Flutter margin with non-linearities: real-time prediction of flutter onset speed

The present article shows a procedure to predict the flutter speed based on real-time tuning of a quasi non-linear aeroelastic model. A two-dimensional non-linear (freeplay) aeroeslastic model is implemented inMatLab/Simulink with incompressible aerodynamic conditions. A comparison with real compressible conditions is provided. Once the numerical validation is accomplished, a parametric aeroelastic model is built in order to describe the proposed procedure and contribute to reduce the number of flight hours needed to expand the flutter envelope.

Statistical mechanics approaches to granular media: between micromechanics and macromechanics

The mechanical behavior of granular materials has been traditionally approached through two theoretical and computational frameworks: macromechanics and micromechanics. Macromechanics focuses on continuum based models. In consequence it is assumed that the matter in the granular material is homogeneous and continuously distributed over its volume so that the smallest element cut from the body possesses the same physical properties as the body. In particular, it has some equivalent mechanical properties, represented by complex and non-linear constitutive relationships. Engineering problems are usually solved using computational methods such as FEM or FDM. On the other hand, micromechanics is the analysis of heterogeneous materials on the level of their individual constituents. In granular materials, if the properties of particles are known, a micromechanical approach can lead to a predictive response of the whole heterogeneous material. Two classes of numerical techniques can be differentiated: computational micromechanics, which consists on applying continuum mechanics on each of the phases of a representative volume element and then solving numerically the equations, and atomistic methods (DEM), which consist on applying rigid body dynamics together with interaction potentials to the particles. Statistical mechanics approaches arise between micro and macromechanics. It tries to state which the expected macroscopic properties of a granular system are, by starting from a micromechanical analysis of the features of the particles and the interactions. The main objective of this paper is to introduce this approach.

Discontinuous solutions and boundary value problems for non-linearly elastic fibre-reinforced materials

En este trabajo se han analizado varios problemas en el contexto de la elasticidad no lineal basándose en modelos constitutivos representativos. En particular, se han analizado problemas relacionados con el fenómeno de perdida de estabilidad asociada con condiciones de contorno en el caso de material reforzados con fibras. Cada problema se ha formulado y se ha analizado por separado en diferentes capítulos. En primer lugar se ha mostrado el análisis del gradiente de deformación discontinuo para un material transversalmente isótropo, en particular, el modelo del material considerado consiste de una base neo-Hookeana isótropa incrustada con fibras de refuerzo direccional caracterizadas con un solo parámetro. La solución de este problema se vincula con instabilidades que dan lugar al mecanismo de fallo conocido como banda de cortante. La perdida de elipticidad de las ecuaciones diferenciales de equilibrio es una condición necesaria para que aparezca este tipo de soluciones y por tanto las inestabilidades asociadas. En segundo lugar se ha analizado una deformación combinada de extensión, inación y torsión de un tubo cilíndrico grueso donde se ha encontrado que la deformación citada anteriormente puede ser controlada solo para determinadas direcciones de las fibras refuerzo. Para entender el comportamiento elástico del tubo considerado se ha ilustrado numéricamente los resultados obtenidos para las direcciones admisibles de las fibras de refuerzo bajo la deformación considerada. En tercer lugar se ha estudiado el caso de un tubo cilíndrico grueso reforzado con dos familias de fibras sometido a cortante en la dirección azimutal para un modelo de refuerzo especial. En este problema se ha encontrado que las inestabilidades que aparecen en el material considerado están asociadas con lo que se llama soluciones múltiples de la ecuación diferencial de equilibrio. Se ha encontrado que el fenómeno de instabilidad ocurre en un estado de deformación previo al estado de deformación donde se pierde la elipticidad de la ecuación diferencial de equilibrio. También se ha demostrado que la condición de perdida de elipticidad y ^W=2 = 0 (la segunda derivada de la función de energía con respecto a la deformación) son dos condiciones necesarias para la existencia de soluciones múltiples. Finalmente, se ha analizado detalladamente en el contexto de elipticidad un problema de un tubo cilíndrico grueso sometido a una deformación combinada en las direcciones helicoidal, axial y radial para distintas geotermias de las fibras de refuerzo . In the present work four main problems have been addressed within the framework of non-linear elasticity based on representative constitutive models. Namely, problems related to the loss of stability phenomena associated with boundary value problems for fibre-reinforced materials. Each of the considered problems is formulated and analysed separately in different chapters. We first start with the analysis of discontinuous deformation gradients for a transversely isotropic material under plane deformation. In particular, the material model is an augmented neo-Hookean base with a simple unidirectional reinforcement characterised by a single parameter. The solution of this problem is related to material instabilities and it is associated with a shear band-type failure mode. The loss of ellipticity of the governing differential equations is a necessary condition for the existence of these material instabilities. The second problem involves a detailed analysis of the combined non-linear extension, inflation and torsion of a thick-walled circular cylindrical tube where it has been found that the aforementioned deformation is controllable only for certain preferred directions of transverse isotropy. Numerical results have been illustrated to understand the elastic behaviour of the tube for the admissible preferred directions under the considered deformation. The third problem deals with the analysis of a doubly fibre-reinforced thickwalled circular cylindrical tube undergoing pure azimuthal shear for a special class of the reinforcing model where multiple non-smooth solutions emerge. The associated instability phenomena are found to occur prior to the point where the nominal stress tensor changes monotonicity in a particular direction. It has been also shown that the loss of ellipticity condition that arises from the equilibrium equation and ^W=2 = 0 (the second derivative of the strain-energy function with respect to the deformation) are equivalent necessary conditions for the emergence of multiple solutions for the considered material. Finally, a detailed analysis in the basis of the loss of ellipticity of the governing differential equations for a combined helical, axial and radial elastic deformations of a fibre-reinforced circular cylindrical tube is carried out.

Caracterización de un robot para aplicaciones de mecanizado con requerimientos de tolerancias

