This thesis analyses the existing types of polyhedral linkages and presents new linkage types for resizing polyhedral shapes. First, the transformation characteristics, most specifically, magnification performances of existing polyhedral linkages are given. Then, methods for synthesizing single degree-of-freedom planar polygonal linkages are described. The polygonal linkages synthesized are used as faces of polyhedral linkages. Next, the derivation of some of the existing linkages using the method given is presented. Finally, some designs of cover panels for the linkages are given. The Cardan Motion is the key point in both analyses of existing linkages and synthesis of new linkages.

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Deployable structures made from ultra-thin composite materials can be folded elastically and are able to self-deploy by releasing the stored strain energy. Their lightness, low cost due to smaller number of components, and friction insensitive behaviour are key attractions for space applications. This dissertation presents a design methodology for lightweight composite booms with multiple tape-spring hinges. The whole process of folding and deployment of the tape-spring hinges under both quasi-static and dynamic loading has been captured in detail through finite element simulations, starting from a micro-mechanical model of the laminate based on the measured geometry and elastic properties of the woven tows. A stress-resultant based six-dimensional failure criterion has been developed for checking if the structure would be damaged. A detailed study of the quasi-static folding and deployment of a tape-spring hinge made from a two-ply plain-weave laminate of carbon-fibre reinforced plastic has been carried out. A particular version of this hinge was constructed and its moment-rotation profile during quasi-static deployment was measured. Folding and deployment simulations of the tape-spring hinge were carried out with the commercial finite element package Abaqus/Explicit, starting from the as-built, unstrained structure. The folding simulation includes the effects of pinching the hinge in the middle to reduce the peak moment required to fold it. The deployment simulation fully captures both the steady-state moment part of the deployment and the final snap back to the deployed configuration. An alternative simulation without pinching the hinge provides an estimate of the maximum moment that could be carried by the hinge during operation. This moment is about double the snap-back moment for the particular hinge design that was considered. The dynamic deployment of a tape-spring hinge boom has been studied both experimentally and by means of detailed finite-element simulations. It has been shown that the deployment of the boom can be divided into three phases: deployment; latching, which may involve buckling of the tape springs and large rotations of the boom; and vibration of the boom in the latched configuration. The second phase is the most critical as the boom can fold backwards and hence interfere with other spacecraft components. A geometric optimisation study was carried out by parameterising the slot geometry in terms of slot length, width and end circle diameter. The stress-resultant based failure criterion was then used to analyse the safety of the structure. The optimisation study was focused on finding a hinge design that can be folded 180 degrees with the shortest possible slot length. Simulations have shown that the strains can be significantly reduced by allowing the end cross-sections to deform freely. Based on the simulations a failure-critical design and a failure-safe design were selected and experimentally verified. The failure-safe optimised design is six times stiffer in torsion, twice stiffer axially and stores two and a half times more strain energy than the previously considered design. Finally, an example of designing a 1 m long self-deployable boom that could be folded around a spacecraft has been presented. The safety of this two-hinge boom has been evaluated during both stowage and dynamic deployment. A safe design that latches without any overshoot was selected and validated by a dynamic deployment experiment.

