Dissertations / Theses on the topic 'Model updating method'

This thesis addresses the problem of identifying the modal properties of a system based only on measurements of the system responses. This situation is frequently encountered in structural dynamics and is particularly relevant for systems where the in-service excitation is not artificially reproducible. The inherent non-linearities in these systems mean that the modal properties estimated using traditional input/output techniques will be different to those exhibited in operation. A common example from the literature is an aircraft in flight where the modal properties are heavily influenced by the operating point, i.e. the combination of load, speed, altitude etc., at which the aircraft is travelling. The process of identifying the modal properties of systems in-service is called Operational Modal Analysis (OMA). Not knowing the input complicates the analysis. Most of the techniques in the literature overcome the lack of knowledge about the unmeasured excitations by assuming they are both spatially and frequentially white, i.e. of equal magnitude and with a flat autospectrum. This thesis presents a new technique for OMA which relaxes these constraints, requiring only that the system is excited by a so called cyclostationary input with a unique cyclic frequency, and that the log spectrum of the second order component of this input is frequentially smooth, as will be explained. Such systems include vehicles with internal combustion engines as the vibration from such an engine exhibits cyclostationary statistics. In this thesis, the technique is applied to a laboratory test rig and a passenger train both using an artificial input, and to a race car using the engine as the excitation. By combining cyclostationary signal processing and the concept of the cepstrum, the technique identifies the resonances and anti-resonances in the transfer functions between each response and the cyclostationary source. These resonances and antiresonances can be used to regenerate Frequency Response Functions (FRFs) and it is shown how the unknown scaling of the system can be recovered by employing finite element model updating in conjunction with this regeneration. In addition, the contribution made to model updating by the anti-resonances is also investigated. Finally, the potential of OMA to inform a model updating process is demonstrated using an experimental case study on a diesel railcar.

This thesis presents a novel method for the correlation of FEM results to experimental test results known as the "Load updating method." Specifically, the load updating method uses the math model from the FEM and the strains measured from experimental or flight test data as inputs and then predicts the loads in the FEM which would result in strains that would correlate best to the measured strains in the least squared sense. In this research, the load updating method is applied to the analysis of a complex frame structure whose validation is challenging due to the complex nature of its structural behavior, load distributions, and error derived from residual strains. A FEM created for this structure is used to generate strain data for thirty-two different load cases. These same thirty-two load cases are replicated in an experimental setup consisting of the frame, supporting structure, and thirty actuators which are used to load the frame according to the specifications for each of the thirty-two load conditions. A force-strain matrix is created from the math model in NASTRAN using unit loads which are separately applied to each load point in order to extract strain results for each of the locations of the seventy-four strain gages. The strain data from the structural test and the force-strain matrix is then input into a Matlab code which is created to perform the load updating method. This algorithm delivers a set of coefficients which in turn gives the updated loads. These loads are applied to the FEM and the strain values extracted for correlation to the strains from test data. It is found that the load updating method applied to this structure produces strains which correlate well to the experimental strain data. Although the loads found using the load updating method do not perfectly match those which are applied during the test, this error is primarily attributed to residual strains within the structure. In summary, the load updating method provides a way to predict loads which, when applied to the FEM, would result in strains that correlate best to the experimental strains. Ultimately, this method could prove especially useful for predicting loads in experimental and flight test structures and could aid greatly in the Federal Aviation Administration (FAA) certification process.<br>Master of Science

To obtain predictive modeling of a spacecraft, the author investigates the effects of adding cables to a simple structure with the goal of developing an understanding of how cables interacting with a structure. In this research, the author presents predictive and accurate modeling of a cable-harnessed structure by means of the Spectral Element Method (SEM). A double beam model is used to emulate a cable-harnessed structure. SEM modeling can define the location and the number of connections between the two beams in a convenient fashion. The proposed modeling is applied and compared with the conventional FEM. The modeling approach was compared to and validated by measurement data. The validated modeling was implemented to investigate the effect of the number of connections, of the spring stiffness of interconnections, and of mass portion of an attached cable. Damping has an important role in structural design because it reduces the dynamic response, thereby avoiding excessive deflection or stress, fatigue loads, and settling times. Experimental results with some specimens indicate a clear change of damping on the main structure with the inclusion of cable dynamics. The author investigated the modification of the damping of the host structure induced by various attached cables. The identification of a damping matrix is performed using measured data. The effect of the flexibility of a cable harness on damping is observed through experiments with various types of cables. The effect of the number of connections on damping is also investigated by changing the number of connections. Moreover, to overcome the sensitivity to noise in measured data of damping matrix identification approach, various methods are compared with a simulated lumped model and real test results. An improved damping matrix identification approach is proposed and can generate the unique damping matrix over the full frequency range of interest.<br>Ph. D.

Reliable flutter analysis of aircraft structures is a major requirement to determine safe flight envelops. Dynamically equivalent finite element model of an aircraft structure correlating well with experimental modal is a major requirement for a reliable flutter analysis. Currently available model updating techniques require enormous time and engineering work to achieve appropriate finite element models of aircraft structures. The method developed within the scope of this thesis work aims to remove important disadvantages of common model updating procedures. In doing this, the method starts with a simple finite element mesh obtained by connecting measurement points, used in the Ground Vibration Test of an aircraft structure, with 3 D Euler-Bernoulli beam elements. Initial estimates of the geometric and material properties are determined by solving structural identification equations derived from the mass and stiffness orthogonality of experimental modes. By using those initial estimates, an initial finite element model is constructed. Starting from this initial finite element model, structural identification equations are updated and solved iteratively by using experimental natural frequencies and eigenvectors of the v updated finite element model representing the same mode shapes with measured normal modes. Iterations are continued until eigen solution of the updated finite element model closely correlates with experimental modal data. The applicability of the method is illustrated on a scaled aircraft model and a real aircraft structure. The results are quite satisfactory but the method requires further improvements to achieve a much better correlation level in case of real aircraft structures.

Science and engineering problems frequently require solving a sequence of single linear systems or a sequence of dual linear systems. We develop algorithms that recycle Krylov subspaces and preconditioners from one system (or pair of systems) in the sequence to the next, leading to efficient solutions. Besides the benefit of only having to store few Lanczos vectors, using BiConjugate Gradients (BiCG) to solve dual linear systems may have application-specific advantages. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in the variational Monte Carlo (VMC) algorithm for electronic structure calculations -- leads to a quadratic error bound. Since one of our focus areas is sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods like CGS, BiCGSTAB, BiCGSTAB2, BiCGSTAB(l), QMR, and TFQMR. We develop a generalized bi-Lanczos algorithm, where the two matrices of the bi-Lanczos procedure are not each other's conjugate transpose but satisfy this relation over the generated Krylov subspaces. This is sufficient for a short term recurrence. Next, we derive an augmented bi-Lanczos algorithm with recycling and show that this algorithm is a special case of generalized bi-Lanczos. The Petrov-Galerkin approximation that includes recycling in the iteration leads to modified two-term recurrences for the solution and residual updates. We generalize and extend the framework of our recycling BiCG to CGS, BiCGSTAB and BiCGSTAB2. We perform extensive numerical experiments and analyze the generated recycle space. We test all of our recycling algorithms on a discretized partial differential equation (PDE) of convection-diffusion type. This PDE problem provides well-known test cases that are easy to analyze further. We use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction and in the VMC algorithm. For a model reduction problem, we show up to 70% savings in iterations, and we also demonstrate that solving the problem without recycling leads to (about) a 50% increase in runtime. Experiments with recycling BiCG for VMC gives promising results. We also present an algorithm that recycles preconditioners, leading to a dramatic reduction in the cost of VMC for large(r) systems. The main cost of the VMC method is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators, so that the cost of constructing Slater matrices in these systems is now linear in the number of particles. However, the cost of computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers. The main contribution here is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of the VMC algorithm from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors.<br>Ph. D.

