Periodic waves and solitons in a nonlinear fibre with resonant impurities

We shall consider a coupled nonlinear Schrodinger equation- Bloch system of equations describing the propagation of a single pulse through a nonlinear dispersive waveguide in the presence of resonances; this could be, for example, a doped optical fibre. By making use of the integrability of the dynamic equations, we shall apply the finite-gap integration method to obtain periodic solutions for this system. Next, we consider the problem of the formation of solitons at a sharp front pulse and, by means of the Whitham modulational theory, we derive the amplitude and velocity of the largest soliton.

Automation and cybernetics: Control of a flexible one-link manipulator

States that control is of the essence in cybernetics. Summarizes the dynamic equations for a flexible one-link manipulator moving in the horizontal plane. Employs the finite element method, based on elementary beam theory, during the process of formulation. Develops and instruments a one-link flexible manipulator in order to control its vibration modes. Uses a simple second-order vibration model which permits vibrations on the rod to be estimated using the hub angle. The validation of the dynamic model and the structural analysis of the flexible manipulator is reached using proper infrared cameras and active light sources for determining actual positions of objects in space. Shows that the performance of the control is satisfactory, even under perturbation action.

Variáveis canônicas não singulares e o movimento rotacional de satélites artificiais

Pós-graduação em Física - FEG

Estudo de multirefringência na dinâmica de um feixe de luz considerando o bilhar anular

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES)

Coupling collective modes in a trapped superfluid

We consider a superfluid cloud composed of a Bose-Einstein condensate oscillating within a magnetic trap (dipole mode) where, due to the existence of a Feshbach resonance, an effective periodic time-dependent modulation in the scattering length is introduced. Under this condition, collective excitations such as the quadrupole mode can take place. We approach this problem by employing both the Gaussian and the Thomas-Fermi variational Ansatze. The resulting dynamic equations are analyzed by considering both linear approximations and numerical solutions, where we observe coupling between dipole and quadrupole modes. Aspects of this coupling related to the variation of the dipole oscillation amplitude are analyzed. This may be a relevant effect in situations where oscillation in a magnetic field in the presence of a bias field B takes place, and should be considered in the interpretation of experimental results.

The Dynamics of Passive Torsional Fatigue Test Rigs: Innovative Applications of Universal Joints

The dynamics of a passive back-to-back test rig have been characterised, leading to a multi-coordinate approach for the analysis of arbitrary test configurations. Universal joints have been introduced into a typical pre-loaded back-to-back system in order to produce an oscillating torsional moment in a test specimen. Two different arrangements have been investigated using a frequency-based sub-structuring approach: the receptance method. A numerical model has been developed in accordance with this theory, allowing interconnection of systems with two-coordinates and closed multi-loop schemes. The model calculates the receptance functions and modal and deflected shapes of a general system. Closed form expressions of the following individual elements have been developed: a servomotor, damped continuous shaft and a universal joint. Numerical results for specific cases have been compared with published data in literature and experimental measurements undertaken in the present work. Due to the complexity of the universal joint and its oscillating dynamic effects, a more detailed analysis of this component has been developed. Two models have been presented. The first represents the joint as two inertias connected by a massless cross-piece. The second, derived by the dynamic analysis of a spherical four-link mechanism, considers the contribution of the floating element and its gyroscopic effects. An investigation into non-linear behaviour has led to a time domain model that utilises the Runge-Kutta fourth order method for resolution of the dynamic equations. It has been demonstrated that the torsional receptances of a universal joint, derived using the simple model, result in representation of the joint as an equivalent variable inertia. In order to verify the model, a test rig has been built and experimental validation undertaken. The variable inertia of a universal joint has lead to a novel application of the component as a passive device for the balancing of inertia variations in slider-crank mechanisms.

