The results of an experimental study of phenomena which occur in the flow of a viscous fluid in closed domains with discrete symmetries are presented. The purpose is to investigate the role which ideas from low-dimensional dynamical systems have to play in describing qualitative changes that take place with variation of the governing parameters. Such a descriptive framework already exists for the case of the Taylor-Couette system, where the domain possesses a continuous azimuthal symmetry group. The present investigation is aimed at establishing the typicality of previously reported behaviour under progressive reductions of azimuthal symmetry. In the first investigations, the fixed outer circular cylinder of the standard system is replaced with one of square cross-section. Thus there is now discrete Ζ₄ symmetry in the azimuthal direction. Knowledge of the two-dimensional flow field is used to establish the nature of the steady three-dimensional motion equivalent to Taylor vortex flow. It is shown that similar bifurcation sequences exist in both standard and square systems for the case of very small aspect ratio where a single Taylor cell is formed. This flow develops as the result of a bifurcation which breaks the Ζ₂ symmetry that is imposed on the annulus by two solid stationary ends. The study is then extended to consider time-dependent effects in the square system. Two different oscillatory single-cell flows are identified, and it is shown that each is the result of a Hopf bifurcation. Selection of a particular dynamic mode is found to depend on the aspect ratio of the system. A low-dimensional bifurcation structure is uncovered which connects the two modes in parameter space, and involves a novel type of steady single-cell flow. Finally, observations are reported of a nontrivial type of dynamical behaviour which bears strong resemblance to motion found in a circularly symmetric Taylor-Couette system that is related to the Šilnikov mechanism for finite-dimensional chaos. A second variant on the Taylor-Couette system is considered where the outer cylinder is shaped like a stadium. The effect is to reduce further the overall symmetry of the domain to a Ζ₂ × Ζ₂ group. The two-dimensional flow field is investigated using both numerical and experimental techniques. Time-dependent phenomena are then investigated in the three-dimensional flow over a relatively wide range of aspect ratio. It is found that a sequence of a Hopf bifurcation followed by period-doubling bifurcations exists up to a certain aspect ratio, beyond which there is an apparently sudden and reversible transition between regular and irregular dynamical behaviour. Although this transition is not of a low-dimensional nature, the experimental results suggest that it exists as the result of a coalescence of the bifurcations which are found at lower values of aspect ratio.

No description available.

Excessive vibrations continue to be a major hurdle in improving machining efficiency and achieving stable high speed cutting. To overcome detrimental vibrations, an enhanced understanding of the underlying nonlinear dynamics is required. Cutting instability is commonly studied through modeling and analysis which incorporates linearization that obscures the true nonlinear characteristics of the system which are prominent at high speeds. Thus to enhance cutting dynamics knowledge, a comprehensive nonlinear turning model that includes tool-workpiece interaction is experimentally validated using a commercial laser vibrometer to capture tool and workpiece vibrations. A procedure is developed to use instantaneous frequency for experimental time-frequency analysis and is shown to thoroughly characterize the underlying dynamics and identify chatter. For the tests performed, chatter is associated with changing spectral components and bifurcations which provides a view of the underlying dynamics not experimentally observed before. Validation of the turning model revealed that the underlying dynamics observed experimentally are accurately captured, and the coupled tool-workpiece chatter vibrations are simulated. The stability diagram shows an increase in the chatter-free limit as the spindle speed increases until 1500rpm where it begins to level out. At high speeds the workpiece dominates the dynamics, and excessive workpiece vibrations create another stability limit to consider. Thus, workpiece dynamics should not be neglected in analyses for the design of machine tools and robust control laws. The chip formation mechanisms and high speeds make micro-milling highly non-linear and capable of producing broadband frequencies that negatively affect the tool. A nonlinear dynamic micro-milling model is developed to study the effect of parameters on tool performance through spectral analysis using instantaneous frequency. A lumped mass-spring-damper system is assumed for modeling the tool, and a slip-line force mechanism is adopted. The effective rake angle, helical angle, and instantaneous chip thickness are accounted for. The model produced the high frequency force components seen experimentally in literature. It is found that increasing the helical angle decreased the forces, and an increase in system stiffness improved the dynamic response. Also, dynamic instability had the largest effect on tool performance with the spindle speed being the most critical parameter.

