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The position-dependent effective mass (PDEM) problem is of enormous importance to the realization of the extended Wentzel-Kramers-Brillouin (WKB) approximation in bound state calculations for semiconductor heterostructures. By studying some model problems, we show that the extended WKB method provides good approximations for the bound states with the high eigenenergies. In addition, the effect of the smoothness of the effective mass distribution functions and potential barrier in the PDEM problems is discussed in our work. We found the precision can be affected by the effective mass and potential barrier in the PDEM.

In the field of aeronautics (Aircraft, helicopters, space launchers...), as in the field of the automobile, reducing the harmful effects of acoustics constitutes a major concern at the international level and justifies the call for further research. With these challenges in mind, the manufacturers have expressed the need for simple models immediately available as early as the stage of preliminary drafts. This Master's Thesis presents the study and the validation of three different approaches to meet this industrial need. The first approach is based on the propagation of waves in the structure; the second approach uses the technique of SEA by calculating the equivalent damping; and the last approach uses the modal technique.In the three approaches, the transfer matrix method has been used to study the succession of layers. Applying these approaches to some aircraft structures confirm their relevance in relation to more exact and costly methods, such as the finite elements method.

Lattice simulations are a popular tool for studying the non-perturbative physics of nonlinear field theories. To perform accurate lattice simulations, a careful account of the discretization error is necessary. Spatial discretization error as a result of lattice spacing dependence in Langevin simulations of anisotropic (2 + 1)-dimensional classical scalar field theories is studied. A transfer integral operator (TIO) method and a one-loop renormalization (1LR) procedure are used to formulate effective potentials. The effective potentials contain counterterms which are intended to suppress the lattice spacing dependence. The two effective potentials were tested numerically in the case of a phi-4 model. A high accuracy modified Euler method was used to evolve a phenomenological Langevin equation. Large scale Langevin simulations were performed in parameter ranges determined to be appropriate. Attempts at extracting correlation lengths as a means of determining effectiveness of each method were not successful. Lattice sizes used in this study were not of a sufficient size to obtain an accurate representation of thermal equilibrium. As an alternative, the initial behaviour of the ensemble field average was observed. Results for the TIO method showed that it was successful at suppressing lattice spacing dependence in a mean field limit. Results for the 1LR method showed that it performed poorly.

Utilizando uma técnica de matriz de transferência, expandimos em série a probabilidade de percolação P(q) para o problema da percolação por sítio na rede quadrada direcionada. Nosso método revela uma inesperada conexão entre este problema e o da enumeração dos modos de se dissecar uma bola. Mostramos que o método pode também ser usado para se expandir em série o tamanho médio do cluster S (p) . Uma análise baseada nos aproximantes de Padé fornece estimativas do valor crítico pc, e também do expoente crítico &#946. / Using a transfer matrix technique we obtain an extended series expansion of the percolation probability P(q) for the directed site percolation problem on the square lattice. Our method reveals an up to now unsuspected connection between this problem and the enumeration of the ways of dissecting a ball. We show that the method can also be used to determine a series expansion for the mean cluster size S(p). An analysis based on Padé approximants gives estimates of the critical threshold pc, and also of the critical exponent &#946.

Control design for flexible robots is a challenging problem. Part of the difficulty comes from a lack of controls-focused modeling tools. Practical flexible robots have several aspects that make them difficult to model: continuous elements, complicated actuators, multiple feedback loops, non-collocated sensors and actuators, and the ability to take on arbitrary three-dimensional poses. Even if existing techniques for modeling flexible structures could model the closed-loop response of a hydraulically-actuated flexible robot with a vibration suppression controller, how would such a model be used for control design? This work presents the development of a modeling approach that meets the needs of a controls engineer. The approach is based on the transfer matrix method (TMM). The TMM has been expanded in several ways to enable it to accurately model practical flexible robots. Quantitative agreement is shown between model and experiment for the interaction of a hydraulic actuator and a flexible structure as well as for the closed-loop response of a system with vibration suppression. Once the ability to model the closed-loop response of the system has been demonstrated, this work focuses on using the model for control design. Control design is facilitated by symbolic implementation of the TMM, which allows closed-form expressions for the closed-loop response of the system to be found without discretization. These closed-form expressions will be transcendental transfer functions for systems with continuous elements. These transfer functions can then be used in various optimization approaches for designing the closed-loop system response.