Simulación eficiente de mecanismos flexibles mediante formulaciones topológicas

El objetivo de esta Tesis es presentar un método eficiente para la evaluación de sistemas multi-cuerpo con elementos flexibles con pequeñas deformaciones, basado en métodos topológicos para la simulación de sistemas tan complejos como los que se utilizan en la práctica y en tiempo real o próximo al real. Se ha puesto un especial énfasis en la resolución eficiente de aquellos aspectos que conllevan mayor coste computacional, tales como la evaluación de las ecuaciones dinámicas y el cálculo de los términos de inercia. Las ecuaciones dinámicas se establecen en función de las variables independientes del sistema, y la integración de las mismas se realiza mediante formulaciones implícitas de index-3. Esta Tesis se articula en seis Capítulos. En el Capítulo 1 se realiza una revisión bibliográfica de la simulación de sistemas flexibles y los métodos más relevantes de integración de las ecuaciones diferenciales del movimiento. Asimismo, se presentan los objetivos de esta Tesis. En el Capítulo 2 se presenta un método semi-recursivo para la evaluación de las ecuaciones de los sistemas multi-cuerpo con elementos flexibles basado en formulaciones topológicas y síntesis modal. Esta Tesis determina la posición de cada punto del cuerpo flexible en función de un sistema de referencia flotante que se mueve con dicho cuerpo y de las amplitudes de ciertos modos de deformación calculados a partir de un mallado obtenido mediante el Método de Elementos Finitos. Se presta especial atención en las condiciones de contorno que se han de tener en cuenta a la hora de establecer las variables que definen la deformación del cuerpo flexible. El Capítulo 3 se centra en la evaluación de los términos de inercia de los sistemas flexibles que generalmente conllevan un alto coste computacional. Se presenta un método que permite el cálculo de dichos términos basado en el uso de 24 matrices constantes que pueden ser calculadas previamente al proceso de integración. Estas matrices permiten evaluar la matriz de masas y el vector de fuerzas de inercia dependientes de la velocidad sin que sea necesario evaluar la posición deformada de todos los puntos del cuerpo flexible. Se realiza un análisis pormenorizado de dichas matrices con el objetivo de optimizar su cálculo estableciendo aproximaciones que permitan reducir el número de dichos términos y optimizar aún más su evaluación. Se analizan dos posibles simplificaciones: la primera utiliza una discretización no-consistente basada en elementos finitos en los que se definen únicamente los desplazamientos axiales de los nodos; en la segunda propuesta se hace uso de una matriz de masas concentradas (Lumped Mass). Basándose en la formulación presentada, el Capítulo 4 aborda la integración eficiente de las ecuaciones dinámicas. Se presenta un método iterativo para la integración con fórmulas de index-3 basado en la proyección de las ecuaciones dinámicas según las variables independientes del sistema multi-cuerpo. El cálculo del residuo del sistema de ecuaciones no lineales que se ha de resolver de modo iterativo se realiza mediante un proceso recursivo muy eficiente que aprovecha la estructura topológica del sistema. Se analizan tres formas de evaluar la matriz tangente del citado sistema no lineal: evaluación aproximada, numérica y recursiva. El método de integración presentado permite el uso de distintas fórmulas. En esta Tesis se analizan la Regla Trapezoidal, la fórmula BDF de segundo orden y un método híbrido TR-BDF2. Para este último caso se presenta un algoritmo de paso variable. En el Capítulo 5 plantea la implementación del método propuesto en un programa general de simulación de mecanismos que permita la resolución de cualquier sistema multi-cuerpo definiéndolo mediante un fichero de datos. La implementación de este programa se ha realizado tanto en C++ como en Java. Se muestran los resultados de las formulaciones presentadas en esta Tesis mediante la simulación de cuatro ejemplos de distinta complejidad. Mediante análisis concretos se comparan la formulación presentada con otras existentes. También se analiza el efecto del lenguaje de programación utilizado en la implementación y los efectos de las posibles simplificaciones planteadas. Por último, el Capítulo 6 resume las principales conclusiones alcanzadas en la Tesis y las futuras líneas de investigación que con ella se abren. ABSTRACT This Thesis presents an efficient method for solving the forward dynamics of a multi-body sys-tem formed by rigid and flexible bodies with small strains for real-time simulation of real-life models. It is based on topological formulations. The presented work focuses on the efficient solution of the most time-consuming tasks of the simulation process, such as the numerical integration of the motion differential equations and in particular the evaluation of the inertia terms corresponding to the flexible bodies. The dynamic equations are formulated in terms of independent variables of the muti-body system, and they are integrated by means of implicit index-3 formulae. The Thesis is arranged in six chapters. Chapter 1 presents a review of the most relevant and recent contributions related to the modelization of flexible multi-body systems and the integration of the corresponding dynamic equations. The main objectives of the Thesis are also presented in detail. Chapter 2 presents a semi-recursive method for solving the equations of a multi-body system with flexible bodies based on topological formulations and modal synthesis. This Thesis uses the floating frame approach and the modal amplitudes to define the position of any point at the flexible body. These modal deformed shapes are obtained by means of the Finite Element Method. Particular attention has been taken to the boundary conditions used to define the deformation of the flexible bodies. Chapter 3 focuses on the evaluation of the inertia terms, which is usually a very time-consuming task. A new method based on the use of 24 constant matrices is presented. These matrices are evaluated during the set-up step, before the integration process. They allow the calculation of the inertia terms in terms of the position and orientation of the local coordinate system and the deformation variables, and there is no need to evaluate the position and velocities of all the nodes of the FEM mesh. A deep analysis of the inertia terms is performed in order to optimize the evaluation process, reducing both the terms used and the number of arithmetic operations. Two possible simplifications are presented: the first one uses a non-consistent approach in order to define the inertia terms respect to the Cartesian coordinates of the FEM mesh, rejecting those corresponding to the angular rotations; the second approach makes use of lumped mass matrices. Based on the previously presented formulation, Chapter 4 is focused on the numerical integration of the motion differential equations. A new predictor-corrector method based on index-3 formulae and on the use of multi-body independent variables is presented. The evaluation of the dynamic equations in a new time step needs the solution of a set on nonlinear equations by a Newton-Raphson iterative process. The computation of the corresponding residual vector is performed efficiently by taking advantage of the system’s topological structure. Three methods to compute the tangent matrix are presented: an approximated evaluation that considers only the most relevant terms, a numerical approach based on finite differences and a recursive method that uses the topological structure. The method presented for integrating the dynamic equations can use a variety of integration formulae. This Thesis analyses the use of the trapezoidal rule, the 2nd order BDF formula and the hybrid TR-BDF2 method. A variable-time step strategy is presented for the last one. Chapter 5 describes the implementation of the proposed method in a general purpose pro-gram for solving any multibody defined by a data file. This program is implemented both in C++ and Java. Four examples are used to check the validity of the formulation and to compare this method with other methods commonly used to solve the dynamic equations of multi-body systems containing flexible bodies. The efficiency of the programming methodology used and the effect of the possible simplifications proposed are also analyzed. Chapter 6 summarizes the main Conclusions obtained in this Thesis and the new lines of research that have been opened.

