature indices defined.

M. Martarelli, P. Castellini, A. Annessi

Chapter 2. The Need for Credibility Guidance for Analyses Quantifying Margin and Uncertainty

Current quantification of margin and uncertainty (QMU) guidance lacks a consistent framework for communicating the credibility of analysis results. Recent efforts at providing QMU guidance have pushed for broadening the analyses supporting QMU results beyond extrapolative statistical models to include a more holistic picture of risk, including information garnered from both experimental campaigns and computational simulations. Credibility guidance would assist in the consideration of belief-based aspects of an analysis. Such guidance exists for presenting computational simulation-based analyses and is under development for the integration of experimental data into computational simulations (calibration or validation), but is absent for the ultimate QMU product resulting from experimental or computational analyses. A QMU credibility assessment framework comprised of five elements is proposed: requirement definitions and quantity of interest selection, data quality, model uncertainty, calibration/parameter estimation, and validation. Through considering and reporting on these elements during a QMU analysis, the decision-maker will receive a more complete description of the analysis and be better positioned to understand the risks involved with using the analysis to support a decision. A molten salt battery application is used to demonstrate the proposed QMU credibility framework.

Benjamin B. Schroeder, Lauren Hund, Robert S. Kittinger

Chapter 3. Failure Behaviour of Composites Under Both Vibration Loading and Environmental Conditions

The study focuses on the understanding of failure behaviour of composites which are subjected to vibration fatigue under environmental temperature conditions. The study of vibration fatigue failure in composites can be challenging because of the coupling between mechanical and thermal properties. In fact, stiffness distribution and self-heating are typically occurring under vibration conditions. As the problem stands, the sole use of either testing or simulation would not be adequate to understand the failure behaviour fully. This paper will present both an experimental and numerical work, based on a component designed with a ply-drop feature to enhance and localise the damage occurrence. The vibration testing experiments were carried while an environmental chamber was used to control the exposure temperature. Similar experimental conditions are simulated in a finite element multi-physics environment, where the crack opening is modelled by VCCT method. The simulation environment is very challenging because both mechanical (dynamics) and thermal behaviours need to be incorporated to study the failure of a given vibration loading. Both experimental and numerical results will be qualitatively compared.

Georgios Voudouris, Dario Di Maio, Ibrahim Sever

Chapter 4. Verification and Validation for a Finite Element Model of a Hyperloop Pod Space Frame

This paper discusses the verification and validation of a finite element (FE) model of the space frame built by the Hyperloop University of Cincinnati (HUC) team for the first student design competition organized by SpaceX. For the purpose of studying the frame performance in various dynamic scenarios, development of a reliable FE model was crucial. A verification and validation (V&V) strategy utilizing physical modal tests and torsional stiffness tests was adopted to ensure that the FE model was capable of accurately capturing the dynamic characteristics of the constructed space frame. This work aims to present the details of the V&V activities as performed on the main frame.

Vignesh Jayakumar, T. S. Indraneel, Rohan Chawla, Sudeshna Mohanty, Shishir Shetty, Dhaval Shiyani, Shabaan Abdallah

Chapter 5. Investigating Nonlinearities in a Demo Aircraft Structure Under Sine Excitation

Developing on the basic idea behind parametric and non-parametric identification of nonlinear systems, another case study on integrating system identification and finite element modelling of nonlinear structures is presented. The first step, which is the focus of this paper, involves using acquired input and output data to derive an experimental model for both the underlying linear model and nonlinear model of the proposed structure, no information about the system is required and only the applied excitations and corresponding accelerations are implemented in the nonlinear identification step. The proposed case study is demonstrated on a nonlinear simple metallic plane assembly with localized stiffness and damping nonlinearities; in this case, an updated linear finite element model of the structure is derived and the nonlinearities experimentally characterised.

S. B. Cooper, S. Manzato, A. Borzacchiello, L. Bregant, B. Peeters

Chapter 6. Sensor Placement for Multi-Fidelity Dynamics Model Calibration

This paper studies a multi-fidelity resource optimization methodology for sensor location in the calibration of dynamics model parameters. Effective calibration can only be achieved if the information collection in the experiments is successful. This requires a thoughtful study of the sensor configuration to maximize information gain in the calibration of system parameters. This paper proposes a framework for optimizing the sensor number and locations to maximize information gain in the calibration of damping parameters for non-linear dynamics problems. Further, we extend the basic framework to the case of multi-fidelity modeling. In the presence of models of multiple fidelity, runs from the high-fidelity model can be used to correct the low-fidelity surrogate and result in stronger physics-informed priors for calibration with experimental data. This multi-fidelity calibration allows the fusion of information from low and high-fidelity models in inverse problems. The proposed sensor optimization methodology is illustrated for a curved panel subjected to acoustic and non-uniform thermal loading. Two models of different fidelity (a time history analysis and a frequency domain analysis) are employed to calibrate the structure’s damping parameters and model errors. The optimization methodology considers two complicating factors: (1) the damping behavior is input-dependent, and (2) the sensor uncertainty is affected by temperature.

