Este trabalho apresenta uma análise paramétrica da reposta dinâmica de colunas de perfuração de poços de petróleo com controle proporcional-integral de velocidade não colocalizado. A operação de perfuração de poços de petróleo e gás em águas profundas consiste na abertura de poços em solo rochoso através de uma broca cuja rotação é controlada por uma mesa rotativa na superfície. O torque imposto pela mesa é transmitido à broca por meio de uma coluna de perfuração. Particularmente no caso de perfuração em águas profundas, as colunas de perfuração podem ser muito extensas e, portanto, bastante flexíveis. As vibrações ocasionadas pela grande flexibilidade das colunas de perfuração são as principais responsáveis por falhas no processo de perfuração. Em particular, o fenômeno não-linear conhecido como stick-slip e relacionado às vibrações torcionais da coluna de perfuração, faz com que um sistema de controle projetado para manter a velocidade da mesa constante dê origem a grandes oscilações na velocidade da broca. Na prática, este fenômeno é amplificado pela inerente não-linearidade do contato entre broca e formação rochosa e pela forte não colocalização entre mesa rotativa e broca. Este trabalho tem por principal objetivo realizar uma análise paramétrica da dinâmica do processo de perfuração, usando um modelo de dois graus de liberdade para representar o conjunto mesa rotativa, coluna de perfuração e broca, para identificar condições nas quais uma lei de controle simples do tipo linear proporcional-integral pode fornecer um desempenho de perfuração estável e satisfatório. / This paper presents a parametric analysis of the dynamics of oilwell drillstrings with non-collocated proportional-integral velocity control. The drilling operation for oil and gas in deep waters consists of opening wells in rocky ground formation by a drill, whose angular speed is controlled by a rotary table at the surface. The torque applied by the table is transmitted to the drill-bit through the drillstring. Particularly in the deepwater drilling case, the drillstring can be very long and therefore very flexible. The vibrations caused by the great flexibility of drilling columns are mainly responsible for the failures in the drilling process. In particular, the nonlinear phenomenon known as stick-slip and related to the torsional vibration of the drillstring, makes that a control system designed to maintain a constant angular velocity at the table yield large variations at the drill-bit angular velocity. In practice, this phenomenon is amplified by the inherent nonlinearity of the contact between drill bit and rock formation and by the strong non-colocalization between rotary table and drill-bit. The main objective of this work is to perform a parametric analysis of the dynamics of the drilling process, using a two degrees of freedom model in order to represent the rotary table assembly, the drilling column and drill-bit, to identify conditions in which a simple control law, such as a linear proportional-integral velocity control, can provide a stable and satisfactory drilling performance.

Les pieds d’aubes de soufflantes de turboréacteurs font face à des sollicitations de type fretting. Il en résulte deux types d'endommagements: (i) l'amorçage et la propagation de fissures, (ii) l'usure des surfaces en contact. OBJECTIF: Afin de fournir les outils permettant de répondre à la problématique industrielle, une méthode semi-analytique permettant de traiter des problèmes de contacts élasto-plastiques revêtus et/ou hétérogènes est développé à partir d'éléments existants et de solutions analytiques et numériques novatrices. METHODE: La structure est simplifiée en supposant un contact entre deux massifs élastiques semi-infinis. Des solutions analytiques donnant pour: + les contributions élémentaires de chargements normaux et tangentiels constants sur une surface rectangulaire + les contributions élémentaires de déformations plastiques supposées constantes sur un volume parallélépipédique + les contributions élémentaires de déformations d’incompatibilité liées à un problème hétérogène (inclusions, revêtements, endommagement,...) supposées constantes sur un volume parallélépipédique. >> les déplacements en surface ou les contraintes dans le volume. Les déplacements en surface ou les contraintes dans le volume sont alors exprimés en utilisant des produits de convolution discrets entre des coefficients d'influence et la source surfacique (chargements en surface) ou volumique (déformations plastiques ou d'incompatibilité). Le problème normal et le problème tangentiel en glissement total ou en glissement partiel peuvent alors être résolus, en prenant en compte les effets plastiques et hétérogènes. L'algorithme d'optimisation sous contrainte utilisé (contact elastique) est celui développé par L.Gallego tandis que la base du solver plastique utilisé a été développé par C. Jacq (contact elasto-plastique sans frottement) RESULTATS: De nouvelles solutions analytiques sont obtenues pour le calcul des déplacements résiduels tangentiels. / Fan blade roots of jet engines are operating under fretting sollicitations. This results in two types of damage: (i) initiation and crack propagation, and (ii) wear of contact surfaces. OBJECTIVE: A semi-analytical method has been developed in order to solve the industrial problem. This method is fast-computing and able to deal with elastic-plastic coated contacts and / or heterogeneous, and