[pt] Em geral, o movimento de corpos se dá em ambiente com barreiras podendo ocorrer colisões. Para que seja possível fazer previsões da dinâmica é necessário saber o que acontece quando um corpo colide. O problema é portanto: conhecida a dinâmica do corpo pré-colisão e as propriedades dos corpos que colidem, prever a dinâmica pós-colisão. Os primeiros trabalhos publicados sobre o assunto datam de 1668 e, até 1984,os modelos existentes pareciam satisfatórios. Porém a aplicação de um desses modelos a um problema simples apresentou geração de energia. Desde então, um grande número de trabalhos tem aparecido na literatura. A tese trata de problema de colisões planas, discute criticamente os modelos da literatura comparando-os através de uma generalização por nós desenvolvida e propõe um novo modelo que engloba alguns dos modelos da literatura. Mostramos os principais problemas de alguns dos modelos e discutimos questões de existência e unicidade. Simulações feitas num programa por nós desenvolvido são apresentadas e ajudam a entender a influência dos coeficientes constitutivos. A validação dos resultados é realizada através de resultados experimentais colhidos da literatura. / [en] In general the motion of a body takes place in a confined environment and collision of the body with the containing wall is possible. To predict the dynamics of a body in this conditions one must know what happens in a collision. The problem is then: known the pre-collision dynamics of the body and the properties of the body and the wall one want to predict the post-collision dynamics. This problem is quite old and it appeared in the literature in 1968. Up to 1984 it seemed that Newton model was enough to solve the problem. But it was found that this was not the case and a renewed interest in the problem appeared. This thesis treats the problem of plan collisions of rigid bodies and tries to classify the differents models and to compare them. A new model is presented and old results are shown in a new framework.

[pt] Este trabalho tem por objetivo apresentar uma introdução à dinâmica de sistemas de multicorpos compostos por partes rígidas e flexíveis, através da exposição das diversas etapas: Modelagem, Simulação e Controle. A modelagem de sistemas de multicorpos é apresentada, atentando para os problemas de representação de rotações, caracterização de deformações dos corpos flexíveis e manipulação simbólica para formulação das equações do movimento. A parametrização de rotações é apresentada utilizando parâmetros clássicos como ângulos de Euler e Bryant, parâmetros de Euler e Rodrigues, assim como, vetor rotação, vetor rotação conforme e quaternios. O problema de singularidade das parametrizações é estudado, através da comparação de diferentes parametrizações. Para a caracterização de deformações dos corpos flexíveis é apresentado o método de modos supostos. A formulação das equações do movimento é apresentada utilizando as equações de Lagrange e Maggi-Kane. O toolbox de manipulação simbólica do MATLAB é utilizado para derivar as equações do movimento. O controle linear de sistemas de multicorpos é apresentado utilizando a representação no espaço de estados. Duas metodologias de projeto de controle são apresentadas: controle via imposição de pólos e controle ótimo. A simulação de sistemas de multicorpos é apresentada por meio de alguns exemplos ilustrativos da dinâmica e do controle de multicorpos, atentando para a escolha do método de integração. Todas as etapas são realizadas no ambiente do MATLAB, utilizando suas funções de manipulação simbólica para a modelagem, suas funções de linearização e controle para o controle e seus algoritmos de integração e funções gráficas para a simulação. / [en] This work intends to present an introduction to the Dynamics of Multibody Systems, with rigid and flexible bodies, by presenting the following stages: Modelization, Control and Simulation. The modelization of multibody systems is presented, exploring finite rotation parametrization, description of deformation of the flexible bodies and symbolic derivation of the equations of motion. Finite rotations parametrization is presented using classical systems of parametrization such as Euler`s and Bryant`s angles, Euler`s and Rodrigues` parameters and conformal rotation vector, rotation vector and quaternions. The problem of singularity of parametrization is studied by the comparison of the various systems of parametrization. The method of assumed modes is presented to describe the deformation of flexible bodies. The formulation of the equations of motion is done using Lagrange`s and Maggi-Kane`s equations. The equations of motion are derived using the MATLAB`s Symbolic Math Toolbox. The state-space linear control of multibody systems is presented. Two different methods are presented to design the control system: eigenvalues imposition and optimal control. The simulation of some numerical examples of multibody systems is presented. An analysis of the integration methods is done. All the computations are done in MATLAB, using the Symbolic Math Toolbox functions to the modelization, the Control Toolbox to the control and the OdeSuite to the integration of the equations of motion.