G. N. Absi, S. Mahadevan

Chapter 7. Application of Cumulative Prospect Theory to Optimal Inspection Decision-Making for Ship Structures

The selection of optimal maintenance solutions under uncertainty is affected by the risk perception of decision-makers. The solution predicted by the minimum expected cost criterion may not conform to the preferences of decision-makers. The aim of this paper is to develop a risk-informed maintenance decision-making framework for corroding ship structures considering risk perceptions. Cumulative prospect theory is employed to model the choice preferences under uncertainty. The optimal ship maintenance strategy is developed as a single goal to maximize the expected prospect value. The uniform inspection interval is assumed to be the only design variable and a condition-based repair policy is considered. Monte Carlo simulations are employed to obtain the distribution of the maintenance and failure costs within the considered service life. The application of the developed framework is demonstrated on a ship hull girder structure.

Changqing Gong, Dan M. Frangopol, Minghui Cheng

Chapter 8. Establishing an RMS von Mises Stress Error Bound for Random Vibration Analysis

The root mean square (RMS) von Mises stress is a criterion used for assessing the reliability of structures subject to stationary random loading. This work investigates error in RMS von Mises stress and its relationship to the error in acceleration for random vibration analysis. First, a theoretical development of stress-acceleration error is introduced for a simplified problem based on modal stress analysis. Using results from the example as a basis, a similar error relationship is determined for random vibration problems. Finite element analyses of test structures subject to an input acceleration auto-spectral density are performed and results from parametric studies are used to determine error. For a given error in acceleration, a relationship to the error in RMS von Mises stress is established. The resulting relation is used to calculate a bound on the RMS von Mises stress based on the computed accelerations. This error bound is useful in vibration analysis, especially where uncertainty and variability must be thoroughly considered.

David Day, Moheimin Khan, Michael Ross, Brian Stevens

Chapter 9. A Neural Network Surrogate Model for Structural Health Monitoring of Miter Gates in Navigation Locks

Structural health monitoring (SHM) of miter gates of navigation locks is crucial for facilitating cargo ship navigation. Closure of these inland waterway structures causes considerable economical loss to the marine cargo and associated industries. In practice, strain gauges are often mounted 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. Arguably, these models are computationally expensive and sometimes they are not suitable for real-time health monitoring or for monitoring confounding environmental effects. In this work, a Multi-Layer Artificial Neural Network (MANN) is designed to serve as a “run time” surrogate model that links data (from the strain gages) to damage classification (gaps in the miter gate contact). Three cases of complexity, combining hydrostatic and thermal loading scenarios with varying gap scenarios, are considered to design the MANN. A confusion matrix is used to evaluate the performance of the networks and derive probabilities. Results show the potential of MANNs as a reliable surrogate model for computationally expensive inverse finite element modeling in damage classification for this application.

Manuel Vega, Ramin Madarshahian, Michael D. Todd

Chapter 10. Model Validation Strategy and Estimation of Response Uncertainty for a Bolted Structure with Model-Form Errors

The model-form uncertainty and the connection parameter uncertainty are difficult to separate in structural dynamics. In this paper, we study a model-form uncertainty substructure identification method. Through the precision evaluation of the hyper-model of the substructure, and the experimental modal analysis of the substructure, the method of identifying the model-form uncertainty in the overall structure is given. Through the experimental modal analysis of the connection substructure, the uncertainty quantification is conducted and the joints parameters of the structure are identified. Finally, the response error estimation of the overall structure is given by combining the uncertainty of model form and of parameters. By comparing with the response errors estimated of the overall experiment, the proposed method is validated. In this paper, using modal parameters and frequency response as the response feature. A general definition of validation metrics was conducted which related to frequency response function. By using a bolted frame the framework of model validation is illustrated and the validation of the method in this paper is proved.

Huijie Li, Qintao Guo, Ming Zhan, Yanhe Tao

Chapter 11. Characteristic Analysis of Modified Dolly Test: A Sensitivity Study of Initial Conditions on Rollover Outcomes

Rollover crashes are known as the most dangerous type of accidents throughout the world. They are associated with multi-directional velocities and accelerations which creates a complex dynamic behavior of the vehicle. Several experimental and numerical methods have been used to gain a better understanding of kinematics of the vehicle and occupants during the rollover crashes. Due to the complex nature of the rollover, any change in the initial conditions may significantly influence the rollover outcomes. The main goal of this study is to assess the effects of initial conditions on dynamic responses of the bus using a modified dolly rollover test procedure. Since the experimental rollover test is very expensive and to decrease the computational costs, the numerical model of cutaway bus was developed using lumped mass-spring-damper in PC-Crash software. First, the model was validated using the experimental data. Then, a series of simulations have been conducted with considering various initial conditions (inputs) such as initial velocity, friction of rollover surface, height of bus’s CG, and initial roll angle. The range of initial variables were selected based on Latin Hypercube Sampling (LHS) with uniform distribution. The simulation results were used to build the surrogate model using Kriging model for each rollover outcomes (outputs) including number of quarter turns, roll distance, deceleration rate, and maximum impact force. The sensitivity of the model to 400 set of input data was computed in MATLAB. The results of the sensitivity analysis indicate that the number of quarter turns and roll distance was highly affected by the initial velocity of the bus. Furthermore, the deceleration rate controlled mainly by friction and initial velocity and a negative correlation with the initial roll rate. No strong correlation between maximum impact force and any input parameters was observed. This led us to perform further research on exploring the effects of dynamic characteristics of the bus on maximum impact force.