This thesis investigates an alternative solution to deal with the civil structure vibration. Non-contact electromagnetic or Eddy current damping is selected as a score of vibration suppression. Electromagnetic damping relies on the interaction between a permanent magnet and conductor. An electromagnetic damper (EMD) is applied both to a laboratory footbridge structure and 6-storey model-scale aluminium moment resisting frame (AMRF). In this first study the EMD is connected in series with an electronic shunt circuit to construct an electromagnetic shunt damper (EMSD). A robust optimisation method is applied to develop the corresponding optimal design formula of the EMSD. The principle of an EMSD is to convert mechanical energy to electrical energy. Hence, the induced electromotive force (emf) is generated by electromagnetic induction. This emf induces an amount of shunt damping, which is fedback to the structure to achieve vibration suppression. It was found that when the impedance was applied, the shunt damping feature was of a similar nature to viscous dampers. In contrast, when an RLC (resistance-inductance-capacitance) circuit is connected, the shunt damping is analogous to a tuned mass damper. A second form of EMD is Eddy current damper (ECD), which relies on a geometrical arrangement of permanent magnets and conductors to produce damping forces. The vertical and horizontal orientation of the magnet, unidirectional and alternative pole projection and moving different direction of the conductor are investigated. A theoretical study involving the infinite boundary and finite boundary (the method of images current) is carried out to obtain an analytical calculation of the damping force. On the basis of this analysis, one type of ECD prototype was physically built. A performance test was carried out to determine the damping characteristics of the ECD, which agreed with the results of the numerical analysis. In addition, the ECD was applied to control the dynamics of the 6-storey AMRF. It was found that, the ECD can effectively increase system damping and have a satisfactory control effect.

The technological advancements of the last decades are making dynamic monitoring an efficient and widespread resource to investigate the safety and health of engineering structures. In the wake of these developments, the thesis proposes methodological tools supporting the seismic assessment of existing buildings through the use of ambient vibration tests. In this context, the literature highlights considerable room to broaden the ongoing research, especially regarding masonry buildings. The recent earthquakes, once again, highlighted the significant vulnerability of this structural typology as an important part of our built heritage, remarking the importance of risk mitigation strategies for the territorial scale. The thesis builds upon a simplified methodology recently proposed in the literature, conceived to assess the post-seismic serviceability of strategic buildings based on their operational modal parameters. The original contributions of the work pursue the theoretical and numerical validation of its basic simplifying assumptions, in structural modelling – such as the in-plane rigid behaving floor diaphragms – and seismic analysis – related to the nonlinear fundamental frequency variations induced by earthquakes. These strategies are commonly employed in the seismic assessment of existing buildings, but require further developments for masonry buildings. The novel proposal of the thesis takes advantage of ambient vibration data to establish direct and inverse mechanical problems in the frequency domain targeted at, first, qualitatively distinguishing between rigid and nonrigid behaving diaphragms and, second, quantitatively identifying their in-plane shear stiffness, mechanical feature playing a primary role in the seismic behaviour of masonry buildings. The application of these tools to real case studies points out their relevance in the updating and validation of structural models for seismic assessment purposes. In the light of these achievements, a model-based computational framework is proposed to develop frequency decay-damage control charts for masonry buildings, which exploit ambient vibration measurements for quick damage evaluations in post-earthquake scenarios. The results of the simulations, finally, highlight the generally conservative nature of ambient vibration-based simplified methodologies, confirming their suitability for the serviceability assessment of existing masonry buildings.

The need for dynamic positioning to function in ice-infested waters is growing as the offshore oil and gas industry enters the Arctic. The ice introduces great challenges, some of which can be resolved through proper ice management. This requires good knowledge of the surrounding ice-environment.This thesis deals with the question of achieving a good state estimator for a sea-ice model. The dynamic thermodynamic sea-ice model of Hilber III (1979) is implemented, and it is shown through simulations that it reacts in a realistic manner to varying air temperature. The states of this model are estimated with an ensemble Kalman filter, and it is shown that different states can be estimated very well by ensemble Kalman filters based on different measurement configurations. This implemented nonlinear sea-ice model and state estimator is meant to serve as a platform where methods designed to select measurement configurations best suited for state estimation can be tested.A suggestion for a method which chooses measurement configurations on-line is presented. The idea is that this method allows for different measurement configurations to be applied at different time steps, all based on which one that provides the best estimate at the current time. Unfortunately there was no time to implement this method and test it on the previously mentioned test platform; it must be kept in mind that it is merely a theoretical suggestion which must be further tested.

This thesis is mostly about the analysis of second order linear vibrating systems. The main purpose of this study is to extend methods which have previously been developed for either undamped or proportionally damped or classically damped systems to the general case. These methods are commonly used in aerospace industries. Ground vibration testing of aircraft is performed to identify the dynamic behaviour of the structure. New aircraft materials and joining methods - composite materials and/or novel adhesive bonding approaches in place of riveted or welded joints - cause higher levels of damping that have not been seen before in aircraft structure. Any change occurring in an original structure causes associated changes of the dynamic behaviour of the structure. Analytical finite element analyses and experimental modal testing have become essential tools for engineers. These techniques are used to determine the dynamic characteristics of mechanical structures. In Chapters 3 and 4, structural analysis and modal testing have been carried out an aircraft-like structure. Modal analysis techniques are used to extract modal data which are identified from a single column of the frequency response matrix. The proposed method is presented for fitting modal peaks one by one. This technique overcomes the difficulty due to the conventional methods which require a series of measured FRFs at different points of excitation. New methods presented in this thesis are developed and implemented initially for undamped systems in all cases. These ideas are subsequently extended for generally damped linear systems. The equations of motion of second order damped systems are represented in state space. These methods have been developed based on Lancaster Augmented Matrices (LAMs) and diagonalising structure preserving equivalences (DSPEs). In Chapter 5, new methods are developed for computing the derivatives of the non-zeros of the diagonalised system and the derivatives of the diagonalising SPEs with respect to modifications in the system matrices. These methods have provided a new approach to the evaluation and the understanding of eigenvalue and eigenvector derivatives. This approach resolves the quandary where eigenvalue and eigenvector derivatives become undefined when a pair of complex eigenvalues turns into a pair of real eigenvalues or vice-versa. They also have resolved when any one or more of the system matrices is singular. Numerical examples have illustrated the new methods and they have shown that the method results overcome certain difficulties of conventional methods. In Chapter 6, Möbius transformations are used to address a problem where the mass matrix is singular. Two new transformations are investigated called system spectral transformation SSTNQ and diagonalising spectral/similarity transformation DSTOQ. The transformation SSTNQ maps between matrices of two systems having the same short eigenvectors and their diagonalised system matrices. The transformation DSTOQ maps between two diagonalising SPE‟s having identical eigenvalues. Modal correlation methods are implemented to evaluate and quantify the differences between the output results from these techniques. Different cross orthogonality measures represent a class of methods which are recently performed as modal correlation for damped systems. In Chapter 7, cross orthogonality measures and mutual orthogonality measures are developed for undamped systems. These measures are defined in terms of real matrices - the diagonalising structure preserving equivalences (DSPEs). New methods are well developed for ill-conditioned system such that they work for all occasions and not only for cases where mass matrix is non-singular. Also a measure of the residuals is introduced which does not demand invertibility of diagonalised system matrices. Model updating methods are used in order to update models of systems by matching the output results from analytical system models with the experimentally obtained values. In Chapter 8, both cross-orthogonality measures and mutual-orthogonality measures are developed and used in the model updating of generally damped linear systems. Model updating based on the mutual orthogonality measures exhibits monotonic convergence from every starting position. That is to say, the ball of convergence has an infinite radius whereas updating procedures based on comparing eigenvectors exhibit a finite ball of convergence. Craig Bampton transformations are one of component methods which are used to reduce and decouple large structure systems. In Chapter 9 Craig Bampton transformations are developed for undamped systems and extended for damped second order systems in state space. Craig Bampton transformations are generalised and presented in SPEs forms. The two parts of the Craig Bampton transformations are extended in the full sizes of the substructure. The extended Craig Bampton transformations are modified to format each block of transformed substructure matrices as LAMs matrices format. This thesis generalises and develops the methods mentioned above and illustrates these concepts with an experimental modal test and some examples. The thesis also contains brief information about basic vibration properties of general linear structures and literature review relevant to this project.