[pt] As estruturas ajustáveis ou dobráveis (deployable structures) consistem em um grupo de estruturas capazes de modificar sua forma e volume para atender a uma variedade de aplicações. Geralmente, são estruturas pré-fabricadas compostas por barras retas ou curvas unidas por meio de articulações que permitem que se movimentem para formar um arranjo compacto e se desdobrem em formas estruturais de grande porte. Essas estruturas atuam como mecanismos durante sua implantação e se tornam estruturas capazes de suportar cargas externas durante a fase de serviço. Além disso, elas devem ser leves e compactas a fim de serem facilmente transportadas e simples e rápidas de montar. Todas essas restrições dificultam a escolha dos melhores parâmetros que levem ao menor peso, maior rigidez e que permitam à estrutura desempenhar suas duas funcionalidades e garantir sua reutilização. Entre os tipos de estruturas dobráveis, os elementos pantográficos (tesouras) têm despertado grande interesse de engenheiros e arquitetos nos últimos anos. Este estudo avalia o comportamento não linear geométrico de arcos planos constituídos por dois tipos clássicos de elementos pantográficos: polar e translacional. Para isso, uma análise geométrica não linear detalhada é conduzida por meio de software de elemento finito elaborado com formulação corrotacional, a fim de avaliar a influência dos parâmetros geométricos da estrutura, como o tipo de elemento patográfico, o tipo de apoio e número de elementos no comportamento não linear e na estabilidade da estrutura. Os resultados obtidos por nossas análises revelam, na maioria dos casos, um comportamento eminentemente não linear característico de arcos esbeltos, com o caminho de equilíbrio exibindo vários pontos limites de carga e deslocamento, onde podem ocorrer saltos para configurações remotas e indesejáveis. Com base nestes resultadods, a influência dos parâmetros do sistema na capacidade de carga do arco é quantificada. / [en] Deployable structures consist of a group of structures capable of modifying their shape and volume in order to meet a range of conditions and needs. They are usually prefabricated structures consisting of straight or curved bars linked together in a compact bundle, which can then be unfolded into large-span, load bearing structural shapes. These structures have dual functionality since they act as mechanisms during its deployment and become immovable structures capable of supporting external loads during the service phase. In addition, they should be lightweight and compact to be easily transported and simple and quick to deploy. All these restrictions make it difficult to choose the best parameters regarding the shape and material of the structure, since many analyzes must be performed in order to find parameters that give lowest weight, highest stiffness, and that allow the structure to perform its two functions and ensure its reuse. Among the types of folding structure, those made of pantographic elements (scissors) have attracted great interest from engineers and architects in recent years. This study evaluates the geometric nonlinear behavior of plane arches constituted by two classic type of pantographic element, namely: polar and translational. For this, a detailed nonlinear geometrical analysis is conducted through tailored corotational finite element software in order to evaluate the influence of the structure s geometrical parameters, type of scissor units and supports on the nonlinear behavior and stability of the structure. The results obtained by our analyzes reveal, in most cases, a characteristic non-linear behavior of these structures with the nonlinear equilibrium path exhibiting several load and displacement limit points where jumps to remote and undesirable configurations may occur. Based on them, the influence of system parameters on the load carrying capacity of the arch is quantified.

Les mètres rubans sont utilisés comme dispositif de déploiement car ils sont légers, compacts, se déploient de manière autonome et ont une capacité d'auto-blocage en position déployée. Ces structures élancées de forme cylindrique présentent un comportement complexe avec formation de plis localisés. Leur modélisation est donc difficile : bien que des modèles de poutre à section flexible (RFleXS) aient été développés. Les travaux réalisés consistent à développer des outils numériques d'aide au dimensionnement de structures déployées par des rubans. Un modèle RFleXS adimensionné dédié aux rubans peu profonds est introduit et analysé, mettant en évidence des liens avec le modèle de barre d'Ericksen régularisé. Ces liens expliquent la formation de plis et caractérisent les trois zones constitutives d'un pli. On détermine de façon analytique le nombre et la position des points de bifurcation des branches de solution obtenues pour un essai de flexion pure d'un ruban. Un enrichissement de la cinématique de section est intégré dans les modèles RFleXS. Les simulations de flexion de ruban montrent alors une bonne corrélation avec les modèles de coque. Une nouvelle formulation des modèles RFleXS est implémentée et conduisant au développement de deux outils numériques : un code de calcul par éléments finis complet et un élément à deux noeuds intégré dans un code commercial. Des essais de flexion réalisés sur des rubans composites viennent compléter ces travaux afin de confronter les simulations numériques à des essais réels. Bien que des écarts soient observés, le comportement global du ruban est bien retranscrit par les modèles de poutre à section flexible. / Due to their lightness, compactness, their autonomous deployment and their ability to self-locking while deployed, tape-springs are considered to deploy structures. These slender and cylindrical structures highlight a complex behaviour because of the formation of localised folds. Tape-springs are then difficult to model but a rod model with flexible cross-section (RFleXS) has been developed in order to characterise the tape-spring behaviour.The purpose of this PhD was to develop numerical tools dedicated to design structures deployed by tape-spings. A dimensionless form of the RFleXS model dedicated to shallow tape spring has been developed and links with a regularised Ericksen's bar have been made. These links help to explain folds creation and to determine characteristics of the three constitutive areas of a fold. Analysis of the dimensionless model leads to determine the finite number and the position of bifurcation points for the pure bending of a tape-spring. The cross-section kinematic is enriched; simulations of bending tests then show a good correlation with shell models. A new implementation of RFleXS models is introduced, leading to the creation of two numerical tools: a full finite element software and a one-dimensional element with two nodes incorporated in Abaqus. Some bending experiments have been performed in order to compare simulations with measured data. Even if discrepancies are observed, these comparisons show that the tape-spring overall behaviour is well predicted by rod models with flexible cross-section.