Mohammad Reza Seyedi, Sungmoon Jung, Jerzy Wekezer

Chapter 12. Input Estimation of a Full-Scale Concrete Frame Structure with Experimental Measurements

This paper studies input estimation of a full-scale concrete frame structure, which is modeled with over a thousand degrees-of-freedom (DOFs). With acceleration response measured from dynamic testing, the natural frequencies and mode shapes of the concrete frame are first identified. The experimentally identified modal properties are compared with those obtained from a finite element (FE) model using nominal material properties. The FE model is then used to construct state-space system matrices for input estimation. With only acceleration measurements, an unbiased minimum-variance estimator combined with an online drift filter is used to estimate the dynamic input generated by a shaker. The estimation results show acceptable performance of the proposed algorithms for application on the full-scale two-story two-bay concrete frame with both simulated and experimental measurements. The effect of sensor locations on input estimation performance is also discussed.

Xi Liu, Yang Wang

Chapter 13. Bayesian Estimation of Acoustic Emission Arrival Times for Source Localization

The onset time of an Acoustic Emission (AE) signal is an important feature for source localization. Due to the large volume of data, manually identifying the onset times of AE signals is not possible when AE sensors are used for health monitoring of a structure. Numerous algorithms have been proposed to autonomously obtain the onset time of an AE signal, with differing levels of accuracy. While some methods generally seem to outperform others (even compared to traditional visual inspection of the time signals), this is not true for all signals, even within the same experiment. In this paper, we propose the use of an inverse Bayesian source localization model to develop an autonomous framework to select the most accurate onset time among several competitors. Without loss of generality, three algorithms of Akaike Information Criterion (AIC), Floating Threshold, and Reciprocal-based picker are used to illustrate the capabilities of the proposed method.Data collected from a concrete specimen are used as an input of the proposed technique. Results show that the proposed technique can select the best onset time candidates from the three mentioned algorithms, automatically. The picked onset time is comparable with manual selection, and accordingly has better accuracy for source localization when compared to any of the single methods.

Ramin Madarshahian, Paul Ziehl, Michael D. Todd

Chapter 14. Quantification and Evaluation of Parameter and Model Uncertainty for Passive and Active Vibration Isolation

Vibration isolation is a common method used for minimizing the vibration of dynamic load-bearing structures in a region past the resonance frequency, when excited by disturbances. The vibration reduction mainly results from the tuning of stiffness and damping during the early design stage. High vibration reduction over a broad bandwidth can be achieved with additional and controlled forces, the active vibration isolation. In this context, “active” does not mean the common understanding that the surroundings are isolated against the machine vibrations. Also in this context, “passive” means that no additional and controlled force is present, other than the common understanding that the machine is isolated against the surroundings. For active vibration isolation, a signal processing chain and an actuator are included in the system. Typically, a controller is designed to enable a force of an actuator that reduces the system’s excitation response. In both passive and active vibration isolation, uncertainty is an issue for adequate tuning of stiffness and damping in early design stage. The two types of uncertainty investigated in this contribution are parametric uncertainty, i.e. the variation of model parameters resulting in the variation of the systems output, and model uncertainty, the uncertainty from discrepancies between model output and experimentally measured output. For this investigation, a simple one mass oscillator under displacement excitation is used to quantify the parameter and model uncertainty in passive and active vibration isolation. A linear mathematical model of the one mass oscillator is used to numerically simulate the transfer behavior for both passive and active vibration isolation, thus predicting the behavior of an experimental test rig of the one mass oscillator under displacement excitation. The models’ parameters that are assumed to be uncertain are mass and stiffness as well as damping for the passive vibration isolation and an additional gain factor for the velocity feedback control in case of active vibration isolation. Stochastic uncertainty is assumed for the parameter uncertainty when conducting a Monte Carlo Simulation to investigate the variation of the numerically simulated transfer functions. The experimental test rig enables purposefully adjustable insertion of parameter uncertainty in the assumed value range of the model parameters in order to validate the model. The discrepancy between model and system output results from model uncertainty and is quantified by the Area Validation Metric and an Bayesian model validation approach. The novelty of this contribution is the application of the Area Validation Metric and Bayes’ approach to evaluate and to compare the two different passive and active approaches for vibration isolation numerically and experimentally. Furthermore, both model validation approaches are compared.