Cette thèse traite de rupture localisée de structures construites en matériau composite hétérogène, comme le béton, à deux échelles différentes. Ces deux échelles sont connectées par le biais de la mise à l'échelle stochastique, où toute information obtenue à l'échelle méso est utilisée comme connaissance préalable à l'échelle macro. À l'échelle méso, le modèle de réseau est utilisé pour représenter la structure multiphasique du béton, à savoir le ciment et les granulats. L'élément de poutre représenté par une poutre Timoshenko 3D intégrée avec de fortes discontinuités assure un maillage complet indépendance de la propagation des fissures. La géométrie de la taille des agrégats est prise en accord avec la courbe EMPA et Fuller tandis que la distribution de Poisson est utilisée pour la distribution spatiale. Les propriétés des matériaux de chaque phase sont obtenues avec une distribution gaussienne qui prend en compte la zone de transition d'interface (ITZ) par l'affaiblissement du béton. À l'échelle macro, un modèle de plasticité multisurface est choisi qui prend en compte à la fois la contribution d'un écrouissage sous contrainte avec une règle d'écoulement non associative ainsi que des composants d'un modèle d'adoucissement de déformation pour un ensemble complet de différents modes de défaillance 3D. Le modèle de plasticité est représenté par le critère de rendement Drucker-Prager, avec une fonction potentielle plastique similaire régissant le comportement de durcissement tandis que le comportement de ramollissement des contraintes est représenté par le critère de St. Venant. La procédure d'identification du modèle macro-échelle est réalisée de manière séquentielle. En raison du fait que tous les ingrédients du modèle à l'échelle macro ont une interprétation physique, nous avons fait l'étalonnage des paramètres du matériau en fonction de l'étape particulière. Cette approche est utilisée pour la réduction du modèle du modèle méso-échelle au modèle macro-échelle où toutes les échelles sont considérées comme incertaines et un calcul de probabilité est effectué. Lorsque nous modélisons un matériau homogène, chaque paramètre inconnu du modèle réduit est modélisé comme une variable aléatoire tandis que pour un matériau hétérogène, ces paramètres de matériau sont décrits comme des champs aléatoires. Afin de faire des discrétisations appropriées, nous choisissons le raffinement du maillage de méthode p sur le domaine de probabilité et la méthode h sur le domaine spatial. Les sorties du modèle avancé sont construites en utilisant la méthode de Galerkin stochastique fournissant des sorties plus rapidement le modèle avancé complet. La procédure probabiliste d'identification est réalisée avec deux méthodes différentes basées sur le théorème de Bayes qui permet d'incorporer de nouvelles bservations générées dans un programme de chargement particulier. La première méthode Markov Chain Monte Carlo (MCMC) est identifiée comme mettant à jour la mesure, tandis que la deuxième méthode Polynomial Chaos Kalman Filter (PceKF) met à jour la fonction mesurable. Les aspects de mise en œuvre des modèles présentés sont donnés en détail ainsi que leur validation à travers les exemples numériques par rapport aux résultats expérimentaux ou par rapport aux références disponibles dans la littérature<br>This thesis deals with the localized failure for structures built of heterogeneous composite material, such as concrete, at two different scale. These two scale are latter connected through the stochastic upscaling, where any information obtained at meso-scale are used as prior knowledge at macro-scale. At meso scale, lattice model is used to represent the multi-phase structure of concrete, namely cement and aggregates. The beam element represented by 3D Timoshenko beam embedded with strong discontinuities ensures complete mesh independency of crack propagation. Geometry of aggregate size is taken in agreement with EMPA and Fuller curve while Poisson distribution is used for spatial distribution. Material properties of each phase is obtained with Gaussian distribution which takes into account the Interface Transition Zone (ITZ) through the weakening of concrete. At macro scale multisurface plasticity model is chosen that takes into account both the contribution of a strain hardening with non-associative flow rule as well as a strain softening model components for full set of different 3D failure modes. The plasticity model is represented with Drucker-Prager yield criterion, with similar plastic potential function governing hardening behavior while strain softening behavior is represented with St. Venant criterion. The identification procedure for macro-scale model is perfomed in sequential way. Due to the fact that all ingredients of macro-scale model have physical interpretation we made calibration of material parameters relevant to particular stage. This approach is latter used for model reduction from meso-scale model to macro-scale model where all scales are considered as uncertain and probability computation is performed. When we are modeling homogeneous material each unknown parameter of reduced model is modeled as a random variable while for heterogeneous material, these material parameters are described as random fields. In order to make appropriate discretizations we choose p-method mesh refinement over probability domain and h-method over spatial domain. The forward model outputs are constructed by using Stochastic Galerkin method providing outputs more quickly the the full forward model. The probabilistic procedure of identification is performed with two different methods based on Bayes’s theorem that allows incorporating new observation generated in a particular loading program. The first method Markov Chain Monte Carlo (MCMC) is identified as updating the measure, whereas the second method Polynomial Chaos Kalman Filter (PceKF) is updating the measurable function. The implementation aspects of presented models are given in full detail as well as their validation throughthe numerical examples against the experimental results or against the benchmarks available from literature

The research followed consist mainly on the use of passive methods for dynamic characterization of soil and building. Local Seismic Response of the construction site and Calibration of FE model of a reinforced concrete building have been performed. Passive seismic measurements have been acquired by two portable seismographs synchronized by each other with GPS antenna. Subsequently, structural verifications (global structural response) have been realized using once the response spectrum estimated from Local Seismic Response and once from technical standards. Finally, confronts between results have been realized at the aim to evaluate the importance of local seismic response of soil.

Among the experimental methods commonly used to define the behaviour of a full scale system, dynamic tests are the most complete and efficient procedures. A dynamic test is an experimental process, which would define a set of characteristic parameters of the dynamic behaviour of the system, such as natural frequencies of the structure, mode shapes and the corresponding modal damping values associated. An assessment of these modal characteristics can be used both to verify the theoretical assumptions of the project, to monitor the performance of the structural system during its operational use. The thesis is structured in the following chapters: The first introductive chapter recalls some basic notions of dynamics of structure, focusing the discussion on the problem of systems with multiply degrees of freedom (MDOF), which can represent a generic real system under study, when it is excited with harmonic force or in free vibration. The second chapter is entirely centred on to the problem of dynamic identification process of a structure, if it is subjected to an experimental test in forced vibrations. It first describes the construction of FRF through classical FFT of the recorded signal. A different method, also in the frequency domain, is subsequently introduced; it allows accurately to compute the FRF using the geometric characteristics of the ellipse that represents the direct input-output comparison. The two methods are compared and then the attention is focused on some advantages of the proposed methodology. The third chapter focuses on the study of real structures when they are subjected to experimental test, where the force is not known, like in an ambient or impact test. In this analysis we decided to use the CWT, which allows a simultaneous investigation in the time and frequency domain of a generic signal x(t). The CWT is first introduced to process free oscillations, with excellent results both in terms of frequencies, dampings and vibration modes. The application in the case of ambient vibrations defines accurate modal parameters of the system, although on the damping some important observations should be made. The fourth chapter is still on the problem of post processing data acquired after a vibration test, but this time through the application of discrete wavelet transform (DWT). In the first part the results obtained by the DWT are compared with those obtained by the application of CWT. Particular attention is given to the use of DWT as a tool for filtering the recorded signal, in fact in case of ambient vibrations the signals are often affected by the presence of a significant level of noise. The fifth chapter focuses on another important aspect of the identification process: the model updating. In this chapter, starting from the modal parameters obtained from some environmental vibration tests, performed by the University of Porto in 2008 and the University of Sheffild on the Humber Bridge in England, a FE model of the bridge is defined, in order to define what type of model is able to capture more accurately the real dynamic behaviour of the bridge. The sixth chapter outlines the necessary conclusions of the presented research. They concern the application of a method in the frequency domain in order to evaluate the modal parameters of a structure and its advantages, the advantages in applying a procedure based on the use of wavelet transforms in the process of identification in tests with unknown input and finally the problem of 3D modeling of systems with many degrees of freedom and with different types of uncertainty.

