Topics in Model Validation and Uncertainty Quantification, Volume 4

As Structural Health Monitoring (SHM) continues to gain popularity, both as an area of research and as a tool for use in industrial applications, the number of technologies associated with SHM will also continue to grow. As a result, the engineer tasked with developing a SHM system is faced with myriad hardware and software technologies from which to choose, often adopting an ad hoc qualitative approach based on physical intuition or past experience to making such decisions, and offering little in the way of justification for a particular decision. The present paper offers a framework that aims to provide the engineer with a qualitative approach for choosing from among a suite of candidate SHM technologies. The framework is outlined for the general case, where a supervised learning approach to SHM is adopted, and is then demonstrated on a problem commonly encountered when developing SHM systems: selection of a damage classifier, where the engineer must select from among a suite of candidate classifiers, the one most appropriate for the task at hand. The data employed for these problems are taken from a preliminary study that examined the feasibility of applying SHM technologies to the RAPid Telescopes for Optical Response observatory network. (Approved for unlimited public release on September 20, 2011, LA-UR 11-05398, Unclassified)

Uncertainty quantification is an integral part of the model validation process and is important to take into account during the design of mechanical systems. Sources of uncertainty are diverse but generally fall into two categories: aleatory uncertainties due to random processes and epistemic uncertainty resulting from a lack of knowledge or erroneous assumptions.This work focuses on the impact of uncertain levels of prestress on the behavior of solar arrays in their stowed configuration. In this context, snubbers are inserted between two adjacent panels to maintain contact and absorb vibrations during launch. However, under high excitation loads, a loss of contact between the two panels may occur. This results in impacts that can cause extensive damages to fragile elements.

In practice, the specific load configuration for which the separation of the two panels occurs is difficult to determine precisely since the exact level of prestress applied to the structure is unknown. An info-gap robustness analysis is applied to study the impact of this lack of knowledge on the prestress safety factor required to avoid loss of contact. The proposed methodology is illustrated using a simplified model of a solar array.

Robustness to parametric uncertainty is fundamental to successful control system design and as such it has been at the core of many design methods developed over the decades. Despite its prominence, most of the work on robust control design has focused on linear models and uncertainties that are non-probabilistic in nature. Recently, researchers have acknowledged this disparity and have been developing theory to address a broader class of uncertainties. This paper presents an experimental application of robust control design for a hybrid class of probabilistic and non-probabilistic parametric uncertainties. The experimental apparatus is based upon the classic inverted pendulum on a cart. The physical uncertainty is realized by a known additional lumped mass at an unknown location on the pendulum. This unknown location has the effect of substantially altering the nominal frequency and controllability of the nonlinear system, and in the limit has the capability to make the system neutrally stable and uncontrollable. Another uncertainty to be considered is a direct current motor parameter. The control design objective is to design a controller that satisfies stability, tracking error, control power, and transient behavior requirements for the largest range of parametric uncertainties. This paper presents an overview of the theory behind the robust control design methodology and the experimental results.

This paper concerns the estimation of the location (and properties) of damage in structures using Bayesian methods and Markov Chain Monte Carlo (MCMC). It is widely recognised that the consideration of uncertainty in structural dynamic systems may be essential, for example from an economic point of view (“Does is make sense to add expensive damping if it will only affect a small proportion of the vehicles we produce?”) or for critical safety purposes (“What is the risk of failure of an airplane engine due to bladed disk mistuning?”). The use of Bayesian methods appears to be a viable approach to obtain inferences about the parameters of such uncertain systems. Here we report on numerical experiments on the use of MCMC to locate a frequency dependent damage in a one-dimensional structure. Transfer function measurements subject to a Gaussian process measurement error are available. The particular structure of the resulting system matrices is then seen to have a special form which results in a semi-analytic solution method being available and a much reduced computational cost. We discuss the characteristics and efficiency of the Bayesian model and MCMC computation and highlight features in the analysis of structural dynamic systems such as higher-frequency multimodality.

Although there has been considerable work on the identification of hysteretic systems over the years, there has been comparatively little using discrete NARX or NARMAX models. One of the reasons for this may be that many of the common continuous-time models for hysteresis, like the Bouc-Wen model are nonlinear in the parameters and incorporate unmeasured states, and this makes a direct analytical discretisation somewhat opaque. Because NARX models are universal in the sense that they can model any input–output process, they can be applied directly without consideration of the hysteretic nature; however, if the polynomial form of NARX were to be used for a Bouc-Wen system, the result would be input-dependent because of the non-polynomial (indeed discontinuous) nature of the original model. The objective of the current paper is to investigate the use of NARX models for Bouc-Wen systems and to consider the use of non-polynomial basis functions as a potential means of alleviating any input-dependence. As the title suggests, the parameter estimation scheme adopted will be an evolutionary one based on Self-Adaptive Differential Evolution (SADE). The paper will present results for simulated data.