Jonathan Lenz, Roland Platz

Chapter 15. Bayesian Model Updating of a Five-Story Building Using Zero-Variance Sampling Method

This study presents the Bayesian model updating and stochastic seismic response prediction of a reinforced concrete frame building with masonry infill panels. After the 2015 Gorkha earthquake, some of the authors visited the building and recorded ambient vibration data using a set of accelerometers. The seismic response of the building was also recorded during one of the moderate aftershocks, using a set of sensors at the basement and the roof. In this study, the ambient vibration data is used to calibrate a model and the earthquake data is used to validate it. Natural frequencies and mode shapes of the building are extracted through an output-only system identification process. An initial finite elementmodel of the building is developed using a recently proposed modeling framework for masonry-infilled RC frames. Bayesian model updating is then performed to update the stiffness of selected structural elements and evaluate their respective uncertainties, given the available data. A novel sampling approach, namely Zero-Variance MCMC, is implemented to address the computational challenges of stochastic simulation when estimating the joint posterior probability distribution of the model’s parameters. This sampling approach has been shown to drastically improve computational efficiency while preserving adequate accuracy. The calibrated model is used for the probabilistic prediction of the seismic response of the building to a moderate earthquake. This predicted response is shown to be in good agreement with the available recorded response of the building at the roof.

Mehdi M. Akhlaghi, Supratik Bose, Peter L. Green, Babak Moaveni, Andreas Stavridis

Chapter 16. Input Estimation and Dimension Reduction for Material Models

Computer models for applications such as climate or materials have become increasingly complex. In particular, the input and output dimensions for these types of models has grown steadily larger, which has increased the computational burden of comparing these models with experimental data. This has spurred the development of statistical techniques for estimating outputs and reducing the dimension. This paper will show an example of these approaches applied to modeling and experiments for Tantalum, a material of interest for the Departments of Defense and Energy. We obtain results from a number of small-scale tests of Tantalum single crystals and use these results in a Bayesian statistical procedure to constrain the range and dimensionality of a Tantalum model.

Sam Myren, Emilio Herrera, Andrew Shoats, Earl Lawrence, Emily Casleton, D. J. Luscher, Saryu Fensin

Chapter 17. Augmented Sequential Bayesian Filtering for Parameter and Modeling Error Estimation of Linear Dynamic Systems

In this paper an augmented sequential Bayesian filtering approach is proposed for parameter and modeling error estimation of linear dynamic systems of civil structures using time domain input-output data through a sequential maximum a posteriori (MAP) estimation approach, which is similar to Kalman filtering method. However, in the application of existing Kalman filters, the estimation of modeling errors is rarely considered. Unlike traditional Kalman filter which provides state estimation at every time step, the proposed filtering approach estimates the parameter and modeling error on a windowing basis, i.e., the input and output data are divided into windows for estimation which would save computation burden. The analytical derivation of the proposed augmented sequential Bayesian filtering method is first presented, and then the method is verified through a numerical case study of a 3-story building model. An earthquake excitation is used as the input and the acceleration time history response of the building model is simulated. The simulated response is then polluted with different levels of Gaussian white noise to account for the measurement noise. The simulated response is used as the measured data for calibrating another 3-story shear building model which is different from the original model for simulation. Modeling errors are introduced in this shear building model including the shear building assumption, grouping strategy and boundary conditions. The augmented sequential Bayesian filtering approach is applied to estimate the model parameters and modeling error. The performance of the proposed method is studied with respect to modeling errors, the number of sensors and the level of noise.

Mingming Song, Hamed Ebrahimian, Babak Moaveni

Chapter 18. On-Board Monitoring of Rail Roughness via Axle Box Accelerations of Revenue Trains with Uncertain Dynamics

In addressing recent demands for increasing loads and speed of Railways, as well as increasing railway network usage, inexpensive and frequent monitoring of the infrastructure may be adopted to ensure safe and reliable operation. Presently, inspection of railway infrastructure is carried out visually or with dedicated Track Recording Vehicles (TRV), which are equipped with a number of optical and inert sensors and periodically collect geometric data. However, such inspection is costly, may only be carried out at periodic, and thereby infrequent intervals, and may disrupt other services and regular operation. As an alternative, relatively low-cost on-board monitoring data collected from revenue-making trains, could offer a cost-effective and more robust approach to monitor railway tracks. This approach relies on accelerometers mounted either on the axle box or on the car body, with the potential of almost continuous monitoring, earlier fault detection and thereby serving as a natural fit for predictive maintenance. However, the dynamics collected from revenue service trains, via low-cost sensors, are inevitably described by uncertainties (speed, weight, rolling stock condition). To this end, we propose an approach relying on model-based system identification for increasing the estimation capacity. A vehicle–rail interaction model is coupled with a dual Kalman filter (KF) on measured axle box vibration data from in-service trains, in order to estimate the input excitation (e.g. rail roughness). Via estimation of the input, we may distinguish between isolated defects (e.g. squats, turnout frogs, welded joints) and effects distributed over a certain track length (e.g. concrete sleeper, wood sleeper, ballast, slab track).

