<p>The aim of this work has been to develop new knowledge about macromodelling of microsystems. Doing that, we have followed two different approaches for generating macromodels, namely model order reduction and lumped modelling. The latter is a rather mature method that has been widely recognized and used for a relatively long period of time. Model order reduction, on the other hand, is a relatively new area still in rapid development. Due to this, the parts considering reduced order modelling is strongly biased towards methodology and concepts, whereas parts on lumped modelling are biased towards systems and devices.</p><p>In the first part of this thesis, we focus on model order reduction. We introduce some approaches for reducing model order for linear systems, and we give an example related to squeeze-film damping. We then move on to investigate model order reduction of nonlinear systems, where we present and use the concept of invariant manifolds. While the concept of invariant manifolds is general, we utilize it for reducing models. An obvious advantage of using invariant manifold theory is that it offers a conceptually clear understanding of effects and behaviour of nonlinear system.</p><p>We exemplify and investigate the accuracy of one method for identifying invariant manifolds. The example is based on an industrialized dual-axis accelerometer.</p><p>A new geometrical interpretation of external forcing, relating to invariant manifolds, is presented. We show how this can be utilized to deal with external forcing in a manner consistent with the invariance property of the manifold. The interpretation also aids in reducing errors for reduce models.</p><p>We extend the asymptotic approach in a manner that makes it possible to create design-parameter sensitive models. We investigate an industrialized dual-axis accelerometer by means of the method and demonstrate capabilities of the method. We also discuss how manifolds for nonlinear dissipative systems can be found.</p><p>Focusing on lumped modelling, we analyse a microresonator. We also discuss the two analogies that can be used to build electrical equivalents of mechanical systems. It is shown how the f → V analogy, linking velocity to voltage, is the natural choice. General properties of lumped modelling are investigated using models with varying degrees of freedom.</p><p>Finally, we analyse an electromagnetic system, intended for levitating objects, and we demonstrate the scaling effects of the system. Furthermore, we prove the intrinsic stability of the system, although the floating disc will be slightly tilted. This is the first analysis done assessing the stability criterions of such a systems. The knowledge arising from the analysis gives strong indications on how such a system can be utilized, designed, and improved.</p>

The aim of this work has been to develop new knowledge about macromodelling of microsystems. Doing that, we have followed two different approaches for generating macromodels, namely model order reduction and lumped modelling. The latter is a rather mature method that has been widely recognized and used for a relatively long period of time. Model order reduction, on the other hand, is a relatively new area still in rapid development. Due to this, the parts considering reduced order modelling is strongly biased towards methodology and concepts, whereas parts on lumped modelling are biased towards systems and devices. In the first part of this thesis, we focus on model order reduction. We introduce some approaches for reducing model order for linear systems, and we give an example related to squeeze-film damping. We then move on to investigate model order reduction of nonlinear systems, where we present and use the concept of invariant manifolds. While the concept of invariant manifolds is general, we utilize it for reducing models. An obvious advantage of using invariant manifold theory is that it offers a conceptually clear understanding of effects and behaviour of nonlinear system. We exemplify and investigate the accuracy of one method for identifying invariant manifolds. The example is based on an industrialized dual-axis accelerometer. A new geometrical interpretation of external forcing, relating to invariant manifolds, is presented. We show how this can be utilized to deal with external forcing in a manner consistent with the invariance property of the manifold. The interpretation also aids in reducing errors for reduce models. We extend the asymptotic approach in a manner that makes it possible to create design-parameter sensitive models. We investigate an industrialized dual-axis accelerometer by means of the method and demonstrate capabilities of the method. We also discuss how manifolds for nonlinear dissipative systems can be found. Focusing on lumped modelling, we analyse a microresonator. We also discuss the two analogies that can be used to build electrical equivalents of mechanical systems. It is shown how the f → V analogy, linking velocity to voltage, is the natural choice. General properties of lumped modelling are investigated using models with varying degrees of freedom. Finally, we analyse an electromagnetic system, intended for levitating objects, and we demonstrate the scaling effects of the system. Furthermore, we prove the intrinsic stability of the system, although the floating disc will be slightly tilted. This is the first analysis done assessing the stability criterions of such a systems. The knowledge arising from the analysis gives strong indications on how such a system can be utilized, designed, and improved.