A contribution on modeling methodologies for multibody systems.

Multibody System Dynamics has been responsible for revolutionizing Mechanical Engineering Design by using mathematical models to simulate and optimize the dynamic behavior of a wide range of mechanical systems. These mathematical models not only can provide valuable informations about a system that could otherwise be obtained only by experiments with prototypes, but also have been responsible for the development of many model-based control systems. This work represents a contribution for dynamic modeling of multibody mechanical systems by developing a novel recursive modular methodology that unifies the main contributions of several Classical Mechanics formalisms. The reason for proposing such a methodology is to motivate the implementation of computational routines for modeling complex multibody mechanical systems without being dependent on closed source software and, consequently, to contribute for the teaching of Multibody System Dynamics in undergraduate and graduate levels. All the theoretical developments are based on and motivated by a critical literature review, leading to a general matrix form of the dynamic equations of motion of a multibody mechanical system (that can be expressed in terms of any set of variables adopted for the description of motions performed by the system, even if such a set includes redundant variables) and to a general recursive methodology for obtaining mathematical models of complex systems given a set of equations describing the dynamics of each of its uncoupled subsystems and another set describing the constraints among these subsystems in the assembled system. This work also includes some discussions on the description of motion (using any possible set of motion variables and admitting any kind of constraint that can be expressed by an invariant), and on the conditions for solving forward and inverse dynamics problems given a mathematical model of a multibody system. Finally, some examples of computational packages based on the novel methodology, along with some case studies, are presented, highlighting the contributions that can be achieved by using the proposed methodology.