V. K. Dertimanis, M. Zimmermann, F. Corman, E. N. Chatzi

Chapter 19. Bayesian Identification of a Nonlinear Energy Sink Device: Method Comparison

Nonlinear energy sink (NES) devices have recently been introduced in civil engineering for structural control. Because of the essential geometric nonlinearities governing these devices, identification must be performed in the time domain. Such methods can be challenging due to processing requirements, sensitivity to noise, and the presence of nonlinearity. Bayesian analysis methods have been shown to overcome these challenges, providing robust identification of nonlinear models. In this study we compare the unscented Kalman filter and the particle filter for the identification of a prototype NES device. Simulated responses developed using a device model and a sample set of parameters are used here to demonstrate and evaluate the identification process. Analysis of the identification results is conducted by varying the identification technique used and the selection of the prior distributions on the parameters. These preliminary numerical results will inform a later implementation on experimental response data.

Alana Lund, Shirley J. Dyke, Wei Song, Ilias Bilionis

Chapter 20. Calibration of a Large Nonlinear Finite Element Model of a Highway Bridge with Many Uncertain Parameters

Finite element (FE) model updating has emerged as a powerful technique for structural health monitoring (SHM) and damage identification (DID) of civil structures. Updating mechanics-based nonlinear FE models allows for a complete and comprehensive damage diagnosis of large and complex structures. Recursive Bayesian estimation methods, such as the Unscented Kalman filter (UKF), have been used to update nonlinear FE models of civil structures; however, their use have been limited to models with a relatively low number of degrees of freedom and with a limited number of unknown model parameters, because it is otherwise impractical for computationally demanding models with many uncertain parameters. In this paper, a FE model of the Marga-Marga bridge, an eight-span seismically-isolated bridge located in Viña del Mar-Chile, is updated based on numerically simulated response data. Initially, 95 model parameters are considered unknown, and then, based on a simplified sensitivity analysis, a total of 27 model parameters are considered in the estimation. Different measurement sets, including absolute accelerations, relative displacements, strains, and shear deformations of the isolators, are analyzed to investigate the effects of considering heterogeneous responses on the estimation results. In addition, a non-recursive estimation procedure is presented and its effectiveness in reducing the computational cost, while maintaining accuracy and robustness in the estimation, is demonstrated.

Rodrigo Astroza, Nicolás Barrientos, Yong Li, Erick Saavedra Flores

Chapter 21. Deep Unsupervised Learning for Condition Monitoring and Prediction of High Dimensional Data with Application on Windfarm SCADA Data

In this work we are addressing the problem of statistical modeling of the joint distribution of data collected from wind turbines interacting due to collective effect of their placement in a wind-farm, the wind characteristics (speed/orientation) and the turbine control. Operating wind turbines extract energy from the wind and at the same time produce wakes on the down-wind turbines in a park, causing reduced power production and increased vibrations, potentially contributing in a detrimental manner to fatigue life. This work presents a Variational Auto-Encoder (VAE) Neural Network architecture capable of mapping the high dimensional correlated stochastic variables over the wind-farm, such as power production and wind speed, to a parametric probability distribution of much lower dimensionality. We demonstrate how a trained VAE can be used in order to quantify levels of statistical deviation on condition monitoring data. Moreover, we demonstrate how the VAE can be used for pre-training an inference model, capable of predicting the power production of the farm together with bounds on the uncertainty of the predictions.Examples employing simulated wind-farm Supervisory Control And Data Acquisition (SCADA) data are presented. The simulated farm data are acquired from a Dynamic Wake Meandering (DWM) simulation of a small wind farm comprised of nine 5 MW turbines in close spacing using OpenFAST.The contribution of this work lies in the introduction of state-of-the-art machine learning techniques in the general context of condition monitoring and uncertainty quantification. We show how the high dimensional joint probability distribution of condition monitoring parameters can be analyzed by exploiting the underlying lower dimensional structure of the data imposed by the physics of the problem. The process of making use of the trained joint distribution for the purposes of inference under uncertainty and condition monitoring is clearly exposed.

C. Mylonas, I. Abdallah, E. N. Chatzi

Chapter 22. Influence of Furniture on the Modal Properties of Wooden Floors

Structure-borne vibration and low-frequency re-radiated noise from internal and external sources cause annoyance for inhabitants in dwellings. A key parameter in the prediction of vibration and noise levels is the modal parameters of the floors in a building, since vibration and sound levels increase when natural frequencies of the floor coincide with the excitation frequencies of a source, e.g. monoharmonic vibration of unbalanced rotating machinery and appliances or HVAC system—or traffic induced ground vibration propagating into the building. This paper has focus on wooden floors built as an assembly of particleboard and timber joists. Such floors constitute horizontal divisions in many dwellings—both older, traditional buildings and new lightweight buildings. The analysis concerns the impact of furniture placed on a floor with otherwise known properties. Given the small mass of a traditional wooden floor, the presence of furniture can be expected to change the modal properties of the floor significantly. The finite-element model, developed for the present analyses, accounts for uncertainty in the position of the furniture, and the analysis addresses the importance of the elevation of the mass above the floor regarding the natural frequencies and the related modes of vibration.