El presente Trabajo fin Fin de Máster, versa sobre una caracterización preliminar del comportamiento de un robot de tipo industrial, configurado por 4 eslabones y 4 grados de libertad, y sometido a fuerzas de mecanizado en su extremo. El entorno de trabajo planteado es el de plantas de fabricación de piezas de aleaciones de aluminio para automoción. Este tipo de componentes parte de un primer proceso de fundición que saca la pieza en bruto. Para series medias y altas, en función de las propiedades mecánicas y plásticas requeridas y los costes de producción, la inyección a alta presión (HPDC) y la fundición a baja presión (LPC) son las dos tecnologías más usadas en esta primera fase. Para inyección a alta presión, las aleaciones de aluminio más empleadas son, en designación simbólica según norma EN 1706 (entre paréntesis su designación numérica); EN AC AlSi9Cu3(Fe) (EN AC 46000) , EN AC AlSi9Cu3(Fe)(Zn) (EN AC 46500), y EN AC AlSi12Cu1(Fe) (EN AC 47100). Para baja presión, EN AC AlSi7Mg0,3 (EN AC 42100). En los 3 primeros casos, los límites de Silicio permitidos pueden superan el 10%. En el cuarto caso, es inferior al 10% por lo que, a los efectos de ser sometidas a mecanizados, las piezas fabricadas en aleaciones con Si superior al 10%, se puede considerar que son equivalentes, diferenciándolas de la cuarta. Las tolerancias geométricas y dimensionales conseguibles directamente de fundición, recogidas en normas como ISO 8062 o DIN 1688-1, establecen límites para este proceso. Fuera de esos límites, las garantías en conseguir producciones con los objetivos de ppms aceptados en la actualidad por el mercado, obligan a ir a fases posteriores de mecanizado. Aquellas geometrías que, funcionalmente, necesitan disponer de unas tolerancias geométricas y/o dimensionales definidas acorde a ISO 1101, y no capaces por este proceso inicial de moldeado a presión, deben ser procesadas en una fase posterior en células de mecanizado. En este caso, las tolerancias alcanzables para procesos de arranque de viruta se recogen en normas como ISO 2768. Las células de mecanizado se componen, por lo general, de varios centros de control numérico interrelacionados y comunicados entre sí por robots que manipulan las piezas en proceso de uno a otro. Dichos robots, disponen en su extremo de una pinza utillada para poder coger y soltar las piezas en los útiles de mecanizado, las mesas de intercambio para cambiar la pieza de posición o en utillajes de equipos de medición y prueba, o en cintas de entrada o salida. La repetibilidad es alta, de centésimas incluso, definida según norma ISO 9283. El problema es que, estos rangos de repetibilidad sólo se garantizan si no se hacen esfuerzos o éstos son despreciables (caso de mover piezas). Aunque las inercias de mover piezas a altas velocidades hacen que la trayectoria intermedia tenga poca precisión, al inicio y al final (al coger y dejar pieza, p.e.) se hacen a velocidades relativamente bajas que hacen que el efecto de las fuerzas de inercia sean menores y que permiten garantizar la repetibilidad anteriormente indicada. No ocurre así si se quitara la garra y se intercambia con un cabezal motorizado con una herramienta como broca, mandrino, plato de cuchillas, fresas frontales o tangenciales… Las fuerzas ejercidas de mecanizado generarían unos pares en las uniones tan grandes y tan variables que el control del robot no sería capaz de responder (o no está preparado, en un principio) y generaría una desviación en la trayectoria, realizada a baja velocidad, que desencadenaría en un error de posición (ver norma ISO 5458) no asumible para la funcionalidad deseada. Se podría llegar al caso de que la tolerancia alcanzada por un pretendido proceso más exacto diera una dimensión peor que la que daría el proceso de fundición, en principio con mayor variabilidad dimensional en proceso (y por ende con mayor intervalo de tolerancia garantizable). De hecho, en los CNCs, la precisión es muy elevada, (pudiéndose despreciar en la mayoría de los casos) y no es la responsable de, por ejemplo la tolerancia de posición al taladrar un agujero. Factores como, temperatura de la sala y de la pieza, calidad constructiva de los utillajes y rigidez en el amarre, error en el giro de mesas y de colocación de pieza, si lleva agujeros previos o no, si la herramienta está bien equilibrada y el cono es el adecuado para el tipo de mecanizado… influyen más. Es interesante que, un elemento no específico tan común en una planta industrial, en el entorno anteriormente descrito, como es un robot, el cual no sería necesario añadir por disponer de él ya (y por lo tanto la inversión sería muy pequeña), puede mejorar la cadena de valor disminuyendo el costo de fabricación. Y si se pudiera conjugar que ese robot destinado a tareas de manipulación, en los muchos tiempos de espera que va a disfrutar mientras el CNC arranca viruta, pudiese coger un cabezal y apoyar ese mecanizado; sería doblemente interesante. Por lo tanto, se antoja sugestivo poder conocer su comportamiento e intentar explicar qué sería necesario para llevar esto a cabo, motivo de este trabajo. La arquitectura de robot seleccionada es de tipo SCARA. La búsqueda de un robot cómodo de modelar y de analizar cinemática y dinámicamente, sin limitaciones relevantes en la multifuncionalidad de trabajos solicitados, ha llevado a esta elección, frente a otras arquitecturas como por ejemplo los robots antropomórficos de 6 grados de libertad, muy populares a nivel industrial. Este robot dispone de 3 uniones, de las cuales 2 son de tipo par de revolución (1 grado de libertad cada una) y la tercera es de tipo corredera o par cilíndrico (2 grados de libertad). La primera unión, de tipo par de revolución, sirve para unir el suelo (considerado como eslabón número 1) con el eslabón número 2. La segunda unión, también de ese tipo, une el eslabón número 2 con el eslabón número 3. Estos 2 brazos, pueden describir un movimiento horizontal, en el plano X-Y. El tercer eslabón, está unido al eslabón número 4 por la unión de tipo corredera. El movimiento que puede describir es paralelo al eje Z. El robot es de 4 grados de libertad (4 motores). En relación a los posibles trabajos que puede realizar este tipo de robot, su versatilidad abarca tanto operaciones típicas de manipulación como operaciones de arranque de viruta. Uno de los mecanizados más usuales es el taladrado, por lo cual se elige éste para su modelización y análisis. Dentro del taladrado se elegirá para acotar las fuerzas, taladrado en macizo con broca de diámetro 9 mm. El robot se ha considerado por el momento que tenga comportamiento de sólido rígido, por ser el mayor efecto esperado el de los pares en las uniones. Para modelar el robot se utiliza el método de los sistemas multicuerpos. Dentro de este método existen diversos tipos de formulaciones (p.e. Denavit-Hartenberg). D-H genera una cantidad muy grande de ecuaciones e incógnitas. Esas incógnitas son de difícil comprensión y, para cada posición, hay que detenerse a pensar qué significado tienen. Se ha optado por la formulación de coordenadas naturales. Este sistema utiliza puntos y vectores unitarios para definir la posición de los distintos cuerpos, y permite compartir, cuando es posible y se quiere, para definir los pares cinemáticos y reducir al mismo tiempo el número de variables. Las incógnitas son intuitivas, las ecuaciones de restricción muy sencillas y se reduce considerablemente el número de ecuaciones e incógnitas. Sin embargo, las coordenadas naturales “puras” tienen 2 problemas. El primero, que 2 elementos con un ángulo de 0 o 180 grados, dan lugar a puntos singulares que pueden crear problemas en las ecuaciones de restricción y por lo tanto han de evitarse. El segundo, que tampoco inciden directamente sobre la definición o el origen de los movimientos. Por lo tanto, es muy conveniente complementar esta formulación con ángulos y distancias (coordenadas relativas). Esto da lugar a las coordenadas naturales mixtas, que es la formulación final elegida para este TFM. Las coordenadas naturales mixtas no tienen el problema de los puntos singulares. Y la ventaja más importante reside en su utilidad a la hora de aplicar fuerzas motrices, momentos o evaluar errores. Al incidir sobre la incógnita origen (ángulos o distancias) controla los motores de manera directa. El algoritmo, la simulación y la obtención de resultados se ha programado mediante Matlab. Para realizar el modelo en coordenadas naturales mixtas, es preciso modelar en 2 pasos el robot a estudio. El primer modelo se basa en coordenadas naturales. Para su validación, se plantea una trayectoria definida y se analiza cinemáticamente si el robot satisface el movimiento solicitado, manteniendo su integridad como sistema multicuerpo. Se cuantifican los puntos (en este caso inicial y final) que configuran el robot. Al tratarse de sólidos rígidos, cada eslabón queda definido por sus respectivos puntos inicial y final (que son los más interesantes para la cinemática y la dinámica) y por un vector unitario no colineal a esos 2 puntos. Los vectores unitarios se colocan en los lugares en los que se tenga un eje de rotación o cuando se desee obtener información de un ángulo. No son necesarios vectores unitarios para medir distancias. Tampoco tienen por qué coincidir los grados de libertad con el número de vectores unitarios. Las longitudes de cada eslabón quedan definidas como constantes geométricas. Se establecen las restricciones que definen la naturaleza del robot y las relaciones entre los diferentes elementos y su entorno. La trayectoria se genera por una nube de puntos continua, definidos en coordenadas independientes. Cada conjunto de coordenadas independientes define, en un instante concreto, una posición y postura de robot determinada. Para conocerla, es necesario saber qué coordenadas dependientes hay en ese instante, y se obtienen resolviendo por el método de Newton-Rhapson las ecuaciones de restricción en función de las coordenadas independientes. El motivo de hacerlo así es porque las coordenadas dependientes deben satisfacer las restricciones, cosa que no ocurre con las coordenadas independientes. Cuando la validez del modelo se ha probado (primera