has bee developed from existing items and innovative analytical and numerical solutions. METHOD: The structure is simplified by assuming two semi-infinite elastic solid in contact, and analytical solutions for: + Contributions of elementary normal and tangential loads constant over a rectangular area + Contributions of elementary plastic strains constant on a parallelepiped + Contributions basic incompatibility of deformations related to a heterogeneous (inclusions, coatings, damage ,...) constant on a parallelepiped. >> on surface displacements and stresses in the volume. Surface displacements and stresses in the volume are then expressed using discrete convolution products (2D or 3D-FFT) between the coefficients of influence and the surfacic (loading surface) or volumic (plastic deformation or inconsistency) source. The normal and tangential problems in both the gross-slip and stick-slip regime can be solved, taking into account the effect of plasticity and heterogeneities. The contact algorithm used (elastic contact) has been developed by L. Gallego while the base of the plastic solver has been developed by C. Jacq (frictionless elastic-plastic contact) RESULTS: New analytical solutions are given for calculating the residual tangential displacements.

La fatigue à nombre de cycles élevé (HCF) est un mode de défaillance courant et dangereux pour les aubages de turbomachines. Elle est induite par les efforts dynamiques élevés générés lors de résonances présentes dans la plage de fonctionnement de ces machines. Les dispositifs amortisseurs basés sur l’utilisation du frottement sec, tels que les nageoires ou les frotteurs sous-plateformes, permettent de réduire les amplitudes vibratoires, voire de repousser les fréquences de résonance hors des zones de fonctionnement. Cependant la conception de ces dispositifs reste encore largement basée sur l’empirisme et ils peuvent être la source d’un effet de désaccordage potentiellement nuisible. L’objectif ici est de développer des modélisations adaptées au traitement du problème de vibration des aubages en présence de frottement sec, ceci afin de mieux maîtriser les comportements physiques mis en jeu et donc, de mieux maîtriser leur processus de conception. Plusieurs modélisations numériques sont testées et confrontées à des résultats de référence. Une comparaison entre procédures de résolution temporelle et fréquentielle est menée et montre l’efficacité des méthodes fréquentielles. La méthode fréquentielle de la balance harmonique à plusieurs harmoniques est adaptée au problème et exploitée dans le cadre d’une étude énergétique. Cette étude conduit à une meilleure compréhension des phénomènes mis en jeu lors de l’aplatissement des pics en fonctionnement ; elle permet de démontrer que l’alternance des états de contact glissant et bloqué est à l’origine de cet aplatissement et non la dissipation d’énergie comme souvent avancé dans la littérature. Enfin, la méthode est exploitée pour décrire le comportement des disques aubés désaccordés. La méthode de Monte Carlo est utilisée pour obtenir les caractéristiques statistiques de la réponse forcée d’un système discret, en tenant compte des variations stochastiques des paramètres du contact notamment, la charge normale, la raideur du contact et le coefficient de frottement. Les résultats obtenus permettent de mieux comprendre les effets de la nature variable de ces paramètres fondamentaux sur la dynamique d’ensemble du système non linéaire. / Fatigue with high number of cycles (HCF) is a current and dangerous mode of failure for the blades of turbo shaft engines. It is induced by the high dynamic stresses generated at resonance in the operating range of these machines. The rubbing devices based on use of dry friction, such as shrouds or under-platform dampers, make it possible to reduce the vibratory amplitudes, to even push back the resonance frequencies out of the operation zones. However the design of these devices remains still largely based on empiricism and they can be the source of a potentially harmful effect of mistuning. The goal is to develop adapted modelling for the treatment of blades vibration problem in the presence of dry friction, this in order to better control the concerned physical behaviours and thus, to better control their process of design. Several numerical modelling are tested and confronted with reference results. A comparison between procedures of time and frequency domains resolution is carried out and shows the effectiveness of frequential methods. The frequential method of the Harmonic Balance including several harmonics is adapted to the problem and is used within an energy study. This study leads to a better comprehension of the phenomena of peaks flattening and it shows that the alternation of the states of slipping and sticking contact is the real cause of this flattening and not energy dissipation like often advanced in the literature. Lastly, the method is exploited to describe the behaviour of mistuned bladed disks. The Monte Carlo method is used to obtain the statistical characteristics of the forced response for a lumped system, by taking account of the stochastic variations of parameters in the contact, specifically the normal load, the stiffness of the contact and the coefficient of friction. Results obtained make it possible to better include/understand the effects of the variable nature of these fundamental parameters on dynamics of nonlinear system.

