Topics in Model Validation and Uncertainty Quantification, Volume 5

A challenging problem encountered in engineering applications is the estimation of a probability-of-failure based on incomplete knowledge of the sources of uncertainty and/or limited sampling. Theories formulated to derive upper probability bounds offer an attractive alternative because first, they avoid postulating the probability laws that are often unknown and second, they substitute numerical optimization for statistical sampling. A critical assessment of one such technique is presented. It derives upper probability bounds from the McDiarmid concentration-of-measure theory, which postulates that fluctuations of a function are more-or-less concentrated about its mean value. Two applications of this theory are presented. The first application analyzes a “toy” polynomial function defined in two dimensions. The upper bounds of probability are calculated and compared to sampling-based estimates of the true-but-unknown probabilities. For this function, the upper bounds obtained are too broad to be useful. These results are confirmed by conducting a similar analysis on a real engineering system, where upper bounds of probability associated with resonant frequencies of a structural system are estimated. A high-fidelity finite element model, previously validated using vibration measurements, is used to predict the frequencies. In this application, the uncertainty is introduced by way of material properties and the effective preload of a beam-to-column connection, modeled explicitly. These applications suggest that the theory not only leads to upper bounds that are inefficient but that can also be sub-optimal if their numerical estimation is based on too few model runs. It is concluded that this particular theory, while mathematically attractive, may not be well suited for engineering applications.

Fatigue-driven damage propagation is one of the most unpredictable failure mechanisms for a large variety of mechanical and structural systems subjected to cyclic and/or random operational loads during their service life. Therefore, monitoring the critical components of these systems, assessing their structural integrity, recursively predicting their remaining fatigue life (RFL), and providing a cost-efficient reliability-based inspection and maintenance (RBIM) plan are crucial tasks. In contribution to these objectives, the authors developed a comprehensive reliability-based fatigue damage prognosis methodology for recursively predicting and updating the RFL of critical structural systems and/or sub-assemblies. An overview of the proposed framework is provided in the first part of the paper. Subsequently, a set of experimental fatigue test data is used to validate the proposed methodology at the reliability component level. The proposed application example analyzes the fatigue-driven crack propagation process in a center-cracked 2024-T3 aluminum plate subjected to a sinusoidal load with random amplitude. Four probabilistic models of increasing load amplitude uncertainty together with damage evolution model parameter uncertainty and measurement uncertainty are considered in this study. The results obtained demonstrate the efficiency of the proposed framework in recursively updating and improving the RFL estimations and the benefits provided by a nearly continuous monitoring system.

A Bayesian probabilistic framework for uncertainty quantification and propagation in structural dynamics is reviewed. Fast computing techniques are integrated with the Bayesian framework to efficiently handle large-order models of hundreds of thousands or millions degrees of freedom and localized nonlinear actions activated during system operation. Fast and accurate component mode synthesis (CMS) techniques are proposed, consistent with the finite element (FE) model parameterization, to achieve drastic reductions in computational effort when performing a system analysis. Additional substantial computational savings are also obtained by adopting surrogate models to drastically reduce the number of full system re-analyses and parallel computing algorithms to efficiently distribute the computations in available multi-core CPUs. The computational efficiency of the proposed approach is demonstrated by updating a high-fidelity finite element model of a bridge involving hundreds of thousands of degrees of freedom.

A Bayesian uncertainty quantification and propagation (UQ&P) framework is presented for identifying nonlinear models of dynamic systems using vibration measurements of their components. The measurements are taken to be either response time histories or frequency response functions of linear and nonlinear components of the system. For such nonlinear models, stochastic simulation algorithms are suitable Bayesian tools to be used for identifying system and uncertainty models as well as perform robust prediction analyses. The UQ&P framework is applied to a small scale experimental model of a vehicle with nonlinear wheel and suspension components. Uncertainty models of the nonlinear wheel and suspension components are identified using the experimentally obtained response spectra for each of the components tested separately. These uncertainties, integrated with uncertainties in the body of the experimental vehicle, are propagated to estimate the uncertainties of output quantities of interest for the combined wheel-suspension-frame system. The computational challenges are outlined and the effectiveness of the Bayesian UQ&P framework on the specific example structure is demonstrated.