validación), se pasa al modelo 2. El modelo número 2, incorpora a las coordenadas naturales del modelo número 1, las coordenadas relativas en forma de ángulos en los pares de revolución (3 ángulos; ϕ1, ϕ 2 y ϕ3) y distancias en los pares prismáticos (1 distancia; s). Estas coordenadas relativas pasan a ser las nuevas coordenadas independientes (sustituyendo a las coordenadas independientes cartesianas del modelo primero, que eran coordenadas naturales). Es necesario revisar si el sistema de vectores unitarios del modelo 1 es suficiente o no. Para este caso concreto, se han necesitado añadir 1 vector unitario adicional con objeto de que los ángulos queden perfectamente determinados con las correspondientes ecuaciones de producto escalar y/o vectorial. Las restricciones habrán de ser incrementadas en, al menos, 4 ecuaciones; una por cada nueva incógnita. La validación del modelo número 2, tiene 2 fases. La primera, al igual que se hizo en el modelo número 1, a través del análisis cinemático del comportamiento con una trayectoria definida. Podrían obtenerse del modelo 2 en este análisis, velocidades y aceleraciones, pero no son necesarios. Tan sólo interesan los movimientos o desplazamientos finitos. Comprobada la coherencia de movimientos (segunda validación), se pasa a analizar cinemáticamente el comportamiento con trayectorias interpoladas. El análisis cinemático con trayectorias interpoladas, trabaja con un número mínimo de 3 puntos máster. En este caso se han elegido 3; punto inicial, punto intermedio y punto final. El número de interpolaciones con el que se actúa es de 50 interpolaciones en cada tramo (cada 2 puntos máster hay un tramo), resultando un total de 100 interpolaciones. El método de interpolación utilizado es el de splines cúbicas con condición de aceleración inicial y final constantes, que genera las coordenadas independientes de los puntos interpolados de cada tramo. Las coordenadas dependientes se obtienen resolviendo las ecuaciones de restricción no lineales con el método de Newton-Rhapson. El método de las splines cúbicas es muy continuo, por lo que si se desea modelar una trayectoria en el que haya al menos 2 movimientos claramente diferenciados, es preciso hacerlo en 2 tramos y unirlos posteriormente. Sería el caso en el que alguno de los motores se desee expresamente que esté parado durante el primer movimiento y otro distinto lo esté durante el segundo movimiento (y así sucesivamente). Obtenido el movimiento, se calculan, también mediante fórmulas de diferenciación numérica, las velocidades y aceleraciones independientes. El proceso es análogo al anteriormente explicado, recordando la condición impuesta de que la aceleración en el instante t= 0 y en instante t= final, se ha tomado como 0. Las velocidades y aceleraciones dependientes se calculan resolviendo las correspondientes derivadas de las ecuaciones de restricción. Se comprueba, de nuevo, en una tercera validación del modelo, la coherencia del movimiento interpolado. La dinámica inversa calcula, para un movimiento definido -conocidas la posición, velocidad y la aceleración en cada instante de tiempo-, y conocidas las fuerzas externas que actúan (por ejemplo el peso); qué fuerzas hay que aplicar en los motores (donde hay control) para que se obtenga el citado movimiento. En la dinámica inversa, cada instante del tiempo es independiente de los demás y tiene una posición, una velocidad y una aceleración y unas fuerzas conocidas. En este caso concreto, se desean aplicar, de momento, sólo las fuerzas debidas al peso, aunque se podrían haber incorporado fuerzas de otra naturaleza si se hubiese deseado. Las posiciones, velocidades y aceleraciones, proceden del cálculo cinemático. El efecto inercial de las fuerzas tenidas en cuenta (el peso) es calculado. Como resultado final del análisis dinámico inverso, se obtienen los pares que han de ejercer los cuatro motores para replicar el movimiento prescrito con las fuerzas que estaban actuando. La cuarta validación del modelo consiste en confirmar que el movimiento obtenido por aplicar los pares obtenidos en la dinámica inversa, coinciden con el obtenido en el análisis cinemático (movimiento teórico). Para ello, es necesario acudir a la dinámica directa. La dinámica directa se encarga de calcular el movimiento del robot, resultante de aplicar unos pares en motores y unas fuerzas en el robot. Por lo tanto, el movimiento real resultante, al no haber cambiado ninguna condición de las obtenidas en la dinámica inversa (pares de motor y fuerzas inerciales debidas al peso de los eslabones) ha de ser el mismo al movimiento teórico. Siendo así, se considera que el robot está listo para trabajar. Si se introduce una fuerza exterior de mecanizado no contemplada en la dinámica inversa y se asigna en los motores los mismos pares resultantes de la resolución del problema dinámico inverso, el movimiento real obtenido no es igual al movimiento teórico. El control de lazo cerrado se basa en ir comparando el movimiento real con el deseado e introducir las correcciones necesarias para minimizar o anular las diferencias. Se aplican ganancias en forma de correcciones en posición y/o velocidad para eliminar esas diferencias. Se evalúa el error de posición como la diferencia, en cada punto, entre el movimiento teórico deseado en el análisis cinemático y el movimiento real obtenido para cada fuerza de mecanizado y una ganancia concreta. Finalmente, se mapea el error de posición obtenido para cada fuerza de mecanizado y las diferentes ganancias previstas, graficando la mejor precisión que puede dar el robot para cada operación que se le requiere, y en qué condiciones. -------------- This Master´s Thesis deals with a preliminary characterization of the behaviour for an industrial robot, configured with 4 elements and 4 degrees of freedoms, and subjected to machining forces at its end. Proposed working conditions are those typical from manufacturing plants with aluminium alloys for automotive industry. This type of components comes from a first casting process that produces rough parts. For medium and high volumes, high pressure die casting (HPDC) and low pressure die casting (LPC) are the most used technologies in this first phase. For high pressure die casting processes, most used aluminium alloys are, in simbolic designation according EN 1706 standard (between brackets, its numerical designation); EN AC AlSi9Cu3(Fe) (EN AC 46000) , EN AC AlSi9Cu3(Fe)(Zn) (EN AC 46500), y EN AC AlSi12Cu1(Fe) (EN AC 47100). For low pressure, EN AC AlSi7Mg0,3 (EN AC 42100). For the 3 first alloys, Si allowed limits can exceed 10% content. Fourth alloy has admisible limits under 10% Si. That means, from the point of view of machining, that components made of alloys with Si content above 10% can be considered as equivalent, and the fourth one must be studied separately. Geometrical and dimensional tolerances directly achievables from casting, gathered in standards such as ISO 8062 or DIN 1688-1, establish a limit for this process. Out from those limits, guarantees to achieve batches with objetive ppms currently accepted by market, force to go to subsequent machining process. Those geometries that functionally require a geometrical and/or dimensional tolerance defined according ISO 1101, not capable with initial moulding process, must be obtained afterwards in a machining phase with machining cells. In this case, tolerances achievables with cutting processes are gathered in standards such as ISO 2768. In general terms, machining cells contain several CNCs that they are interrelated and connected by robots that handle parts in process among them. Those robots have at their end a gripper in order to take/remove parts in machining fixtures, in interchange tables to modify position of part, in measurement and control tooling devices, or in entrance/exit conveyors. Repeatibility for robot is tight, even few hundredths of mm, defined according ISO 9283. Problem is like this; those repeatibilty ranks are only guaranteed when there are no stresses or they are not significant (f.e. due to only movement of parts). Although inertias due to moving parts at a high speed make that intermediate paths have little accuracy, at the beginning and at the end of trajectories (f.e, when picking part or leaving it) movement is made with very slow speeds that make lower the effect of inertias forces and allow to achieve repeatibility before mentioned. It does not happens the same if gripper is removed and it is exchanged by an spindle with a machining tool such as a drilling tool, a pcd boring tool, a face or a tangential milling cutter… Forces due to machining would create such big and variable torques in joints that control from the robot would not be able to react (or it is not prepared in principle) and would produce a deviation in working trajectory, made at a low speed, that would trigger a position error (see ISO 5458 standard) not assumable for requested function. Then it could be possible that tolerance achieved by a more exact expected process would turn out into a worst dimension than the one that could be achieved with casting process, in principle with a larger dimensional variability in process (and hence with a larger tolerance range reachable). As a matter of fact, accuracy is very tight in CNC, (its influence can be ignored in most cases) and it is not the responsible of, for example position tolerance when drilling a hole. Factors as, room and part temperature, manufacturing quality of machining fixtures, stiffness at clamping system, rotating error in 4th axis and part positioning error, if there are previous holes, if machining tool is properly balanced, if shank is suitable for that machining type… have more influence. It is interesting to know that, a non specific element as common, at a manufacturing plant in the enviroment above described, as a robot (not needed to be added, therefore with an additional minimum investment), can improve value chain decreasing manufacturing costs. And when it would be possible to combine that the robot dedicated to handling works could support CNCs´ works in its many waiting time while CNCs cut, and could take an spindle and help to cut; it