The benefits of intentional friction damping to reduce the occurrence of wear and premature failure of turbomachinery bladed-disk assemblies are well known and many studies on this topic have focused on the analysis and prediction of the complicated nonlinear forced response exhibited by these structures. In this research, extensions of the recently introduced multi-harmonic Hybrid Frequency-Time method are proposed for the efficient analysis of the response of realistic structures featuring displacement-dependent nonlinearities, such as the friction and impact phenomena that may occur in the presence of friction dampers or when two parts of the same structure periodically contact each other. These theoretical extensions are adapted to the study of large scale, industrial bladed-disk structures that may feature cyclic symmetry or mistuning. Two analysis techniques are developed for the modeling of displacement-dependent nonlinearities. In the first technique friction dampers are modeled as nonlinear operators representing the contact forces acting on the blades, from the simple case of monodirectional friction with constant normal load to the more complex case of three dimensional contact with variable normal load. The analysis of the forced response of several nonlinear systems illustrates the capabilities of this approach as well as the complexity of the typical behavior exhibited by friction damped structures. The second technique introduced helps analyze structures experiencing intermittent contact or friction between two parts or sub-components of the same assembly. This method is applied to the study of the forced response of several simple systems and is used with great efficiency to predict the nonlinear behavior of a beam with a crack. This approach also allows the dampers to be modeled realistically as stand-alone components appended to the bladed disk assembly. In this case the bladed disk assembly as well as the friction dampers are modeled as independent structures that interact at their contacting interfaces. This allows the use of detailed finite element models of dampers rather than having to make simplifying assumptions regarding their geometry. These two methods are applied to the study of the nonlinear forced response a realistic bladed-disk assembly featuring a wedge damper model and a structure-like damper model.

La thèse a pour objet la simulation numérique de divers problèmes fortement non linéaires de la mécanique des milieux continus, en particulier en formage des métaux et en géomécanique. Le formalisme théorique puis numérique de la mécanique non linéaire des milieux continus, des couplages hydromécaniques et thermomécaniques et des modèles de comportement élastoplastique et élastoviscoplastique est développé étape par étape, permettant la construction du code aux éléments finis LAGAMINE. Celui-ci est ensuite utilisé pour simuler quelques problèmes spécifiques.

Previous work in the Railway Technology Laboratory at Virginia Tech focused on better capturing the dynamics of the friction wedge, modeled as a 3D rigid body. The current study extends that work to a half-truck model treated as an application of multibody dynamics with unilateral contact to model the friction wedge interactions with the bolster and the sideframe. The half-truck model created in MATLAB is a 3D, dynamic, multibody dynamics model comprised of four rigid bodies: a bolster, two friction wedges, and a sideframe assembly. The model allows each wedge four degrees of freedom: vertical displacement, longitudinal displacement (between the bolster and sideframe), pitch (rotation around the lateral axis), and yaw (rotation around the vertical axis). The bolster and the sideframe have only the vertical degree of freedom. The geometry of these bodies can be adjusted for various simulation scenarios. The bolster can be initialized with a pre-defined yaw (rotation around the vertical axis) and the sideframe may be initialized with a pre-defined pitch/toe (rotation around the lateral axis). The multibody dynamics half-truck model simulation results have been compared with results from NUCARS®, an industry standard train modeling software, for similar inputs. The multibody dynamics models have also been extended to a variably damped full-truck model and a variably damped half-truck warping model. These models were reformulated to react dynamically to simulated truck warp inputs. The ability to better characterize truck warping properties can prevent train roll over and derailments from truck hunting. In a quarter-truck variably damped configuration the effects of a curved wedge surface has also been explored. Actual friction wedges have surfaces which are slightly curved, this iteration in the multibody dynamics friction wedge modeling attempts to draw one step closer to actual friction wedge geometry. This model lays the ground work for a contact dependant wedge wearing model based on material properties and tribology. / Master of Science

