The purpose of this paper is to present the results of a study of linear spaces with special emphasis of linear transformations, norms, and inner products.

The first objective of the dissertation is to develop a suitable reduced order model capable of investigating the nonlinear mechanical behavior of von-Karman plates under electrostatic actuation. The second objective is to investigate the nonlinear static and dynamic behavior of rectangular microplates under small and large actuating forces. In the first part, we present and compare various approaches to develop reduced order models for the nonlinear von-Karman rectangular microplates actuated by nonlinear electrostatic forces. The reduced-order models aim to investigate the static and dynamic behavior of the plate under small and large actuation forces. A fully clamped microplate is considered. Different types of basis functions are used in conjunction with the Galerkin method to discretize the governing equations. First we investigate the convergence with the number of modes retained in the model. Then for validation purpose, a comparison of the static results is made with the results calculated by a nonlinear finite element model. The linear eigenvalue problem for the plate under the electrostatic force is solved for a wide range of voltages up to pull-in. In the second part, we present an investigation of the static and dynamic behavior of a fully clamped microplate. We investigate the effect of different non-dimensional design parameters on the static response. The forced-vibration response of the plate is then investigated when the plate is excited by a harmonic AC load superimposed to a DC load. The dynamic behavior is examined near the primary and secondary (superharmonic and subharmonic) resonances. The microplate shows a strong hardening behavior due to the cubic nonlinearity of midplane stretching. However, the behavior switches to softening as the DC load is increased. Next, near-square plates are studied to understand the effect of geometric imperfections of microplates. In the final part of the dissertation, we investigate the mechanical behavior of initially curved microplates. Microplates often experience an initial curvature imperfection, due to the micro fabrication process, which affects significantly their mechanical behavior. In this case a clamped-free-clamped-free microplate is considered. We validate the reduced order model by comparing the calculated static behavior and the fundamental natural frequency with those computed by a finite element model. As case studies, we consider two commonly encountered profiles of the initial curvature imperfection and study their effects on both the static and dynamic responses of the microplates. Next, an initially curved microplate made of silicon nitride is studied. The static behavior of the microplate is investigated when applying a DC voltage. Then, the dynamic behavior of the microplate is examined under the application of a harmonic AC voltage, superimposed to a DC voltage. Simulation results calculated by the reduced order model are compared with experimental data for model validation purpose, which show good agreement.

The turbomachinery related industry relies heavily on numerical tools for the design and development of modern turbomachines. In order to be competitive turbomachines ought to be highly efficient and robust. This has lead engineers to develop more aggressive designs, which often leads to lower margins of structural reliability.  One of the strongest threats to turbomachines are high cycle fatigue problems which arise from aeroelastic phenomena such as flutter. According to Kielb R. (2013) many of such problems are detected at developing testing stage. This implies that the prediction capabilities for aeroelastic phenomena are in need of further development and/or tuning. This is especially evident for unsteady flow phenomena at transonic regimes. A very important step for the improvement of unsteady aerodynamic solvers is the validation and comparison of such solvers. The present thesis concerns with the validation and comparison of a non-linear time marching (ANSYS CFX) and the GKN’s in-house linearised solvers used for flutter analysis. The former has recently implemented a new feature called Transient Blade Row TBR, which drastically reduces the simulation domain to a maximum of two blades.  In order to be included in the deign process, such tool need to be validated. In the same way, the recently launched in-house code LINNEA needs to be validated in order to be considered as a design tool. Experimental data from the aeroelastic standard configuration 4, and the FUTURE project were used for the validation purposes. The validation process showed that the solvers agreed very well between them for the standard configuration. Such agreement was less clear for the FUTURE compressor; nonetheless, the solutions still sit within the bulk of solutions provided from the different FUTURE partners. The validation showed that these tools provide with similar results as the state of the art tools from different companies. This indicates that they can be used in the design process. At the same time it was observed that there is room for improvement in the solvers, as these still present some considerable differences with the experimental results.

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[pt] Este trabalho formula um método para a determinação das funções de rede de um circuito linear genérico, utilizando a interpolação, em pontos do circuito de raio unitário no plano complexo, de funções racionais reais. O método desenvolvido independe de considerações de ordem topológica podendo, ser usado com programas de análise de circuitos baseados na análise nodal, o que o torna bastante rápido, apesar de menos precioso, que os métodos baseados em técnicas de gráficos de fluxo de sinal. / [en] This work describes a method for generating the coeficients of a network function of a general linear circuit, using the interpolation of a real rational function on points of the unit circle in the s plane. The method is independent of topological considerations and suitable for use with network analysis programs based on the nodal formulation of network equations, beeing faster but less precise, than methods using flow graph techniques.

<p> Of greatest interest in the field of linear motion electrical machines are, the flat linear induction motor and liquid metal pump. The former has feasibility in high speed transportation while the latter is becoming of increasing importance in metallurgical processes. </p> <p> This thesis examines and extends the traditional representation for the usual sheet secondary classification and then presents a common theory which also allows composite secondary machines to be analysed. An idealized model is developed consisting of a number of regions representing air gaps, iron segments, and secondary conductors. A general solution for the field quantities is obtained. The concept of wave impedance and a transfer matrix approach allow a wide variety of configurations to be analysed. Unlike previous work the approach developed here allows for simultaneous investigation of lateral variation and skin effect. In the limiting cases where in tum skin effect and lateral variation are neglected the solutions are in agreement with known results. </p> / Thesis / Master of Engineering (MEngr)

Extending endofunctors on the category of sets and functions to the category of sets and relations requires one to introduce a certain amount of laxness. This in turn requires us to consider bicategories rather than ordinary categories. The subject of lax extensions of Set-based functors is one of the fundamental components of monoidal topology, an active area of research in categorical algebra. The recent theory of linear bicategories, due to Cockett, Koslowski and Seely, is an extension of the usual notion of bicategory to include a second composition in a way analogous to the two connectives of multiplicative linear logic. It turns out that the category of sets and relations has a second composition making it a linear bicategory. The goal of this thesis is first to define the notion of lax extension of Set-based functors to linear bicategories, and then demonstrate crucial properties of our definition.

Since the first recorded theatrical events, theatre has been presented in a linear fashion. It is no surprise that the majority of American Musical Theatre also follows a linear plot conception. Musicals that blatantly defy linearity are often called "concept" musicals. However, there is a small portion of concept musicals that do not abandon their linear plot altogether, but instead choose to skew the line in order to give the audience a unique perspective of the story being told. This mass categorization does not accurately take into account the extreme difference in structure of the two types of concept musicals. For this purpose, I explored the notion of a new structural category, the Less Linear Concept Musical, in hopes of identifying the unique characteristics and challenges associated with this type of musical. Beginning with Aristotle's "The Poetics" and ending in modern day Musical Theatre, I examine key events in the shaping of this musical theatre form, as well as define the form itself. The application of this research culminated in a classroom presentation of Jason Robert Brown's The Last Five Years, which I directed. I pose the question, "Can sound direction conquer the pitfalls found within the Less Linear Concept Musical?"

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LD2668 .T4 1940 S52 / Master of Science