would be double interesting. So according to all this, it would be interesting to be able to know its behaviour and try to explain what would be necessary to make this possible, reason of this work. Selected robot architecture is SCARA type. The search for a robot easy to be modeled and kinematically and dinamically analyzed, without significant limits in the multifunctionality of requested operations, has lead to this choice. Due to that, other very popular architectures in the industry, f.e. 6 DOFs anthropomorphic robots, have been discarded. This robot has 3 joints, 2 of them are revolute joints (1 DOF each one) and the third one is a cylindrical joint (2 DOFs). The first joint, a revolute one, is used to join floor (body 1) with body 2. The second one, a revolute joint too, joins body 2 with body 3. These 2 bodies can move horizontally in X-Y plane. Body 3 is linked to body 4 with a cylindrical joint. Movement that can be made is paralell to Z axis. The robt has 4 degrees of freedom (4 motors). Regarding potential works that this type of robot can make, its versatility covers either typical handling operations or cutting operations. One of the most common machinings is to drill. That is the reason why it has been chosen for the model and analysis. Within drilling, in order to enclose spectrum force, a typical solid drilling with 9 mm diameter. The robot is considered, at the moment, to have a behaviour as rigid body, as biggest expected influence is the one due to torques at joints. In order to modelize robot, it is used multibodies system method. There are under this heading different sorts of formulations (f.e. Denavit-Hartenberg). D-H creates a great amount of equations and unknown quantities. Those unknown quatities are of a difficult understanding and, for each position, one must stop to think about which meaning they have. The choice made is therefore one of formulation in natural coordinates. This system uses points and unit vectors to define position of each different elements, and allow to share, when it is possible and wished, to define kinematic torques and reduce number of variables at the same time. Unknown quantities are intuitive, constrain equations are easy and number of equations and variables are strongly reduced. However, “pure” natural coordinates suffer 2 problems. The first one is that 2 elements with an angle of 0° or 180°, give rise to singular positions that can create problems in constrain equations and therefore they must be avoided. The second problem is that they do not work directly over the definition or the origin of movements. Given that, it is highly recommended to complement this formulation with angles and distances (relative coordinates). This leads to mixed natural coordinates, and they are the final formulation chosen for this MTh. Mixed natural coordinates have not the problem of singular positions. And the most important advantage lies in their usefulness when applying driving forces, torques or evaluating errors. As they influence directly over origin variable (angles or distances), they control motors directly. The algorithm, simulation and obtaining of results has been programmed with Matlab. To design the model in mixed natural coordinates, it is necessary to model the robot to be studied in 2 steps. The first model is based in natural coordinates. To validate it, it is raised a defined trajectory and it is kinematically analyzed if robot fulfils requested movement, keeping its integrity as multibody system. The points (in this case starting and ending points) that configure the robot are quantified. As the elements are considered as rigid bodies, each of them is defined by its respectively starting and ending point (those points are the most interesting ones from the point of view of kinematics and dynamics) and by a non-colinear unit vector to those points. Unit vectors are placed where there is a rotating axis or when it is needed information of an angle. Unit vectors are not needed to measure distances. Neither DOFs must coincide with the number of unit vectors. Lengths of each arm are defined as geometrical constants. The constrains that define the nature of the robot and relationships among different elements and its enviroment are set. Path is generated by a cloud of continuous points, defined in independent coordinates. Each group of independent coordinates define, in an specific instant, a defined position and posture for the robot. In order to know it, it is needed to know which dependent coordinates there are in that instant, and they are obtained solving the constraint equations with Newton-Rhapson method according to independent coordinates. The reason to make it like this is because dependent coordinates must meet constraints, and this is not the case with independent coordinates. When suitability of model is checked (first approval), it is given next step to model 2. Model 2 adds to natural coordinates from model 1, the relative coordinates in the shape of angles in revoluting torques (3 angles; ϕ1, ϕ 2 and ϕ3) and distances in prismatic torques (1 distance; s). These relative coordinates become the new independent coordinates (replacing to cartesian independent coordinates from model 1, that they were natural coordinates). It is needed to review if unit vector system from model 1 is enough or not . For this specific case, it was necessary to add 1 additional unit vector to define perfectly angles with their related equations of dot and/or cross product. Constrains must be increased in, at least, 4 equations; one per each new variable. The approval of model 2 has two phases. The first one, same as made with model 1, through kinematic analysis of behaviour with a defined path. During this analysis, it could be obtained from model 2, velocities and accelerations, but they are not needed. They are only interesting movements and finite displacements. Once that the consistence of movements has been checked (second approval), it comes when the behaviour with interpolated trajectories must be kinematically analyzed. Kinematic analysis with interpolated trajectories work with a minimum number of 3 master points. In this case, 3 points have been chosen; starting point, middle point and ending point. The number of interpolations has been of 50 ones in each strecht (each 2 master points there is an strecht), turning into a total of 100 interpolations. The interpolation method used is the cubic splines one with condition of constant acceleration both at the starting and at the ending point. This method creates the independent coordinates of interpolated points of each strecht. The dependent coordinates are achieved solving the non-linear constrain equations with Newton-Rhapson method. The method of cubic splines is very continuous, therefore when it is needed to design a trajectory in which there are at least 2 movements clearly differents, it is required to make it in 2 steps and join them later. That would be the case when any of the motors would keep stopped during the first movement, and another different motor would remain stopped during the second movement (and so on). Once that movement is obtained, they are calculated, also with numerical differenciation formulas, the independent velocities and accelerations. This process is analogous to the one before explained, reminding condition that acceleration when t=0 and t=end are 0. Dependent velocities and accelerations are calculated solving related derivatives of constrain equations. In a third approval of the model it is checked, again, consistence of interpolated movement. Inverse dynamics calculates, for a defined movement –knowing position, velocity and acceleration in each instant of time-, and knowing external forces that act (f.e. weights); which forces must be applied in motors (where there is control) in order to obtain requested movement. In inverse dynamics, each instant of time is independent of the others and it has a position, a velocity, an acceleration and known forces. In this specific case, it is intended to apply, at the moment, only forces due to the weight, though forces of another nature could have been added if it would have been preferred. The positions, velocities and accelerations, come from kinematic calculation. The inertial effect of forces taken into account (weight) is calculated. As final result of the inverse dynamic analysis, the are obtained torques that the 4 motors must apply to repeat requested movement with the forces that were acting. The fourth approval of the model consists on confirming that the achieved movement due to the use of the torques obtained in the inverse dynamics, are in accordance with movements from kinematic analysis (theoretical movement). For this, it is necessary to work with direct dynamics. Direct dynamic is in charge of calculating the movements of robot that results from applying torques at motors and forces at the robot. Therefore, the resultant real movement, as there was no change in any condition of the ones obtained at the inverse dynamics (motor torques and inertial forces due to weight of elements) must be the same than theoretical movement. When these results are achieved, it is considered that robot is ready to work. When a machining external force is introduced and it was not taken into account before during the inverse dynamics, and torques at motors considered are the ones of the inverse dynamics, the real movement obtained is not the same than the theoretical movement. Closed loop control is based on comparing real movement with expected movement and introducing required corrrections to minimize or cancel differences. They are applied gains in the way of corrections for position and/or tolerance to remove those differences. Position error is evaluated as the difference, in each point, between theoretical movemment (calculated in the kinematic analysis) and the real movement achieved for each machining force and for an specific gain. Finally, the position error obtained for each machining force and gains are mapped, giving a chart with the best accuracy that the robot can give for each operation that has been requested and which conditions must be provided.