The use of small satellites is rapidly growing, especially satellites with masses between 1 and 10 kg and few litres of volume. The main reasons are due to the low development time and cost. Electronics miniaturization and high density integration is enabling the small satellites class to perform more and better tasks and at a lower cost. When deployable structures are required for the missions, the actual paradigm is that there are very few that have been successfully developed and flown. It is usually not possible to scale down existing deployable structures from larger satellites. Power and attitude control is also very limited in small satellites thus, completely new deployable structures, low mass and with high packaging ratio (yet large and with adequate mechanical properties when deployed) must be developed. Furthermore, such new structures are usually made of very thin and light materials which complicates the on-ground tests prior the launch. Therefore, advances in modelling and simulation deployable structures such as booms are also of great interest for the scientific community. This thesis and the papers included herein focus on the finite element modelling of a meter-class passively deployable boom – based on the SIMPLE boom by Thomas W. Murphey – and deployment simulations. Experimental tests were also carried on a boom prototype suspended from a gravity off-loading system. An analytical model produced certain parameters which are used for validation of the finite element model. The strain energy stored in the boom prior to deployment and spacecraft displacements during deployment agreed well. The deployment time, however, have discrepancies: the models predicted a deployment time six times faster than the experimental tests. For that reason the deployment simulations cannot be compared with the tests. The reason of the discrepancies are believed to be due to the actual material model and the contacts formulation used in the finite element model. The finite element simulations, however, shows a reasonable behaviour given the nature of the deployment thus, despite the necessary improvements, we believe that future improvements in the material and friction models will provide us more realistic results. / Användningen av små satelliter ökar snabbt, särskilt satelliter med en vikt på mellan 1 och 10 kg och bara några liters volym. De främsta orsakerna till detta är den korta utvecklingstiden och den låga kostnaden. Elektronikminiatyrisering och hög integreringsdensitet möjliggör för små satelliter att utföra fler och bättre uppgifter till en lägre kostnad. När utfällbara strukturer krävs för uppdragen är nuvarande läge att det är få som utvecklats och flugits framgångsrikt. Det är inte heller alltid möjligt att skala ner utfällbara strukturer som utformats för användning i större satelliter. I små satelliter är den tillgängliga elektriska energin och volymen starkt begränsade faktorer och därmed måste helt nya passivt utfällbara strukturer med låg vikt och liten packningsvolym, men ändå rätt storlek och mekaniska egenskaper när de är utfällda, utvecklas. Dessa strukturer är vanligen tillverkade av mycket tunna och lätta material, som komplicerar tester innan uppskjutningen p.g.a. tyngdkraften. Därför är det av stort intresse att noggrant kunna modellera och simulera ett tyngdlöst utfällningsförlopp. Denna licentiatuppsats och bilagda artiklar i fokuserar på finit elementmodellering och utfällningssimuleringar av en 1 meter lång passivt utfällbar bom baserad på SIMPLE-bommen som utformats av Thomas W. Murphey. Utfällningsexperiment har utförts på en prototyp av bommen upphängd i ett tyngdkraftskompenserande system. Analytiska modeller har använts för att validera