Thesis (Ph.D.)--Boston University / This thesis presents methods for interpolating transfer functions of damped vibrating systems. Primary applications lie in the design and control of damped structures. The interpolations reduce the number of frequencies at which the transfer function must be computed or measured. The transfer functions are assumed to have impulse responses that are real-valued and causal, so a method is developed for constructing interpolations that implicitly satisfy these conditions. The method is applied to a particular choice of basis function that corresponds to a Fourier series in the time domain. Numerical results indicate that satisfaction of the causality condition increases the accuracy of the interpolation. A detailed investigation is made into interpolations for viscously damped systems, whose transfer functions are linear combinations of basis functions derived from the complex-valued eigenpairs of the system. Since the estimation of all eigenpairs is computationally expensive, a method is developed to estimate only those eigenpairs that significantly contribute to the transfer function in the specified frequency band. The method uses eigenvalues of the corresponding undamped system, which are much easier to compute, as starting guesses in an iterative algorithm. One advantage of the method is the assurance that it finds all eigenvalues in a specified region of the complex plane.

Problems in the control and identification of structural dynamic systems can lead to multimodal optimization problems, which are difficult to solve using classical gradient based methods. In this work, optimization problems pertaining to the vibration control of smart structures and the exploration of isospectral systems are addressed. Isospectral vibrating systems have identical natural frequencies, and existence of the isospectral systems proves non-uniqueness in system identification. For the smart structure problem, the optimal location(s) of collocated actuator(s)/sensor(s) and the optimal feedback gain matrix are obtained by maximizing the energy dissipated by the feedback control system. For the isospectral system problem, both discrete and continuous systems are considered. An error function is designed to calculate the error between the spectra of two distinct structural dynamic systems. For the discrete system, the Jacobi matrix, derived from the given system, is modified and the problem is posed as an optimization problem where the objective is to minimize the non-negative error function. Isospectral spring-mass systems are obtained. For the continuous system, finite element modeling is used and an error function is designed to calculate the error between the spectra of the uniform beam and the non-uniform beam. Non-uniform cantilever beams which are isospectral to a given uniform cantilever beam are obtained by minimizing the non-negative error function. Numerical studies reveal several isospectral systems, and optimal gain matrices and sensor/actuator locations for the smart structure. New evolutionary algorithms, which do not need genetic operators such as crossover and mutation, are used for the optimization. These algorithms are: Artificial bee colony (ABC) algorithm, Glowworm swarm optimization (GSO) algorithm, Firefly algorithm (FA) and Electromagnetism inspired optimization (EIO) algorithm.

Monte Carlo simulation techniques have emerged as widely accepted computing tools in tackling many problems in modern structural mechanics. Apart from developments in computational hardware, which have undoubtedly made simulation strategies practically feasible, the success of Monte Carlo simulations has also resulted equally significantly from the methodological developments aimed at controlling sampling variance of the Monte Carlo estimates. The study reported in the present thesis is aimed at developing and validating Monte Carlo simulation based approaches with inbuilt variance reduction capabilities to deal with problems of time variant reliability modeling, random vibration testing, and updating reliability models for statically/dynamically loaded instrumented structures. The relevant literature has been reviewed in Chapter 1. Time variant reliability analysis of randomly parametered and randomly driven non-linear vibrating systems has been tackled by combining two Monte Carlo variance reduction strategies into a single framework (Chapter 2). The first of these strategies is based on the application of the Girsanov transformation to account for the randomness in dynamic excitations and, the second approach is fashioned after the subset simulation method to deal with randomness in system parameters. A novel experimental test procedure to estimate the reliability of structural dynamical systems under excitations specified via random process models has been proposed (Chapter 3). The samples of random excitations to be used in the test are modified by the addition of an artificial control force. An unbiased estimator for the reliability is derived based on measured ensemble of responses under these modified inputs based on the tenets of Girsanov’s transformation. The study observes that an acceptable choice for the control force (that can reduce the sampling variance of the estimator) can be made solely based on experimental techniques. This permits the proposed procedure to be applied in the experimental study of time variant reliability of complex structural systems which are difficult to model mathematically. Illustrative example consists of a multi-axes shake table study on bending-torsion coupled, geometrically non-linear, five-storey frame under uni/bi-axial, non-stationary, random base excitation. The first order reliability method (FORM) and inverse FORM have been extended to handle the problem of updating reliability models for existing, statically loaded structures based on measured responses (Chapter 4). The proposed procedures are implemented by combining Matlab based reliability modules with finite element models residing on the Abaqus software. Numerical illustrations on linear and non-linear frames are presented. A solution strategy within the framework of Monte Carlo simulation based dynamic state estimation method and Girsanov’s transformation for variance reduction has been developed to tackle the problem of updating the reliability of instrumented structures based on measured response under random dynamic loading (Chapter 5). For linear Gaussian state space models, the solution is developed based on continuous version of the Kalman filter, while, for non-linear and (or) non-Gaussian state space models, bootstrap particle filters are adopted. Results from laboratory testing of an archetypal five storey bending-torsion coupled frame under seismic base motions form the basis of one of the illustrative examples. A set of three annexures contain details of numerical methods for discretizing Ito’s differential equations (Annexure 1), working of the Girsanov transformation through Kolmogorov’s equations (Annexure 2) and tools for interfacing Matlab and Abaqus codes (Annexure 3).