The influence of axial microgravity on the breakage of axisymmetric slender liquid bridges

The dynamics of inviscid, axisymmetric liquid bridges permits a simplified treatment if the bridge is long enough. Under such condition the evolution of the liquid zone is satisfactorily explained through a non-linear one-dimensional model. In the case of breaking, the one-dimensional model fails when the neck radius of the liquid column is close to zero; however, the model allows the calculation of the time variation of the liquid-bridge interface as well as of the fluid velocity field and, because the last part of the evolution is not needed, the overall results such as the breaking time and the volume of each of the two drops resulting after breakage can be calculated. In this paper numerical results concerning the behavior of clinical liquid bridges subjected to a small axial gravitational field are presented.

Analysis of Combined Bending and Punching Tests of Reinforced Concrete Slabs within IMPACT III Project

The paper reports on a collaborative effort between the Swiss Federal Nuclear Safety Inspectorate (ENSI) and their consultants Principia and Stangenberg. As part of the IMPACT III project, reduced scale impact tests of reinforced concrete structures were carried out. The simulation of test X3 is presented here and the numerical results are compared with those obtained in the test, carried out in August 2013. The general object is to improve the safety of nuclear facilities and, more specifically, to demonstrate the capabilities of current simulation techniques to reproduce the behaviour of a reinforced concrete structure impacted by a soft missile. The missile is a steel tube with a mass of 50 kg and travelling at 140 m/s. The target is a 250 mm thick, 2,1 m by 2,1 m reinforced concrete wall, held in a stiff supporting frame. The reinforcement includes both longitudinal and transverse rebars. Calculations were carried out before and after the test with Abaqus (Principia) and SOFiSTiK (Stangenberg). In the Abaqus simulation the concrete is modelled using solid elements and a damaged plasticity formulation, the rebars with embedded beam elements, and the missile with shell elements. In SOFiSTiK the target is modelled with non-linear, layered shell elements for the reinforcement on both sides; non-linear shear deformations of shell/plate elements are approximately included. The results generally indicate a good agreement between calculations and measurements.

A gradient-enhanced continuum damage model with application to fibre-reinforced tissues at finite strains

A non-local gradient-based damage formulation within a geometrically non-linear set- ting is presented. The hyperelastic constitutive response at local material point level is governed by a strain energy function which is additively composed by an isotropic neo-Hookean matrix and by an anisotropic fibre-reinforced material based on the model proposed by T. Gasser, R. Ogden, and G. Holzapfel.

Dynamic stability of long axisymmetric liquid bridges

This paper deals with the non-linear forced oscillations of axisymmetric long liquid bridges between equal disks. The dynamics of the liquid bridge has been analyzed by using a self-similar, one-dimensional model already used in similar problems. The influence of the dynamics on the static stability limits, as well as the main characteristics of the non-linear behaviour of long liquid bridges, have been studied with in the range of validity of the mathematical model used here.

Optimization of the Integrated Guidance and Control for a Dual Aerodynamic Control Missile

