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Ventilação natural por efeito chaminé : estudo em modelo reduzido de pavilhões industriais / Natural ventilation for buoyancy - study of model reduced of industrial pavilionsChiarello, Juliana Ana January 2006 (has links)
A ventilação nas edificações tem por objetivo principal criar um ambiente interno confortável e saudável, preservando a qualidade do ar interior. Um dos principais sistemas de renovação do ar, e praticamente a única opção economicamente viável, utilizado no projeto de ambientes industriais é a ventilação natural. Esta pode ser promovida por dois mecanismos: o denominado efeito chaminé e o efeito dos ventos, porém também podem ocorrer pelos dois simultaneamente. O objetivo deste trabalho é pesquisar a viabilidade do emprego de modelos reduzidos no projeto da ventilação natural em pavilhões industriais, considerando unicamente o efeito chaminé. Para atingir esse objetivo foi ensaiado um modelo, com escala geométrica 1/100, de um pavilhão industrial real situado na cidade de Passo Fundo – RS, que será utilizado como uma fábrica para injeção de alumínio em carcaças e rotores elétricos. Nos ensaios foram medidas velocidades na abertura de saída, com anemômetro de fio quente, e temperaturas com termopares, tanto dentro como fora do modelo. Os resultados alcançados permitem avaliar quais são as leis de semelhança relevantes neste tipo de ensaio com modelo reduzido, as limitações do procedimento experimental e, ainda, determinar valores de vazões de ar nas aberturas de saída e compará-los com vazões obtidas através da utilização de alguns modelos teóricos. / The ventilation in the constructions has for main objective to create a comfortable and healthful internal environment, preserving the quality of the interior air. One of the main systems of renewal of air, and practically the only economically viable option, used in the industrial environment project is the natural ventilation. This can be promoted by two mechanisms: the called buoyancy and the effect of the winds, however also can occur for the two simultaneously. The objective of this work is to search the viability of the job of models reduced in the project of the natural ventilation in industrial pavilions, considering solely the buoyancy. To reach this objective a model was assayed, with geometric scale 1/100, of a situated real industrial pavilion in the city of Passo Fundo – RS, that will be used as a plant for injection of aluminum in carcasses and electric rotors. In the assays speeds in the exit opening had been measured, with anemometer of hot wire, and temperatures with thermocouples, in such a way inside as it are of the model. The reached results allow to evaluate which are the excellent laws of similarity in this type of assay with reduced model, the limitations of the experimental procedure and still to determine values of air outflows in the exit openings and to compare them with outflows gotten through the use of some theoretical models.
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Modelling tools for quieter vehicles : Waves in poro– and visco–elastic continuaÖstberg, Martin January 2012 (has links)
New modelling tools intended to contribute to the development of components for quieter vehicles are developed. The tools are based on continuum models for wave propagation in poro– and visco–elastic media. By using geometric attributes of the studied components, the computational cost may be radically decreased compared to traditional methods. By assigning known analytical functions for one or two of the spatial directions, the spatial dimension of the remaining numerical problem is reduced. This reduction of spatial dimensions is performed in two di↵erent ways. The first one treats wave propagation in infinitely extended homogeneous and hollowed cylindrical rods, or wave guides, consisting of visco–elastic media. The wave solutions obtained are then used to model rubber vibration isolators of finite length by mode–matching the fields to the radial boundary conditions of interest. The second one is a method for modelling rotationally symmetric multilayered structures consisting of poro–elastic, elastic and fluid domains. By using a harmonic expansion for the azimuthal spatial dependence, the original three–dimensional problem is split up into several, much smaller, two– dimensional ones, radically decreasing the computational load. Moreover, using a mixed measurement/modelling approach, the audible frequency range characteristics of a viscous damper from a truck is studied, illustrating the influence of the rubber bushings by which it is attached to surrounding structures. The modelling approaches presented in this thesis are intended as tools aiding the design process of new vehicles, enabling new technology striving for more sustainable vehicle concepts. More specifically, the tools aim to improve the modelling of sound and vibration properties which are often penalised when seeking new, more sustainable vehicle designs. / <p>QC 20120522</p> / Centre for Eco2 Vehicle Design