This paper presents a probabilistic damage identification study on a full-scale structure, the Dowling Hall Footbridge, through Bayesian finite element (FE) model updating. The footbridge is located at Tufts University campus and is equipped with a continuous monitoring system that measures the ambient acceleration response of the bridge. A set of data is recorded once an hour or when triggered by large vibrations. The modal parameters of the footbridge are extracted based on each set of measured ambient vibration data and are used for model updating. In this study, effects of physical damage are simulated by loading a small segment of footbridge’s deck with concrete blocks. The footbridge deck is divided into five segments and the added mass on each segment is considered as an updating parameter. Overall, 72 sets of data are collected during the loading period (i.e., damaged state of the bridge) and different subsets of these data are used to find the location and extent of the damage (added mass). Adaptive Metropolis Hasting algorithm with adaption on the proposal probability density function is successfully used to generate Markov Chains for sampling the posterior probability distributions of the five updating parameters. Effect of the number of data sets used in the identification process is investigated on the posterior probability distributions of the updating parameters.

Long-span bridges usually experience different input excitations at their ground supports that emanate from differences in wave arrival times, and soil conditions, as well as loss of coherency in arriving waves. These spatial variations can drastically influence the dynamic response; hence, this phenomenon must be considered in any vibration-based identification method. There are numerous Multi-Input Multi-Output (MIMO) identification techniques that may be applied to data recorded at long-span bridges that experience spatial variations in their input motions. However, inertial soil-structure interaction effects severely reduce the accuracy of these techniques because the actual Foundation Input Motion (FIM) cannot be recorded during earthquakes. In this study, we present an extension to a novel blind identification method that we had developed earlier, which enables the method to handle multiple input motions. For the sake of simplicity, we only consider wave passage effects—that is, all unknown input motions are assumed to be identical except for a known/unknown phase-delay. This method comprises two steps. In the first step, the spatial time-frequency distributions of recorded responses are used for extracting the mode shapes and the modal coordinates. This is achieved through a Blind Source Separation (BSS) technique. In the second step, cross relations among the extracted modal coordinates are used for identifying the natural frequencies, damping ratios, modal contribution factors, along with the unknown input motions through a least-squares technique. Both simulated and experimental examples are provided, which suggest that the method is capable of accurately identifying the dynamic characteristics of long-span bridges from recorded response signals without the knowledge of input motions, even in the presence of wave passage effects due to phase-delays.

Analytical and numerical models that simulate the physical processes inevitably contain errors due to the mathematical simplifications and the lack of knowledge about the physical parameters that control the actual behavior. In this sense, parametric identification of civil engineering structures using uncertain numerical models should be subject to a particular interest in terms of accuracy and reliability of identified models. In this study, model uncertainties are modeled by fuzzy numbers and quantified using fuzzy model updating approach. In order to find the possible variation range of the response parameters (e.g. natural frequencies, mode shapes and strains) using uncertain finite element model, successive updating is employed. A simplified approach is proposed in order to facilitate the time consuming successive model updating phase. The identified variation range of the response parameters is employed to construct the fuzzy membership functions for each response parameter. Finally, fuzzy finite element model updating method (FFEMU) is used to obtain the membership functions of the model parameters. Different sets of model parameters are chosen to represent different models in terms of accuracy and these parameters are identified in the same way to investigate the model complexity. A two span laboratory grid structure developed for simulating bridge structures is used to validate and demonstrate the proposed approaches. The results show that the proposed approaches can efficiently be utilized to quantify the modeling uncertainties for more realizable and quantitative condition assessment and decision making purposes.

This paper derives exact results for the maximum achievable accuracy when estimating modal parameters from free vibration signals contaminated by Gaussian white noise. These limits are found through the Cramer-Rao lower bound. The paper compares the exact findings with previous approximate results found in the literature. Comparisons are drawn with results from stochastic simulations of a Fourier domain approach for estimation of natural frequency and damping in single degree of freedom system.

This paper proposes the use of probability bounds with the Pseudo-inverse Finite Element (PiFE) method for structural model updating. The technique estimates the probability bound of structural parameters based on dynamic or static features such as modal parameters or static displacements. Two methods are explored for the calculation of the probability bounds: (i) Naïve method and (ii) all possible combinations. The capabilities of the technique are explored using a two degree of freedom structural system where the stiffness is considered uncertain. Results indicate that both the Naïve and all possible combination techniques are applicable with PiFE and produce bounds that include the cumulative distribution function of the structural parameters. The probability bounds found with the all possible combinations method was narrower for this particular example.