Among the experimental methods commonly used to define the behaviour of a full scale system, dynamic tests are the most complete and efficient procedures. A dynamic test is an experimental process, which would define a set of characteristic parameters of the dynamic behaviour of the system, such as natural frequencies of the structure, mode shapes and the corresponding modal damping values associated. An assessment of these modal characteristics can be used both to verify the theoretical assumptions of the project, to monitor the performance of the structural system during its operational use. The thesis is structured in the following chapters: The first introductive chapter recalls some basic notions of dynamics of structure, focusing the discussion on the problem of systems with multiply degrees of freedom (MDOF), which can represent a generic real system under study, when it is excited with harmonic force or in free vibration. The second chapter is entirely centred on to the problem of dynamic identification process of a structure, if it is subjected to an experimental test in forced vibrations. It first describes the construction of FRF through classical FFT of the recorded signal. A different method, also in the frequency domain, is subsequently introduced; it allows accurately to compute the FRF using the geometric characteristics of the ellipse that represents the direct input-output comparison. The two methods are compared and then the attention is focused on some advantages of the proposed methodology. The third chapter focuses on the study of real structures when they are subjected to experimental test, where the force is not known, like in an ambient or impact test. In this analysis we decided to use the CWT, which allows a simultaneous investigation in the time and frequency domain of a generic signal x(t). The CWT is first introduced to process free oscillations, with excellent results both in terms of frequencies, dampings and vibration modes. The application in the case of ambient vibrations defines accurate modal parameters of the system, although on the damping some important observations should be made. The fourth chapter is still on the problem of post processing data acquired after a vibration test, but this time through the application of discrete wavelet transform (DWT). In the first part the results obtained by the DWT are compared with those obtained by the application of CWT. Particular attention is given to the use of DWT as a tool for filtering the recorded signal, in fact in case of ambient vibrations the signals are often affected by the presence of a significant level of noise. The fifth chapter focuses on another important aspect of the identification process: the model updating. In this chapter, starting from the modal parameters obtained from some environmental vibration tests, performed by the University of Porto in 2008 and the University of Sheffild on the Humber Bridge in England, a FE model of the bridge is defined, in order to define what type of model is able to capture more accurately the real dynamic behaviour of the bridge. The sixth chapter outlines the necessary conclusions of the presented research. They concern the application of a method in the frequency domain in order to evaluate the modal parameters of a structure and its advantages, the advantages in applying a procedure based on the use of wavelet transforms in the process of identification in tests with unknown input and finally the problem of 3D modeling of systems with many degrees of freedom and with different types of uncertainty.

In the recent years, vibration-based structural damage identification has been subject of significant research in structural engineering. The basic idea of vibration-based methods is that damage induces mechanical properties changes that cause anomalies in the dynamic response of the structure, which measures allow to localize damage and its extension. Vibration measured data, such as frequencies and mode shapes, can be used in the Finite Element Model Updating in order to adjust structural parameters sensible at damage (e.g. Young’s Modulus). The novel aspect of this thesis is the introduction into the objective function of accurate measures of strains mode shapes, evaluated through FBG sensors. After a review of the relevant literature, the case of study, i.e. an irregular prestressed concrete beam destined for roofing of industrial structures, will be presented. The mathematical model was built through FE models, studying static and dynamic behaviour of the element. Another analytical model was developed, based on the ‘Ritz method’, in order to investigate the possible interaction between the RC beam and the steel supporting table used for testing. Experimental data, recorded through the contemporary use of different measurement techniques (optical fibers, accelerometers, LVDTs) were compared whit theoretical data, allowing to detect the best model, for which have been outlined the settings for the updating procedure.

Fractures and fracture networks govern the mechanical and fluid flow behavior of rock masses. Tunneling and other rock mechanics applications therefore require the characterization of rock fractures based on geological data. Field investigations produce only a limited amount of data from boreholes, outcrops, cut slopes, and geophysical surveys. In tunneling, the process of excavation creates a priceless opportunity to gather more data during construction. Typically, however, these data are not utilized due to the impedance of sampling and analysis on the flow of construction, and safety concerns with sampling within unlined tunnel sections. However, the use of this additional data would increase the overall safety, quality, and cost savings of tunneling. This study deals with several aspects of the above, with the goal of creating methods and tools to allow engineers and geologists to gather and analysis fracture data in tunnels without interrupting the excavation and without compromising safety. Distribution-independent trace density and mean trace length estimators are developed using principles of stereology. An optimization technique is developed utilizing Differential Evolution to infer fracture size and shape from trace data obtained on two or more nonparallel sampling planes. A method of producing nearly bias free empirical trace length CDF's is also introduced. These new methods and tools were validated using Monte Carlo simulations. A field study was conducted in an existing tunnel allowing the above methods and tools to be further validated and tested. A relational database was developed to aid in storage, retrieval, and analysis of field data. Fracture models were built and updated using fracture data from within the tunnel. Utilization of state of the art imaging techniques allowed for remote sampling and analysis, which were enhanced by the use of 3d visualization techniques.<br>Ph. D.

Questa tesi concerne lo sviluppo di un sistema di processamento di dati fisici e sperimentali per test ad elevata velocità di deformazione tramite la barra di Hopkinson. La barra di Hopkison è la tipologia di test più utilizzata comunemente per svolgere test dinamici, determinando le proprietà dei materiali caratterizzandone la risposta dinamica ad elevate velocità di deformazione. Utilizzando una barra di Hopkinson progettata e realizzata recentemente dal gruppo di costruzione di macchine della Università Politecnica delle Marche, dati sperimentali accurati e precisi cosi come immagini catturate ad alta velocità di provini in alluminio AA6061-T6 possono essere ottenuti in test svolti ad elevate velocità di deformazione. Provini di dimensioni diverse sia di trazione che di compressione dello stesso materiale sono stati utilizzati e i risultati analizzati col fine di calibrare le proprietà del materiale utilizzando il modello plastico constitutivo di Johnson-Cook. I coefficienti dei materiali secondo Johnson-Cook sono stati caratterizzati utilizzando approcci differenti. Un primo approccio coinvolge il cosiddetto metodo FEMU (Finite Element Method Updating) dove simulazioni numeriche sono ripetute cambiando di volta in volta i parametri del materiale sino a che la deviazione sperimentale e numerica non scenda sotto una soglia ritenuta accettabile. Il secondo metodo, denominato FastCAM, si basa sulla analisi digitale delle immagini ottenute grazie ad una camera con alto rateo di acquisizione immagini ottenute durante il test e calibrando il modello di Johnson-Cook con minimizzazioni analitiche senza la necessità di simulazioni ad elementi finiti. Inoltre, una terza tecnica inversa è stata implementata, la quale consiste nell'applicare il FEMU ma utilizzando una funzione di costo dove anche una differenza tra i profili dei provini numerici e quelli ottenuti sperimentalmente è considerata e minimizzata. In aggiunta, due provini in acciaio con una ben definita composizione (ottenuta da un centro di ricerca) sono stati testati. Per concludere, i vantaggi e svantaggi delle diverse tecniche sono stati valutati e confrontati tra loro.<br>This thesis work reports on a mainly accurate and precise physical and experimental data capture as well as data processing system for high strain rate tests using the split Hopkinson bar. The split Hopkinson bar is the most commonly used method of employing dynamic tests, determining material properties and characterize the dynamic response of materials at high strain rate. Using a recent well equipped split Hopkinson bar developed by mechanical engineering design department of Ancona University, accurate and precise experimental data as well as high-speed image capturing of aluminum alloy AA6061-T6 samples are obtained for high strain rate testing. Different sizes of tensile and compression samples of same material were used and data were studied in order to calibrate material properties using Johnson-Cook constitutive plasticity model. Johnson-Cook material coefficients have been characterized using different approaches. The first approach involves classical Finite Element Method Updating (FEMU) where numerical simulations are repeated with different material parameters until the deviation of experimental, as well as numerical data, falls below an acceptable threshold. The second method, commonly known as FastCAM, is based on the analysis of the digital images attained by a fast camera during the tests and calibrates the Johnson-Cook model by an analytical minimization procedure in the absent of FE simulation. Furthermore, the third inverse technique was also implemented, consisting of applying the FE model updating but using an enriched cost function, where also the mismatch between the numerical and acquired specimen shape profiles is included and minimized. Furthermore, two additional steel based materials with confidential alloy composition (cast by a research center) are tested. To finalized, the advantages and drawbacks of the different techniques are assessed and compared.