Magnetically levitated microrobotic systems have shown a great deal of promise for micromanipulation tasks. A new large-gap magnetic suspension system has recently been developed at the University of Waterloo in order to develop microrobotic systems for various applications. In order to achieve motion with the system, a model is needed in order to facilitate the design of various aspects of the system, such as the microrobot and the controller. In order to derive equations of motion for the system attempts were made to characterize the force produced by the magnetic drive unit in terms of a simple analytical equation. The force produced by the magnetic drive unit was estimated with the aid of a finite element model. The derived equations were able to predict the general trend of the force curves, and with sufficient parameter tweaking the error between the force estimated by the finite element model and the force estimated by the analytical equation could be minimized. System models describing the motion of the system in the horizontal and vertical directions are identified and compared to the actual system response. The vertical position response is identified through a least squares parameter estimate of the closed loop response combined with a partial reconstruction of the root locus diagram, with the model structure based on the known dynamics of a simplified form of magnetic levitation. This model was able to provide a reasonable prediction of the system response for a variety of PID controllers under a variety of input conditions. The horizontal models are identified using a least-squares parameter estimate of the open loop characteristics of the system. The horizontal models are able to provide a reasonable prediction of the system response under PD and PID control. Full spatial motion of a microrobot prototype is demonstrated over a working range of 20x22x30 mm³, with PID controller parameters and reference trajectories adjusted to minimize disturbances. The RMS error at steady state is on the order of 0. 020 mm for vertical positioning and 0. 008 mm for horizontal positioning. A linear quadratic regulator implemented for vertical position control was able to reduce the vertical position RMS error to 0. 014 mm.

Magnetically levitated microrobotic systems have shown a great deal of promise for micromanipulation tasks. A new large-gap magnetic suspension system has recently been developed at the University of Waterloo in order to develop microrobotic systems for various applications. In order to achieve motion with the system, a model is needed in order to facilitate the design of various aspects of the system, such as the microrobot and the controller. In order to derive equations of motion for the system attempts were made to characterize the force produced by the magnetic drive unit in terms of a simple analytical equation. The force produced by the magnetic drive unit was estimated with the aid of a finite element model. The derived equations were able to predict the general trend of the force curves, and with sufficient parameter tweaking the error between the force estimated by the finite element model and the force estimated by the analytical equation could be minimized. System models describing the motion of the system in the horizontal and vertical directions are identified and compared to the actual system response. The vertical position response is identified through a least squares parameter estimate of the closed loop response combined with a partial reconstruction of the root locus diagram, with the model structure based on the known dynamics of a simplified form of magnetic levitation. This model was able to provide a reasonable prediction of the system response for a variety of PID controllers under a variety of input conditions. The horizontal models are identified using a least-squares parameter estimate of the open loop characteristics of the system. The horizontal models are able to provide a reasonable prediction of the system response under PD and PID control. Full spatial motion of a microrobot prototype is demonstrated over a working range of 20x22x30 mm³, with PID controller parameters and reference trajectories adjusted to minimize disturbances. The RMS error at steady state is on the order of 0. 020 mm for vertical positioning and 0. 008 mm for horizontal positioning. A linear quadratic regulator implemented for vertical position control was able to reduce the vertical position RMS error to 0. 014 mm.

Konstanz, Univ., Diplomarbeit, 2006.

Thesis (Ph. D.)--University of Texas at Austin, 2007. / Vita. Includes bibliographical references.

Σκοπός της διπλωματικής εργασίας, είναι η κατασκευή ενός κυκλώματος με ανάδραση για τον έλεγχο ενός ηλεκτρομαγνήτη με στόχο την αιώρηση ενός σταθερού μαγνητικού αντικειμένου. / The purpose of the thesis is to build a circuit with feedback in order to control a solenoid to levitate a constant magnetic object.

