Asymmetric bending is explored as a means of inducing bend-twist coupling in an isotropic, fixed-wing airfoil. An analytical model describing the bend-twist coupling behavior of a constant-section airfoil undergoing steady wind loading is derived from Euler-Bernoulli beam theory, and evaluated over a range of structural and material stiffness. Finite element analysis is carried out in the ANSYS Parametric Design Language environment for an asymmetric, two-dimensional beam. Three-dimensional finite element analysis is carried out for two candidate blade models created in Pro/Engineer based on the NACA 64618 airfoil. Deformation results for the two- and three-dimensional finite element models are compared with analytical solutions. Results of this investigation highlight the dependency between the cross-sectional properties of a spar support and its tendency to exhibit twist-coupling under transverse loading.

Ce mémoire de maîtrise porte sur les billards mathématiques et la distribution des points de réflexion des trajectoires périodiques d’une table de billard strictement convexe. Un billard mathématique est un système dynamique généré par le mouvement libre d’une particule à l’intérieur d’un domaine dont la frontière est parfaitement réfléchissante. Une question d’intérêt particulier dans l’étude des billards mathématiques est celle de ses trajectoires périodiques. Nous considérons le cas des billards planaires strictement convexes. Il est connu que les points de réflexion des trajectoires périodiques de période n faisant un tour de table sont équidistribués par rapport à une mesure naturelle sur la frontière. Nous montrons ce résultat par une méthode nouvelle et relativement élémentaire utilisant la théorie de Lazuktin [12]. Dans le premier chapitre, nous donnons une description précise de la dynamique des billards et une brève introduction à la théorie de Lazuktin, aux applications de torsion et aux caustiques. Dans les chapitres 2 à 4, nous développons chacun des concepts précédents et expliquons comment ceux-ci sont liés aux billards. Le chapitre 5 est consacré à la preuve de notre résultat principal, divisée en deux parties. Nous concluons en donnant une annexe sur la théorie de la mesure. / This master’s thesis is concerned with mathematical billiards and distribution of reflection points of periodic trajectories of a strictly convex billiard table. A mathematical billiard is a dynamical system generated by the free motion of a particle inside of a domain with a perfectly reflecting boundary. A question of particular interest in the study of mathematical billiards is that of its periodic trajectories. We consider the case of planar strictly convex billiards. It is known that the reflection points of periodic trajectories of period n making one turn around the table are equidistributed with respect to a natural measure on the boundary. We show this result by a new and relatively elementary method using Lazuktin’s theory [12]. In the first chapter, we give a precise description of billiard dynamics and a brief introduction of Lazuktin’s theory, twist mappings and caustics. In Chapter 2 to 4, we elaborate each of the previous concepts and explain how they are related to billiards. Chapter 5 is dedicated to the proof of our main result, divided into two parts. We conclude by giving an appendix about measure theory.

Optical testing in adverse environments, ophthalmology and applications where characterization by curvature is leveraged all have a common goal: accurately estimate wavefront shape. This dissertation investigates wavefront sensing techniques as applied to optical testing based on gradient and curvature measurements. Wavefront sensing involves the ability to accurately estimate shape over any aperture geometry, which requires establishing a sampling grid and estimation scheme, quantifying estimation errors caused by measurement noise propagation, and designing an instrument with sufficient accuracy and sensitivity for the application. Starting with gradient-based wavefront sensing, a zonal least-squares wavefront estimation algorithm for any irregular pupil shape and size is presented, for which the normal matrix equation sets share a pre-defined matrix. A Gerchberg–Saxton iterative method is employed to reduce the deviation errors in the estimated wavefront caused by the pre-defined matrix across discontinuous boundary. The results show that the RMS deviation error of the estimated wavefront from the original wavefront can be less than λ/130~ λ/150 (for λ equals 632.8nm) after about twelve iterations and less than λ/100 after as few as four iterations. The presented approach to handling irregular pupil shapes applies equally well to wavefront estimation from curvature data. A defining characteristic for a wavefront estimation algorithm is its error propagation behavior. The error propagation coefficient can be formulated as a function of the eigenvalues of the wavefront estimation-related matrices, and such functions are established for each of the basic estimation geometries (i.e. Fried, Hudgin and Southwell) with a serial