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Modelling tools for quieter vehicles : Waves in poro-and visco-elastic continuaOstberg, Martin 07 June 2012 (has links) (PDF)
New modelling tools intended to contribute to the development of components for quieter vehicles are developed. The tools are based on continuum models for wave propagation in poro- and visco-elastic media. By using geometric attributes of the studied components, the computational cost may be radically decreased compared to traditional methods. By assigning known analytical functions for one or two of the spatial directions, the spatial dimension of the remaining numerical problem is reduced. This reduction of spatial dimensions is performed in two di↵erent ways. The first one treats wave propagation in infinitely extended homogeneous and hollowed cylindrical rods, or wave guides, consisting of visco-elastic media. The wave solutions obtained are then used to model rubber vibration isolators of finite length by mode-matching the fields to the radial boundary conditions of interest. The second one is a method for modelling rotationally symmetric multilayered structures consisting of poro-elastic, elastic and fluid domains. By using a harmonic expansion for the azimuthal spatial dependence, the original three-dimensional problem is split up into several, much smaller, two- dimensional ones, radically decreasing the computational load.Moreover, using a mixed measurement/modelling approach, the audible frequency range characteristics of a viscous damper from a truck is studied, illustrating the influence of the rubber bushings by which it is attached to surrounding structures.The modelling approaches presented in this thesis are intended as tools aiding the design process of new vehicles, enabling new technology striving for more sustainable vehicle concepts. More specifically, the tools aim to improve the modelling of sound and vibration properties which are often penalised when seeking new, more sustainable vehicle designs.
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Ventilação natural por efeito chaminé : estudo em modelo reduzido de pavilhões industriais / Natural ventilation for buoyancy - study of model reduced of industrial pavilionsChiarello, Juliana Ana January 2006 (has links)
A ventilação nas edificações tem por objetivo principal criar um ambiente interno confortável e saudável, preservando a qualidade do ar interior. Um dos principais sistemas de renovação do ar, e praticamente a única opção economicamente viável, utilizado no projeto de ambientes industriais é a ventilação natural. Esta pode ser promovida por dois mecanismos: o denominado efeito chaminé e o efeito dos ventos, porém também podem ocorrer pelos dois simultaneamente. O objetivo deste trabalho é pesquisar a viabilidade do emprego de modelos reduzidos no projeto da ventilação natural em pavilhões industriais, considerando unicamente o efeito chaminé. Para atingir esse objetivo foi ensaiado um modelo, com escala geométrica 1/100, de um pavilhão industrial real situado na cidade de Passo Fundo – RS, que será utilizado como uma fábrica para injeção de alumínio em carcaças e rotores elétricos. Nos ensaios foram medidas velocidades na abertura de saída, com anemômetro de fio quente, e temperaturas com termopares, tanto dentro como fora do modelo. Os resultados alcançados permitem avaliar quais são as leis de semelhança relevantes neste tipo de ensaio com modelo reduzido, as limitações do procedimento experimental e, ainda, determinar valores de vazões de ar nas aberturas de saída e compará-los com vazões obtidas através da utilização de alguns modelos teóricos. / The ventilation in the constructions has for main objective to create a comfortable and healthful internal environment, preserving the quality of the interior air. One of the main systems of renewal of air, and practically the only economically viable option, used in the industrial environment project is the natural ventilation. This can be promoted by two mechanisms: the called buoyancy and the effect of the winds, however also can occur for the two simultaneously. The objective of this work is to search the viability of the job of models reduced in the project of the natural ventilation in industrial pavilions, considering solely the buoyancy. To reach this objective a model was assayed, with geometric scale 1/100, of a situated real industrial pavilion in the city of Passo Fundo – RS, that will be used as a plant for injection of aluminum in carcasses and electric rotors. In the assays speeds in the exit opening had been measured, with anemometer of hot wire, and temperatures with thermocouples, in such a way inside as it are of the model. The reached results allow to evaluate which are the excellent laws of similarity in this type of assay with reduced model, the limitations of the experimental procedure and still to determine values of air outflows in the exit openings and to compare them with outflows gotten through the use of some theoretical models.