Las futuras misiones para misiles aire-aire operando dentro de la atmósfera requieren la interceptación de blancos a mayores velocidades y más maniobrables, incluyendo los esperados vehículos aéreos de combate no tripulados. La intercepción tiene que lograrse desde cualquier ángulo de lanzamiento. Una de las principales discusiones en la tecnología de misiles en la actualidad es cómo satisfacer estos nuevos requisitos incrementando la capacidad de maniobra del misil y en paralelo, a través de mejoras en los métodos de guiado y control modernos. Esta Tesis aborda estos dos objetivos simultáneamente, al proponer un diseño integrando el guiado y el control de vuelo (autopiloto) y aplicarlo a misiles con control aerodinámico simultáneo en canard y cola. Un primer avance de los resultados obtenidos ha sido publicado recientemente en el Journal of Aerospace Engineering, en Abril de 2015, [Ibarrondo y Sanz-Aranguez, 2015]. El valor del diseño integrado obtenido es que permite al misil cumplir con los requisitos operacionales mencionados empleando únicamente control aerodinámico. El diseño propuesto se compara favorablemente con esquemas más tradicionales, consiguiendo menores distancias de paso al blanco y necesitando de menores esfuerzos de control incluso en presencia de ruidos. En esta Tesis se demostrará cómo la introducción del doble mando, donde tanto el canard como las aletas de cola son móviles, puede mejorar las actuaciones de un misil existente. Comparado con un misil con control en cola, el doble control requiere sólo introducir dos servos adicionales para accionar los canards también en guiñada y cabeceo. La sección de cola será responsable de controlar el misil en balanceo mediante deflexiones diferenciales de los controles. En el caso del doble mando, la complicación añadida es que los vórtices desprendidos de los canards se propagan corriente abajo y pueden incidir sobre las superficies de cola, alterando sus características de control. Como un primer aporte, se ha desarrollado un modelo analítico completo para la aerodinámica no lineal de un misil con doble control, incluyendo la caracterización de este efecto de acoplamiento aerodinámico. Hay dos modos de funcionamiento en picado y guiñada para un misil de doble mando: ”desviación” y ”opuesto”. En modo ”desviación”, los controles actúan en la misma dirección, generando un cambio inmediato en la sustentación y produciendo un movimiento de translación en el misil. La respuesta es rápida, pero en el modo ”desviación” los misiles con doble control pueden tener dificultades para alcanzar grandes ángulos de ataque y altas aceleraciones laterales. Cuando los controles actúan en direcciones opuestas, el misil rota y el ángulo de ataque del fuselaje se incrementa para generar mayores aceleraciones en estado estacionario, aunque el tiempo de respuesta es mayor. Con el modelo aerodinámico completo, es posible obtener una parametrización dependiente de los estados de la dinámica de corto periodo del misil. Debido al efecto de acoplamiento entre los controles, la respuesta en bucle abierto no depende linealmente de los controles. El autopiloto se optimiza para obtener la maniobra requerida por la ley de guiado sin exceder ninguno de los límites aerodinámicos o mecánicos del misil. Una segunda contribución de la tesis es el desarrollo de un autopiloto con múltiples entradas de control y que integra la aerodinámica no lineal, controlando los tres canales de picado, guiñada y cabeceo de forma simultánea. Las ganancias del autopiloto dependen de los estados del misil y se calculan a cada paso de integración mediante la resolución de una ecuación de Riccati de orden 21x21. Las ganancias obtenidas son sub-óptimas, debido a que una solución completa de la ecuación de Hamilton-Jacobi-Bellman no puede obtenerse de manera práctica, y se asumen ciertas simplificaciones. Se incorpora asimismo un mecanismo que permite acelerar la respuesta en caso necesario. Como parte del autopiloto, se define una estrategia para repartir el esfuerzo de control entre el canard y la cola. Esto se consigue mediante un controlador aumentado situado antes del bucle de optimización, que minimiza el esfuerzo total de control para maniobrar. Esta ley de alimentación directa mantiene al misil cerca de sus condiciones de equilibrio, garantizando una respuesta transitoria adecuada. El controlador no lineal elimina la respuesta de fase no-mínima característica de la cola. En esta Tesis se consideran dos diseños para el guiado y control, el control en Doble-Lazo y el control Integrado. En la aproximación de Doble-Lazo, el autopiloto se sitúa dentro de un bucle interior y se diseña independientemente del guiado, que conforma el bucle más exterior del control. Esta estructura asume que existe separación espectral entre los dos, esto es, que los tiempos de respuesta del autopiloto son mucho mayores que los tiempos característicos del guiado. En el estudio se combina el autopiloto desarrollado con una ley de guiado óptimo. Los resultados obtenidos demuestran que se consiguen aumentos muy importantes en las actuaciones frente a misiles con control canard o control en cola, y que la interceptación, cuando se lanza cerca del curso de colisión, se consigue desde cualquier ángulo alrededor del blanco. Para el misil de doble mando, la estrategia óptima resulta en utilizar el modo de control opuesto en la aproximación al blanco y utilizar el modo de desviación justo antes del impacto. Sin embargo la lógica de doble bucle no consigue el impacto cuando hay desviaciones importantes con respecto al curso de colisión. Una de las razones es que parte de la demanda de guiado se pierde, ya que el misil solo es capaz de modificar su aceleración lateral, y no tiene control sobre su aceleración axial, a no ser que incorpore un motor de empuje regulable. La hipótesis de separación mencionada, y que constituye la base del Doble-Bucle, puede no ser aplicable cuando la dinámica del misil es muy alta en las proximidades del blanco. Si se combinan el guiado y el autopiloto en un único bucle, la información de los estados del misil está disponible para el cálculo de la ley de guiado, y puede calcularse la estrategia optima de guiado considerando las capacidades y la actitud del misil. Una tercera contribución de la Tesis es la resolución de este segundo diseño, la integración no lineal del guiado y del autopiloto (IGA) para el misil de doble control. Aproximaciones anteriores en la literatura han planteado este sistema en ejes cuerpo, resultando en un sistema muy inestable debido al bajo amortiguamiento del misil en cabeceo y guiñada. Las simplificaciones que se tomaron también causan que el misil se deslice alrededor del blanco y no consiga la intercepción. En nuestra aproximación el problema se plantea en ejes inerciales y se recurre a la dinámica de los cuaterniones, eliminado estos inconvenientes. No se limita a la dinámica de corto periodo del misil, porque se construye incluyendo de modo explícito la velocidad dentro del bucle de optimización. La formulación resultante en el IGA es independiente de la maniobra del blanco, que sin embargo se ha de incluir en el cálculo del modelo en Doble-bucle. Un típico inconveniente de los sistemas integrados con controlador proporcional, es el problema de las escalas. Los errores de guiado dominan sobre los errores de posición del misil y saturan el controlador, provocando la pérdida del misil. Este problema se ha tratado aquí con un controlador aumentado previo al bucle de optimización, que define un estado de equilibrio local para el sistema integrado, que pasa a actuar como un regulador. Los criterios de actuaciones para el IGA son los mismos que para el sistema de Doble-Bucle. Sin embargo el problema matemático resultante es muy complejo. El problema óptimo para tiempo finito resulta en una ecuación diferencial de Riccati con condiciones terminales, que no puede resolverse. Mediante un cambio de variable y la introducción de una matriz de transición, este problema se transforma en una ecuación diferencial de Lyapunov que puede resolverse mediante métodos numéricos. La solución resultante solo es aplicable en un entorno cercano del blanco. Cuando la distancia entre misil y blanco es mayor, se desarrolla una solución aproximada basada en la solución de una ecuación algebraica de Riccati para cada paso de integración. Los resultados que se han obtenido demuestran, a través de análisis numéricos en distintos escenarios, que la solución integrada es mejor que el sistema de Doble-Bucle. Las trayectorias resultantes son muy distintas. El IGA preserva el guiado del misil y consigue maximizar el uso de la propulsión, consiguiendo la interceptación del blanco en menores tiempos de vuelo. El sistema es capaz de lograr el impacto donde el Doble-Bucle falla, y además requiere un orden menos de magnitud en la cantidad de cálculos necesarios. El efecto de los ruidos radar, datos discretos y errores del radomo se investigan. El IGA es más robusto, resultando menos afectado por perturbaciones que el Doble- Bucle, especialmente porque el núcleo de optimización en el IGA es independiente de la maniobra del blanco. La estimación de la maniobra del blanco es siempre imprecisa y contaminada por ruido, y degrada la precisión de la solución de Doble-Bucle. Finalmente, como una cuarta contribución, se demuestra que el misil con guiado IGA es capaz de realizar una maniobra de defensa contra un blanco que ataque por su cola, sólo con control aerodinámico. Las trayectorias estudiadas consideran una fase pre-programada de alta velocidad de giro, manteniendo siempre el misil dentro de su envuelta de vuelo. Este procedimiento no necesita recurrir a soluciones técnicamente más complejas como el control vectorial del empuje o control por chorro para ejecutar esta maniobra. En todas las demostraciones matemáticas se utiliza el producto de Kronecker como una herramienta practica para manejar las parametrizaciones dependientes de variables, que resultan en matrices de grandes dimensiones. ABSTRACT Future missions for air to air endo-atmospheric missiles require the interception of targets with higher speeds and more maneuverable, including forthcoming unmanned supersonic combat vehicles. The interception will need to