<p>I dagens spelindustri baseras nästan samtliga 3D-spel på fysiksimuleringar med stela kroppar (rigid bodies). Examensarbetet undersöker vilka alternativa modeller som finns för att simulera mjuka deformerbara objekt, deformerbara i det avseendet att de kan ändra form och inte nödvändigtvis att de kan gå sönder i flera bitar. Rapporten inleds med en undersökande del som tar upp några existerande metoder för att hantera dynamiken inom ett mjukt objekt för att sedan beskriva en metod i detalj som dessutom implementeras i ett kodbibliotek. Ett deformerbart objekt är inte så spännande om det inte finns något sätt att deformera det på, därför undersöks även hur kollisionshantering kan gå till. Även här har rapporten först en undersökande del för att sedan beskriva en specifik metod i detalj som implementeras i kodbiblioteket. Examensarbetet resulterar i slutändan i en grundläggande interaktiv simuleringsmiljö för mjuka deformerbara objekt i form av ett kodbibliotek.</p>

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Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο. / Σκοπός της εργασίας είναι η παρουσίαση των εξισώσεων κίνησης του στερεού σώματος και η μελέτη δύο σημαντικών επιλύσιμων περιπτώσεων κίνησης στρόβου (Lagrange, Euler) . Στo πρώτο κεφάλαιο περιγράφουμε την κίνηση ενός στερεού σώματος χρησιμοποιώντας την ομάδα στροφών. Αποδεικνύουμε το θεώρημα Chasles το οποίο μας δείχνει πως η μετακίνηση ενός στερεού μπορεί να αποσυντεθεί σε περιστροφή γύρω από έναν άξονα και μεταφορά πάνω σε αυτόν. Στη συνέχεια σκοπός μας είναι η κατανόηση της γωνιακής ταχύτητας ενός στερεού σώματος. Σημαντικό ρόλο σε αυτή την πορεία παίζει τόσο το αδρανειακό όσο και το ενσωματωμένο στο στερεό σύστημα αναφοράς. Έπειτα δίνονται οι ορισμοί της ενέργειας, της στροφορμής, της ροπής και οι εκφράσεις τους συναρτήσει γνωστών πλέον εννοιών από τα προηγούμενα. Το κεφάλαιο ολοκληρώνεται με την Δυναμική που έχει ως αντικείμενο μελέτης και έρευνας τη κίνηση των σωμάτων υπό την επίδραση δυνάμεων, και καταλήγει στην παρουσίαση των εξισώσεων Euler. Στο δεύτερο κεφάλαιο στρέφουμε το ενδιαφέρον μας στις εφαρμογές και παρουσιάζουμε την επίλυση δύο σημαντικών προβλημάτων της μηχανικής: η κίνηση ενός συμμετρικού στρόβου που κινείται υπό την επίδραση του βάρους του έχοντας ένα σταθερό σημείο (ο στρόβος του Lagrange) και η κίνηση ενός στερεού που κινείται χωρίς την επίδραση εξωτερικών ροπών (ο στρόβος του Euler). Οι λύσεις εκφράζονται μέσω Ελλειπτικών Συναρτήσεων. Τέλος, στο τρίτο κεφάλαιο παρατίθενται σχόλια στις εργασίες των Holmes - Marsden και των Heijden - Yagasaki που αφορούν την ύπαρξη χαοτικής συμπεριφοράς στην διαταραγμένη περίπτωση Lagrange, που αναφέρεται σε σχεδόν συμμετρικό στρόβο.