Lars Vabbersgaard Andersen, Christian Frier, Lars Pedersen, Peter Persson

Chapter 23. Optimal Sensor Placement for Response Reconstruction in Structural Dynamics

A framework for optimal sensor placement (OSP) for response reconstruction under uncertainty is presented based on information theory. The OSP is selected as the one that maximizes an expected utility function taken as the mutual information between data and response quantities of interest (QoI). The expected utility function is extended to make the OSP design robust to uncertainties in structural model parameter and modelling errors. The resulting utility function is a multidimensional integral of the information entropy for each possible value of the model parameters, weighted by the prior or posterior probability distribution of the model parameters. The formulation uses the Gaussian nature of the response QoI given the measurements to simplify the expected utility function in terms of the covariance matrix of the uncertainty in the response output QoI given the values of modeling parameters. Methods to compute the multidimensional integrals and to optimize the sensor placement are discussed. The implementation is presented for two cases used to predict response time histories from output-only measured data: modal expansion techniques and filter-based techniques.

Costas Papadimitriou

Chapter 24. Finite Element Model Updating Accounting for Modeling Uncertainty

A novel approach to deal with modeling uncertainty when updating mechanics-based finite element (FE) models is presented. In this method, a dual adaptive filtering approach is adopted, where the Unscented Kalman filter (UKF) is used to estimate the unknown parameters of the nonlinear FE model and a linear Kalman filter (KF) is employed to estimate the diagonal terms of the covariance matrix of the simulation error vector based on a covariance-matching technique. Numerically simulated response data of a two-dimensional three-story three-bay steel frame structure with eight unknown material model parameters subjected to seismic base excitation is employed to illustrate and validate the proposed methodology. The results of the validation studies show that the proposed approach significantly outperforms the parameter-only estimation approach widely investigated and used in the literature.

Rodrigo Astroza, Andres Alessandri, Joel P. Conte

Chapter 25. Model-Based Decision Support Methods Applied to the Conservation of Musical Instruments: Application to an Antique Cello

In musical instrument making and restoration domains, the variability of the materials and the irreversibility of the changes are issues for the experimental study of the impact of design changes and restorations on musical instruments. In addition, the analytical methods based on simplified geometries and models are not sufficiently detailed for the study of complex structures and phenomena. The virtual prototyping, and its different capabilities, can be a powerful method for instrument makers and museum curators as a decision support tool. Nevertheless, the accuracy of the model is an important matter to assess good predictions. In the case of antique and unique instruments, it is sometimes hard to obtain exhaustive geometrical properties. Similarly, it is also difficult to evaluate the material properties of full instruments, and this uncertainty may have a strong impact on the output features of the numerical models. In this study, a numerical model of cello is developed using finite element method. It is used to evaluate the impact of a modification of a geometrical property on dynamical features. It is shown that the lack of knowledge on the arching height of the top and back plates of a cello has a strong impact on the computed dynamical properties of the cello. Secondly, the model is considered with and without repair cleats and defects like galleries excavated by wood-boring insects. It is observed that the bridge admittance exhibits discrepancies above 220 Hz which is in the low frequencies domain of the model and quantify the impact of repairs. This model capability is a starting point for further simulations accounting for material and geometrical uncertainties and to assess the confidence level of a model for restoration issues.

R. Viala, V. Placet, S. Le Conte, S. Vaiedelich, S. Cogan

Chapter 26. Optimal Sensor Placement for Response Predictions Using Local and Global Methods

A Bayesian framework for model-based optimal sensor placement for response predictions is presented. Our interest lies in determining the parameters of the model in order to make predictions about a particular response quantity of interest. This problem is not adequately explored since the majority of currently available literature is focused on parameter inference, rather than prediction inference. The model parameters are inferred by collecting experimental data which depends on the chosen sensor locations. The parameter values are uncertain and their uncertainty is described by a prior probability density function. The measured quantity, or data, is a quantity that can be predicted by the model which depends on both parameters and sensor locations. A prediction error equation is used to describe the discrepancy between the model-predicted measured quantity and the actual data collected from the experiment. The sensor locations are optimized with respect to prediction inference, while the case of parameter inference is derived as a special case under a more general framework. The posterior covariance matrix is used as a measure of uncertainty in the predictions. Two approaches are developed for its calculation, one global and one local. The local approach is based on sensitivities at a fixed value of the parameters, while the global approach uses Monte Carlo sampling and explores the full range of uncertainty in the parameters. A simple numerical example is presented in order to illustrate and verify the two approaches.

Costas Argyris, Costas Papadimitriou, Geert Lombaert

Chapter 27. Incorporating Uncertainty in the Physical Substructure During Hybrid Substructuring

In hybrid substructuring, a structural system is partitioned into a numerical substructure and a physical substructure. Typically, the physical substructure consists of a system component whose behavior is difficult to model while the numerical substructure consists of a computational model of the remainder of the system. Hybrid substructuring has previously been shown to be an effective method to quantify the effect of parametric uncertainties in the numerical substructure on the response of the system. This paper proposes and implements a methodology where the effect of parametric uncertainty can also be incorporated into the physical substructure. This idea is implemented in a series of small-scale Real-Time Hybrid Substructuring (RTHS) tests on a magneto-rheological fluid damper used to control a two degree-of-freedom mass-spring system. The physical current supplied to the damper is treated as a random variable. Using the RTHS test results, a metamodel of the system’s frequency domain behavior is developed using Principal Component Analysis and Kriging. This metamodel is then used to evaluate probabilistic system performance.

