This master's thesis deals with measurement of torsional rigidity of a vehicle body with the application of TRITOP optical measuring system. The measurement was made in dynamic test-room of Škoda Auto a.s. together with a methodology used by this company. In the light of the objective comparison of both measuring methods and their accuracy, the optimal method of measurment and evaluation was made.

This thesis deals with the measurement of the torsional stiffness of the frame using Tritop system. There is described development of the frame due to the complexity of torsional stiffness. Further there is outlined a brief overview of the types of measurement and the effect of torsional stiffness on the driving behavior of the vehicle. The aim of the work is to develop a method for measuring torsional stiffness of the vehicle Formula Student and from the results of repeated measurements to assess the accuracy and repeatability of measurements.

Nous étudions la clôture algébrique et définissable dans les groupes libres. Les résultats principaux peuvent être résumés comme suit. Nous montrons un résultat de constructibilité des groupes hyperboliques sans torsion au-dessus de la clôture algébrique d'un sous-ensemble engendrant un groupe non abélien. Nous avons cherché à comprendre la place qu'occupe la clôture algébrique acl_G(A) dans certaines décompositions de G. Nous avons étudié la possibilité de la généralisation de la méthode de Bestvina-Paulin dans d'autres directions, en considérant les groupes de type fini qui agissent d'une manière acylindrique (au sens de Bowditch) sur les graphes hyperboliques. Enfin, nous avons étudié les relations qui existent entre les différentes notions de clôture algébrique et entre la clôture algébrique et la clôture définissable / In Chapter 1 we give basics on combinatorial group theory, starting from free groups and proceeding with the fundamental constructions: free products, amalgamated free products and HNN extensions. We outline a synthesis of Bass-Serre theory, preceded by a survey on Cayley graphs and graphs of groups. After proving the main theorem of Bass-Serre theory, we present its application to the proof of Kurosh subgroup theorem. Subsequently we recall main definitions and properties of hyperbolic spaces. In Section 1.4 we define algebraic and definable closures and recall a few other notions of model theory related to saturation and homogeneity. The last section of Chapter 1 is devoted to asymptotic cones. In Chapter 2 we prove a theorem similar to Bestvina-Paulin theorem on the limit of a sequence of actions on hyperbolic graphs. Our setting is more general: we consider Bowditch-acylindrical actions on arbitrary hyperbolic graphs. We prove that edge stabilizers are (finite bounded)-by-abelian, that tripod stabilizers are finite bounded and that unstable edge stabilizers are finite bounded. In Chapter 3 we introduce the essential notions on limit groups, shortening argument and JSJ decompositions. In Chapter 4 we present the results on constructibility of a torsion-free hyperbolic group from the algebraic closure of a subgroup. Also we discuss constructibility of a free group from the existential algebraic closure of a subgroup. We obtain a bound to the rank of the algebraic and definable closures of subgroups in torsion-free hyperbolic groups. In Section 4.2 we prove some results about the position of algebraic closures in JSJ decompositions of torsion-free hyperbolic groups and other results for free groups. Finally, in Chapter 5 we answer the question about equality between algebraic and definable closure in a free group. A positive answer has been given for a free group F of rank smaller than 3. Instead, for free groups of rank strictly greater than 3 we found some counterexample. For the free group of rank 3 we found a necessary condition on the form of a possible counterexample.