simuleringarna och töjningsenergin som lagrats i bommen innan utfällning och rymdfarkostens förflyttning efter utfällning överensstämmer väl. Utfällningstiden avviker dock och båda modellerna predikterar en utfällningstid som är sex gånger snabbare än den tiden som observeras i experimenten. Anledningen till skillnaderna antas delvis bero på begränsningar i den använda materialmodellen och i algoritmer för hantering av kontakt i den finita elementmodellen. De finite elementsimuleringarna visar dock ett rimligt dynamisk beteende hos bommen baserat på vad som observerats i experimenten och även om modellen är i behov av förbättring så finns det stora förhoppningar att åstadkomma en mer realistisk modell genom införande av förbättrade kontakalgoritmer och nogrannare modellering av dämpning och friktion. / <p>QC 20130506</p>

[pt] Estruturas reticuladas espaciais com grandes vãos são encontradas em uma variedade de aplicações em engenharia. Muitas dessas estruturas apresentam um comportamento eminentemente não linear, envolvendo tanto não linearidades físicas quanto geométricas, o que leva em muitos casos a múltiplas configurações de equilíbrio. Em particular, estruturas biestáveis estão usualmente sujeitas a instabilidades por ponto limite (snap-through), bifurcações simétrica instável ao longo do caminho não linear de equilíbrio, instabilidade elástica de elementos individuais, devido à plastificação destes elementos ou a interação destes fenômenos. O presente trabalho tem como objetivo a análise detalhada de duas classes de estruturas biestáveis: treliças piramidais (instabilidade indesejada) e estruturas ajustáveis com elementos de tesoura (instabilidade desejada). Ferramentas teóricas e computacionais são desenvolvidas para a investigação da influência das medidas de deformação quadrática e logarítmica, deformações elasto-plásticas e instabilidades na resposta estática e dinâmica não linear de um módulo de treliça piramidal. Uma formulação corrotacional em elementos finitos é proposta para descrever a ligação espacial flexível encontrada nas estruturas ajustáveis biestáveis aqui estudadas. A análise de estruturas com grandes vãos formadas pela junção de módulos de treliças piramidais ou módulos ajustáveis é apresentada. Os resultados obtidos mostram que a presença e interação das diversas fontes de instabilidade têm uma grande influência no comportamento destas estruturas e pode determinar ou não a sua viabilidade em aplicações práticas. / [en] Large span reticulated structures are applied in a variety of engineering applications. Many of these structures present a nonlinear behavior involving both geometric and material nonlinearities with multistable configurations. Particularly, bistable structures are often subjected to instability phenomena, such as snap-through and bifurcations of the whole structure, individual units or single bars. The present work, focuses on two classes of bistable structural systems: pyramidal trusses (undesired instability) and deployable scissor structures (desired design instability). Theoretical and computational tools are developed to investigate the influence of the strain measure, elasto-plastic deformations and instability phenomena on the nonlinear static and dynamic response of bistable pyramidal trusses. A compliant corrotational spatial joint finite element formulation with finite size is developed and applied to study bistable deployable scissor modules. The analysis of bistable large span structures formed by the assembly of modules is also carried out. It s shown that the presence and interaction of the studied buckling sources have deep influence on the systems behavior and can ultimately determine their viability in practical applications.