Connor Ligeikis, Richard Christenson

Chapter 28. Applying Uncertainty Quantification to Structural Systems: Parameter Reduction for Evaluating Model Complexity

Different mathematical models can be developed to represent the dynamic behavior of structural systems and assess properties, such as risk of failure and reliability. Selecting an adequate model requires choosing a model of sufficient complexity to accurately capture the output responses under various operational conditions. However, as model complexity increases, the functional relationship between input parameters varies and the number of parameters required to represent the physical system increases, reducing computational efficiency and increasing modeling difficulty. The process of model selection is further exacerbated by uncertainty introduced from input parameters, noise in experimental measurements, numerical solutions, and model form. The purpose of this research is to evaluate the acceptable level of uncertainty that can be present within numerical models, while reliably capturing the fundamental physics of a subject system. However, before uncertainty quantification can be performed, a sensitivity analysis study is required to prevent numerical ill-conditioning from parameters that contribute insignificant variability to the output response features of interest. The main focus of this paper, therefore, is to employ sensitivity analysis tools on models to remove low sensitivity parameters from the calibration space. The subject system in this study is a modular spring-damper system integrated into a space truss structure. Six different cases of increasing complexity are derived from a mathematical model designed from a two-degree of freedom (2DOF) mass spring-damper that neglects single truss properties, such as geometry and truss member material properties. Model sensitivity analysis is performed using the Analysis of Variation (ANOVA) and the Coefficient of Determination R 2. The global sensitivity results for the parameters in each 2DOF case are determined from the R 2 calculation and compared in performance to evaluate levels of parameter contribution. Parameters with a weighted R 2 value less than .02 account for less than 2% of the variation in the output responses and are removed from the calibration space. This paper concludes with an outlook on implementing Bayesian inference methodologies, delayed-acceptance single-component adaptive Metropolis (DA-SCAM) algorithm and Gaussian Process Models for Simulation Analysis (GPM/SA), to select the most representative mathematical model and set of input parameters that best characterize the system’s dynamic behavior.

Robert Locke, Shyla Kupis, Christopher M. Gehb, Roland Platz, Sez Atamturktur

Chapter 29. Non-unique Estimates in Material Parameter Identification of Nonlinear FE Models Governed by Multiaxial Material Models Using Unscented Kalman Filtering

Bayesian nonlinear finite element (FE) model updating using input and output measurements have emerged as a powerful technique for structural health monitoring (SHM), and damage diagnosis and prognosis of complex civil engineering systems. The Bayesian approach to model updating is attractive because it provides a rigorous framework to account for and quantify modeling and parameter uncertainty. This paper employs the unscented Kalman filter (UKF), an advanced nonlinear Bayesian filtering method, to update, using noisy input and output measurement data, a nonlinear FE model governed by a multiaxial material constitutive law. Compared to uniaxial material constitutive models, multiaxial models are typically characterized by a larger number of material parameters, thus requiring parameter estimation to be performed in a higher dimensional space. In this work, the UKF is applied to a plane strain FE model of Pine Flat dam (a concrete gravity dam on King’s River near Fresno, California) to update the time-invariant material parameters of the cap plasticity model, a three-dimensional non-smooth multi-surface plasticity concrete model, used to represent plain concrete behavior. This study considers seismic input excitation and utilizes numerically simulated measurement response data. Estimates of the multi-axial material model parameters (for the single material model used in this study) are non-unique. All sets of parameter estimates yield very similar and accurate seismic response predictions of both measured and unmeasured response quantities.

Mukesh Kumar Ramancha, Ramin Madarshahian, Rodrigo Astroza, Joel P. Conte

Chapter 30. On Key Technologies for Realising Digital Twins for Structural Dynamics Applications

The term digital twin has gained increasing popularity over the last few years. The concept, loosely based on a virtual model framework that can replicate a particular system for contexts of interest over time, will require the development and integration of several key technologies in order to be fully realised. This paper, focusing on vibration-related problems in mechanical systems, discusses these key technologies as the building blocks of a digital twin. The example of a simulation digital twin that can be used for asset management is then considered. After briefly discussing the building blocks required, the process of data-augmented modelling is selected for detailed investigation. This concept is one of the defining characteristics of the digital twin idea, and using a simple numerical example, it is shown how augmenting a model with data can be used to compensate for the inherent model discrepancy. Finally the implications of this type of data augmentation for future digital twin technology is discussed.

D. J. Wagg, P. Gardner, R. J. Barthorpe, K. Worden

Chapter 31. Hygro-mechanical Modelling of Wood and Glutin-based Bond Lines of Wooden Cultural Heritage Objects

A comprehensive modelling of the transient hygro-mechanical behaviour of complex wooden structures by finite element method is targeted. New methods and material models for glutin-based bond lines are developed, since bond lines proved to have significant influence on moisture transport and fracture behaviour. The models are validated and applied to the structural analysis of wooden music instruments exposed to mechanical and hygric loadings.