Wir konstruieren Beispiele von Kontaktmannigfaltigkeiten in jeder ungeraden Dimension, welche endliche nicht-triviale algebraische Torsion (im Sinne von Latschev-Wendl) aufweisen, somit straff sind und keine starke symplektische Füllung haben. Wir beweisen, dass Giroux Torsion algebraische 1-Torsion in jeder ungeraden Dimension impliziert, womit eine Vermutung von Massot-Niederkrüger-Wendl bewiesen wird. Wir konstruieren unendlich viele nicht diffeomorphe Beispiele von 5-dimensionalen Kontaktmannigfaltigkeiten, welche straff sind, keine starke symplektische Füllung zulassen und keine Giroux Torsion haben. Wir erhalten Obstruktionen für symplektische Kobordismen, ohne für deren Beweis die SFT Maschinerie zu verwenden. Wir geben eine provisorische Definition eines spinalen offenen Buchs in höherer Dimension an, basierend auf der vom 3-dimensionalen Fall aus Lisi-van Horn Morris-Wendl. In einem Anhang geben wir in gemeinsamer Autorenschaft mit Richard Siefring eine wesentliche Zusammenfassung der Schnitttheorie für punktierte holomorphe Kurven und Hyperflächen an, welche die 3-dimensionalen Resultate von Siefring auf höhere Dimensionen verallgemeinert. Mittels der Schnitttheorie erhalten wir eine Anwendung für holomorphe Blätterungen von Kodimension zwei, die wir benutzen um das Verhalten von holomorphem Kurven in unseren Beispielen einzuschränken. / We construct examples in any odd dimension of contact manifolds with finite and non-zero algebraic torsion (in the sense of Latschev-Wendl), which are therefore tight and do not admit strong symplectic fillings. We prove that Giroux torsion implies algebraic 1-torsion in any odd dimension, which proves a conjecture of Massot-Niederkrüger-Wendl. We construct infinitely many non-diffeomorphic examples of 5-dimensional contact manifolds which are tight, admit no strong fillings, and do not have Giroux torsion. We obtain obstruction results for symplectic cobordisms, for which we give a proof not relying on SFT machinery. We give a tentative definition of a higher-dimensional spinal open book decomposition, based on the 3-dimensional one of Lisi-van Horn Morris-Wendl. An appendix written in co-authorship with Richard Siefring gives a basic outline of the intersection theory for punctured holomorphic curves and hypersurfaces, which generalizes his 3-dimensional results to higher dimensions. From the intersection theory we obtain an application to codimension-2 holomorphic foliations, which we use to restrict the behaviour of holomorphic curves in our examples.

Dans la première partie de cette thèse, on présente des bornes supérieures fines pour le nombre de sous-variétés irréductibles de torsion maximales dans une sous-variété du tore complexe algébrique $(\mathbb{C}^{\times})^n$ et d'une variété abélienne. Dans les deux cas, on donne une borne explicite en termes du degré des polynômes définissants et la variété ambiante. De plus, la dépendance en le degré des polynômes est optimale. Dans le cas du tore complexe, on donne aussi une borne explicite en termes du degré torique de la sous-variété. En conséquence de ce dernier résultat, on démontre les conjectures de Ruppert, et Aliev et Smyth pour le nombre de points de torsion isolés dans une hypersurface. Ces conjectures bornent ce nombre en terme, respectivement, du multi-degré et du volume du polytope de Newton d'un polynôme définissant l'hypersurface.Dans la deuxième partie de cette thèse, on présente une borne supérieure pour la hauteur des zéros isolés, dans le tore, d'un système de polynômes de Laurent sur un corps adélique qui satisfait la formule du produit. Cette borne s'exprime en termes des intégrales mixtes des fonctions toit locales associées à la hauteur choisie et le système des polynômes de Laurent. On montre aussi que cette borne est presque optimale dans quelques familles d'exemples. Ce résultat est un analogue arithmétique du théorème de Bern\v{s}tein-Ku\v{s}nirenko. / In the first part of this thesis we present sharp bounds on the number of maximal torsion cosets in a subvariety of a complex algebraic torus $(\mathbb{C}^{\times})^n$ and of an Abelian variety. In both cases, we give an explicit bound in terms of the degree of the defining polynomials and the ambient variety. Moreover, the dependence on the degree of the polynomials is sharp. In the case of the complex torus, we also give an effective bound in terms of the toric degree of the subvariety. As a consequence of the latter result, we prove the conjectures of Ruppert, and Aliev and Smyth on the number of isolated torsion points of a hypersurface. These conjectures bound this number in terms of the multidegree and the volume of the Newton polytope of a polynomial defining the hypersurface, respectively.In the second part of the thesis, we present an upper bound for the height of isolated zeros, in the torus, of a system of Laurent polynomials over an adelic field satisfying the product formula. This upper bound is expressed in terms of the mixed integrals of the local roof functions associated to the chosen height function and to the system of Laurent polynomials. We also show that this bound is close to optimal in some families of examples. This result is an arithmetic analogue of the classical Bern\v{s}tein-Ku\v{s}nirenko theorem.

Ce mémoire porte sur la segmentation des fibres de la matière blanche et sur le développement d'outils visuels permettant d'interagir avec les résultats. Pour y parvenir, une métrique innovatrice permettant de quantifier la différence entre deux fibres de la matière blanche est créée. Cette mesure fait appel à des notions de multirésolution, de courbure, de torsion afin de caractériser la forme géométrique d'une fibre. Elle regroupe également des mesures plus simples telles la distance du cosinus, la distance euclidienne entre les centres de masse et la différence des longueurs d'arc pour capter respectivement l'orientation, la translation et la taille d'une fibre. Ensuite, une nouvelle technique de segmentation permettant de gérer des quantités importantes de données est développée. Finalement, ces nouvelles méthodes sont validées sur différents jeux de données.