L’objectif principal de cette thèse consiste à étudier plusieurs thématiques : l’étude de l’observation et la commande d’un système de structure flexible et l’étude de la stabilité asymptotique d’un système d’échangeurs thermiques. Ce travail s’inscrit dans le domaine du contrôle des systèmes décrits par des équations aux dérivées partielles (EDP). On s’intéresse au système du corps-poutre en rotation dont la dynamique est physiquement non mesurable. On présente un observateur du type Luenberger de dimension infinie exponentiellement convergent afin d’estimer les variables d’état. L’observateur est valable pour une vitesse angulaire en temps variant autour d’une constante. La vitesse de convergence de l’observateur peut être accélérée en tenant compte d’une seconde étape de conception. La contribution principale de ce travail consiste à construire un simulateur fiable basé sur la méthode des éléments finis. Une étude numérique est effectuée pour le système avec la vitesse angulaire constante ou variante en fonction du temps. L’influence du choix de gain est examinée sur la vitesse de convergence de l’observateur. La robustesse de l’observateur est testée face à la mesure corrompue par du bruit. En mettant en cascade notre observateur et une loi de commande stabilisante par retour d’état, on souhaite obtenir une stabilisation globale du système. Des résultats numériques pertinents permettent de conjecturer la stabilité asymptotique du système en boucle fermée. Dans la seconde partie, l’étude est effectuée sur la stabilité exponentielle des systèmes d’échangeurs thermiques avec diffusion et sans diffusion. On établit la stabilité exponentielle du modèle avec diffusion dans un espace de Banach. Le taux de décroissance optimal du système est calculé pour le modèle avec diffusion. On prouve la stabilité exponentielle dans l’espace Lp pour le modèle sans diffusion. Le taux de décroissance n’est pas encore explicité dans ce dernier cas. / The main objective of this thesis consists to investigate the following themes : observation and control of a flexible structure system and asymptotic stability of a heat exchangers system. This work is placed in the field of the control of systems described by partial differential equations (PDEs). We consider a rotating body-beam system whose dynamics are not physically measurable. An infinite-dimensional exponentially convergent Luenberger-like observer is presented in order to estimate the state variables. The observer is also valid for a time-varying angular velocity around some constant. We can accelerate the decay rate of the observer by a second step design. The main contribution of this work consists in building a numerical simulator based on the finite element method (FEM). A numerical investigation is carried out for the system with constant or time-varying angular velocity. We examine the influence of the gain choice on the decay rate of the observer. The robustness of the observer is tested with the measurement corrupted by noise. By cascading our observer and a feedback control law, we wish to obtain a global stabilization of the rotating bodybeam system. The relevant numerical results make it possible for us to conjecture that the closed-loop system is locally asymptotically stable. We investigate the exponential stability of the heat exchangers systems with diffusion or without diffusion. We establish the exponential stability of the model with diffusion in a Banach space. Moreover, the optimal decay rate of the system is computed for the model with diffusion. We prove exponential stability in (C[0, 1])4 space for the model without diffusion. The optimal decay rate in the latter case is not yet found.