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Ventilação natural por efeito chaminé : estudo em modelo reduzido de pavilhões industriais / Natural ventilation for buoyancy - study of model reduced of industrial pavilionsChiarello, Juliana Ana January 2006 (has links)
A ventilação nas edificações tem por objetivo principal criar um ambiente interno confortável e saudável, preservando a qualidade do ar interior. Um dos principais sistemas de renovação do ar, e praticamente a única opção economicamente viável, utilizado no projeto de ambientes industriais é a ventilação natural. Esta pode ser promovida por dois mecanismos: o denominado efeito chaminé e o efeito dos ventos, porém também podem ocorrer pelos dois simultaneamente. O objetivo deste trabalho é pesquisar a viabilidade do emprego de modelos reduzidos no projeto da ventilação natural em pavilhões industriais, considerando unicamente o efeito chaminé. Para atingir esse objetivo foi ensaiado um modelo, com escala geométrica 1/100, de um pavilhão industrial real situado na cidade de Passo Fundo – RS, que será utilizado como uma fábrica para injeção de alumínio em carcaças e rotores elétricos. Nos ensaios foram medidas velocidades na abertura de saída, com anemômetro de fio quente, e temperaturas com termopares, tanto dentro como fora do modelo. Os resultados alcançados permitem avaliar quais são as leis de semelhança relevantes neste tipo de ensaio com modelo reduzido, as limitações do procedimento experimental e, ainda, determinar valores de vazões de ar nas aberturas de saída e compará-los com vazões obtidas através da utilização de alguns modelos teóricos. / The ventilation in the constructions has for main objective to create a comfortable and healthful internal environment, preserving the quality of the interior air. One of the main systems of renewal of air, and practically the only economically viable option, used in the industrial environment project is the natural ventilation. This can be promoted by two mechanisms: the called buoyancy and the effect of the winds, however also can occur for the two simultaneously. The objective of this work is to search the viability of the job of models reduced in the project of the natural ventilation in industrial pavilions, considering solely the buoyancy. To reach this objective a model was assayed, with geometric scale 1/100, of a situated real industrial pavilion in the city of Passo Fundo – RS, that will be used as a plant for injection of aluminum in carcasses and electric rotors. In the assays speeds in the exit opening had been measured, with anemometer of hot wire, and temperatures with thermocouples, in such a way inside as it are of the model. The reached results allow to evaluate which are the excellent laws of similarity in this type of assay with reduced model, the limitations of the experimental procedure and still to determine values of air outflows in the exit openings and to compare them with outflows gotten through the use of some theoretical models.
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Modélisation et instrumentation d'un bâtiment et de ses systèmes pour optimiser sa gestion énergétique / Modeling and instrumentation of a building and its systems to optimize energy managementViot, Hugo 29 November 2016 (has links)
Dans un contexte de raréfaction des énergies à bas coût et de lutte contre le changement climatique, le secteur du bâtiment est contraint à des objectifs de diminution de ses consommations d'énergie. Un contrôle adapté des systèmes énergétiques du bâtiment peut constituer un levier important pour tendre vers cet objectif. Le but premier de ces travaux est la construction de modèles de faibles dimensions afin de pouvoir les embarquer dans un contrôleur pour améliorer la gestion énergétique. Ces modèles doivent être capables de s’enrichir des mesures disponibles sur site. Un bâtiment démonstrateur situé à l'Institut Universitaire de Technologie Génie Civil Construction Durable de Bordeaux sert de support au projet. Ce travail se décline en quatre volets. Le premier concerne la modélisation puisque des modèles légers basés sur l'analogie électrique et la représentation d'état sont proposés afin de décrire la dynamique du bâtiment sur un horizon de contrôle de quelques jours. Le deuxième volet concerne l'instrumentation du bâtiment puisque des campagnes de mesures courtes sont réalisées afin d'identifier les valeurs des paramètres du modèle de façon à minimiser l'écart modèle-mesure. Certains capteurs servent par la suite pour la gestion énergétique du bâtiment ; ces travaux posent donc également la question du jeu minimal de capteurs. Le troisième volet concerne la caractérisation des systèmes servant à piloter l'ambiance puisque dans une logique de contrôle optimal il faut être capable de relier l'effet de la commande sur la grandeur d'intérêt (température de confort). Le bâtiment démonstrateur comprend deux systèmes de chauffage : un plancher chauffant et des ventilo-convecteurs ainsi qu'une centrale de traitement d'air double flux pour le renouvellement d'air. Le dernier volet concerne la gestion énergétique avec l'utilisation d'un contrôleur prédictif embarquant l'un des modèles identifiés. Ainsi l’objectif est de chercher à anticiper le pilotage d'un système capacitif à long temps de réponse, tel que le plancher chauffant, grâce à la connaissance des perturbations futures (occupation, météo) sur un horizon de prédiction de quelques heures. Un contrôle réactif est assuré par les ventilo-convecteurs en appoint. La gestion prédictive est comparée à des stratégies de gestion plus classiques en simulation et in situ avec le bâtiment démonstrateur. L'originalité de ces travaux est de proposer une démarche pour la mise en place d'une boucle de contrôle complète (contrôleur/capteur /actionneurs) et de montrer