numbering scheme, where a square sampling grid array is sequentially indexed row by row. The results show that with the wavefront piston-value fixed, the odd-number grid sizes yield lower error propagation than the even-number grid sizes for all geometries. The Fried geometry either allows sub-sized wavefront estimations within the testing domain or yields a two-rank deficient estimation matrix over the full aperture; but the latter usually suffers from high error propagation and the waffle mode problem. Hudgin geometry offers an error propagator between those of the Southwell and the Fried geometries. For both wavefront gradient-based and wavefront difference-based estimations, the Southwell geometry is shown to offer the lowest error propagation with the minimum-norm least-squares solution. Noll’s theoretical result, which was extensively used as a reference in the previous literature for error propagation estimate, corresponds to the Southwell geometry with an odd-number grid size. For curvature-based wavefront sensing, a concept for a differential Shack-Hartmann (DSH) curvature sensor is proposed. This curvature sensor is derived from the basic Shack-Hartmann sensor with the collimated beam split into three output channels, along each of which a lenslet array is located. Three Hartmann grid arrays are generated by three lenslet arrays. Two of the lenslets shear in two perpendicular directions relative to the third one. By quantitatively comparing the Shack-Hartmann grid coordinates of the three channels, the differentials of the wavefront slope at each Shack-Hartmann grid point can be obtained, so the Laplacian curvatures and twist terms will be available. The acquisition of the twist terms using a Hartmann-based sensor allows us to uniquely determine the principal curvatures and directions more accurately than prior methods. Measurement of local curvatures as opposed to slopes is unique because curvature is intrinsic to the wavefront under test, and it is an absolute as opposed to a relative measurement. A zonal least-squares-based wavefront estimation algorithm was developed to estimate the wavefront shape from the Laplacian curvature data, and validated. An implementation of the DSH curvature sensor is proposed and an experimental system for this implementation was initiated. The DSH curvature sensor shares the important features of both the Shack-Hartmann slope sensor and Roddier’s curvature sensor. It is a two-dimensional parallel curvature sensor. Because it is a curvature sensor, it provides absolute measurements which are thus insensitive to vibrations, tip/tilts, and whole body movements. Because it is a two-dimensional sensor, it does not suffer from other sources of errors, such as scanning noise. Combined with sufficient sampling and a zonal wavefront estimation algorithm, both low and mid frequencies of the wavefront may be recovered. Notice that the DSH curvature sensor operates at the pupil of the system under test, therefore the difficulty associated with operation close to the caustic zone is avoided. Finally, the DSH-curvature-sensor-based wavefront estimation does not suffer from the 2π-ambiguity problem, so potentially both small and large aberrations may be measured.

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In dieser Dissertation werden Aspekte von superkonformen Quantenfeldtheorien untersucht, die für die sogenannte AdS/CFT Korrespondenz relevant sind. Die AdS/CFT Korrespondenz beschreibt eine Dualität zwischen Stringtheorien im Anti-de Sitter Raum und superkonformen Quantenfeldtheorien im Minkowskiraum. In diesem Kontext wurde die sog. Wilsonschleifen / Amplituden Dualität entdeckt, die die Übereinstimmung von n-Gluon MHV Amplituden und n-seitigen polygonalen Wilsonschleifen in der N=4 supersymmetrischen Yang-Mills (SYM) Theorie beschreibt. Im ersten Teil dieser Dissertation wird die Wilsonschleifenseite einer solchen möglichen Dualität in der N=6 superkonformen Chern-Simons (ABJM) Theorie untersucht. Das Hauptergebnis dieser Untersuchungen ist, dass der Erwartungswert der n-seitigen polygonalen Wilsonschleifen auf Einschleifenebene verschwindet, während er auf Zweischleifenebene in seiner funktionalen Form identisch zu der analogen Wilsonschleife in N=4 SYM auf Einschleifenniveau ist. Außerdem wird eine anomale konforme Wardidentität für Wilsonschleifen in Chern-Simons Theorie berechnet. Zudem werden die damit im Zusammenhang stehenden Entwicklungen für Amplituden und Korrelatoren in der ABJM Theorie diskutiert. Im zweiten Teil dieser Dissertation werden Dreipunktfunktionen von zwei geschützten Operatoren und einem Twist-Zwei Operator mit beleibigem Spin j in der N=4 SYM Theorie berechnet. Dafür werden die Indizes des Spin j Operators auf den Lichtkegel projiziert und der Korrelator wird in einem Grenzfall untersucht in dem der Impuls der bei dem Spin j Operator einfließt verschwindet. Dieser Grenzfall vereinfacht die perturbative Berechnung erheblich, da alle Dreipunktdiagramme effektiv auf Zweipunktdiagramme reduziert werden und die Abhängigkeit der Mischungsmatrix auf Einschleifenebene