O controle e otimização da operação da suspensão eletromagnética de um protótipo de veículo MAGLEV são apresentados neste trabalho. Este tipo de sistema é inerentemente instável e altamente não linear, sendo um excelente exemplo para o estudo e comparação de diferentes metodologias de controle. Nos últimos anos, experiências significativas com veículos levitados têm ganhado força. A arquitetura do veículo consiste de quatro atuadores eletromagnéticos, quatro sensores de entreferros e quatro acelerômetros, todos instalados nos cantos do veículo. A dinâmica do sistema é descrita através de três modelos matemáticos diferentes: modelo de corpo rígido com três graus de liberdade (movimentos vertical, de rolagem e de arfagem), modelo de corpo não rígido, onde está presente mais um grau de liberdade (movimento de torção), e modelo SISO com quatro movimentos verticais independentes. Blocos de compatibilização de variáveis são utilizados devido à diferença no número de variáveis do modelo matemático com o número de variáveis medidas e de atuação disponíveis no veículo. Os acelerômetros, usados para estimar as velocidades do veículo, foram instalados com o objetivo de estudar-se o controle ótimo da suspensão por realimentação de estados com o emprego da técnica do Regulador Linear Quadrático (RLQ) e utilização do método do lugar das raízes para definição dos pólos de malha fechada. Para efeito de comparação foi implementado um controle H'IND.2'/H'INFINITO' por realimentação da saída. Controladores independentes para cada grau de liberdade foram simulados no software MATLAB e depois implementados e testados no veículo utilizando uma placa de aquisição de dados instalada no microcomputador. A preocupação principal do projeto dos controladores foi a estabilidade do sistema na ocorrência de perturbações que desviem o entreferro de seu valor nominal.Exceto para o sistema com controle RLQ e modelo de corpo rígido, os resultados experimentais obtidos com o protótipo mostram que o sistema em malha fechada é estável e apresenta uma resposta transiente satisfatória. O sistema com controle RLQ e modelo de corpo não rígido apresentou a melhor performance entre todas as alternativas testadas. / The control and optimization of the electromagnetic suspension operation of a MAGLEV vehicle prototype are presented in this work. This kind of system is inherently unstable and highly nonlinear, being an excellent example for the study and comparison of different control methodologies. In the last years significant experience has been gained with levitated vehicles. The vehicle architecture consists of four electromagnetic actuators, four air gap sensors and four accelerometers, all located at the corners of the vehicle. The vehicle dynamics is described through three different mathematical models: rigid body model, with three degrees of freedom (heave, roll and pitch motions), not-rigid body model, with an additional degree of freedom (torsion motion) and SISO model, with four independent heave motions. Blocks to make variables compatible are used in based on the difference between the number of variables of the mathematical model and the number of measured and actuation variables available in the vehicle. The accelerometers, used to estimate the speeds of the vehicle, were installed in order to study the optimal control of the suspension for states feedback. The Linear Quadratic Regulator (LQR) technique combined with the root locus method for definition of the closed-loop poles were used. The H'IND.2'/H'INFINITE' controller with output feedback was implemented for sake of comparison. Independent controllers for each degree of freedom were simulated with MATLAB then implemented and tested in the vehicle using AD/DA converter installed in the microcomputer. The first goal of the controllers' design was the system stability when subject to disturbances that drive the air away from its nominal value. Except for the system with RLQ controller and rigid body model, the experimental results obtained with the prototype show that the closed-loop system is stable and exhibits a satisfactory transient response.The system with the RLQ controller and not-rigid body model exhibits the best performance among all the alternatives tested.

Electromagnetic levitation experiments provide a powerful tool that allows for the study of nucleation, solidification and growth in a containerless processing environment. Containerless processing allows for the study of reactive melts at elevated temperatures without chemical interactions or contamination from a container. Further, by removing the interface between the liquid and its container, this processing technique allows for greater access to the undercooled region for solidification studies. However, in these experiments it is important to understand the magnetohydrodynamic flow within the sample and the effects that this fluid flow has on the experiment. A recent solidification study found that aluminum-nickel alloy sample have an unusual response of the growth rate of the solid to changes in undercooling. This alloy experienced a decrease in the growth velocity as the initial undercooling deepened, instead of the expected increase in solidification velocity with deepening undercoolings. Current work is exploring several different theories to explain this phenomenon. Distinguishing among these theories requires a comprehensive understanding of the behavior of the internal fluid flow. Our project, USTIP, has done flow modeling to support this and multiple other collaborators on ISS-EML. The fluid flow models presented for the aluminum-nickel sample provide critical insights into the nature of the flow within the aluminum-nickel alloy experiments conducted in the ISS-EML facility. These models have found that for this sample the RNG k-ε model should be used with this sample at temperatures greater than 1800 K and the laminar flow model should be used at temperatures lower than 1600 K. Other work in the ISS-EML, has studied the thermophysical properties of liquid germanium and has found the most recent measurements using oscillating drop techniques to have a discrepancy from the expected property measurements taken terrestrially. Investigating this discrepancy required the quantification of the velocity and characterization of the internal fluid flow in the drop. The models have found that the flow within the sample maintains turbulent behavior throughout cooling. This thesis presents the analysis of the internal flow of four additional samples processed in the International Space Station Electromagnetic Levitation facility. These samples consist of the following alloys: Ti39.5Zr39.5Ni21, Cu50Zr50, Vitreloy 106, and Zr64Ni36. Our collaborators work required the internal flow to be characterized and quantified for their work on solidification. In addition to quantifying the velocity of the flow, the Reynolds number was calculated to characterize the flow during processing. Additionally, the shear-strain rate was calculated for the flow during processing up to the recalescence of the melt.

The purpose of this research is to explore the operational characteristics of a Maglev-based Automated Highway System and how it would interact with freeway operations. The extension of traditional traffic flow phenomenon, including weaving, merging, and stopping distance, into the automated system is looked at. These are also extended into platoon operations and their effect on such properties as gap control and ultimately the capacity of such a system. The ability to incorporate an AHS system into the existing Interstate Highway System is investigated. This includes placing the magways in the right-of-way of the highway system and interfacing the AHS with the existing freeways. A model is developed and run to simulate the assignment of traffic between the freeway and the guideway links. Both operational concepts of user equilibrium and system optimal conditions are explored, and equations are found to estimate the amount of traffic which can be found on the links based on the total traffic volume. / Master of Science