Rapidly assembled tent structures are temporary enclosures used to house people or goods. Their uses vary to include recreation, refugee housing, and military shelters. The structural concepts applied in these shelters are as variable as their uses. Some make use of a tensioned fabric and pole system to provide structural strength. Others have a load-bearing frame with attached fabric skin. Further variants make use of inflatable arches or consist of modular containers. Analysis of a number of different types of rapidly assembled tent structures reveals an area where innovation can occur. Conflicts in the last ten years suggest that rapidly assembled shelters for both military purposes and humanitarian relief have the greatest need for innovative solutions. Existing shelters used by the military lack the versatility and speed of deployment necessary in modern conflict. The lack of scalability in the designs makes it difficult to use an existing tent in different situations. They are slow to construct, heavy, and difficult to transport in large numbers. These problems suggest that there is a need for new shelters that better meet the needs of the military. The application of deployable structures technology meets military's needs for structures with the advantages of a small compacted volume, rapid assembly, and ease of deployment. This makes them ideal for application to shelter structures. The aim of this dissertation was to develop a new type of deployable, long-span shelter frame based upon tiled Bennett mechanisms. An overlapping combination of equilateral Bennett mechanisms yields a structure that opens into a half-cylinder shape, providing an enclosed space useful and applicable to the problem of deployable shelters. The specific application considered in the design portion of this process will be a long-span deployable shelter capable of housing military helicopters. This report details the development of the Bennett Shelter concept. Its deployed and compacted geometries are explored, and a procedure for determining structural properties and dimensions is presented. The full concept for the structure, from outer covering to foundation support is then detailed. Loads affecting the structure are determined, and the process of modelling and analysing the structure is then considered. Optimisation of the structure with respect to weight and serviceability requirements is conducted using a number of different materials, and full analysis of the optimal geometries is completed. As no method exists for evaluating the effect of imperfections on the deployment of overconstrained mechanisms, a procedure is derived. The effects of manufacturing imperfections on deployment of the Bennett mechanism are then explored using the method. A full examination of the variation of energy within the Bennett Shelter during deployment provides valuable insight into the performance of the structure. With the above analysis complete, it is shown that the Bennett Shelter is viable as a long-span deployable shelter.

Dans un contexte où l'utilisation de structures déployables s'est généralisée, le département Recherche de la société Thales Alenia Space étudie un nouveau concept de télescope spatial dont le miroir secondaire est déployé grâce au déroulement de six mètres rubans. Des études antérieures ont permis la mise au point d'un prototype constitué de rubans métalliques dont le déploiement s'est avéré trop violent. Dans ce travail de thèse nous proposons d'une part un nouveau type de ruban à la vitesse de déroulement maîtrisable et d'autre part un modèle original décrivant le comportement dynamique de tels rubans, permettant de mieux appréhender les phénomènes complexes pouvant intervenir lors de scénarios de pliage, de déploiement ou de déroulement. La solution envisagée pour contrôler la vitesse de déroulement repose sur l'exploitation des propriétés mécaniques d'une couche de matériau viscoélastique collée à la surface du ruban. Ces propriétés variant avec la température permettent de garantir un maintien de la position enroulée à froid et assurent un déroulement régulé grâce à un réchauffage localisé. Ces phénomènes ont été mis en évidence expérimentalement et numériquement. La lourdeur des méthodes classiques de modélisation et le manque de richesse des méthodes simplifiées nous ont conduit à développer un modèle de poutre à section fortement déformable permettant de décrire le comportement dynamique des rubans en grands déplacements. Partant d'un modèle de coque, l'originalité de la méthode repose essentiellement sur l'introduction d'une cinématique de type elastica pour décrire les grandes variations de forme de la section. Un modèle énergétique 1D est obtenu en intégrant dans la section et le problème est résolu à l'aide du logiciel de modélisation par éléments finis COMSOL. On propose finalement un modèle continu 1D à 4 paramètres cinématiques qui permet de rendre compte d'une large gamme de phénomènes intervenant dans des scénarios complexes de pliage, de déroulement et de déploiement dynamique. / The research department of Thales Alenia Space is studying new concepts of space telescopes