博士<br>國立臺灣大學<br>土木工程學研究所<br>97<br>Discrepancies always exist between the dynamic properties predicted by a finite element model and those measured directly from the structure. In this study, a direct updating method based on the orthogonality constraints is proposed for updating the mass and stiffness matrices of the structure first using a single set of modal data. This method hinges on replacement of the modal vector of concern by the modal matrix in computing the correction matrices to solve the problem of insufficient known conditions. Such a method is then extended to update the structural model for each of the first few sets of modal data that are experimentally made available. Two kinds of updating procedures are proposed, one is to conduct the model updating in a mode-by-mode manner and the other is in a simultaneous manner. In the numerical studies, it was demonstrated that for buildings of the shear type, the cantilever beam, continuous bridges and domes, the natural frequencies predicted by the updated model agree well with the measured ones for those modes that are experimentally made available, while the remaining modes remain basically untouched. In the end, a comparative study is performed for the proposed direct model updating method and the improved inverse eigensenstivity method (IIEM) proposed by Lin et al. (1995) for updating the mass and stiffness matrices of a structure based on the measured modal data. From the comparison study, it is demonstrated that the direct updating method presented herein is superior and more suitable for engineering applications. Since the proposed approach is simple, accurate and robust, it should be favored by engineers for practical applications.

博士<br>國立臺灣大學<br>工程科學及海洋工程學研究所<br>104<br>During structural design and analysis, an appropriate finite element analysis model is required to capture the dynamic characteristics of structures, such as modal parameters (natural frequencies and mode shapes). However, owing to the structural complexity, model variables related to geometry and material properties (for example, moment of inertia and Young’s modulus) should be reasonably simplified. In order to minimize the difference in dynamic characteristics between simulation and actual phenomena, these variables need to be verified through tests or modified if errors exist. In this study, an iterative method of the eigensensitivity matrix, which is an iterative method, is used for updating the structural analysis model. The eigensensitivity matrix is composed of first-order partial differentials of the model variables for natural frequency and mode shape of each modes, which were used to control the regulating of variables in iteration and convergence. The mass matrix and stiffness matrix of the structure were corrected by updating model variables of the analysis model through the iterative procedure, making the analysis results close to the results in the modal test. First, in the present study, a portal beam structure was used to illustrate the updating procedure for the finite element model. The results: (I)Regardless of the noise contamination in the test results (natural frequencies and mode shapes), the model updating effects considering the two convergence indexes (Frobenius norm and Correlation Coefficients) were good. Moreover, the influence of the contaminated test results on the updating effects can be evaluated using model variables as examples. (II)The modal kinetic energy (MKE) in the finite element analysis model was applied to evaluate the appropriate number of measuring points of the modal test; eigensensitivity matrix was used to analyze and obtain the importance order of degrees of freedom in test model; and measured data was set as updating targets, a better updating effect was obtained. (III) We have combined the “residual force vector method” and the “eigensensitivity method” and proposed a two-phase evaluation method to determine the location of elements to be modified in the finite element model. In this study, the test results of a “five-story scaled steel structure” from the National Center for Research on Earthquake Engineering are regarded as an equivalent “shear beam” analysis model to be updated. Based on the sensitivities of characteristic values of the five modes to “model variables,” choosing model variables to which the characteristic value of a mode are more sensitive has better updating effects than using the“same” model variables for each mode as the updating target. Compared with the analysis results of “3D finite element” structure model, the results of the equivalent shear beam structure model after the update show better agreement with the test results. Furthermore, the updated results of model variables meet the engineering requirements. Therefore, the equivalent “shear beam” structure model and its updated results of “model variables” can be used for structural design and analysis.

碩士<br>國立臺灣科技大學<br>營建工程系<br>94<br>This thesis presents a newly developed stochastic simulation for Bayesian model updating, model class selection and model averaging, named the transitional Markov chain Monte Carlo approach (TMCMC). The idea behind TMCMC is to avoid the problem of sampling from difficult posterior probability density functions (PDF) but sampling from a series of PDFs that converge to the posterior PDF and that are easier to sample. The TMCMC approach is based on Markov chain Monte Carlo (MCMC), while it is more versatile and robust than MCMC. It is shown that TMCMC is able to draw samples from some difficult PDFs, e.g. multi-modal and very peaked PDFs. The TMCMC approach can also estimate evidence of the chosen probabilistic model class conditioning on the measured data, a key component for Bayesian model class selection and model averaging. Three examples are used to demonstrate the effectiveness of the TMCMC approach in Bayesian model updating, model class selection and model averaging.

Numerical models are pervasive tools in science and engineering for simulation, design, and assessment of physical systems. In structural engineering, finite element (FE) models are extensively used to predict responses and estimate risk for built structures. While FE models attempt to exactly replicate the physics of their corresponding structures, discrepancies always exist between measured and model output responses. Discrepancies are related to aleatoric uncertainties, such as measurement noise, and epistemic uncertainties, such as modeling errors. Epistemic uncertainties indicate that the FE model may not fully represent the built structure, greatly limiting its utility for simulation and structural assessment. Model updating is used to reduce error between measurement and model-output responses through adjustment of uncertain FE model parameters, typically using data from structural vibration studies. However, the model updating problem is often ill-posed with more unknown parameters than available data, such that parameters cannot be uniquely inferred from the data. This dissertation focuses on two approaches to remedy ill-posedness in FE model updating: parametrization and regularization. Parametrization produces a reduced set of updating parameters to estimate, thereby improving posedness. An ideal parametrization should incorporate model uncertainties, effectively reduce errors, and use as few parameters as possible. This is a challenging task since a large number of candidate parametrizations are available in any model updating problem. To ameliorate this, three new parametrization techniques are proposed: improved parameter clustering with residual-based weighting, singular vector decomposition-based parametrization, and incremental reparametrization. All of these methods utilize local system sensitivity information, providing effective reduced-order parametrizations which incorporate FE model uncertainties. The other focus of this dissertation is regularization, which improves posedness by providing additional constraints on the updating problem, such as a minimum-norm parameter solution constraint. Optimal regularization is proposed for use in model updating to provide an optimal balance between residual reduction and parameter change minimization. This approach links computationally-efficient deterministic model updating with asymptotic Bayesian inference to provide regularization based on maximal model evidence. Estimates are also provided for uncertainties and model evidence, along with an interesting measure of parameter efficiency.

Currently, effective reservoir management systems play a very important part in exploiting reservoirs. Fully exploiting all the possible events for a petroleum reservoir is a challenge because of the infinite combinations of reservoir parameters. There is much unknown about the underlying reservoir model, which has many uncertain parameters. MCMC (Markov Chain Monte Carlo) is a more statistically rigorous sampling method, with a stronger theoretical base than other methods. The performance of the MCMC method on a high dimensional problem is a timely topic in the statistics field. This thesis suggests a way to quantify uncertainty for high dimensional problems by using the MCMC sampling process under the Bayesian frame. Based on the improved method, this thesis reports a new approach in the use of the continuous MCMC method for automatic history matching. The assimilation of the data in a continuous process is done sequentially rather than simultaneously. In addition, by doing a continuous process, the MCMC method becomes more applicable for the industry. Long periods of time to run just one realization will no longer be a big problem during the sampling process. In addition, newly observed data will be considered once it is available, leading to a better estimate. The PUNQ-S3 reservoir model is used to test two methods in this thesis. The methods are: STATIC (traditional) SIMULATION PROCESS and CONTINUOUS SIMULATION PROCESS. The continuous process provides continuously updated probabilistic forecasts of well and reservoir performance, accessible at any time. It can be used to optimize long-term reservoir performance at field scale.