herausfällt. Das Ergebnis stimmt mit der Analyse der Operatorproduktentwicklung von Vierpunktfunktionen geschützter Operatoren von Dolan und Osborn aus dem Jahre 2004 überein. / In this thesis aspects of superconformal field theories that are of interest in the so-called AdS/CFT correspondence are investivated. The AdS/CFT correspondence states a duality between string theories living on Anti-de Sitter space and superconformal quantum field theories in Minkowski space. In the context of the AdS/CFT correspondence the so-called Wilson loop / amplitude duality was discovered, stating the equality of the finite parts of n-gluon MHV amplitudes and n-sided lightlike polygonal Wilson loops in N=4 supersymmetric Yang-Mills (SYM) theory. It is the subject of the first part of this thesis to investigate the Wilson loop side of a possible similar duality in N=6 superconformal Chern-Simons matter (ABJM) theory. The main result is, that the expectation value of n-sided lightlike polygonal Wilson loops vanishes at one-loop order and at two-loop order is identical in its functional form to the Wilson loop in N=4 SYM theory at one-loop order. Furthermore, an anomalous conformal Ward identity for Wilson loops in Chern-Simons theory is derived. Related developments and symmetries of amplitudes and correlators in ABJM theory are discussed as well. In the second part of this thesis we calculate three-point functions of two protected operators and one twist-two operator with arbitrary even spin j in N =4 SYM theory. In order to carry out the calculations, the indices of the spin j operator are projected to the light-cone and the correlator is evaluated in a soft-limit where the momentum coming in at the spin j operator becomes zero. This limit largely simplifies the perturbative calculation, since all three-point diagrams effectively reduce to two-point diagrams and the dependence on the one-loop mixing matrix drops out completely. The result is in agreement with the analysis of the operator product expansion of four-point functions of half-BPS operators by Dolan and Osborn in 2004.

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The aim of this investigation was to review literature for the most suitable aeroelastic tailoring analysis tools for long slender composite structures, and integrate them into an aeroelastic tailoring process. The JS1C Revelation is a high performance sailplane made from modern composites, mostly carbon fibre. This has the advantage of being more rigid than traditional engineering materials, thereby reducing the effects of the twisting deflections on these long slender structures due to aerodynamic loads. The implementing of aeroelastic tailoring can create bend-twist couples for performance improvements. Composites enable the use of aeroelastic tailoring to improve gliding performance. Flaperon 3 of the JS1C 21 m was used as the design problem for aeroelastic tailoring. Aeroelastic tailoring was done by analysing the flaperon structure at the different layup angles to determine the correct design point to tailor the structure to improve aerodynamic performance at thermalling and cruise, but mostly cruise since it accounts for 70% of the flight time. The composite structure analysis tool has the objective to get results during concept design. This directed the line of research of analysis tools to a solution method of two dimensional cross-section mesh properties projected onto a one dimensional beam. The literature of Hodges had good verification and published data on the analysis tools. The analysis tools comprised of three programs that were not very user friendly. Thus the author compiled a Matlab program as a user interface tool to run the three programs together. The aeroelastic tailoring process systematically works through the known design variables and objectives, which are given as inputs to the analysis tool. The analysis tool plots the coupling data versus layup angle. From this the best layup angles for a sought-after bend-twist couple is used to aeroelastically tailor the wing. The composite structure analysis tool’s accuracy was verified by analysing cantilever beam deflections and comparing the results with hand calculations and SolidWorks Simulation FEM results. The analysis tool’s accuracy was further verified by comparing the aerodynamic torsional load’s twist deflections with thin walled tube theory. The analysis tool was validated by applying a torsional load at the tip of a JS1C production Flaperon 3 in an experimental setup and then comparing this result with the Flaperon 3 modelled in the analysis tool. These comparisons also ensured that the model’s composite material properties and the meshing of the flaperon cross-sectional properties were correct. This aeroelastic tailoring was validated with the advantage of then being used to improve the aerodynamic performance of the JS1C Revelation 21 m tip’s flaperon. This improvement could be made by making use of a tailored bend-twist couple to reduce the effect of the aerodynamic load’s twist deflections. A test sample of the JS1C 21 m flaperon 3 was used to