The goal of this work is to propose a model calibration strategy for an industrial problem consisting in a MW class geared wind turbine power train subjected to uncertain loads. Lack of knowledge is commonplace in this kind of engineering system and a realistic model calibration cannot be performed without taking into account this type of uncertainty. The question at stake in this study is how to perform a robust predictive model of a dynamic system given that the excitations are poorly known. The uncertainty in the latter will be represented with an info-gap model. The tradeoff between fidelity to data and robustness to uncertainty is then investigated in order to maximize the robustness of the prediction error at a given horizon of uncertainty. This methodology is illustrated on a simple academic model and on a more complex engineering system representing a wind turbine geared power train.

Several plausible modeling strategies are available to develop finite element (FE) models of ever-increasingly complex phenomena. Expert judgment is typically used to choose which strategy to employ, while the “best” modeling approach remains unknown. This paper proposes a decision analysis methodology that offers a systematic and rigorous methodology for comparing plausible modeling strategies. The proposed methodology departs from the conventional approach that considers only test-analysis correlation to select the model that provides the highest degree of fidelity-to-data. The novelty of the methodology lies in an exploration of the trade-offs between robustness to uncertainty and fidelity-to-data. Exploring robustness to model imprecision and inexactness, in addition to fidelity-to-data, lends credibility to the simulation by guaranteeing that its predictions can be trusted even if some of the modeling assumptions and input parameters are incorrect. To demonstrate this approach, an experimental configuration is analyzed in which large masses are used to load the CX-100 wind turbine blade in bending during vibration testing. Two plausible simulations are developed with differing strategies to implement these large masses using (i) a combination of point-mass and spring elements or (ii) solid elements. In this paper, the authors study the ability of the two FE models to predict the experimentally obtained natural frequencies, and the robustness of these competing models to uncertainties in the input parameters. Considering robustness for model selection provides the extent to which prediction accuracy deteriorates as the lack-of-knowledge increases. Therefore, the preferable modeling strategy is the one that offers the best compromise between fidelity-to-data and robustness to uncertainty. To predict the bending vibration of the CX-100 wind turbine blade, it is observed that the modeling strategy with solid elements is far superior to the other one in its ability to provide a compromise between fidelity-to-data and robustness to the modeling assumptions.

Validation experiments are conducted at discrete settings within the domain of interest to assess the predictive maturity of a model over the entire domain. Satisfactory model performance merely at these discrete tested settings is insufficient to ensure that the model will perform well throughout the domain, particularly at settings far from validation experiments. The goal of coverage metrics is to reveal how well a set of validation experiments represents the entire operational domain. The authors identify the criteria of an exemplary coverage metric, evaluate the ability of existing coverage metrics to fulfill each criterion, and propose a new, improved coverage metric. The proposed metric favors interpolation over extrapolation through a penalty function, causing the metric to prefer a design of validation experiments near the boundaries of the domain, while simultaneously exploring inside the domain. Furthermore, the proposed metric allows the coverage to account for uncertainty associated with validation experiments. Application of the proposed coverage metric on a practical, non-trivial problem is demonstrated on the Viscoplastic Self-Consistent material plasticity code for 5182 aluminum alloy.

This paper reviews the weighted orthogonality property of modal vectors, the many Test-Analysis Model (TAM) transforms that have been developed to reduce Finite Element Model (FEM) based mass matrices and the Modal Assurance Criterion (MAC). These associated technologies have all been developed to try and correlate the FEM and experimentally obtained mode shapes. A case study is presented where the Effective Independence method for Degree-of-Freedom (DOF) selection was used to systematically reduce the DOF’s of the FEM and understand the effects of DOF reduction on MAC, the Guyan TAM and the System Equivalent Reduction/Expansion Process (SEREP) TAM.

CAE models have become more and more critical for decision making in various stages of the product development process with less hardware builds and shorten development time. To serve their purpose, it is important to understand the CAE capability of a model. A metric is essential to quantify this capability as well as track its history for improvement over the time. In particular, the noise and vibration sensitivity of a vehicle to a unit force or torque in a wide frequency range is of interest. In this paper, various correlation analysis techniques are applied to the Frequency Response Spectrum data from a CAE model and hardware test to develop a correlation metric. The comparison between Principal Component Analysis, Canonical Correlation Analysis and the developed metrics in this paper is presented. The correlation metrics developed in this paper also matches the subjective evaluation of our CAE capability for different measurement.