s'il existe un intérêt à la gestion prédictive de systèmes à long temps de réponse dans le domaine du bâtiment. / The building sector is forced to reduce its energy consumption in a context of high energy prices and global warming. Proper control of building energy systems can be an important lever to move towards this goal. The main goal of this work is to obtain small size buildings models in order to use it in a controller to improve energy management. The inputs of these models are fed in real-time with available measurements on site. A demonstration building at the IUT Civil Engineering and Sustainable Construction of Bordeaux serves as experimental support for the project. This work consists of four parts. The first one is to make lightweight models based on the electrical analogy and state-space representation to describe the dynamics of the building on upcoming days. The second part concerns the instrumentation of the building because short measurement campaigns are carried out to identify the model parameter values to minimize the gap between model output and measurement. Some sensors are then used for energy management of the building; thus this work also raise the question of the minimum set of sensors. The third address the characterization of the systems used to control air temperature. For an optimal control logic we must be able to link the effect of the command on the interest variable (air temperature). The demonstration building includes two heating systems : floor heating system (FHS) and fan coil units (FC). A dual flow air handling unit (AHU) is used for air renewal. The last part concerns energy management with the use of a predictive controller boarding one of the identified models. This project intends to anticipate the control of long time response capacitive systems as floor heating through knowledge of future disturbances on a prediction horizon of a few hours (occupation, weather). A reactive control is ensured by the fan coil units. Predictive management is compared to more conventional management strategies in simulation and on-site with the demonstrator building. The originality of this work is to propose a method for the establishment of a full control loop (controller/sensor/actuator) and demonstrate an interest in the predictive management of long response time systems in the building sector.
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Construction de modèles réduits pour la simulation dynamique et optimisation de modèles électromagnétiques numériques / Building Reduced Order Models for Dynamic Simulation and Optimization of Numeric Electromagnetic ModelsAntunes oliveira leite, Mateus 24 April 2018 (has links)
Il y a des nombreuses méthodes capables de produire des modèles numériques très précis des systèmes électromagnétiques. Si la précision demandée est très importante ou la nature du phénomène modélisé est très complexe, il faudra résoudre un nombre élevé d’équations. Si le modèle est utilisé dans des applications où de nombreux paramètres doit être prises en compte, comme c’est le cas de la conception optimale, la résolution de ces équations est souvent très couteuse.Pour éviter ce calcul, des algorithmes de réduction de modèle ont été développés. Il s’agit de procédures permettant de trouver des modèles réduits qui représentent la relation entrée/sortie du modèle fin de manière très précise mais en utilisant un nombre réduit d’équations.Dans cette thèse, les techniques de réduction de modèles sont analysées et améliorées. Une attention particulière est faite à la technique de correspondance des moments. Les problèmes de placement des points d’expansion, de stabilité et de précision sont plus particulièrement étudiés. Cela a permit la simulation et conception optimale de modèles électromagnétiques complexes qu’un jeu de bus bar et un problème de diffusion d’ondes.De plus, nous avons développé une méthode d'échantillonnage adaptatif. Des algorithmes classiques d’optimisation ont été couplés aux modèles réduits permettant l’accélération des calculs. Les méthodes proposées ont été testés dans des problèmes électromagnétiques obtenus par la méthode « Partial Equivalent Circuit Element (PEEC) ». / There are many available methods capable of producing High-Fidelity Models (HFM) of electromagnetic systems. If the required precision is very high or the nature of the phenomenon that is being modeled is complex, a very large number of equations may have to be solved. If the model is used for applications where many different geometries or parameters must be considered, as is the case in design optimization, solving these equations many times can be very time consuming.To avoid the burden of this computation, Model Order Reduction (MOR) algorithms have been developed. They consist in procedures of finding Reduced Order Models (ROMs) that accurately describe the input/output behavior of the High-Fidelity Model but using only a very small number of equations.In this work, MORs techniques are analyzed and improved. Special attention is paid to Moment Matching. Problems like placement of expansion points, stability and numerical robustness are investigated. This has allowed the simulation and optimization of complex electromagnetic device. As examples of application we have presented a laminates bus bar and a wave scattering problem.In addition to that, a smart adaptive way of sampling the design space to allow fast optimization has been developed. The sampled points are used to perform interpolation and approximate the objective function in a very fast manner. Classic optimization techniques have also been coupled with the Reduced Order Models, accelerating the computations. The proposed approaches have been tested mainly in electromagnetic problems obtained by the Partial Equivalent Circuit Element (PEEC) method.