Le comportement local au contact et son interaction avec la dynamique globale du système sont à l'origine d’innombrables problèmes de contact concernant plusieurs disciplines telles que la tribologie, la géophysique, la mécanique de vibration ou la mécanique de la rupture. Lorsque deux corps élastiques sont en mouvement relatif avec une interface de frottement, des vibrations induites se produisent dans le système. Dans un point de vue macroscopique, le scénario macroscopique de stick-slip survenant pendant le mouvement relatif est caractérisé par la chute soudaine de la force de frottement (état de glissement), séparées par des périodes d'accumulation d'énergie élastique (état d’adhérence). Autrement, une instabilité dynamique se produit quand un mode de vibration du système mécanique devient instable en raison des forces de frottement. Ces types d'instabilités, générées par des forces de frottement, ont été principalement objet de publies traitant de problèmes spécifiques dans différents domaines. Dans ce contexte, des analyses expérimentales et numériques ont été ici mis en place pour comprendre comme le comportement de l'interface locale affecte la réponse macroscopique du système et vice-versa, au cours de scénarios d'instabilité. Les scénarios macroscopiques (instabilité de « stick-slip macroscopique », instabilité modale, glissement continu stable), survenant entre deux milieux élastiques simples en mouvement relatif, ont été étudiés numériquement et expérimentalement. Un dispositif expérimental dédié (TRIBOWAVE) a été développé et a permis de reproduire et examiner les différents scénarios de frottement dans des conditions aux limites bien contrôlées. Les mêmes scénarios de frottement ont été reproduits par des simulations numériques transitoires. Une loi de frottement en fonction du temps d’adhérence (stick) a été définie à partir des essais expérimentaux. La loi de frottement obtenue a été mise en œuvre dans le modèle numérique, conduisant à une validation quantitative des scénarios de frottement par les expériences. Les simulations transitoires non linéaires, l’analyse aux valeurs propres complexes et les tests expérimentaux ont permis de dessiner des cartes de scénarios d'instabilité en fonction des paramètres clés du système. Validé par la comparaison avec les mesures des signaux expérimentaux globaux (forces, accélérations / vitesse), le modèle numérique a permis d'étudier le couplage entre le comportement du contact local (distribution de l'état du contact, propagation des ondes et des ruptures, précurseurs) et la réponse dynamique du système au cours du « stick-slip macroscopique », de l’instabilité due au couplage modale et du glissement continu stable. La compréhension du couplage entre le contact et la dynamique des systèmes apportera de nouvelles améliorations sur le contrôle des instabilités de contact et les problèmes d'usure connexes. / Local contact behavior and its interaction with the global dynamics of the system are at the origin of innumerable contact issues concerning several different disciplines like tribology, geophysics, vibration mechanics or fracture mechanics. When two elastic media are in relative motion with a frictional interface, friction induced vibrations arise into the system. By a macroscopic point of view, the “macroscopic stick-slip” scenario occurring during relative motion is characterized by sudden friction force drops (sliding state) along the time, separated by periods of elastic energy accumulation (stick state). Instead, the mode dynamic instability occurs when a vibration mode of the mechanical system becomes unstable, due to frictional contact forces. This kind of instabilities, generated by frictional forces, have been mainly object of papers dealing with specific issues in different domains. In this context, experimental and numerical analyses have been focused here on understanding how the local interface behavior affects the macroscopic frictional response of the system, and, conversely, during instability scenarios. The macroscopic frictional scenarios (macroscopic stick-slip instability, mode coupling instability, stable continuous sliding) arising between two simple elastic media in relative motion have been investigated numerically and experimentally. A newer experimental setup (TRIBOWAVE) has been developed and it allowed to reproduce and to investigate the different scenarios under well-controlled boundary conditions. The same frictional scenarios have been reproduced by transient numerical simulations. A dedicated friction law as a function of adherence (sticking) time has been recovered by means of experimental tests. The obtained friction law has been implemented in the numerical model, leading to a quantitative validation of the simulated scenarios by the experiments. Nonlinear transient simulations, complex eigenvalue analyses and experimental tests allowed for drawing instability maps as a function of system key parameters. The numerical model, validated by the comparison with the experimental global measurements (forces, accelerations/velocity), allowed for investigating the coupling between the local contact behavior (contact status distribution, wave and rupture propagation, precursors) and the system dynamic response during macroscopic stick-slip instability, mode coupling instability and stable continuous sliding. The understanding of the coupling between contact and system dynamics will bring to further improvements on the control of contact instabilities and related wear issues.