be achieved from any angle and off-boresight launch conditions. One of the most significant discussions in missile technology today is how to satisfy these new operational requirements by increasing missile maneuvering capabilities and in parallel, through the development of more advanced guidance and control methods. This Thesis addresses these two objectives by proposing a novel optimal integrated guidance and autopilot design scheme, applicable to more maneuverable missiles with forward and rearward aerodynamic controls. A first insight of these results have been recently published in the Journal of Aerospace Engineering in April 2015, [Ibarrondo and Sanz-Aránguez, 2015]. The value of this integrated solution is that it allows the missile to comply with the aforementioned requirements only by applying aerodynamic control. The proposed design is compared against more traditional guidance and control approaches with positive results, achieving reduced control efforts and lower miss distances with the integrated logic even in the presence of noises. In this Thesis it will be demonstrated how the dual control missile, where canard and tail fins are both movable, can enhance the capabilities of an existing missile airframe. Compared to a tail missile, dual control only requires two additional servos to actuate the canards in pitch and yaw. The tail section will be responsible to maintain the missile stabilized in roll, like in a classic tail missile. The additional complexity is that the vortices shed from the canard propagate downstream where they interact with the tail surfaces, altering the tail expected control characteristics. These aerodynamic phenomena must be properly described, as a preliminary step, with high enough precision for advanced guidance and control studies. As a first contribution we have developed a full analytical model of the nonlinear aerodynamics of a missile with dual control, including the characterization of this cross-control coupling effect. This development has been produced from a theoretical model validated with reliable practical data obtained from wind tunnel experiments available in the scientific literature, complement with computer fluid dynamics and semi-experimental methods. There are two modes of operating a missile with forward and rear controls, ”divert” and ”opposite” modes. In divert mode, controls are deflected in the same direction, generating an increment in direct lift and missile translation. Response is fast, but in this mode, dual control missiles may have difficulties in achieving large angles of attack and high level of lateral accelerations. When controls are deflected in opposite directions (opposite mode) the missile airframe rotates and the body angle of attack is increased to generate greater accelerations in steady-state, although the response time is larger. With the aero-model, a state dependent parametrization of the dual control missile short term dynamics can be obtained. Due to the cross-coupling effect, the open loop dynamics for the dual control missile is not linearly dependent of the fin positions. The short term missile dynamics are blended with the servo system to obtain an extended autopilot model, where the response is linear with the control fins turning rates, that will be the control variables. The flight control loop is optimized to achieve the maneuver required by the guidance law without exceeding any of the missile aerodynamic or mechanical limitations. The specific aero-limitations and relevant performance indicators for the dual control are set as part of the analysis. A second contribution of this Thesis is the development of a step-tracking multi-input autopilot that integrates non-linear aerodynamics. The designed dual control missile autopilot is a full three dimensional autopilot, where roll, pitch and yaw are integrated, calculating command inputs simultaneously. The autopilot control gains are state dependent, and calculated at each integration step solving a matrix Riccati equation of order 21x21. The resulting gains are sub-optimal as a full solution for the Hamilton-Jacobi-Bellman equation cannot be resolved in practical terms and some simplifications are taken. Acceleration mechanisms with an λ-shift is incorporated in the design. As part of the autopilot, a strategy is defined for proper allocation of control effort between canard and tail channels. This is achieved with an augmented feed forward controller that minimizes the total control effort of the missile to maneuver. The feedforward law also maintains the missile near trim conditions, obtaining a well manner response of the missile. The nonlinear controller proves to eliminate the non-minimum phase effect of the tail. Two guidance and control designs have been considered in this Thesis: the Two- Loop and the Integrated approaches. In the Two-Loop approach, the autopilot is placed in an inner loop and designed separately from an outer guidance loop. This structure assumes that spectral separation holds, meaning that the autopilot response times are much higher than the guidance command updates. The developed nonlinear autopilot is linked in the study to an optimal guidance law. Simulations are carried on launching close to collision course against supersonic and highly maneuver targets. Results demonstrate a large boost in performance provided by the dual control versus more traditional canard and tail missiles, where interception with the dual control close to collision course is achieved form 365deg all around the target. It is shown that for the dual control missile the optimal flight strategy results in using opposite control in its approach to target and quick corrections with divert just before impact. However the Two-Loop logic fails to achieve target interception when there are large deviations initially from collision course. One of the reasons is that part of the guidance command is not followed, because the missile is not able to control its axial acceleration without a throttleable engine. Also the separation hypothesis may not be applicable for a high dynamic vehicle like a dual control missile approaching a maneuvering target. If the guidance and autopilot are combined into a single loop, the guidance law will have information of the missile states and could calculate the most optimal approach to the target considering the actual capabilities and attitude of the missile. A third contribution of this Thesis is the resolution of the mentioned second design, the non-linear integrated guidance and autopilot (IGA) problem for the dual control missile. Previous approaches in the literature have posed the problem in body axes, resulting in high unstable behavior due to the low damping of the missile, and have also caused the missile to slide around the target and not actually hitting it. The IGA system is posed here in inertial axes and quaternion dynamics, eliminating these inconveniences. It is not restricted to the missile short term dynamic, and we have explicitly included the missile speed as a state variable. The IGA formulation is also independent of the target maneuver model that is explicitly included in the Two-loop optimal guidance law model. A typical problem of the integrated systems with a proportional control law is the problem of scales. The guidance errors are larger than missile state errors during most of the flight and result in high gains, control saturation and loss of control. It has been addressed here with an integrated feedforward controller that defines a local equilibrium state at each flight point and the controller acts as a regulator to minimize the IGA states excursions versus the defined feedforward state. The performance criteria for the IGA are the same as in the Two-Loop case. However the resulting optimization problem is mathematically very complex. The optimal problem in a finite-time horizon results in an irresoluble state dependent differential Riccati equation with terminal conditions. With a change of variable and the introduction of a transition matrix, the equation is transformed into a time differential Lyapunov equation that can be solved with known numerical methods in real time. This solution results range limited, and applicable when the missile is in a close neighborhood of the target. For larger ranges, an approximate solution is used, obtained from solution of an algebraic matrix Riccati equation at each integration step. The results obtained show, by mean of several comparative numerical tests in diverse homing scenarios, than the integrated approach is a better solution that the Two- Loop scheme. Trajectories obtained are very different in the two cases. The IGA fully preserves the guidance command and it is able to maximize the utilization of the missile propulsion system, achieving interception with lower miss distances and in lower flight times. The IGA can achieve interception against off-boresight targets where the Two- Loop was not able to success. As an additional advantage, the IGA also requires one order of magnitude less calculations than the Two-Loop solution. The effects of radar noises, discrete radar data and radome errors are investigated. IGA solution is robust, and less affected by radar than the Two-Loop, especially because the target maneuvers are not part of the IGA core optimization loop. Estimation of target acceleration is always imprecise and noisy and degrade the performance of the two-Loop solution. The IGA trajectories are such that minimize the impact of radome errors in the guidance loop. Finally, as a fourth contribution, it is demonstrated that the missile with IGA guidance is capable of performing a defense against attacks from its rear hemisphere, as a tail attack, only with aerodynamic control. The studied trajectories have a preprogrammed high rate turn maneuver, maintaining the missile within its controllable envelope. This solution does not recur to more complex features in service today, like vector control of the missile thrust or side thrusters. In all the mathematical treatments and demonstrations, the Kronecker product has been introduced as a practical tool to handle the state dependent parametrizations that have resulted in very high order matrix equations.