Doctor of Philosophy / A detailed theoretical and numerical investigation of the behaviour of reactive systems under the influence of chaotic stirring is presented. These systems exhibit stationary solutions arising from the balance between chaotic advection and diffusion. Excessive stirring of such systems results in the termination of the reaction via a saddle-node bifurcation. The solution behaviour of these systems is analytically described using a recently developed nonperturbative, non-asymptotic variational method. This method involves fitting appropriate parameterised test functions to the solution, and also allows us to describe the bifurcations of these systems. This method is tested against numerical results obtained using a reduced one-dimensional reaction-advection-diffusion model. Four one- and two-component reactive systems with multiple homogeneous steady-states are analysed, namely autocatalytic, bistable, excitable and combustion systems. In addition to the generic stirring-induced saddle-node bifurcation, a rich and complex bifurcation scenario is observed in the excitable system. This includes a previously unreported region of bistability characterised by a hysteresis loop, a supercritical Hopf bifurcation and a saddle-node bifurcation arising from propagation failure. Results obtained with the nonperturbative method provide a good description of the bifurcations and solution behaviour in the various regimes of these chaotically stirred reaction-diffusion systems.

Thesis (M.Sc. (Applied Mathematics))--Mahidol University, 1986.

Thesis (Ph. D.)--Ohio State University, 2005. / Title from first page of PDF file. Document formatted into pages; contains ix, 134 p.; also includes graphics. Includes bibliographical references (p. 130-134). Available online via OhioLINK's ETD Center

Thermoacoustic instability is of fundamental and applied interest in both scientific research and practical applications. This study aims to explore several very important sub-aspects in this field and contribute to a better understanding of thermoacoustic instability as encountered in typical gas turbines and rocket engines. Atomization has been recognized as a key mechanism in driving applied thermoacoustic instability. In this regard, this study mainly focuses on the atomization process relevant for delineation of thermoacoustic instability, contributing to a comprehensive understanding of the effect of acoustics on primary and secondary atomization. Firstly, a tree-based adaptive solver and VOF method are employed to simulate the jet primary atomization. The code is validated by theoretical, numerical and experimental results to demonstrate its capability and accuracy in terms of atomization in both low-speed and high-speed regime. Perturbation frequency and amplitude have shown to affect the atomization significantly. Besides, the effect of acoustic forcing on liquid ligament has also been numerically investigated. A volume source term is introduced to extend the solver to model the compressible effects in the presence of acoustic forcing. The influence of acoustic wave number, amplitude and frequency has been examined in detail. In terms of modelling the thermoacoustic instability, bifurcation analysis is carried out for a time-delayed thermoacoustic system using the Method of Line approach. Good predictions have been obtained to capture the nonlinear behaviors inherent in the system. Moreover, model-based simulation and control of thermoacoustic instability have been conducted. A low-order wave-based network model for acoustics is coupled with nonlinear flame describing function to predict the nonlinear instability characteristics in both frequency and time domain. Furthermore, active feedback control is implemented. Two different controllers have been designed to eliminate the thermoacoustic instability to an acceptably low level and may be employed in a practical manner.

Orientador: Edson Denis Leonel / Resumo: Este estudo objetiva provar que sistemas dinâmicos de dimensão N, de codimensão um e satisfazendo as condições do teorema da bifurcação de Hopf, podem ser expressos em uma forma analítica simplificada que preserva a topologia do espaço de fases da configuração original, na vizinhança do ponto de equilíbrio. A esta forma simplificada é atribuído o nome de forma normal. Para tanto, foi utilizado a teoria da variedade central, necessária para reduzir a dimensão de sistemas à sua variedade bidimensional, e o teorema das formas normais, utilizando-se como método para determinar a forma simplificada da variedade central associada aos sistemas dinâmicos, atendendo as condições do teorema da bifurcação de Hopf. A partir da análise dos resultados aqui encontrados foi possível construir a prova matemática de que sistemas de dimensão N, atendendo as condições do teorema de Hopf, podem ser reescritos em uma expressão analítica geral e simplificada. Enfim, através deste estudo foi possível resumir todos os resultados aqui obtidos em um teorema geral que, além de reduzir a custosa tarefa de obtenção de formas normais, abrange sistemas N-dimensionais com ocorrência da bifurcação de Hopf. / Abstract: In this work we prove the following: consider a N-dimensional system that is reduced to its center manifold. If it is proved the system satisfies the conditions of Hopf bifurcation theorem, then the original system of differential equations is rewritten in a simpler analytical expression that preserves the phase space topology. This last is also known as the normal form. The center manifold is used to derive a reduced order expression, and the normal form theory is applied to simplify the form of the dynamics on the center manifold. The key results here allow constructing a general mathematical proof for the normal form of N-dimensional systems reduced to its center manifold. In the class of dynamical systems under Hopf bifurcations, the present work reduces the work done to obtain normal forms. / Mestre

[pt] Este trabalho visa estudar a teoria das bifurcações associada a sistemas de equações diferenciais não lineares com aplicação aos estudos de estabilidade em sistemas elétricos de potência. Como aplicação desta teoria analisamos o problema de operação ótima em sistemas de potência. Novas restrições são derivadas desta teoria que, incluídas às restrições tradicionais de um fluxo de potência ótimo, garantem a estabilidade do ponto de operação obtido. / [en] This work aims at studying bufurcation theory associated to non-linear differential equations systems with application on stability of electrical power systems. As an application of this theory, the problem of optimal operation in power systems is analyzed. New restriction to optimum power flow are found and together with traditional ones, assure the stability of the operation point.