Dissertação de mestrado integrado em Engenharia de Materiais<br>The Finite Element Method (FEM) is widely used to solve structural analysis problems. In this work, a novel Finite Element Model Updating methodology for static analysis is presented. The aim of the work is to improve the quality of the results using the Finite Element Updating techniques, by optimizing geometric parameters of the models and material properties in order to minimize deflection. Deflection can be minimized by increasing the Inertia moment of the section and/or Young modulus of the material. The Young modulus can be optimized by selecting an adequate material. In this work, material selection charts were used to determine the most reliable material. The selected material was then tested by tensile and extensometry tests to obtain Young modulus, Yield stress, and Poisson coefficient. The Inertia moment can be maximized by improving the geometry of the section, such as adding ribs or webs. A substantial improvement of the deflection can be achieved, but, in order to obtain the best results, optimization must be used. A MATLAB program was used to optimize the ANSYS models using a programming code. In order to know if the results are getting worse or better in relation to the previous iterations, an objective function was defined. The model is optimized when is not possible to further optimize the objective function.<br>O método dos elementos finitos (FEM) é amplamente utilizado para resolver problemas de análise estrutural estática. Neste trabalho é apresentada uma nova metodologia de melhoria de modelos de elementos finitos para análise estática. O objectivo do trabalho é melhorar a qualidade dos resultados utilizando as técnicas de melhoramento de elementos finitos, através da optimização de parâmetros geométricos dos modelos e propriedades do material, de modo a reduzir os deslocamentos. Os deslocamentos podem ser minimizados através do aumento do momento de Inércia da secção e/ou módulo de Young do material. O módulo de Young pode ser optimizado através da escolha de um material adequado. Neste trabalho foram usadas cartas de selecção de materiais para determinar o material mais adequado. Foram feitos ensaios de extensometria e de tracção ao material seleccionado para obter as propriedades relevantes do material: módulo de Young, tensão de Cedência e coeficiente de Poisson. O momento de inércia pode ser maximizado melhorando a geometria da secção com nervuras ou redes longitudinais. Uma melhoria substancial do deslocamento pode ser obtida, mas, de modo a obter os melhores resultados, tem de se usar optimização. O programa MATLAB foi utilizado para optimizar os modelos do ANSYS com um código de programação. De modo a saber se os resultados estão a melhorar ou a piorar em cada iteração, em relação às iterações anteriores, uma função objectivo foi definida. O modelo está optimizado quando não é possível optimizar mais a função objectivo.

碩士<br>國立成功大學<br>交通管理(科學)學系<br>85<br>The purpose of this research is to discuss the factors that affect the updating effect of mode-choice models. Using multinomial logit model, I compared and analyzed the updating effect of several issues, i.e., updating methods, sample size, utility function specification, data type, and alternative choice set. The intercity trip data between Tainan to Taipei werecollected for the empirical study. The empirical results showed: 1. Among the four updating methods, transfer scaling and joint context estimation had the best updating effect. However, the easier updating procedure of transfer scaling made it an effective and efficient updating method. 2. In the case of same alternative choice set, the sample size of 120 was large enough for model updating. When the updating samples used was 180, reestimating the model would yield better results than updating models. However, in the case of different alternative choice set, the sample size must be increased to 180 to get a good updating result. 3. The performance of utility function specification had great influence on the updating models. The specification effect was greater than updating effect. The travel behavior revealed from the exponential value of travel variables showed that the marginal utility of travel cost was diminishing while travel time was increasing. 4. The best updating combination between two data sets was original RP data plus updating RP or JP data. The ratio of 1:2 between RP and SP for the JP data worked well. 5. The updating methods could apply to added new alternative mode situation. However, the underlying assumption of IIA must be satisfied.

碩士<br>國立臺灣大學<br>土木工程學研究所<br>93<br>The greatest characteristics of a civil engineering structure lie in its uncertainty of construction and huge scale. To account for the uncertainty of constructions, an important problem is to update the analysis model so that it can produce results as close as possible to those observed in field. As the scale of a structure is usually huge, a simple and fast method for model updating is the one most desired by engineers. The purpose of this thesis is to develop finite element model updating methods that are suitable for practical use. To facilitate the derivation, the most often used shear buildings are taken as the example of study. In general, two categories of methods are used for model updating, i.e., the direct and parameter updating methods. The parameter updating methods were widely used because they can preserve the physical meanings of the updated matrices, but the procedure for determining the parameters is complicated. For buildings of the shear type, this paper presents a new way for selecting the parameters, by which the dominating parameters are preserved in the updating, while the procedure is greatly simplified. Furthermore, to circumvent the complicated procedures involved in parameter selection of the parameter updating methods, a direct model updating method is proposed. Compared with the existing ones, this method can be easily understood, while the physical meanings of the system matrices can be appreciated from the results of dynamic analyses. To demonstrate the applicability of the proposed updating methods, several typical examples that are likely to be encountered in practice were studied. It is confirmed that the proposed methods are workable.

博士<br>國立臺灣大學<br>土木工程學研究所<br>105<br>In the nonlinear structural response simulations, researchers often need to calibrate the parameters of a nonlinear material model with the experimental data by using the trial and error method. This can be very tedious and time consuming. In order to improve the calibration efficiency, the efficient method of parameter identification is desired. This study is to present two gradient-based parameter identification methods (GBM_MF and GBM_MU) for off-line model fitting and on-line model updating, respectively. The proposed GBM_MF method for off-line model fitting can assist the engineers and researchers, who are engaged in the nonlinear structural analyses, in model calibration. In addition, for the advanced hybrid simulation with on-line model updating, the proposed parameter identification method (GBM_MU) with innovative modification is presented in this dissertation. The shaking table test of a five-story BRB frame (BRBF) conducted in E-defense Japan in 2009 is utilized to verify the effectiveness of the proposed methods in the off-line and on-line applications. Compared with the measured responses, the results of off-line model fitting application can confirm that the proposed gradient-based method (GBM_MF) allows the efficient model calibration for the accurate simulation of the nonlinear responses of the BRBF. Moreover, the advantage of the on-line model updating with the proposed parameter identification method (GBM_MU) is demonstrated through the simulated hybrid tests. As a result, the proposed gradient-based methods of parameter identification for off-line model fitting and on-line model updating can advance the earthquake engineering research and practice.

碩士<br>國立交通大學<br>土木工程系所<br>102<br>A structural system can be functionally experience by structural and its corresponding modal parameters. Likewise, the parameters are the basis of describing a system. The way of model updating structural system parameters has been developed for years; most of studies are analyzed under a presupposition that mode shape is already known. Therefore, the errors and disadvantages are embedded in those studies. Our thesis is designed without measuring mode shape and provides an alternative way to acquire mode shape. Furthermore, the study is accompanied by a suitable way of revising structural system parameter in order to enhance the accuracy of the result. The method is based on two strategies: 1. Concentrating stuffiness matrix and mass matrix by Iterative method for dynamic condensation of structural matrices. 2. Improving the accuracy when measuring mode shape by setting modal parameters in a way of combining frequency information and Genetic Algorithms. Respectively, one reduces the amount of parameters and advances the search in high intensity; and the other, accompanied with the property of Iterative method for dynamic condensation of structural matrices and mode shape from the previous strategy, revises structural parameters by Cross-model cross-mode method for model updating. In conclusion, the effective value of revising structural system parameter can be executed by only a decisive frequency of the whole structure. This solves the disadvantages in traditional studies of measuring mode shape , and fulfills the purpose of revising structural system parameters.

碩士<br>國立高雄海洋科技大學<br>輪機工程研究所<br>102<br>This study developed a three-dimensional orthogonal piping system model construction method. When the layout (including positions and orientations) of equipment sets changed, the piping model connecting the equipment sets can be automatically updated, after using this method to find the new, corresponding equipment-piping relationships. 　　The orthogonal pipe coordinate method developed builds pipe models to connect equipment sets, with each pipe parallel with a coordinate axis and using the smallest number of elbows, based on the orientations and starting/ending positions of pipes at the inlet or outlet of the equipment sets. The main research work completed includes (a) treating the equipment set as a rigid body, use coordinate translation and rotation technique to calculate the new coordinates of equipment set after translation and rotation, (b) developing a method to connect the inlet or outlet of two equipment sets with orthogonal pipes, and (3) saving segments of pipe at the inlet or outlet of equipment sets for inserting valves prior to constructing orthogonal pipes.