Simulation of robot manipulator control strategies

The high capital cost of robots prohibit their economic application. One method of making their application more economic is to increase their operating speed. This can be done in a number of ways e.g. redesign of robot geometry, improving actuators and improving control system design. In this thesis the control system design is considered. It is identified in the literature review that two aspects in relation to robot control system design have not been addressed in any great detail by previous researchers. These are: how significant are the coupling terms in the dynamic equations of the robot and what is the effect of the coupling terms on the performance of a number of typical independent axis control schemes?. The work in this thesis addresses these two questions in detail. A program was designed to automatically calculate the path and trajectory and to calculate the significance of the coupling terms in an example application of a robot manipulator tracking a part on a moving conveyor. The inertial and velocity coupling terms have been shown to be of significance when the manipulator was considered to be directly driven. A simulation of the robot manipulator following the planned trajectory has been established in order to assess the performance of the independent axis control strategies. The inertial coupling was shown to reinforce the control torque at the corner points of the trajectory, where there was an abrupt demand in acceleration in each axis but of opposite sign. This reduced the tracking error however, this effect was not controllable. A second effect was due to the velocity coupling terms. At high trajectory speeds it was shown, by means of a root locus analysis, that the velocity coupling terms caused the system to become unstable.

The study and measurements of the behaviour and characteristics of electromagnetic relays

The thesis presents a theoretical and practical study of the dynamic behaviour of electromagnetic relays. After discussing the problem of solving the dynamicc equations analytically and presenting a historical survey of the earlier works in the relay and its dynamics, the simulation of a relay on the analogue computer is discussed. It is shown that the simulation may be used to obtain specific solutions to the dynamic equations. The computer analysis provides the dynamic characteristics for design purposes and may be used in the study of bouncing, rebound oscillations and stability of the armature motion. An approximate analytical solution to the two dynamic equations is given based on the assumption that the dynamic variation of the pull with the position of the armature is linear. The assumption is supported by the Computer-aided analysis and experimental results. The solution is intended to provide a basis for a rational design. A rigorous method of analysing the dynamic performance by using Ahlberg's theory is also presented. This method may be justified to be the extension of Ahlberg's theory by taking the mass and frictional damping forces into account. While calculating the armature motion mathematically, Ahlberg considers the equilibrium of two kinds of forces, namely pull and load, and disregards the mass and friction forces, whereas the present method deals with the equilibrium of all four kinds of forces. It is shown how this can be utilised to calculate the dynamic characteristics for a specific design. The utility of this method also extends to the study of stability, contact bounce and armature rebound. The magnetic circuit and other related topics which are essential to the study of relay dynamics are discussed and some necessary experimental results are given.

Conception d’un mécanisme déployable à grand ratio d’expansion et de son système d’actionnement par roues d’inertie pour applications spatiales