validate aeroelastic tailoring. The test sample was designed to be 1 m in length and have all the specified tailoring coupling characteristics that could improve the aerodynamic performance of the JS1C 21 m flaperon 3. The test sample was manufactured according to Jonker Sailplanes manufacturing standards and experimentally set up with the same applied deflections as in the analysis tool. The calculated bend-twist values and the experimental setup results were similar with a negligible difference, assuming small displacements and an aspect ratio greater than 13; this confirmed that the PreVABS/VABS/GEBT composite structure analysis tool could be used in aeroelastic tailoring to predict and design the bend-twist couple needed to improve the aerodynamic performance of the JS1C 21 m. While the twist behaviour of Flaperon 3 was improved by the tailored bend-twist couple, it was still necessary to add pre-twist as well, to fully address the effects of twisting by aerodynamic forces. / MIng (Mechanical Engineering), North-West University, Potchefstroom Campus, 2014

The aim of this investigation was to review literature for the most suitable aeroelastic tailoring analysis tools for long slender composite structures, and integrate them into an aeroelastic tailoring process. The JS1C Revelation is a high performance sailplane made from modern composites, mostly carbon fibre. This has the advantage of being more rigid than traditional engineering materials, thereby reducing the effects of the twisting deflections on these long slender structures due to aerodynamic loads. The implementing of aeroelastic tailoring can create bend-twist couples for performance improvements. Composites enable the use of aeroelastic tailoring to improve gliding performance. Flaperon 3 of the JS1C 21 m was used as the design problem for aeroelastic tailoring. Aeroelastic tailoring was done by analysing the flaperon structure at the different layup angles to determine the correct design point to tailor the structure to improve aerodynamic performance at thermalling and cruise, but mostly cruise since it accounts for 70% of the flight time. The composite structure analysis tool has the objective to get results during concept design. This directed the line of research of analysis tools to a solution method of two dimensional cross-section mesh properties projected onto a one dimensional beam. The literature of Hodges had good verification and published data on the analysis tools. The analysis tools comprised of three programs that were not very user friendly. Thus the author compiled a Matlab program as a user interface tool to run the three programs together. The aeroelastic tailoring process systematically works through the known design variables and objectives, which are given as inputs to the analysis tool. The analysis tool plots the coupling data versus layup angle. From this the best layup angles for a sought-after bend-twist couple is used to aeroelastically tailor the wing. The composite structure analysis tool’s accuracy was verified by analysing cantilever beam deflections and comparing the results with hand calculations and SolidWorks Simulation FEM results. The analysis tool’s accuracy was further verified by comparing the aerodynamic torsional load’s twist deflections with thin walled tube theory. The analysis tool was validated by applying a torsional load at the tip of a JS1C production Flaperon 3 in an experimental setup and then comparing this result with the Flaperon 3 modelled in the analysis tool. These comparisons also ensured that the model’s composite material properties and the meshing of the flaperon cross-sectional properties were correct. This aeroelastic tailoring was validated with the advantage of then being used to improve the aerodynamic performance of the JS1C Revelation 21 m tip’s flaperon. This improvement could be made by making use of a tailored bend-twist couple to reduce the effect of the aerodynamic load’s twist deflections. A test sample of the JS1C 21 m flaperon 3 was used to validate aeroelastic tailoring. The test sample was designed to be 1 m in length and have all the specified tailoring coupling characteristics that could improve the aerodynamic performance of the JS1C 21 m flaperon 3. The test sample was manufactured according to Jonker Sailplanes manufacturing standards and experimentally set up with the same applied deflections as in the analysis tool. The calculated bend-twist values and the experimental setup results were similar with a negligible difference, assuming small displacements and an aspect ratio greater than 13; this confirmed that the PreVABS/VABS/GEBT composite structure analysis tool could be used in aeroelastic tailoring to predict and design the bend-twist couple needed to improve the aerodynamic performance of the JS1C 21 m. While the twist behaviour of Flaperon 3 was improved by the tailored bend-twist couple, it was still necessary to add pre-twist as well, to fully address the effects of twisting by aerodynamic forces. / MIng (Mechanical Engineering), North-West University, Potchefstroom Campus, 2014