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Numerically investigating the effects of gasoline surrogate physical and chemical properties in a gasoline compression ignition (GCI) engineAtef, Nour 06 1900 (has links)
Gasoline compression ignition (GCI) engines show promise in meeting stringent new
environmental regulations, as they are characterized by high efficiency and low emissions.
Simulations using chemical kinetic models provide an important platform for investigating the
behaviors of the fuels inside these engines. However, because real fuels are complex, simulations
require surrogate mixtures of small numbers of species that can replicate the properties of real
fuels. Accordingly, the development of high fidelity, well-validated kinetic models for surrogates
is critical in order to accurately replicate the combustion chemistry of different fuels under
engine-related conditions.
This work focuses on the development of combustion kinetic models to better understand
gasoline fuel combustion in GCI engines. An updated iso-octane detailed kinetic model was
developed based on new thermodynamic group values and recently evaluated rate coefficients
from literature. The model was validated against a wide range of experimental data and
conditions.
The iso-octane model was further used in 0D simulations for a homogeneous charge
compression ignition (HCCI) engine. The results showed that the low-temperature heat release in
engines increases with engine boosting when the addition of alky radicals to molecular oxygen is
more favored. Ethanol addition was also found to act as a radical sink which inhibits the radical
pool formation and results in lower reactivity.
Although detailed models provide clarification of the combustion chemistry, their high
computational cost impedes their utilization in 3-D engine simulations. Hence, a reduced model
for toluene primary reference fuels was developed and validated against ignition delay time and
flame speed experiments from literature. The model was then used in numerically investigating
the effects of the fuel’s physical properties using hollow-cone and multi-hole injectors in a
partially premixed compression ignition (PPCI) engine. It was concluded that the effects of
physical properties are evident in multi-hole injection cases, which is attributable to the
differences in mixture stratification.
Finally, reduced models for multi-components surrogates for three full-blend fuels (light
naphtha-Haltermann straight-run naphtha and GCI fuels) were developed. The models were
validated against ignition delay time experiments from the literature and tested in 3D engine
simulations.
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Estudo de painéis com abertura constituídos por alvenaria estrutural de blocos / Study of block masonry walls with openingsNascimento Neto, Joel Araújo do 28 October 2003 (has links)
O estudo abordado nesta tese relaciona-se ao comportamento de painéis com abertura constituídos por alvenaria estrutural de blocos. A motivação pelo tema teve como base um estudo anterior, em que o autor verificou a grande influência dos lintéis no comportamento do sistema de contraventamento de edifícios em alvenaria estrutural. O aspecto mais geral do estudo ora desenvolvido foi avaliar, de forma minuciosa, o efeito de acoplamento causado pela presença de lintéis no painel. Para tanto, foram realizados ensaios com modelos físicos reduzidos na escala 1:3, cujos resultados foram avaliados de modo qualitativo, sem o compromisso de determinar correlações com protótipos. Várias configurações de painéis isolados e painéis com abertura representativa de porta e janela foram ensaiadas, de modo a se avaliar o efeito de acoplamento gerado pelo lintel. Além disso, foram avaliados, também, os efeitos de diferentes intensidades de pré-compressão e de armaduras construtivas no comportamento do painel. A análise dos resultados experimentais foi dividida em duas etapas: aplicação da pré-compressão e aplicação da força horizontal. Na primeira foram verificadas as deformações axiais dos painéis e determinados os correspondentes módulos de deformação longitudinal. Na segunda etapa foram avaliados a rigidez e os efeitos da não-linearidade de forma geral, as deformações ocorridas nos lintéis e nas paredes de cada modelo, assim como em regiões críticas para o surgimento de fissuras, a exemplo dos cantos da abertura e da base das paredes. Além disso, foram realizadas modelagens numéricas de alguns painéis, utilizando-se o método dos elementos finitos, e os resultados obtidos foram comparados com os experimentais. Adicionalmente, foram avaliados alguns modelos simplificados na fase linear do comportamento dos painéis com abertura de porta, assim como na fase não-linear do painel isolado. / The present thesis deals with the behavior of block masonry walls with openings. The development of the work was motivated bya a previous study, where the author verified the large influence of the lintels