Variation in per capita interaction strength: Thresholds due to nonlinear dynamics and nonequilibrium conditions

I measured the strength of interaction between a marine herbivore and its growing resource over a realistic range of absolute and relative abundances. The herbivores (hermit crabs: Pagurus spp.) have slow and/or weak functional and numerical responses to epiphytic diatoms (Isthmia nervosa), which show logistic growth in the absence of consumers. By isolating this interaction in containers in the field, I mimicked many of the physical and biological variables characteristic of the intertidal while controlling the densities of focal species. The per capita effects of consumers on the population dynamics of their resource (i.e., interaction strength) were defined by using the relationship between hermit crab density and proportional change in the resource. When this relationship is fit by a Weibull function, a single parameter distinguishes constant interaction strength from one that varies as a function of density. Constant interaction strength causes the proportion of diatoms to fall linearly or proportionally as hermit crab density increases whereas per capita effects that increase with density cause an accelerating decline. Although many mathematical models of species interactions assume linear dynamics and invariant parameters, at least near equilibrium, the per capita effects of hermit crabs on diatoms varied substantially, apparently crossing a threshold from weak to strong when consumption exceeded resource production. This threshold separates a domain of coexistence from one of local extinction of the resource. Such thresholds may help explain trophic cascades, resource compensation, and context-dependent interaction strengths, while indicating a way to predict trophic effects, despite nonlinearities, as a function of vital rates.

Linear chaos for the Quick-Thinking-Driver model.

In recent years, the topic of car-following has experimented an increased importance in traffic engineering and safety research. This has become a very interesting topic because of the development of driverless cars (Google driverless cars, http://en.wikipedia.org/wiki/Google_driverless_car). Driving models which describe the interaction between adjacent vehicles in the same lane have a big interest in simulation modeling, such as the Quick-Thinking-Driver model. A non-linear version of it can be given using the logistic map, and then chaos appears. We show that an infinite-dimensional version of the linear model presents a chaotic behaviour using the same approach as for studying chaos of death models of cell growth.

Modelagem e controle de posição e orientação de uma Plataforma de Stewart

Este trabalho apresenta o controle de posição e orientação de um modelo não linear de Plataforma de Stewart com seis graus de liberdade construído no ambiente de sistemas multicorpos ADAMS® desenvolvido pela Mechanical Dynamics, Inc. O modelo não linear é exportado para o ambiente SIMULINK® desenvolvido pela MathWorks, Inc., onde o controle de posição e orientação é realizado a partir da linearização do modelo e a aplicação de um sistema seguidor com realimentação de estados. Utililiza-se, também o SIMULINK® para implementar a dinâmica de um sistema servoválvula e cilindro hidráulico com um servocontrole de pressão e assim simular o comportamento dinâmico de um simulador de vôo com acionamento hidráulico. A utilização destes pacotes comerciais visa obter uma economia de tempo e esforço na modelagem de sistemas mecânicos complexos e na programação para obtenção da resposta do sistema no tempo, além de facilitar a análise de várias configurações de Plataformas de Stewart

Mathematical approach for predicting non-negative tristimulus values using the CAT02 chromatic adaptation transform

It has been reported that for certain colour samples, the chromatic adaptation transform CAT02 imbedded in the CIECAM02 colour appearance model predicts corresponding colours with negative tristimulus values (TSVs), which can cause problems in certain applications. To overcome this problem, a mathematical approach is proposed for modifying CAT02. This approach combines a non-negativity constraint for the TSVs of corresponding colours with the minimization of the colour differences between those values for the corresponding colours obtained by visual observations and the TSVs of the corresponding colours predicted by the model, which is a constrained non-linear optimization problem. By solving the non-linear optimization problem, a new matrix is found. The performance of the CAT02 transform with various matrices including the original CAT02 matrix, and the new matrix are tested using visual datasets and the optimum colours. Test results show that the CAT02 with the new matrix predicted corresponding colours without negative TSVs for all optimum colours and the colour matching functions of the two CIE standard observers under the test illuminants considered. However, the accuracy with the new matrix for predicting the visual data is approximately 1 CIELAB colour difference unit worse compared with the original CAT02. This indicates that accuracy has to be sacrificed to achieve the non-negativity constraint for the TSVs of the corresponding colours.

Feedbacks between vegetation pattern and resource loss dramatically decrease ecosystem resilience and restoration potential in a simple dryland model

Conceptual frameworks of dryland degradation commonly include ecohydrological feedbacks between landscape spatial organization and resource loss, so that decreasing cover and size of vegetation patches result in higher water and soil losses, which lead to further vegetation loss. However, the impacts of these feedbacks on dryland dynamics in response to external stress have barely been tested. Using a spatially-explicit model, we represented feedbacks between vegetation pattern and landscape resource loss by establishing a negative dependence of plant establishment on the connectivity of runoff-source areas (e.g., bare soils). We assessed the impact of various feedback strengths on the response of dryland ecosystems to changing external conditions. In general, for a given external pressure, these connectivity-mediated feedbacks decrease vegetation cover at equilibrium, which indicates a decrease in ecosystem resistance. Along a gradient of gradual increase of environmental pressure (e.g., aridity), the connectivity-mediated feedbacks decrease the amount of pressure required to cause a critical shift to a degraded state (ecosystem resilience). If environmental conditions improve, these feedbacks increase the pressure release needed to achieve the ecosystem recovery (restoration potential). The impact of these feedbacks on dryland response to external stress is markedly non-linear, which relies on the non-linear negative relationship between bare-soil connectivity and vegetation cover. Modelling studies on dryland vegetation dynamics not accounting for the connectivity-mediated feedbacks studied here may overestimate the resistance, resilience and restoration potential of drylands in response to environmental and human pressures. Our results also suggest that changes in vegetation pattern and associated hydrological connectivity may be more informative early-warning indicators of dryland degradation than changes in vegetation cover.

Missing Rings in Pinus halepensis – The Missing Link to Relate the Tree-Ring Record to Extreme Climatic Events

Climate predictions for the Mediterranean Basin include increased temperatures, decreased precipitation, and increased frequency of extreme climatic events (ECE). These conditions are associated with decreased tree growth and increased vulnerability to pests and diseases. The anatomy of tree rings responds to these environmental conditions. Quantitatively, the width of a tree ring is largely determined by the rate and duration of cell division by the vascular cambium. In the Mediterranean climate, this division may occur throughout almost the entire year. Alternatively, cell division may cease during relatively cool and dry winters, only to resume in the same calendar year with milder temperatures and increased availability of water. Under particularly adverse conditions, no xylem may be produced in parts of the stem, resulting in a missing ring (MR). A dendrochronological network of Pinus halepensis was used to determine the relationship of MR to ECE. The network consisted of 113 sites, 1,509 trees, 2,593 cores, and 225,428 tree rings throughout the distribution range of the species. A total of 4,150 MR were identified. Binomial logistic regression analysis determined that MR frequency increased with increased cambial age. Spatial analysis indicated that the geographic areas of south-eastern Spain and northern Algeria contained the greatest frequency of MR. Dendroclimatic regression analysis indicated a non-linear relationship of MR to total monthly precipitation and mean temperature. MR are strongly associated with the combination of monthly mean temperature from previous October till current February and total precipitation from previous September till current May. They are likely to occur with total precipitation lower than 50 mm and temperatures higher than 5°C. This conclusion is global and can be applied to every site across the distribution area. Rather than simply being a complication for dendrochronology, MR formation is a fundamental response of trees to adverse environmental conditions. The demonstrated relationship of MR formation to ECE across this dendrochronological network in the Mediterranean basin shows the potential of MR analysis to reconstruct the history of past climatic extremes and to predict future forest dynamics in a changing climate.