The aim of this thesis is to develop a numerical method to find approximations to solutions of the double-slip and double-spin model for the deformation and flow of granular materials. The model incorporates the physical and kinematic concepts of yield, shearing motion on slip lines, dilatation and average grain rotation. The equations governing the model comprise a set of five first order partial differential equations for the five dependent variables comprising two stress variables, two velocity components and the density. For steady state flows, the model is hyperbolic and the characteristic directions and relations along the characteristics are presented. The numerical approximation for the rate of working of the stresses are also presented. The model is then applied to a number of granular flow problems using the numerical method.

This work analyzes partial slip contact problems in the theory of linear viscoelasticity using both the semi-analytical method and nite element method. Such problems arise in metal-polymer contacts in orthopedic implants and similar applications. The boundary conditions of such problems are inherently mixed and vary with time, thus restricting the use of classical correspondence principle, which have been the basic approach for most of the solved problems in viscoelasticity. In the present semi-analytical approach, the governing equations for the vis-coelastic partial-slip contact are formulated as a pair of coupled Singular Integral Equations (SIEs) for a pin-plate geometry using the viscoelastic analogues of Green's functions. The formulation is entirely in the time-domain, avoiding Laplace transforms. Both Coulomb and hysteretic e ects are considered, and arbitrary load histories, including the bidirectional pin loads and remote plate stresses, are allowed. Moreover, the contact patch is allowed to advance and recede with no restrictions. The presence of viscoelastic behavior necessitates the application of the stick zone boundary condition in convolved form, and also introduces additional convolved gap terms in the governing equations, which are not present in the elastic case. Transient, as well as steady-state contact tractions, are obtained under load-hold, unload-hold, unload-reload, cyclic bidirectional (fretting) and remote plate loading for a three-element delayed elastic solid. A wide range of loads, loading rates, friction coeficients and the conforming nature of the contact are considered. The contact size, stick-zone size, indenter approach, maximum pressure, Coulomb energy dissipation are tracked during fretting. The edge-of-contact stresses and the subsurface stresses for the viscoelastic plate due to the contact tractions are determined by solving an equivalent traction boundary value problem. It is found that the viscoelastic fretting contact tractions for materials with delayed elastic nature shakedown just like their elastic counterparts. However, the number of cycles to attain shakedown states is strongly dependent on the ratio of the load cycle time to the relaxation time constant of the viscoelastic material. In monotonic load-hold case, the viscoelastic steady-state tractions agree well with the tractions from an equivalent elastic analysis using the shear modulus at infinite time. Whereas, the viscoelastic fretting tractions in shakedown differ considerably from their elastic counterparts. This is due to the fact that the contact patch does not increase monotonically in fretting-type(cyclic) loading. Hence, an approximate elastic analysis misleads to an incorrect edge-of-contact stresses. During fretting, the edge-of-contact hoop stress also shakedown and reaches its peak value at the trailing edge-of-contact when the horizontal pin load reaches its maximum. Moreover, the peak tensile of the edge-of-contact hoop stress increases with the increase in the Coulomb friction coefficient. In cyclic loading, Coulomb dissipation in a cycle at steady-state is almost independent of the rate at which the load is cycled. However, the viscous energy dissipated in a cycle is a strong function of the ratio of the load cycle time to the relaxation time constant. The steady-state cyclic hysteretic energy dissipation typically dominates the cyclic Coulomb dissipation, with a more pronounced difference at slower load cycling. However, despite this, it is essential to model an accurate viscoelastic fretting contacts including the effects of both viscous and