Michael Kaliske, Daniel Konopka

Chapter 32. Modelling of Sympathetic String Vibrations in the Clavichord Using a Modal Udwadia-Kalaba Formulation

The vibratory and acoustic modeling of musical instruments is important for several purposes in cultural heritage preservation, performance studies and musical creation. On the one hand, building a model helps understanding the key features of an instrument, and then is useful for evaluation, documentation and preservation of historical models. On the other hand, modeling and simulation can help for improving existing instruments, or even designing new instruments by extension of the model. The clavichord is an early keyboard instrument equipped with a very simple mechanics. The strings are excited by small metal wedges or blades (the tangents) placed at the end of the keys. The tangent remains in contact with the strings for the duration of the note, defining the vibrating length of the string. All strings are coupled at a same bridge. A string is divided into three sections: a damped section (DS) between the hitch-pin and the tangent; the played section (PS), excited by the tangents, between the tangent and the bridge; and the resting section (RS) between the bridge and the tuning pin. Because of the coupling through the bridge of the PS and RS, the RS is set into vibration, acting as sympathetic strings. The vibratory responses of the RS is modelled using a modal approach based on the Udwadia-Kalaba formulation. Firstly, a review of the method is presented, accompanied with measurements performed on an instrument (copy of a Hubert 1784 fretted clavichord), which include an experimental modal analysis at the instrument bridge and measurements of string motions. Then, simulation results are reported and compared with experimental measurements.

J.-T. Jiolat, J.-L. Le Carrou, J. Antunes, C. d’Alessandro

Chapter 33. Modeling and Stochastic Dynamic Analysis of a Piezoelectric Shunted Rotating Beam

This work presents a variational based stochastic electromechanical coupling model for response analysis of a rotating cantilever beam with piezoelectric patches surface-mounted. The resonant shunt circuits are connected to the piezoelectric elements to reduce vibrations of some specific resonance frequencies. The deterministic equations of motion are derived by the generalised form of Hamilton’s principle for electromechanical systems and Rayleigh-Ritz modeling method based on the orthogonal polynomial bases, while the Penalty method is adopted to connect the beam and piezoelectric patches. The parameter uncertainties are taken into account in both the structural and electric components. The generalized polynomial chaos expansion (gPCE) is employed to represent propagation of parameter uncertainties and to estimate the statistical characteristics of the responses. Various results are presented and compared with the Monte Carlo simulation (MCS) in order to validate the efficiency of the proposed formulation. Uncertainty analyses are carried out to ascertain the effects of probabilistic parameters on the responses. The results reveal that both the structure and piezoelectric uncertainty can affect the vibration behaviors, and consideration of parameter uncertainties is needed in dynamic designs in order to minimise the vibration response at resonance frequencies.

Zhenguo Zhang, Ningyuan Duan, Jiajin Tian, Hongxing Hua

Chapter 34. On Digital Twins, Mirrors and Virtualisations

A powerful new idea in the computational representation of structures is that of the digital twin. The concept of the digital twin emerged and developed over the last two decades, and has been identified by many industries as a highly-desired technology. The current situation is that individual companies often have their own definitions of a digital twin, and no clear consensus has emerged. In particular, there is no current mathematical formulation of a digital twin. A companion paper to the current one will attempt to present the essential components of the desired formulation. One of those components is identified as a rigorous representation theory of models, how they are validated, and how validation information can be transferred between models. The current paper will outline the basic ingredients of such a theory, based on the introduction of two new concepts: mirrors and virtualisations. The paper is not intended as a passive wish-list; it is intended as a rallying call. The new theory will require the active participation of researchers across a number of domains including: pure and applied mathematics, physics, computer science and engineering. The paper outlines the main objects of the theory and gives examples of the sort of theorems and hypotheses that might be proved in the new framework.

K. Worden, E. J. Cross, P. Gardner, R. J. Barthorpe, D. J. Wagg

Chapter 35. Applications of Reduced Order and Surrogate Modeling in Structural Dynamics

Despite recent advances in computational science, the adoption of computationally intensive, high-fidelity simulation models remains a challenge for many structural dynamics applications, especially those within the domain of uncertainty quantification (UQ), requiring repeated calls to a computationally intensive simulator. Reduced order and surrogate models offer an attractive alternative to circumvent this challenge. This contribution investigates how these modeling principles can be leveraged for different UQ applications. For both types of approximate models, the development of the corresponding (reduced order or surrogate) model is directly informed through simulations of the high-fidelity numerical model. The tuning of the approximate model aims to improve accuracy for the specific UQ task at hand, rather than targeting a globally accurate approximation. The specific applications discussed correspond to seismic loss estimation (for reduced order modeling) and posterior sampling for Bayesian inference (for surrogate modeling).

Alexandros A. Taflanidis, Jize Zhang, Dimitris Patsialis
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