The study of randomly excited nonlinear dynamical systems forms the focus of this thesis. We discuss two classes of problems: first, the characterization of nonlinear random response of the system before it comes into existence and, the second, assimilation of measured responses into the mathematical model of the system after the system comes into existence. The first class of problems constitutes forward problems while the latter belongs to the class of inverse problems. An outstanding feature of these problems is that they are almost always not amenable for exact solutions. We tackle in the present study these two classes of problems using Monte Carlo simulation tools in conjunction with Markov process theory, Bayesian model updating strategies, and particle filtering based dynamic state estimation methods. It is well recognized in literature that any successful application of Monte Carlo simulation methods to practical problems requires the simulation methods to be reinforced with effective means of controlling sampling variance. This can be achieved by incorporating any problem specific qualitative and (or) quantitative information that one might have about system behavior in formulating estimators for response quantities of interest. In the present thesis we outline two such approaches for variance reduction. The first of these approaches employs a substructuring scheme, which partitions the system states into two sets such that the probability distribution of the states in one of the sets conditioned on the other set become amenable for exact analytical solution. In the second approach, results from data based asymptotic extreme value analysis are employed to tackle problems of time variant reliability analysis and updating of this reliability. We exemplify in this thesis the proposed approaches for response estimation and model updating by considering wide ranging problems of interest in structural engineering, namely, nonlinear response and reliability analyses under stationary and (or) nonstationary random excitations, response sensitivity model updating, force identification, residual displacement analysis in instrumented inelastic structures under transient excitations, problems of dynamic state estimation in systems with local nonlinearities, and time variant reliability analysis and reliability model updating. We have organized the thesis into eight chapters and three appendices. A resume of contents of these chapters and appendices follows. In the first chapter we aim to provide an overview of mathematical tools which form the basis for investigations reported in the thesis. The starting point of the study is taken to be a set of coupled stochastic differential equations, which are obtained after discretizing spatial variables, typically, based on application of finite element methods. Accordingly, we provide a summary of the following topics: (a) Markov vector approach for characterizing time evolution of transition probability density functions, which includes the forward and backward Kolmogorov equations, (b) the equations governing the time evolution of response moments and first passage times, (c) numerical discretization of governing stochastic differential equation using Ito-Taylor’s expansion, (d) the partial differential equation governing the time evolution of transition probability density functions conditioned on measurements for the study of existing instrumented structures, (e) the time evolution of response moments conditioned on measurements based on governing equations in (d), and (f) functional recursions for evolution of multidimensional posterior probability density function and posterior filtering density function, when the time variable is also discretized. The objective of the description here is to provide an outline of the theoretical formulations within which the problems of response estimation and model updating are formulated in the subsequent chapters of the present thesis. We briefly state the class of problems, which are amenable for exact solutions. We also list in this chapter major text books, research monographs, and review papers relevant to the topics of nonlinear random vibration analysis and dynamic state estimation. In Chapter 2 we provide a review of literature on solutions of problems of response analysis and model updating in nonlinear dynamical systems. The main focus of the review is on Monte Carlo simulation based methods for tackling these problems. The review accordingly covers numerical methods for approximate solutions of Kolmogorov equations and associated moment equations, variance reduction in simulation based analysis of Markovian systems, dynamic state estimation methods based on Kalman filter and its variants, particle filtering, and variance reduction based on Rao-Blackwellization. In this review we chiefly cover papers that have contributed to the growth of the methodology. We also cover briefly, the efforts made in applying the ideas to structural engineering problems. Based on this review, we identify the problems of variance reduction using substructuring schemes and data based extreme value analysis and, their incorporation into response estimation and model updating strategies, as problems requiring further research attention. We also identify a range of problems where these tools could be applied. We consider the development of a sequential Monte Carlo scheme, which incorporates a substructuring strategy, for the analysis of nonlinear dynamical systems under random excitations in Chapter 3. The proposed substructuring ensures that a part of the system states conditioned on the remaining states becomes Gaussian distributed and is amenable for an exact analytical solution. The use of Monte Carlo simulations is subsequently limited for the analysis of the remaining system states. This clearly results in reduction in sampling variance since a part of the problem is tackled analytically in an exact manner. The successful performance of the proposed approach is illustrated by considering response analysis of a single degree of freedom nonlinear oscillator under random excitations. Arguments based on variance decomposition result and Rao-Blackwell theorems are presented to demonstrate that the proposed variance reduction indeed is effective. In Chapter 4, we modify the sequential Monte Carlo simulation strategy outlined in the preceding chapter to incorporate questions of dynamic state estimation when data on measured responses become available. Here too, the system states are partitioned into two groups such that the states in one group become Gaussian distributed when conditioned on the states in the other group. The conditioned Gaussian states are subsequently analyzed exactly using the Kalman filter and, this is interfaced with the analysis of the remaining states using sequential importance sampling based filtering strategy. The development of this combined Kalman and sequential importance sampling filtering method constitutes one of the novel elements of this study. The proposed strategy is validated by considering the problem of dynamic state estimation in linear single and multi-degree of freedom systems for which exact analytical solutions exist. In Chapter 5, we consider the application of the tools developed in Chapter 4 for a class of wide ranging problems in nonlinear random vibrations of existing systems. The nonlinear systems considered include single and multi-degree of freedom systems, systems with memoryless and hereditary nonlinearities, and stationary and nonstationary random excitations. The specific applications considered include nonlinear dynamic state estimation in systems with local nonlinearities, estimation of residual displacement in instrumented inelastic dynamical system under transient random excitations, response sensitivity model updating, and identification of transient seismic base motions based on measured responses in inelastic systems. Comparisons of solutions from the proposed substructuring scheme with corresponding results from direct application of particle filtering are made and a satisfactory mutual agreement is demonstrated. We consider next questions on time variant reliability analysis and corresponding model updating in Chapters 6 and 7, respectively. The research effort in these studies is focused on exploring the application of data based asymptotic extreme value analysis for problems on hand. Accordingly, we investigate reliability of nonlinear vibrating systems under stochastic excitations in Chapter 6 using a two-stage Monte Carlo simulation strategy. For systems with white noise excitation, the governing equations of motion are interpreted as a set of Ito stochastic differential equations. It is assumed that the probability distribution of the maximum over a specified time duration in the steady state response belongs to the basin of attraction of one of the classical asymptotic extreme value distributions. The first stage of the solution strategy consists of selection of the form of the extreme value distribution based on hypothesis testing, and, the next stage involves the estimation of parameters of the relevant extreme value distribution. Both these stages are implemented using data from limited Monte Carlo simulations of the system response. The proposed procedure is illustrated with examples of linear/nonlinear systems with single/multiple degrees of freedom driven by random excitations. The predictions from the proposed method are compared with the results from large scale Monte Carlo simulations, and also with the classical analytical results, when available, from the theory of out-crossing statistics. Applications of the proposed method for vibration data obtained from laboratory conditions are also discussed. In Chapter 7 we consider the problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations. Here we assume that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes’ theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplified by considering the reliability analysis of a few low dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on limited amount of pertinent Monte Carlo simulations. A summary of the contributions made and a few suggestions for future work are presented in Chapter 8. The thesis also contains three appendices. Appendix A provides details of the order 1.5 strong Taylor scheme that is extensively employed at several places in the thesis. The formulary pertaining to the bootstrap and sequential importance sampling particle filters is provided in Appendix B. Some of the results on characterizing conditional probability density functions that have been used in the development of the combined Kalman and sequential importance sampling filter in Chapter 4 are elaborated in Appendix C.