Coulomb friction dissipation to obtain accurate contact stresses. A 12-element generalized Maxwell solid with long time scales representing a well characterized viscoelastic material like PMMA is also studied. The material chosen is of slowly relaxing nature and the ratio of the instantaneous shear modulus(G0) to the modulus at the infinite time(G1) is almost equal to 1000. In such cases, the material is effectively always in a transient state, with no steady edge-of-contact. As a consequence, the location of the peak hoop stress keeps on shifting when the load cycle is repeated. Interestingly, the rate at which the viscoelastic material relaxes affects the contact tractions. It is observed that the rapidly relaxing materials show qualitatively different tractions in the partial slip, with local traction spikes close to the edges-of-contact and concomitant high-stress gradients. On the other hand, finite element method is also used to analyze the partial slip viscoelastic contacts. In FEA, the pin-plate geometry is modeled using a custom mesh maker, where a 2D-continuum plane strain element is used for the plate and rigid element for the pin. The technique uses 'ABAQUS Standard' solver to solve the contact problem. Finite element analysis for a wide range of loads comparable with the SIE technique is performed. The tractions and contact sizes for various load cases such as unload-reload, fretting-type cyclic loads from both SIE and FEA agrees well. In certain conditions, there exist multiple contact arcs or stick zones that are currently difficult to solve with SIE's. However, such problems are treated using FEA and one such problem is illustrated.

La dynamique granulaire amenant à l'état critique présente un intérêt dans la compréhension de la déstabilisation menant à l'avalanche. Son étude permet d'avoir des pistes de compréhension sur des mécanismes plus complexes telles les catastrophes géophysiques (séismes, glissements de terrain, éboulements). Ainsi, lorsqu'un milieu granulaire tridimensionnel sous gravité est quasi-statiquement incliné, des précurseurs sont observés à partir d'une dizaine de degrés avant l'avalanche. Ces précurseurs correspondent à des réarrangements collectifs de grains observés à la surface qui apparaissent pseudo-périodiquement avec l'angle d'inclinaison. Cette thèse fournit une caractérisation expérimentale des précurseurs détectés à la surface par méthode optique et dans le volume par méthodes acoustiques (linéaire et non linéaire). Tout d'abord, de bonnes corrélations sont trouvées entre les réarrangements à la surface et dans le volume. Dans un second temps, l'étude est poursuivie avec une liste non exhaustive de paramètres influant sur les propriétés des précurseurs. L'état de surface des grains est crucial pour la dynamique des précurseurs. Puis, une tentative de description de la déstabilisation est réalisée avec notamment la mesure de la variation des paramètres élastiques. Les précurseurs d'avalanches correspondent à des pertes successives de rigidité du système, suivies du renforcement de ce dernier. / The granular dynamics leading to the avalanche is of interest in understanding the destabilization conducting to one. Its study implies a good train of throught in the understanding of more complex mechanisms such as geophysical disasters (earthquakes, landslides, rockslides). Thus, when a three-dimensional granular medium under gravity is quasi-statically tilted, precursors are observed from the tilt of ten degrees before the avalanche. These precursors correspond to collective rearrangements of grains observed on the free surface which appear pseudo-periodically with the angle of inclination. In order to understand this phenomenon, the thesis presents an experimental characterisation of the precursors detected on the surface by optical method and in the bulk by acoustic methods (linear and nonlinear). Firstly, good correlations were found between the surface and bulk rearrangements which led to extending the study with a non-exhaustive list of parameters affecting the precursors properties. The surface features of the grains have appeared to be particularly crucial in the precursors dynamics. Finally, an attempt at a description of the destabilization is approached with the measurement of the variation of the elastic parameters revealing that precursors of avalanches correspond to successive loss of rigidity of the system, followed by hardening of the latter.