A bactéria Zymomonas mobilis atraiu considerável interesse nas últimas décadas devido ao seu metabolismo único e a suas eficientes características fermentativas na produção de etanol através de açúcares simples. Além disso, dependendo do substrato utilizado outros produtos podem ser obtidos como ácido lático, ácido acético, ácido fórmico, acetona, levana, e sorbitol. Na literatura, a Z. mobilis tem sido proposta como microrganismo mais promissor que a convencional levedura Sacharomyces cerevisiae para a produção industrial de etanol. Na fermentação em modo contínuo o microrganismo apresenta oscilações (i.e., bifurcações Hopf) em baixas taxas de diluição (Df < 0,1/h). Diversos modelos têm sido propostos para descrever a dinâmica oscilatória do cultivo contínuo de Z. mobilis. Entre tais está o modelo de Jöbses et al. (1986) que foi ajustado experimentalmente em baixas taxas de diluição (Df < 0,1/h) e concentrações médias de substrato alimentado (Cso = 150 kg/m³). Recentemente, o modelo foi extrapolado por Elnashaine et al. (2006) que encontrou uma região operacional muito mais rentável a altas taxas de diluição (Df < 2,0/h) e concentração de substrato (Cso = 200 kg/m³). Embora o modelo de Jöbses não tenha sido validado nesta região, nossa contribuição assumirá que esta extrapolação é aceitável e então uma estratégia de controle foi proposta para manter o sistema trabalhando nesta região operacional. Para uma aplicação industrial bem sucedida da Z. mobilis, é necessário uma estratégia de controle eficiente e simples. Esse trabalho analisa o problema de controle e otimização de um biorreator contínuo de biosíntese de etanol pela bactéria modelado por Jöbses et al. (1986). Esse sistema apresenta multiplicidade de equilíbrios em determinadas condições operacionais. A idéia é manter o processo próximo à região de maior produtividade, localizada nas proximidades de um conjunto de bifurcações sela (onde o sistema torna-se instável). Baseado em uma análise sistemática da controlabilidade do sistema usando o índicie não-linear RPN percebe-se que é possível controlar o processo usando um controlador linear. Finalmente o trabalho aborda algumas características importantes no sistema de controle como a utilização de uma transformada nas ações do controlador com vistas a manter o biorreator no ótimo operacional. / Zymomonas mobilis has attracted considerable interest over the past decades pursuant to its unique metabolism and ability to rapidly and efficiently produce ethanol from simple sugars. In addition to ethanol depending on the substrate other fermentation products can occur, such as lactic acid, acetic acid, formic acid, acetone, and sorbitol. In the literature, Zymomonas mobilis has been proposed as a more promising microorganism than conventional yeast Saccharomyces cerevisiae for industrial production of ethanol. The major drawback of this microorganism is that it exhibits sustained oscillations (i.e., Hopf bifurcation) for low dilution rates (i.e., ,Df <=0.1 h-1) when grown in continuous mode. This leads to decreased ethanol productivity and less efficient use of available substrate. Various models have been proposed to describe the oscillatory dynamics of continuous Zymomonas mobilis cultures. One of them is the Jöbses et al. (1986) model that was fitted to experimental data with low dilution rate (i.e.,Df <= 0.1 h-1) and middle inlet substrate concentration (i.e., Cso~=150 kg/m³). Later, it was extrapolated outside of this operating region by Elnashaie et al. (2006), who have found a much more profitable operating region at higher dilution rates (Df~= 2.0 h -¹) and inlet concentrations (Cso~= 200 kg/m³). Notwithstanding the Jöbses's models has not been validated at this region, our contribution will assume that this extrapolation is acceptable and we will propose a control strategy to maintain the system working at this more profitable operating region. For a successful application of any industrial Z. mobilis facility, it is necessary to have an efficient and simple control strategy. This work analyzes the control and optimization problem of a continuous Z. mobilis bioreactor modeled by Jöbses et al. (1986). This system has steady state multiplicity in part of the operating range. The idea is to maintain the process close to the manifold border where is achievable the highest ethanol production. Based on a systematically analysis of the operational controllability using the nonlinear RPN indices it is identified that the process can be controlled using a linear controller. Finally in this work is proposed a variable transformation that makes easy to maintain the bioreactor close to the optimum.