Following on from the first part of this short sequence, this paper will investigate the use of a Bayesian methodology for the identification of Bouc-Wen hysteretic systems by NARX models. The approach—based on Markov Chain Monte Carlo—offers a number of advantages over the evolutionary approach of the first paper. Among them are the ability to sample from the probability density functions of the parameters in order to develop nonparametric estimators and the possibility of selecting model terms in a principled manner. The paper will investigate the use of the Deviance Information Criterion (DIC) as a means of selecting model terms, specifically the special basis functions developed for the Bouc-Wen system in Part I. Results for simulated data will be given.

This paper develops a Bayesian methodology for uncertainty quantification and test resource allocation in multi-level systems. The various component, subsystem, and system-level models, the corresponding parameters, and the model errors are connected efficiently using a Bayes network. This provides a unified framework for uncertainty analysis where test data can be integrated along with computational models and simulations. The Bayes network is useful in two ways: (1) in a forward problem where the various sources of uncertainty are propagated through multiple levels of modeling to predict the overall uncertainty in the system response; and (2) in an inverse problem where the model parameters of multiple subsystems are calibrated simultaneously using test data. The calibration procedure leads to a decrease in the variance of the model parameters, and hence, in the overall system performance prediction. Then the Bayes network is used for test resource allocation where an optimization-based procedure is used to identify tests that can effectively reduce the uncertainty in the system model prediction are identified. The proposed methods are illustrated using two numerical examples: a multi-level structural dynamics problem and a multi-disciplinary thermally induced vibration problem.

Recent papers in the field of Finite Element Model (FEM) updating have highlighted the benefits of Bayesian techniques. The Bayesian approaches are designed to deal with the uncertainties associated with complex systems, which is the main problem in the development and updating of FEMs. This paper highlights the complexities and challenges of implementing any Bayesian method when the analysis involves a complicated structural dynamic model. In such systems an analytical Bayesian formulation might not be available in an analytic form; therefore this leads to the use of numerical methods, i.e. sampling methods. The main challenge then is to determine an efficient sampling of the model parameter space. In this paper, three sampling techniques, the Metropolis-Hastings (MH) algorithm, Slice Sampling and the Hybrid Monte Carlo (HMC) technique, are tested by updating a structural beam model. The efficiency and limitations of each technique is investigated when the FEM updating problem is implemented using the Bayesian Approach. Both MH and HMC techniques are found to perform better than the Slice sampling when Young’s modulus is chosen as the updating parameter. The HMC method gives better results than MH and Slice sampling techniques, when the area moment of inertias and section areas are updated.

This paper presents a model-based monitoring framework for the detection of fatigue-related crack damages in plate-type structures commonly seen in aluminum ship hulls. The monitoring framework involves vibration-based damage detection methodologies and finite element modeling of continuum plate structures. A Bayesian-based damage detection approach is adopted for locating probable damage areas. Identifying potential damage locations by evaluating all possible combinations of finite elements in the model is computationally infeasible. To reduce the search space and computational efforts, initial knowledge of the probable damage zones and a heuristic-based branch-and-bound scheme are systematically included in the Bayesian damage detection framework. In addition to an overview of the model-based monitoring framework, preliminary results from numerical simulations and experimental tests for a plate specimen with a welded stiffener are presented to illustrate the Bayesian damage detection approach and to demonstrate the potential application of the approach to detect fatigue cracks in metallic plates.

In the study of effect of mistuning on the dynamic response analysis of bladed disk systems, in literature, probabilistic methods are used. Conversely, in this paper, mistuning will be investigated by possibilistic analysis where mistuning parameters are modeled as fuzzy variables possessing possibility distributions. Fuzzy forced response characteristics of mistuned bladed disk systems are determined by mathematical basis of fuzzy sets. In order to do so, extension principle solution of fuzzy functions is used which overcomes the dependency issue problem observed on interval arithmetic solutions; hence, enhancing solution accuracy. Membership function distributions are digitized using alpha-cut methodology, slicing distributions to levels of confidence. Bounds of fuzzy variables in each and every level of confidence are determined using genetic optimization. Using this method, fuzzy forced response characteristics of a cyclically symmetric lumped parameter bladed disk model is determined. The possibilistic interpretation of mistuning is exemplified by determining the bounds of the possible maximum blade forced responses of the system for different orders of engine order excitation and by determining worst-possible case.

Parametric variation of large order finite element models is required for bracketing dynamic response uncertainties of complex systems. It is generally accepted that structural joint interface stiffness parameters are uncertain over parametric ranges that may span orders of magnitude. Therefore incremental sensitivity analysis techniques are not appropriate. Recent development of a vector-based sensitivity analysis technique that utilizes residual shape functions provides an efficient, accurate tool for vibration mode uncertainty studies. This paper introduces an extension of the sensitivity technique for dynamic response uncertainty studies. The key innovation lies in an efficient quasi-static stiffness sensitivity formulation that is closely related to mathematical principles employed in the vibration mode sensitivity technique. An end-to-end sensitivity formulation, appropriate for response to transient, steady-state and random environments, is defined for displacement, member load and stress recovery based on the mode displacement method with quasi-static residual vector augmentation. A simple illustrative example is provided to demonstrate accuracy and efficiency of the technique.