Thesis (MSc)--Stellenbosch University, 2013. / ENGLISH ABSTRACT: The central objective of our study focuses on torsion points on elliptic curves. The case of elliptic curves over finite fields is explored up to giving explicit formulae for the cardinality of the set of points on such curves. For finitely generated fields of characteristic zero, a presentation and discussion of some known results is made. Some applications of elliptic curves are provided. In one particular case of applications, we implement an integer factorization algorithm in a computer algebra system SAGE based on Lenstra’s elliptic curve factorisation method. / AFRIKAANSE OPSOMMING: Die hoofdoel van ons studie is torsiepunte op elliptiese krommes. Ons ondersoek die geval van elliptiese krommes oor ‘n eindige liggaam met die doel om eksplisiete formules vir die aantal punte op sulke krommes te gee. Vir ‘n eindig-voortgebringde liggaam met karakteristiek nul bespreek ons sekere bekende resultate. Sommige toepassings van elliptiese krommes word gegee. In een van hierdie toepassings implementeer ons ‘n heeltallige faktoriseringalgoritme in die rekenaar-algebrastelsel SAGE gebaseer op Lenstra se elliptiese krommefaktoriseeringmetode.

The aim of this thesis is to design and build a torsional Kolsky bar apparatus for testing cylindrical specimens in torsion at high strain-rates. In addition to well-established designs, this testing apparatus will include a conical mirror combined with a high speed camera that allows time-resolved optical observation of the shear deformation on the surface of the specimen. The basic design of a Kolsky bar consists of a loading bar, input bar, specimen, and output bar. The experiment is conducted by storing torque in the loading bar and then releasing the torque by breaking the clamp and sending a shear wave pulse through the apparatus into the specimen. This shear wave pulse is monitored by strain gages mounted on the input and output bars. Analysis of the strain waves in the input and output bar is used to extract the shear stress - shear strain profile of the specimen. Several experiments were conducted on 6061-O and 1100-O aluminum with wall thicknesses ranging from 0.3 to 1.5 mm. / text

......

The considerable progress in the research and development of thin-walled beam structures responds to their growing use in engineering construction and to their increased need for efficiency in strength and cost. The result is a structure that exhibits large shear strains and important non uniform warping under different loadings, such as non uniform torsion, shear bending and distortion... A unified approach is formulated in this thesis for 3D thin walled beam structures with arbitrary profile geometries, loading cases and boundary conditions. A single warping function, defined by a linear combination of longitudinal displacements at cross sectional nodes (derived from Prokic work), is enhanced and adapted in order to qualitatively and quantitatively reflect and capture the nature of a widest possible range of behaviors. Constraints are prescribed at the kinematics level in order to enable the study of arbitrary cross sections for general loading. This approach, differing from most published theories, has the advantage of enabling the study of arbitrary cross sections (closed/opened or mixed) without any restrictions or distinctions related to the geometry of the profile. It generates automatic data and characteristic computations from a kinematical discretization prescribed by the profile geometry. The amount of shear bending, torsional and distortional warping and the magnitude of the shear correction factor is computed for arbitrary profile geometries with this single formulation. The proposed formulation is compared to existing theories with respect to the main assumptions and restrictions. The variation of the location of the torsional center, distortional centers and distortional rotational ratio of a profile is discussed in terms of their dependency on the loading cases and on the boundary conditions. A 3D beam finite element model is developed and validated with several numerical applications. The displacements, rotations, amount of warping, normal and shear stresses are compared with reference solutions for general loading cases involving stretching, bending, torsion and/or distortion. Some examples concern the case of beam assemblies with different shaped profiles where the connection type determines the nature of the warping transmission. Other analyses –for which the straightness assumption of Timoshenko theory is relaxed– investigate shear deformation effects on the deflection of short and thin beams by varying the aspect ratio of the beam. Further applications identify the cross sectional distortion and highlight the importance of the distortion on the stresses when compared to bending and torsion even in simple loading cases. Finally, a non linear finite element based on the updated lagrangian formulation is developed by including torsional warping degrees of freedom. An incremental iterative method using the arc length and the Newton-Raphson methods is used to solve the non linear problem. Examples are given to study the flexural, torsional, flexural torsional and lateral torsional buckling problems for which a coupling between the variables describing the flexural and the torsional degrees of freedom occurs. The finite element results are compared to analytical solutions based on different warping functions and commonly used in linear stability for elastic structures having insufficient lateral or torsional stiffnesses that cause an out of plane buckling.