whose secondary mirror is deployed thanks to the unreeling of six tape-springs. A breadboard using metallic tape-springs has been built during preliminary studies and has exhibited a deployment that is too energetic and induce too important shocks.In this thesis a new kind of tape-spring with a controlled uncoiling speed is introduced. Secondly a rod model with highly deformable thin-walled cross-sections describing the dynamic behaviour of tape-springs is derived.In order to over come the deployment speed of a tape spring, a viscoelastic layer is stuck on its sides. Thanks to its properties varying with the temperature, the viscoelastic layer is used to maintain the tape-spring in a coiled configuration at low temperature whereas a local heating leads to a controlled uncoiling. These phenomenons have been underlined experimentally and numerically.Because of the high complexity of classical shell models and the lack of details of simplified models, smart modelling methods need to be developed to describe the highly non linear behaviour of a tape-spring. A planar rod model with highly deformable thin-walled cross-sections that accounts for large displacements and large rotations in dynamics is proposed. Starting from a classical shellmodel, the main additional assumption consists in introducing an elastica kinematics to describe thelarge changes of the cross-section shape with very few parameters. The expressions of the strain andkinetic energies are derived by performing an analytical integration over the section. The Hamilton principle is directly introduced in a suitable finite element software to solve the problem. Several examples (folding, coiling and deployment of a tape spring) are studied through the FEM software COMSOL to demonstrate the ability of the 4-parameter model to account for several phenomena: creation of a single fold and associated snap-through behaviour, splitting of a fold into two, motion of a fold along the tape during a dynamic deployment, scenarios of coiling and uncoiling of a bistable tape-spring.

Ce travail a pour cadre une collaboration entre le LMA et Thales Alenia Space. Nous nous intéressons au comportement des structures flexibles et plus particulièrement des mètres rubans qui ont la particularité de pouvoir, grâce à l’aplatissement de la section, s’enrouler ou développer des pliages localisés. Une première thèse a permis d’une part la mise au point d’un nouveau type de mètre ruban au déroulement maîtrisable thermiquement et d’autre part le développement d’un modèle plan de poutre à section flexible. Dans le travail de thèse présenté ici, nous proposons une version étendue de ce modèle adaptée à la simulation du comportement dynamique tridimensionnel des mètres rubans en grands déplacements et en grandes rotations. Ce modèle est dérivé de la théorie des coques et repose sur l’introduction d’hypothèses cinématiques et sthéniques adaptées. La déformation de la section est caractérisée par celle de sa ligne moyenne qui peut se déformer dans son plan par flexion et torsion mais non par extension, ainsi que hors de son plan par gauchissement de torsion. Les fortes variations de forme de la section dans son plan peuvent alors être décrites par une cinématique de type Elastica, tandis qu’une cinématique de type Vlassov est utilisée pour définir le gauchissement dans le repère local attaché à la section. Le modèle unidimensionnel est obtenu par intégration sur la section des expressions de la théorie des coques, une approche énergétique permet ensuite de formuler le problème associé qui est résolu grâce au logiciel de modélisation par éléments finis COMSOL. / This work was carried out within the framework of a collaboration between the LMA and Thales Alenia Space. We focus on the behaviour of flexible structures and more specifically of tape springs, whose particularity lies in their capacity to coil up or to form localized folds through the flattening of their cross-section. A first thesis led to the development of a new type of tape spring whose uncoiling is controlled thermically on one hand and of a planar rod model with a flexible thin-walled cross-section on the other hand. In this thesis, we offer an extended version of this model dedicated to the simulation of three-dimensional dynamic behavior of tape springs in large displacements and large rotations. This model is derived from shell theory and is based on the introduction of adapted kinematic and sthenic hypotheses. The deformation of the cross-section is characterized by that of its average line which can deform in its own plane by flexion and twisting but not by extension, as well as out of its plane through torsional warping. The large changes of the cross-section shape in its plane can then be described by an Elastica kinematics, whereas a Vlassov kinematics is used to define the warping in the local frame attached to the section. The unidimensionnal model is obtained by integration over the cross-section of the expressions of the shell theory, an energetic approach then allows to express the associated problem which is solved thanks to the finite element modeling software COMSOL.