Mechanical systems under vibration excitation are prime candidate for being modeled by linear time invariant systems. Damage localization in such systems, when the excitation is not measurable, can be carried out using the Stochastic Dynamic Damage Locating Vector (SDDLV) approach, a method that interrogates changes in a matrix that has the same kernel as the change in the transfer matrix at the sensor locations. Damage location is related to some residual derived from the kernel. Deciding that this residual is zero is up to now done using empirically defined thresholds. In this paper, we describe how the uncertainty of the state space system can be used to derive uncertainty on the damage localization residuals to decide about the damage location. The results are illustrated in finite element models of a truss and of a plate.

Tolerance is a major source of uncertainty and contributes significantly to the variation of dynamic responses, worsening the repeatability of assembled products. A systematic tolerance design strategy is required to control this kind of variation, and an approach for tolerance design based on robust design theory is proposed in this paper with a focus on the optimization of the dynamic response. The approach is based on Taguchi’s method, and performed by the following steps: (1) define the input and output parameters for the problem; (2) determine the effects of the control factors on the dynamic responses of interest; (3) identify factors to be adjusted and transform the problem into a multi-objective optimization. A benchmark tolerance design of a joint assembly of aero engine casings is used to verify the feasibility of the approach.

The FRF-based substructuring method considering elastic coupling is investigated for predicting vibration transmission and uncertainty propagation in floating raft system. The effects of various commonly encountered errors of substructure FRFs, such as pole shifting, reciprocity violation, negative imaginary part of driving-point FRFs, on the modeling results are investigated. The simulation results demonstrate that the noise on FRFs of raft can influence the assembly FRFs of base and raft; while, some kinds of noises cannot be propagated to the assembly FRFs of machines for the rigid machines are coupled through elastic isolators, bringing about impedance mismatch. The SVD cannot eliminate the studied noises. The uncertainty in substructure FRFs can be amplified due to the coupling process. Propagation of uncertainties of FRFs is quantified based on an analytical approximate moment method using the derivatives of substructure FRFs and variance defined on these parameters. The moment method is of high efficiency and accuracy compared with Monte Carlo probabilistic method.

Modal analysis is widely used both on single components and mechanical complex assemblies and it is recognized to be a fundamental step on the functional design process. From experimental point of view, a change in a system parameter due to the need of describing a different assembly configuration, can require iterative measurements, and can be quite time consuming. On the other hand, by evaluating the dynamic behaviour of the single component instead of the whole system, it is not straightforward to forecast the general dynamics of the entire assembly: inertia and stiffness couplings give rise to curious dynamic phenomena, namely crossing and veering of eigenvalue loci. Many theoretical studies on eigenvalue curve crossing and curve veering, i.e. the coincidence of two eigenfrequencies or the abrupt divergence of natural frequencies trends, have been carried out in recent years, but only few references on detailed test sessions and practical applications are available. The present paper wants to give a better overview on the change of the dynamic properties of a system by comparing global mode shapes to single component mode shapes. The examined structure is a crank mechanism, made of a crankshaft joined to four connecting rods and four pistons. The chosen control parameter that is responsible of a change in the dynamic properties of the system is the crank angle. Numerical models have been used to compute eigenvalues and eigenvectors of the analysed structure, considering both FEM models and multibody approach. Finally, an original graphical interpretation of the transition from component to system dynamics is presented by means of the MAC index.

Occupant footfalls are often the most critical source of floor vibration on the elevated floors of buildings. Floor motions can disturb occupants, leading to frequent complaints and loss of functionality. In laboratory and healthcare facilities, this issue can be more critical, as high-resolution imaging equipment with stringent vibration criteria is often employed. Achieving these criteria requires sufficiently stiff and massive floor structures to effectively resist the forces exerted from user traffic. The difficulty for engineers is predicting these low levels of vibration. Two commonly used analysis methods to predict footfall vibration levels in steel buildings are the American Institute of Steel Construction (AISC) Design Guide 11, and The Steel Construction Institute (SCI) P354. The latter is more robust, as it can predict multi-modal time history responses at any point on the floor. Dynamic footfall loading is determined by considering walkers moving along reasonable pathways identified in the architectural floor plans. For concrete structures, The Concrete Centre (CCIP-016) proposes a methodology similar to the SCI. In this study, three steel and one concrete building are instrumented to measure footfall-induced vibrations. The measured values are compared to the predictions of the aforementioned methods, and the superiority of the SCI-P354 and CCIP-016 methods is shown.