Cette thèse propose de développer des mécanismes déployables pour applications spatiales ainsi que des modes d’actionnement permettant leur déploiement et le contrôle de l’orientation en orbite de l’engin spatial les supportant. L’objectif étant de permettre le déploiement de surfaces larges pour des panneaux solaires, coupoles de télécommunication ou sections de station spatiale, une géométrie plane simple en triangle est retenue afin de pouvoir être assemblée en différents types de surfaces. Les configurations à membrures rigides proposées dans la littérature pour le déploiement de solides symétriques sont optimisées et adaptées à l’expansion d’une géométrie ouverte, telle une coupole. L’optimisation permet d’atteindre un ratio d’expansion plan pour une seule unité de plus de 5, mais présente des instabilités lors de l’actionnement d’un prototype. Le principe de transmission du mouvement d’un étage à l’autre du mécanisme est revu afin de diminuer la sensibilité des performances du mécanisme à la géométrie de ses membrures internes. Le nouveau modèle, basé sur des courroies crantées, permet d’atteindre des ratios d’expansion plans supérieurs à 20 dans certaines configurations. L’effet des principaux facteurs géométriques de conception est étudié afin d’obtenir une relation simple d’optimisation du mécanisme plan pour adapter ce dernier à différents contextes d’applications. La géométrie identique des faces triangulaires de chaque surface déployée permet aussi l’empilement de ces faces pour augmenter la compacité du mécanisme. Une articulation spécialisée est conçue afin de permettre le dépliage des faces puis leur déploiement successivement. Le déploiement de grandes surfaces ne se fait pas sans influencer lourdement l’orientation et potentiellement la trajectoire de l’engin spatial, aussi, différentes stratégies de contrôle de l’orientation novatrices sont proposées. Afin de tirer profit d’une grande surface, l’actionnement par masses ponctuelles en périphérie du mécanisme est présentée, ses équations dynamiques sont dérivées et simulées pour en observer les performances. Celles-ci démontrent le potentiel de cette stratégie de réorientation, sans obstruction de l’espace central du satellite de base, mais les performances restent en deçà de l’effet d’une roue d’inertie de masse équivalente. Une stratégie d’actionnement redondant par roue d’inertie est alors présentée pour différents niveaux de complexité de mécanismes dont toutes les articulations sont passives, c’est-à-dire non actionnées. Un mécanisme à quatre barres plan est simulé en boucle fermée avec un contrôleur simple pour valider le contrôle d’un mécanisme ciseau commun. Ces résultats sont étendus à la dérivation des équations dynamiques d’un mécanisme sphérique à quatre barres, qui démontre le potentiel de l’actionnement par roue d’inertie pour le contrôle de la configuration et de l’orientation spatiale d’un tel mécanisme. Un prototype à deux corps ayant chacun une roue d’inertie et une seule articulation passive les reliant est réalisé et contrôlé grâce à un suivi par caméra des modules. Le banc d’essai est détaillé, ainsi que les défis que l’élimination des forces externes ont représenté dans sa conception. Les résultats montrent que le système est contrôlable en orientation et en configuration. La thèse se termine par une étude de cas pour l’application des principaux systèmes développés dans cette recherche. La collecte de débris orbitaux de petite et moyenne taille est présentée comme un problème n’ayant pas encore eu de solution adéquate et posant un réel danger aux missions spatiales à venir. L’unité déployable triangulaire entraînée par courroies est dupliquée de manière à former une coupole de plusieurs centaines de mètres de diamètre et est proposée comme solution pour capturer et ralentir ces catégories de débris. Les paramètres d’une mission à cette fin sont détaillés, ainsi que le potentiel de réorientation que les roues d’inertie permettent en plus du contrôle de son déploiement. Près de 2000 débris pourraient être retirés en moins d’un an en orbite basse à 819 km d’altitude.

A deformation model of flexible, high area-to-mass ratio debris under perturbations and validation method