[pt] Durante a perfuração de poços de petróleo, a coluna de perfuração apresenta um comportamento dinâmico complexo, esta tese foca no estudo experimental e na modelagem matemática deste comportamento. Neste trabalho, destaca-se as vibrações autoexcitadas axiais, laterais e torcionais, que podem levar a efeitos como o bit bouncing, o whirling e stick-slip torcional. A primeira contribuição desta tese é a análise experimental de um bancada de testes, que fornece informações sobre a dinâmica de sistemas torcionais. A influência dos parâmetros de controle não lineares na resposta do sistema é investigada, identificando as condições sob as quais o fenômeno stick-slip ocorre. Em segundo lugar, a tese propõe estratégias de identificação de sistemas para sistemas não lineares, utilizando a mesma bancada de testes supracitada. Uma abordagem híbrida para a identificação é proposta, onde técnicas de modelagem de caixa cinza e caixa preta são combinadas para calibrar os parâmetros do sistema, particularmente aqueles associados ao atrito. Essa abordagem aumenta a precisão das estimativas em comparação com os métodos tradicionais de caixa cinza, mantendo a interpretabilidade. Além disso, a pesquisa emprega physics-informed deep learning para estimar os parâmetros mecânicos e de atrito do modelo de dois graus de liberdade. A calibração usando dados experimentais obtidos de uma bancada de testes fornece informações sobre o comportamento de sistemas de perfuração. Finalmente, a tese apresenta investigações experimentais sobre o acoplamento entre oscilações torcionais e axiais utilizando uma bancada experimental de perfuração em escala de laboratório modificada e adaptada equipada com brocas e amostras de rocha reais. Em resumo, esta tese aumenta a compreensão da dinâmica de colunas de perfuração e apresenta aplicações úteis para técnicas de identificação de sistemas na análise de oscilações torcionais e axiais. / [en] During drilling for oil extraction purposes, the drill string experiences complex dynamic behavior, and this work delves into the experimental study and the mathematical modeling of such behavior. Self-excited vibrations, such as axial, lateral, and torsional vibrations, which can lead to detrimental effects such as bit bouncing, whirling, and torsional stick-slip are highlighted in this thesis. Distinct aspects of drilling dynamics are considered in this investigation to enhance the understanding of various phenomena. Initially, an experimental analysis of a lab-scale rig is conducted, providing valuable insights into the dynamics of such systems. And the influence of control parameters on the system’s response is examined, particularly in identifying the conditions under which the stick-slip phenomenon is likely to occur. Secondly, the thesis proposes system identification strategies for nonlinear systems, specifically focusing on the same laboratory test rig. An innovative ensemble approach is proposed, which combines gray and black-box modeling techniques to effectively calibrate the parameters of a dynamical system, particularly those associated with friction. This approach improves prediction accuracy compared to traditional gray-box methods while maintaining interpretability in the dynamic responses. Furthermore, the research employs physics-informed deep learning to estimate the low-dimensional model mechanical and friction parameters. Calibration using experimental data obtained from a specialized setup provides insights into the drill-string system s behavior. Finally, the thesis involves experimental investigations on the coupling between torsional and axial oscillations using a modified and adapted lab-scale drilling rig equipped with real drill bits and rock samples. In summary, this thesis advances our understanding of drill-string dynamics and presents helpful applications for system identification techniques.

[pt] Problemas dinâmicos governados por problemas de valor inicial (PVI) não lineares, em geral, despertam grande interesse da comunidade científica. O conhecimento da solução desses PVI facilita o entendimento das características dinâmicas do problema. Porém, infelizmente, muitos dos PVI de interesse não têm solução conhecida. Nesse caso, uma alternativa é o cálculo de aproximações para a solução. Métodos numéricos e analíticos são eficientes nessa tarefa e podem fornecer aproximações com a precisão desejada. Os métodos numéricos foram muito desenvolvidos nos últimos anos e amplamente aplicados em problemas de diversas áreas da engenharia. Pacotes computacionais de fácil utilização foram criados e hoje fazem parte dos mais tradicionais programas de simulação numérica. Entretanto, as aproximações numéricas têm uma desvantagem em relação às aproximações analíticas. Elas não permitem o entendimento de como a solução depende dos parâmetros do problema. Visto isso, esta dissertação foca na análise e implementação de técnicas analíticas denominadas métodos de perturbação. Foram estudados os métodos de Lindstedt-Poincaré e de múltiplas escalas de tempo. As metodologias foram aplicadas em um PVI envolvendo a equação de Duffing não amortecida. Programas em álgebra simbólica foram desenvolvidos com objetivo de calcular aproximações analíticas hierárquicas para a solução desse problema. Foi feita