This paper presents an uncertainty analysis for a new methodology to predict reaction forces in weapon store connections of naval aircraft. The proposed methodology utilizes a strain-gage-based measurement technique in which a series of sensors are affixed to the connector and calibrated with a set of known loading configurations. The calibration matrix relating the measured strains to the loads can then be used to estimate unknown loads from measured strains. In this way, the system can be used to monitor the force through the connector when the aircraft is in service.

The primary objective in this work is to quantify the inherent uncertainties due to noise, miscalibration, or the estimation process itself in this loads estimation approach. First, the relationship between variance on the sensor measurements and the variance on the predicted force is characterized. A technique of sensor fusion called the weighted residual method is used to minimize the variance on force components that are the most important to the user of the system. The effect of thermal loading on the system is also explored via finite element simulation.

The motivation of this paper is to achieve a rapid condition assessment and identification for aero-space structures. For such a circumstance, transmissibility is a good vibration-domain feature for damage detection and localization, highlighting the sensitivity of the relative dynamics between any two interest points on the structure to many forms of damage (such as connection losses). With a quantified uncertainty model of transmissibility available, test results may be compared to baseline (undamaged) statistics, and statistical significance under certain confidence level may be found. Receiver operating characteristic curves suggest the optimized threshold of detection that balances detections and false alarms. In this paper, three such models are presented, and the aforementioned statistical identifications are validated through a plate structure with a single-edge clamped boundary condition. The benchmark test results show very good consistency with the model prediction and the robustness to different noise contamination situations.

Unbalance is among the most common mechanical faults in rotating machinery, and is of particular interest to the aviation industry. A state of the art machine fault simulator has been used in order to recreate a range of unbalance faults which have been studied in detail from the perspective of fault localisation. High fidelity finite element models have been created in NASTRAN NX and experimentally validated against results for the machine fault simulator. The applicability of such simulations is discussed from an IVHM perspective, along with the potential for such research to influence the development of future engine health management with respect to improved safety and maintenance.

This paper presents an application of a well-known model updating approach based on the inverse eigensensitivity method. The methodology is applied to an avionic equipment that supports different electronic instruments. In order to predict the operative dynamic behavior, a modal analysis is necessary, but high modal density and effects of nonlinearities of links, joints or rivets in the structure cannot allow an adequate correlation from test and numerical analysis. The structure is composed of two plates of aluminum connected by frame links with different sections and thickness to improve the stiffness on the borders and in the centre where the instruments are positioned. The different parts are connected with rivets. The target is the gap minimization between measured and computed modal parameters through the adjustment of a small number of physical parameters in the FE model. The original contribution of the paper consists of substructuring and numerical strategies of coupling and select eigenfrequencies and eigenvectors in the iterative parameter updating. The approach demonstrates that few iterations are required to get a good correlation between EMA and FEA.

The design of structures and machines must consider the restrictions imposed by the boundary conditions. Such conditions can be of dynamic nature, thus limiting the frequency ranges that the structure can operate. Among the different design tools available for dealing with dynamic restrictions, shape optimization is a way of deviating natural frequencies from problematic ranges during the design process. In this work, a cantilever plate is optmized aiming at desired natural frequencies. A coarse grid finite element model is correlated to results from experimental modal analysis, and the optimization is done with help of Nastran software. The main contribution is showing that an interpolation of the coarse grid optimization results leads to the solution of the fine grid optimization problem. Hence, it is possible to obtain the smooth geometry that optimizes natural frequencies in plates from coarse grid models, thus not requiring high computational costs.

To study the dynamic behavior of complex assembled structures consisting of several substructures and real joints connecting them, an updated finite element model of the associated structure is required. This paper presents a new technique to create an accurate updated finite element model of such structures. Given the fact that modal testing of real joints (such as bolt with some washers) are almost impossible; in this research the updated model of the assembled structures is constructed by utilizing parametric finite element model of the joint in conjunction with modal testing of the assembly structure and its substructures. In this paper, eigen-sensitivity method (used for characterizing cost function) and genetic algorithm (used for minimization scheme) are employed to update the assembled structure as well as substructures. A laboratory-scale unsymmetrical cross-beam is employed as the case study. The actual dynamic properties of the joint (including stiffness, mass and damping matrix) of this structure were estimated. The accuracy of the estimated parameters of the model was examined by comparison of the FRFs of the real assembled structure with the ones of the updated model. By achieving full compliance between these FRFs, the accuracy and efficiency of the proposed method, in a wide frequency range, is demonstrated.