In this paper, the finite element model updating technique based on first-order optimization is investigated and applied to a jointed aero engine casing. Vibration modal testing is conducted and modal data, i.e. natural frequencies and mode shapes, are obtained to update the FE models of two aero engine casings and their jointed structure. A two-step strategy is proposed in the updating process. In the first step, model updating is carried out on the two single casings separately in order to validate the casing component models. In the second step, the assembled casing is updated with emphasis on updating joint parameters. The joints are modeled using a layer of continuous solid elements that have material properties to be adjusted. The final updated FE model for the aero-engine casing is able to predict natural frequencies and mode shapes close to the measured ones.

Traction properties of agricultural tires on deformable grounds are strongly dependent on the interaction between tread elements with the soil, which is in turn function of parameters like tread design and inflating pressure. The present paper presents a numerical model aimed at investigating tire performance and soil compaction under different conditions, considering effects due to tangential and normal compliance of the terrain surface as well as tread design and inflating pressure. In particular, the ground has been modelled as a deformable springs layer and the tire as a flexible ring reproducing the tread geometry. Longitudinal and vertical dynamics of the tire are taken into account. Moreover, effect of tire carcass deformation was included considering the in-plane eigenmodes of the tire identified through a series of experimental tests.

Modal analyses, model validations and correlations are performed for the different configurations of the International Space Station (ISS). Three Dedicated Thruster Firings (DTF) tests were conducted during ISS Stage ULF4; this paper will focus on the analysis and results of the DTF S4-1A, which occurred on October 11, 2010. The objective of this analysis is to validate and correlate analytical models used to verify the ISS critical interface dynamic loads.

During the S4-1A Dedicated Thruster Firing test, on-orbit dynamic measurements were collected using four main ISS instrumentation systems along with a Russian high rate sensor; Internal Wireless Instrumentation System (IWIS), External Wireless Instrumentation System (EWIS), Structural Dynamic Measurement System (SDMS), Space Acceleration Measurement System (SAMS) and Internal Measurement Unit (IMU). ISS external cameras also recorded the movement of one of the main solar array tips, array 1A.

Modal analyses were performed on the measured data to extract modal parameters including frequency, damping, and mode shape information. Correlation and comparisons between test and analytical frequencies and mode shapes were performed to assess the accuracy of the analytical models for the configuration under consideration. Based on the frequency comparisons, the accuracy of the mathematical model is assessed and model refinement recommendations are given.

The numerical modeling of vibration induced atomization of liquids validates existing operating parameters of known systems. A computational fluid dynamics analysis is performed which assists in model verification and reveals a critical configuration of driving amplitude and liquid depth that must be fulfilled for atomization. In this configuration the droplet kinetic energy exceeds the fluid resistance energy and the atomization process initiates. Verification and identification of the parameters for atomization of a thin film of water is accomplished.

Existing literature on the operation of vibrating mesh nebulizers does not entirely explain the principles by which these devices atomize liquids. Many previous studies assume a spray or extrusion mode of droplet generation, but it can be demonstrated that the high frequency vibration of these devices is sufficient to produce aerosol droplets. A thin film of liquid vibrated under the correct conditions will produce a fountain of atomized liquid droplets. The formation of standing waves on the surface of a thin film have an oscillating frequency that is half the driving frequency, a wavelength that is equal to a function of the driving frequency, a mean droplet diameter one-third the standing wavelength dimension, and are also dependent of fluid density and interfacial surface tension.

Sound generated by different sources in water can be determined by underwater acoustic systems. In order to detect sound in water, the underwater acoustic system should be designed with respect to level and frequency characteristics of sound. Therefore, resonance frequency of the underwater acoustic system is the most important design parameter and it should be obtained by design iterations. During these iterations, resonance frequency of the underwater acoustic system can be found by finite element method with related analysis. Also, this analysis should be verified by experimental techniques. In this study, dynamical design and analysis of underwater acoustic system, which has 5 ×5 arrays of identical tonpilz transducers and these transducers are attached to the carbon fiber reinforced composite, are detailed. Moreover, this dynamical model of the underwater acoustic system is verified by experimental techniques, which are admittance, transmit voltage response and received voltage sensitivity measurements.