uma análise paramétrica, ou seja, estudo de como as condições iniciais e os valores de parâmetros influem nas aproximações. Além disso, as aproximações analíticas obtidas foram comparadas com aproximações numéricas calculadas através do método do Runge- Kutta. O método de múltiplas escalas de tempo também foi aplicado em um PVI que representa a dinâmica de um sistema massa-mola-amortecedor com atrito seco. Devido ao atrito, a resposta do sistema pode ser caracterizada em duas fases alternadas, a fase de stick e a fase de slip, compondo um fenômeno chamado stick-slip. Verificou-se que as aproximações obtidas para resposta do sistema pelo método de múltiplas escalas de tempo têm boa acurácia na representação da dinâmica do stick-slip. / [en] Dynamical problems governed by non-linear initial value problems (IVP), in general, are of great interest of the scientific community. The knowledge of the solution of these IVPs facilitates the understanding of the dynamic characteristics of the problem. However, unfortunately, many of the IVPs of interest does not present a known solution. In this case, an alternative is to calculate approximations for the solution. Numerical and analytical methods are efficient in this assignment and can provide approximations with the desired precision. Numerical methods have been developed over the last years and have been widely applied to dynamical problems in various engineering areas. Computational packages, easy to use, were created and today are part of the most traditional numerical simulation programs. However, numerical approximations have a disadvantage in relation to analytical approaches. They do not allow the understanding of how the solution depends on the problem parameters. Given this, this dissertation focuses on the analysis and implementation of analytical techniques called perturbation methods. The Lindstedt-Poincaré method and multiple time scales method were studied. The methodologies were applied in an IVP involving the non-damped Duffing equation. Symbolic algebra programs were developed with the purpose of calculating hierarchical analytical approximations to the solution of this problem. A parametric analysis was performed, in other words, a study of how the approximations are influenced by initial conditions and parameter values. In addition, the analytical approximations obtained were compared with numerical approximations calculated using the Runge-Kutta method. The multiple scales method was also applied in a IVP that represents the dynamics of a mass-spring-damper oscillator with dry friction. Due to friction, the system response can be characterized in two alternating phases, the stick phase and the slip phase, composing a phenomenon called stick-slip. It was verified that the approximations obtained for system response by the multiple scales method represent the stick-slip dynamics with good accuracy.

[pt] Este trabalho apresenta um estudo do comportamento dinâmico de uma bancada de testes representativa do sistema real de perfuração, composta por um motor CC acoplado a um sistema físico torcional, sujeita a fontes de atrito que induzem um regime de stick-slip no sistema em duas regiões distintas. O estudo incluiu a identificação de parâmetros da bancada de testes por meio de uma série de ensaios experimentais; e a caracterização do atrito, por meio do levantamento experimental da curva do coeficiente de atrito, em função da velocidade angular dos rotores principais. O intuito inicial foi a obtenção de um modelo numérico que fosse o mais simples possível e que representasse bem a bancada de testes. Uma vez obtido o modelo numérico, prosseguiu-se com uma série de simulações que permitissem uma caracterização indireta do regime de atrito ao qual estivessem submetidos os rotores principais, partindo-se apenas de medições de parâmetros no motor. Esse estudo é de grande relevância para a compreensão qualitativa da dinâmica do sistema real de perfuração, uma vez que ainda hoje não há técnicas totalmente confiáveis para caracterização do comportamento da coluna no fundo do poço a partir de dados da superfície somente. / [en] This paper presents a study of the dynamic behavior of a representative test bench of a real rotary drilling system, comprising a DC motor coupled to a very exible torsional system subjected to sources of friction which can induce self-excitation into two distinct regions of the system. The study includes the identification of parameter settings from the test bench by means of a series of experimental tests and characterization of friction, by obtaining the experimental curve of the friction coefficient as a function of the angular speed of the main rotor. The initial aim was to obtain a numerical model as simple as possible, capable of representing the test bench. Once obtained the numerical model, a series of numerical simulations were done, which allow an indirect characterization of the friction condition to which main rotors were subjected, starting only with the parameters measured at the drive. This study is of great importance for a qualitative understanding of the dynamics of the real drilling system, since today there is no fully reliable techniques to characterize the behavior of the column in the deep from surface data only.