This work focuses on the simulation of fluids as they transition between a solid and a liquid state, and as they interact with rigid bodies in a realistic fashion. There is an underlying theme to my work that I did not recognize until I examined my body of research as a whole. The equations of motion that are generally considered appropriate only for liquids or gas can also be used to model solids. Without adding extra constraints, one can model a solid simply as a fluid with a high viscosity. Admittedly, this representation will only get you so far, but this simple representation can create some very nice animations of objects that start as solids, and then melt into liquid over time. Another way to represent solids with the fluid equations is to add extra constraints to the equations. I use this representation in the parts of this work that focus on the two-way coupling of liquids with rigid bodies. The coupling affects both how the liquid moves the rigid bodies, and how the rigid bodies in turn affect the motion of the fluid. There are three components that are needed to allow solids and fluids to interact: a rigid body solver, a fluid solver, and a mechanism for the coupling of the two solvers. The fluid solver used in this work was presented in [8]. This Melting and Flowing solver is a fast and stable system for animating materials that melt, flow, and solidify. Examples of realworld materials that exhibit these phenomena include melting candles, lava flow, the hardening of cement, icicle formation, and limestone deposition. Key to this fluid solver is the idea that we can plausibly simulate such phenomena by simply varying the viscosity inside a standard fluid solver, treating solid and nearly-solid materials as very high viscosity fluids. The computational method modifies the Marker-And-Cell algorithm [99] in order to rapidly simulate fluids with variable and arbitrarily high viscosity. The modifications allow the viscosity of the material to change in space and time according to variation in temperature, water content, or any other spatial variable. This in turn allows different locations in the same continuous material to exhibit states ranging from the absolute rigidity or slight bending of hardened wax to the splashing and sloshing of water. The coupling that ties together the rigid body and fluid solvers was presented in [7], and is known as the Rigid Fluid method. It is a technique for animating the interplay between rigid bodies and viscous incompressible fluid with free surfaces. Distributed Lagrange multipliers are used to ensure two-way coupling that generates realistic motion for both the solid objects and the fluid as they interact with one another. The rigid fluid method is so named because the simulator treats the rigid objects as if they were made of fluid. The rigidity of such an object is maintained by identifying the region of the velocity field that is inside the object and constraining those velocities to be rigid body motion. The rigid fluid method is straightforward to implement, incurs very little computational overhead, and can be added as a bridge between current fluid simulators and rigid body solvers. Many solid objects of different densities (e.g., wood or lead) can be combined in the same animation. The rigid body solver used in this work is the impulse based solver, with shock propagation introduced by Guendelman et al. in [36]. The rigid body solver allows for collisions ranging from completely elastic, where an object can bounce around forever without loss of energy, to completely inelastic where all energy is spent in the collision. Static and dynamic frictional forces are also incorporated. The details of this rigid body solver will not be discussed, but the small changes needed to couple this solver to interact with fluid will be. When simulating fluids, the fluid-air interface (free surface) is an important part of the simulation. In [8], the free surface is modelled by a set of marker particles, and after running a simulation we create detailed polygonal models of the fluid by splatting particles into a volumetric grid and then render these models using ray tracing with sub-surface scattering. In [7], I model the free surface with a particle level set technique [14]. The surface is then rendered by first extracting a triangulated surface from the level set and then ray tracing that surface with the Persistence of Vision Raytracer (http://povray.org).

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[pt] Um problema bem conhecido da Mecânica Clássica consiste no estudo do movimento de um corpo no espaço, especialmente quando o problema é conservativo e livre de forças. Este trabalho utiliza ferramentas modernas da Dinâmica para interpretar os movimentos com grande amplitude, ultrapassando os limites de estabilidade obtidos pelo conceito de Lyapunov. O problema da singularidade numérica que ocorre utilizando-se ângulos de Cardan pode ser eliminado com a descrição por quatérnios. A versatilidade dos quatérnios na Dinâmica é discutida, assim como a dificuldade do estudo do movimento próximo aos pontos de singularidade usando ângulos cardânicos. Enfatiza-se a influência dos momentos principais de inércia na estabilidade do movimento. Obtém-se um valor numérico da energia cinética mínima necessária para que o movimento atravesse o limite de estabilidade. O giroscópio Magnus é um instrumento educacional muito conveniente no estudo do movimento de um corpo livre no espaço. O rotor desse giroscópio representa um corpo em uma suspensão cardânica com anel externo e interno, o que dá ao corpo a liberdade de movimento necessária. Desenvolve-se nesta tese o modelo matemático de um corpo em suspensão cardânica, incluindo-se o atrito existente entre os componentes do sistema mecânico (além de considerar as inércias do rotor e dos anéis ou quadros). O problema da singularidade na descrição com rotações seqüenciais, que existe no caso de um corpo no espaço, é eliminado quando se considera a inércia dos quadros. Estuda-se o comportamento do giroscópio ao longo do tempo, sem outras restrições, considerando a perda de energia cinética devido ao atrito. Avalia-se também como a mudança dos momentos de inércia influencia a estabilidade do movimento do sistema. / [en] A well known conservative problem in Classical Mechanics consists in the force free motion of a body in space. This work uses modern tools from Dynamics to interpret great amplitude movements crossing the limits of stability in the concept of Lyapunov. The numerical singularity that arises with the use of Cadan angles can be eliminated with quaternion representation. The versatility of quaternions in Dynamics is discussed, as well as the difficulty in investigating the motion near to singularity points when using cardanic angles. The influence of the principal moments of inertia on the stability of the motion is discussed. A numerical value for the minimal kinetic energy to cross the stability border is obtained. The Magnus Gyroscope is an educational instrument, very convenient in the study of the motion of a free body in outer space. The rotor of this gyroscope represents the body on a cardanic suspension with outer and inner ring, which gives the body the necessary freedom of motion. In this work a mathematical model of a body in cardanic suspension is generated, including friction between gimbals and rotor (besides considering the inertia of these components). The singularity problem in the free body solution is eliminated when the inertia of the gimbals is considered. Long term behavior of the unrestricted motion is investigated, considering the loss of kinetic energy due to friction. It is also shown how the change of moments of inertia due to the gimbals influences the stability of the motion of system.