A new type of space debris was recently discovered by Schildknecht in near -geosynchronous orbit (GEO). These objects were later identified as exhibiting properties associated with High Area-to-Mass ratio (HAMR) objects. According to their brightness magnitudes (light curve), high rotation rates and composition properties (albedo, amount of specular and diffuse reflection, colour, etc), it is thought that these objects are multilayer insulation (MLI). Observations have shown that this debris type is very sensitive to environmental disturbances, particularly solar radiation pressure, due to the fact that their shapes are easily deformed leading to changes in the Area-to-Mass ratio (AMR) over time. This thesis proposes a simple effective flexible model of the thin, deformable membrane with two different methods. Firstly, this debris is modelled with Finite Element Analysis (FEA) by using Bernoulli-Euler theory called “Bernoulli model”. The Bernoulli model is constructed with beam elements consisting 2 nodes and each node has six degrees of freedom (DoF). The mass of membrane is distributed in beam elements. Secondly, the debris based on multibody dynamics theory call “Multibody model” is modelled as a series of lump masses, connected through flexible joints, representing the flexibility of the membrane itself. The mass of the membrane, albeit low, is taken into account with lump masses in the joints. The dynamic equations for the masses, including the constraints defined by the connecting rigid rod, are derived using fundamental Newtonian mechanics. The physical properties of both flexible models required by the models (membrane density, reflectivity, composition, etc.), are assumed to be those of multilayer insulation. Both flexible membrane models are then propagated together with classical orbital and attitude equations of motion near GEO region to predict the orbital evolution under the perturbations of solar radiation pressure, Earth’s gravity field, luni-solar gravitational fields and self-shadowing effect. These results are then compared to two rigid body models (cannonball and flat rigid plate). In this investigation, when comparing with a rigid model, the evolutions of orbital elements of the flexible models indicate the difference of inclination and secular eccentricity evolutions, rapid irregular attitude motion and unstable cross-section area due to a deformation over time. Then, the Monte Carlo simulations by varying initial attitude dynamics and deformed angle are investigated and compared with rigid models over 100 days. As the results of the simulations, the different initial conditions provide unique orbital motions, which is significantly different in term of orbital motions of both rigid models. Furthermore, this thesis presents a methodology to determine the material dynamic properties of thin membranes and validates the deformation of the multibody model with real MLI materials. Experiments are performed in a high vacuum chamber (10-4 mbar) replicating space environment. A thin membrane is hinged at one end but free at the other. The free motion experiment, the first experiment, is a free vibration test to determine the damping coefficient and natural frequency of the thin membrane. In this test, the membrane is allowed to fall freely in the chamber with the motion tracked and captured through high velocity video frames. A Kalman filter technique is implemented in the tracking algorithm to reduce noise and increase the tracking accuracy of the oscillating motion. The forced motion experiment, the last test, is performed to determine the deformation characteristics of the object. A high power spotlight (500-2000W) is used to illuminate the MLI and the displacements are measured by means of a high resolution laser sensor. Finite Element Analysis (FEA) and multibody dynamics of the experimental setups are used for the validation of the flexible model by comparing with the experimental results of displacements and natural frequencies.

Object-Oriented Modeling Simulation and Control of Activated Sludge Process

Object-oriented modeling is spreading in current simulation of wastewater treatments plants through the use of the individual components of the process and its relations to define the underlying dynamic equations. In this paper, we describe the use of the free-software OpenModelica simulation environment for the object-oriented modeling of an activated sludge process under feedback control. The performance of the controlled system was analyzed both under normal conditions and in the presence of disturbances. The object-oriented described approach represents a valuable tool in teaching provides a practical insight in wastewater process control field.

Experimentally and coupled CFD/CSD assisted structural and aeroelastic modelling of the Javelin UAV outboard wing section

This study presents the procedure followed to make a prediction of the critical flutter speed for a composite UAV wing. At the beginning of the study, there was no information available on the materials used for the construction of the wing, and the wing internal structure was unknown. Ground vibration tests were performed in order to detect the structure’s natural frequencies and mode shapes. From tests, it was found that the wing possesses a high stiffness, presenting well separated first bending and torsional natural frequencies. Two finite element models were developed and matched to experimental results. It has been necessary to introduce some assumptions, due to the uncertainties regarding the structure. The matching process was based on natural frequencies’ sensitivity with respect to a change in the mechanical properties of the materials. Once experimental results were met, average material properties were also found. Aerodynamic coefficients for the wing were obtained by means of a CFD software. The same analysis was also conducted when the wing is deformed in its first four mode shapes. A first approximation for flutter critical speed was made with the classical V - g technique. Finally, wing’s aeroelastic behavior was simulated using a coupled CFD/CSD method, obtaining a more accurate flutter prediction. The CSD solver is based on the time integration of modal dynamic equations, requiring the extraction of mode shapes from the previously performed finite-element analysis. Results show that flutter onset is not a risk for the UAV, occurring at velocities well beyond its operative range.

A numerical method for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field

A finite difference technique, based on a projection method, is developed for solving the dynamic three-dimensional Ericksen-Leslie equations for nematic liquid crystals subject to a strong magnetic field. The governing equations in this situation are derived using primitive variables and are solved using the ideas behind the GENSMAC methodology (Tome and McKee [32]; Tome et al. [34]). The resulting numerical technique is then validated by comparing the numerical solution against an analytic solution for steady three-dimensional flow between two-parallel plates subject to a strong magnetic field. The validated code is then employed to solve channel flow for which there is no analytic solution. (C) 2009 Elsevier B.V. All rights reserved.