The study of randomly excited nonlinear dynamical systems forms the focus of this thesis. We discuss two classes of problems: first, the characterization of nonlinear random response of the system before it comes into existence and, the second, assimilation of measured responses into the mathematical model of the system after the system comes into existence. The first class of problems constitutes forward problems while the latter belongs to the class of inverse problems. An outstanding feature of these problems is that they are almost always not amenable for exact solutions. We tackle in the present study these two classes of problems using Monte Carlo simulation tools in conjunction with Markov process theory, Bayesian model updating strategies, and particle filtering based dynamic state estimation methods. It is well recognized in literature that any successful application of Monte Carlo simulation methods to practical problems requires the simulation methods to be reinforced with effective means of controlling sampling variance. This can be achieved by incorporating any problem specific qualitative and (or) quantitative information that one might have about system behavior in formulating estimators for response quantities of interest. In the present thesis we outline two such approaches for variance reduction. The first of these approaches employs a substructuring scheme, which partitions the system states into two sets such that the probability distribution of the states in one of the sets conditioned on the other set become amenable for exact analytical solution. In the second approach, results from data based asymptotic extreme value analysis are employed to tackle problems of time variant reliability analysis and updating of this reliability. We exemplify in this thesis the proposed approaches for response estimation and model updating by considering wide ranging problems of interest in structural engineering, namely, nonlinear response and reliability analyses under stationary and (or) nonstationary random excitations, response sensitivity model updating, force identification, residual displacement analysis in instrumented inelastic structures under transient excitations, problems of dynamic state estimation in systems with local nonlinearities, and time variant reliability analysis and reliability model updating. We have organized the thesis into eight chapters and three appendices. A resume of contents of these chapters and appendices follows. In the first chapter we aim to provide an overview of mathematical tools which form the basis for investigations reported in the thesis. The starting point of the study is taken to be a set of coupled stochastic differential equations, which are obtained after discretizing spatial variables, typically, based on application of finite element methods. Accordingly, we provide a summary of the following topics: (a) Markov vector approach for characterizing time evolution of transition probability density functions, which includes the forward and backward Kolmogorov equations, (b) the equations governing the time evolution of response moments and first passage times, (c) numerical discretization of governing stochastic differential equation using Ito-Taylor’s expansion, (d) the partial differential equation governing the time evolution of transition probability density functions conditioned on measurements for the study of existing instrumented structures, (e) the time evolution of response moments conditioned on measurements based on governing equations in (d), and (f) functional recursions for evolution of multidimensional posterior probability density function and posterior filtering density function, when the time variable is also discretized. The objective of the description here is to provide an outline of the theoretical formulations within which the problems of response estimation and model updating are formulated in the subsequent chapters of the present thesis. We briefly state the class of problems, which are amenable for exact solutions. We also list in this chapter major text books, research monographs, and review papers relevant to the topics of nonlinear random vibration analysis and dynamic state estimation. In Chapter 2 we provide a review of literature on solutions of problems of response analysis and model updating in nonlinear dynamical systems. The main focus of the review is on Monte Carlo simulation based methods for tackling these problems. The review accordingly covers numerical methods for approximate solutions of Kolmogorov equations and associated moment equations, variance reduction in simulation based analysis of Markovian systems, dynamic state estimation methods based on Kalman filter and its variants, particle filtering, and variance reduction based on Rao-Blackwellization. In this review we chiefly cover papers that have contributed to the growth of the methodology. We also cover briefly, the efforts made in applying the ideas to structural engineering problems. Based on this review, we identify the problems of variance reduction using substructuring schemes and data based extreme value analysis and, their incorporation into response estimation and model updating strategies, as problems requiring further research attention. We also identify a range of problems where these tools could be applied. We consider the development of a sequential Monte Carlo scheme, which incorporates a substructuring strategy, for the analysis of nonlinear dynamical systems under random excitations in Chapter 3. The proposed substructuring ensures that a part of the system states conditioned on the remaining states becomes Gaussian distributed and is amenable for an exact analytical solution. The use of Monte Carlo simulations is subsequently limited for the analysis of the remaining system states. This clearly results in reduction in sampling variance since a part of the problem is tackled analytically in an exact manner. The successful performance of the proposed approach is illustrated by considering response analysis of a single degree of freedom nonlinear oscillator under random excitations. Arguments based on variance decomposition result and Rao-Blackwell theorems are presented to demonstrate that the proposed variance reduction indeed is effective. In Chapter 4, we modify the sequential Monte Carlo simulation strategy outlined in the preceding chapter to incorporate questions of dynamic state estimation when data on measured responses become available. Here too, the system states are partitioned into two groups such that the states in one group become Gaussian distributed when conditioned on the states in the other group. The conditioned Gaussian states are subsequently analyzed exactly using the Kalman filter and, this is interfaced with the analysis of the remaining states using sequential importance sampling based filtering strategy. The development of this combined Kalman and sequential importance sampling filtering method constitutes one of the novel elements of this study. The proposed strategy is validated by considering the problem of dynamic state estimation in linear single and multi-degree of freedom systems for which exact analytical solutions exist. In Chapter 5, we consider the application of the tools developed in Chapter 4 for a class of wide ranging problems in nonlinear random vibrations of existing systems. The nonlinear systems considered include single and multi-degree of freedom systems, systems with memoryless and hereditary nonlinearities, and stationary and nonstationary random excitations. The specific applications considered include nonlinear dynamic state estimation in systems with local nonlinearities, estimation of residual displacement in instrumented inelastic dynamical system under transient random excitations, response sensitivity model updating, and identification of transient seismic base motions based on measured responses in inelastic systems. Comparisons of solutions from the proposed substructuring scheme with corresponding results from direct application of particle filtering are made and a satisfactory mutual agreement is demonstrated. We consider next questions on time variant reliability analysis and corresponding model updating in Chapters 6 and 7, respectively. The research effort in these studies is focused on exploring the application of data based asymptotic extreme value analysis for problems on hand. Accordingly, we investigate reliability of nonlinear vibrating systems under stochastic excitations in Chapter 6 using a two-stage Monte Carlo simulation strategy. For systems with white noise excitation, the governing equations of motion are interpreted as a set of Ito stochastic differential equations. It is assumed that the probability distribution of the maximum over a specified time duration in the steady state response belongs to the basin of attraction of one of the classical asymptotic extreme value distributions. The first stage of the solution strategy consists of selection of the form of the extreme value distribution based on hypothesis testing, and, the next stage involves the estimation of parameters of the relevant extreme value distribution. Both these stages are implemented using data from limited Monte Carlo simulations of the system response. The proposed procedure is illustrated with examples of linear/nonlinear systems with single/multiple degrees of freedom driven by random excitations. The predictions from the proposed method are compared with the results from large scale Monte Carlo simulations, and also with the classical analytical results, when available, from the theory of out-crossing statistics. Applications of the proposed method for vibration data obtained from laboratory conditions are also discussed. In Chapter 7 we consider the problem of time variant reliability analysis of existing structures subjected to stationary random dynamic excitations. Here we assume that samples of dynamic response of the structure, under the action of external excitations, have been measured at a set of sparse points on the structure. The utilization of these measurements in updating reliability models, postulated prior to making any measurements, is considered. This is achieved by using dynamic state estimation methods which combine results from Markov process theory and Bayes’ theorem. The uncertainties present in measurements as well as in the postulated model for the structural behaviour are accounted for. The samples of external excitations are taken to emanate from known stochastic models and allowance is made for ability (or lack of it) to measure the applied excitations. The future reliability of the structure is modeled using expected structural response conditioned on all the measurements made. This expected response is shown to have a time varying mean and a random component that can be treated as being weakly stationary. For linear systems, an approximate analytical solution for the problem of reliability model updating is obtained by combining theories of discrete Kalman filter and level crossing statistics. For the case of nonlinear systems, the problem is tackled by combining particle filtering strategies with data based extreme value analysis. The possibility of using conditional simulation strategies, when applied external actions are measured, is also considered. The proposed procedures are exemplified by considering the reliability analysis of a few low dimensional dynamical systems based on synthetically generated measurement data. The performance of the procedures developed is also assessed based on limited amount of pertinent Monte Carlo simulations. A summary of the contributions made and a few suggestions for future work are presented in Chapter 8. The thesis also contains three appendices. Appendix A provides details of the order 1.5 strong Taylor scheme that is extensively employed at several places in the thesis. The formulary pertaining to the bootstrap and sequential importance sampling particle filters is provided in Appendix B. Some of the results on characterizing conditional probability density functions that have been used in the development of the combined Kalman and sequential importance sampling filter in Chapter 4 are elaborated in Appendix C.

Mestrado em Gestão (MBA).<br>Neste trabalho é apresentado o processo de construção dos modelos input-output para os Açores, ilha de São Miguel e ilha Terceira, relativo ao ano de 2001. Após algumas considerações teóricas, são descritas as três fases de implementação dos modelos. Primeiro, o quadro de empregos disponível é ampliado para aumentar o nível de detalhe em alguns ramos de actividade. Segundo, o modelo input-output dos Açores é derivado a partir do quadro de empregos e da matriz de produção. Finalmente, é proposto um método de regionalização, sendo aplicado ao caso Açores, obtendo-se os modelos input-output para as ilhas de São Miguel e Terceira.<br>ABSTRACT: In this work we present the building process of the input-output models for Açores and the islands of São Miguel and Terceira, year 2001. After some theoretical considerations, the three different stages of the model building process are presented. First, the available use table is expanded in order to achieve a higher detail in certain industries. Second, the input-output model for Açores is derived from the use and supply table. Finally, a regionalization methodology is proposed and applied to the case of Açores, having as a result the models for São Miguel and Terceira.