The McTronX Research Group at the North-West University is conducting research in the field of Active Magnetic Bearings (AMBs) with the aim of establishing a knowledge base for future industry consultation. AMBs are environmentally friendly and are a necessity in the pebble bed modular reactor (PBMR), a South-African initiated project, which is predicted to be the means of supplying Africa and many other countries with modular energy in the future. Aside from the PBMR, there are numerous other AMB industrial applications. The aim of this project is to develop a comprehensive AMB model that considers the effect that rotor dynamics has on an AMB system. This model is used to analyse a double radial AMB, capable of suspending a rigid- and flexible rotor, to explain previously noticed phenomena. Two modelling methods are focussed on namely the System Matrix Method and Transfer Matrix Method (TMM) both of which are implemented in MATLAB®. The rigid rotor model is firstly implemented as a point mass in state-space form followed by use of the TMM to analyse its bending modes. The stability and critical speeds of the system are analysed due to a change in the supports' properties along with rotor gyroscopy and its effect on the system. During analysis of the flexible rotor the TMM was used via a similar approach as was followed with the rigid rotor. The results indicate that the system is experiencing lower than expected damping due to the model that is used within the control loop. The previously assumed rotor model in the control loop is not sufficient to describe its complex behaviour. This causes the unexpected damping characteristics. This project suggests future work to be conducted in expanding the frequency domain model of the rotor within the control loop to account for its physical shape. / Thesis (M.Ing. (Computer and Electronical Engineering))--North-West University, Potchefstroom Campus, 2009.

The McTronX Research Group at the North-West University is conducting research in the field of Active Magnetic Bearings (AMBs) with the aim of establishing a knowledge base for future industry consultation. AMBs are environmentally friendly and are a necessity in the pebble bed modular reactor (PBMR), a South-African initiated project, which is predicted to be the means of supplying Africa and many other countries with modular energy in the future. Aside from the PBMR, there are numerous other AMB industrial applications. The aim of this project is to develop a comprehensive AMB model that considers the effect that rotor dynamics has on an AMB system. This model is used to analyse a double radial AMB, capable of suspending a rigid- and flexible rotor, to explain previously noticed phenomena. Two modelling methods are focussed on namely the System Matrix Method and Transfer Matrix Method (TMM) both of which are implemented in MATLAB®. The rigid rotor model is firstly implemented as a point mass in state-space form followed by use of the TMM to analyse its bending modes. The stability and critical speeds of the system are analysed due to a change in the supports' properties along with rotor gyroscopy and its effect on the system. During analysis of the flexible rotor the TMM was used via a similar approach as was followed with the rigid rotor. The results indicate that the system is experiencing lower than expected damping due to the model that is used within the control loop. The previously assumed rotor model in the control loop is not sufficient to describe its complex behaviour. This causes the unexpected damping characteristics. This project suggests future work to be conducted in expanding the frequency domain model of the rotor within the control loop to account for its physical shape. / Thesis (M.Ing. (Computer and Electronical Engineering))--North-West University, Potchefstroom Campus, 2009.

Lattice simulations are a popular tool for studying the non-perturbative physics of nonlinear field theories. To perform accurate lattice simulations, a careful account of the discretization error is necessary. Spatial discretization error as a result of lattice spacing dependence in Langevin simulations of anisotropic (2 + 1)-dimensional classical scalar field theories is studied. A transfer integral operator (TIO) method and a one-loop renormalization (1LR) procedure are used to formulate effective potentials. The effective potentials contain counterterms which are intended to suppress the lattice spacing dependence. The two effective potentials were tested numerically in the case of a phi-4 model. A high accuracy modified Euler method was used to evolve a phenomenological Langevin equation. Large scale Langevin simulations were performed in parameter ranges determined to be appropriate. Attempts at extracting correlation lengths as a means of determining effectiveness of each method were not successful. Lattice sizes used in this study were not of a sufficient size to obtain an accurate representation of thermal equilibrium. As an alternative, the initial behaviour of the ensemble field average was observed. Results for the TIO method showed that it was successful at suppressing lattice spacing dependence in a mean field limit. Results for the 1LR method showed that it performed poorly.

Utilizando uma técnica de matriz de transferência, expandimos em série a probabilidade de percolação P(q) para o problema da percolação por sítio na rede quadrada direcionada. Nosso método revela uma inesperada conexão entre este problema e o da enumeração dos modos de se dissecar uma bola. Mostramos que o método pode também ser usado para se expandir em série o tamanho médio do cluster S (p) . Uma análise baseada nos aproximantes de Padé fornece estimativas do valor crítico pc, e também do expoente crítico &#946. / Using a transfer matrix technique we obtain an extended series expansion of the percolation probability P(q) for the directed site percolation problem on the square lattice. Our method reveals an up to now unsuspected connection between this problem and the enumeration of the ways of dissecting a ball. We show that the method can also be used to determine a series expansion for the mean cluster size S(p). An analysis based on Padé approximants gives estimates of the critical threshold pc, and also of the critical exponent &#946.