on the behavior of the bracing system of structural masonry buildings. The main aspect of the study was the meticulous evaluation of the coupled effect provided by the lintels. Many experimental tests on third-scale models were carried out to analyze the phenomenon in a qualitative way, without the compromise of determinging correlations to the prototypes. Several configurations of isolated walls and panels with door and window openings were tested, evaluating the coupling action provided by the lintels. The effects of different pre-compression values and the presence f constructive steel bars were include on the study. The analaysis of the experimental results was performed in two stages: application of the vertical force before the application of horizontal force at the top. In the first stage axial strains were measured with the corresponding assessment of the longitudinal Young modulus of the panels. Te second stage included the stiffness computation and the evaluation of the non-linearity in a general sense, the strains on the lintel and the walls, and the critical cracking areas, such as wall's bases and corners. Experimental results were compared with those obtained with Finite Element modeling of the panels. Finally simplified models were presented for the linear phase of walls with openings and the non-linear behavior of the isolated panels.
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Modélisation de liaisons flexibles amortissantes en élastomères pour la prédiction du comportement dynamique de systèmes complexes / Flexible and dissipative rubber mounts modelisation for the prediction of complex systems dynamic behaviorMorin, Benjamin 03 November 2016 (has links)
Dans le cadre de l’amortissement passif de structure, les élastomères sont employés dans les industries du transport sous la forme de liaisons amortissantes. Ces matériaux ont un comportement dépendant de la fréquence, de la température et de l’amplitude d’excitation. Les modèles numériques associés peuvent être coûteux en temps de calcul, notamment en phase d’optimisation. Le but de cette thèse est de proposer un modèle réduit efficace de ces liaisons amortissantes, qui prenne en compte la dissipation viscoélastique et les précharges non-linéaires dans les liaisons. La première partie de ce mémoire se concentre sur la représentation de la dissipation par le modèle réduit. Une loi de comportement viscoélastique, basée sur un modèle rhéologique identifié expérimentalement, est utilisée avec la méthode des éléments finis pour obtenir un modèle numérique des liaisons amortissantes. Un premier modèle réduit prédictif, prenant en compte la dissipation en est dérivé en utilisant une extension originale des méthodes de sous-structuration. La deuxième partie traite de l’influence des précharges statiques non-linéaires sur le comportement dynamique et la dissipation dans les liaisons. Pour cela, une loi de comportement hyper-visco-élastique linéarisée autour d’un état précontraint statique non-linéaire est développée. Les méthodes de sous-structuration introduites dans la première partie sont alors enrichies afin de tenir compte de la dissipation et les non-linéarités géométriques dans les liaisons. Finalement, ces modèles réduits à 2 nœuds permettent des gains en temps de calcul d’un facteur 50 à 100 et sont facilement utilisables par l’ingénieur en phase de conception. / In the context of passive damping, various mechanical systems from the space, aeronautic or auto-mobile industry use elastomer components (shock absorbers, silent blocks, flexible joints...).These materials have frequency, temperature and amplitude dependentcharacteristics. The associated numerical models may become computationally too expensive during an optimization process.The aim of this work is to propose an efficient reduced model of rubber devices that account for the viscoelastic damping and the non-linear pre-stress in the dampers.The first part of this thesis is about how to include the viscoelasticdamping in the reduced model. It starts by using a viscoelastic constitutive relation, based on experimental identification, within the frame of the finite element method to obtain a numerical model of the rubber dampers. A first efficient reduced model is then derived from this FE model by using an original extension of sub-structuring methods in the case of viscoelastic damping.In the second part, the influence of non-linear static pre-stress overthe dynamic behavior and the dissipation in the dampers is studied. An hyper-visco-elastic constitutive relation, linearized in the neighbourhood of a pre-stressed state, is developed. The sub-structuring methods presented in the first part are then upgraded to account for the damping and the geometrical non-linearities in the dampers. Finally, these 2-node reduced models give access to greatly reduced computation times (50 to 100 times faster) and are easy to use for the engineer.
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