This thesis presents a theoretical application of multibody dynamics with unilateral contact to model the interaction of the damping element in a freight train suspension, the friction wedge, with the bolster and the side frame. The objective of the proposed approach is to produce a stand-alone model that can better characterize the interaction between the bolster, the friction wedge, and the side frame subsystems. The new model allows the wedge four degrees of freedom: vertical displacement, longitudinal (between the bolster and the side frame) displacement, pitch (rotation about the lateral axis), and yaw (rotation about the vertical axis). The new model also allows for toe variation. The stand-alone model shows the capability of capturing dynamics of the wedge which were not possible to simulate using previous models. The inclusion of unilateral contact conditions is integral in quantifying the behavior during lift-off and the stick-slip phenomena. The resulting friction wedge model is a 3D, dynamic, stand-alone model of a bolster-friction wedge-side frame assembly. The new stand-alone model was validated through simulation using simple inputs. The dedicated train modeling software NUCARS® has been used to run simulations with similar inputs and to compare — when possible — the results with those obtained from the new stand-alone MATLAB friction wedge model. The stand-alone model shows improvement in capturing the transient dynamics of the wedge better. Also, it can predict not only normal forces going into the side frame and bolster, but also the associated moments. Significant simulation results are presented and the main differences between the current NUCARS® models and the new stand-alone MATLAB models are highlighted. / Master of Science

[pt] Neste trabalho apresenta-se uma metodologia pelo método dos elementos finitos para a análise estática-dinâmica bi-dimensional de linhas marítimas considerando-se o contato com o leito marinho. Para o cabo considera-se o elemento de dois nós do modelo de viga de Euler-Bernoulli com as grandezas estáticas e dinâmicas referidas a um sistema co- rotacionado. Considera-se a nãolinearidade geométrica associada ao movimento de corpo rígido da estrutura, supondo pequenas deformações. No movimento da linha, são consideradas as influências do peso próprio, empuxo, carregamentos hidrodinâmicos das correntes marinhas, deslocamentos prescritos no ponto superior da linha, forças de inercia e ação de flutuadores. Os efeitos sobre a linha associados ao contato com o fundo marinho são considerados empregando-se a técnica dos multiplicadores de Lagrange. Estes representam o carregamento (forças e momentos) requerido para garantir as restrições geométricas. No modelo não são considerados os efeitos das forças de atrito com o fundo marinho, cuja rigidez é considerada muito superior à do cabo. A matriz de rigidez usual do modelo de vigas é aumentada de forma a acomodar as novas incógnitas resultantes da imposição das condições de contato. Desta forma, condições de contato, separação repetidos (contato variável) e grandes movimentos relativos do cabo com o leito são contemplados pelo modelo. A solução das equações algébricas não-lineares, resultantes da integração temporal passo-a-passo de Newmark, é obtida com a técnica iterativa de Newton Raphson. A metodologia numérica foi implementada e resultados de alguns testes para situações da engenharia offshore são mostrados e comparados com os apresentados por outros autores independentes. / [en] In this work a finite element methodology for two- dimensional static and/or dynamic analysis of maritime risers including contact with the seabed is considered. The line is modeled by a two node beam element based on the Euler- Bernoulli theory with a co-rotational coordinate system, defined from the structure space coordinate nodes, to refer the element equilibrium and motion equations. Geometric non-linear effects are considered, all associated with rigid body motions from the structure, under small strain hypothesis. In the riser motion, the influences of its own weight, buoyancy forces, hydrodynamic loadings due to the maritime currents, the prescribed displacements at the top point line, the inertia forces and the floating action are considered. Effects on the line associated to the seabed contact are considered using Lagrange multipliers. They represent, the required loading (forces and moments) to ensure the cable geometric restrictions. In the model, friction forces effects on the seabed are not considered and the seabed stiffness is taken much greater than the riser s. The beam model stiffness matrix results into an increased matrix to accommodate the new model unknowns reactions, which are represented by the contact forces and moments. Similarly, conditions for variable contact and/or separation, as well as large relative motions between the flexible line and the seabed, are contemplated by the model. Solutions of the resulting nonlinear motion equations are obtained using Newmark s step-by-step time integration procedure with the Newton-Raphson iterative technique, for the numerical solution convergence. The methodology has been implemented and some representative offshore engineering testing analysis results are presented compared to other independently published solutions.

The present work deals with the numerical solution of dynamic contact and fracture problems. The contact problem is a Signorini problem with or without Coulomb friction. The fracture problem uses a cohesive zone model with a prescribed crack path. These problems are characterized by a non-regular boundary condition and can be formulated with evolutionary variational inequations or differential inclusions. For the numerical solution, we combine, as usual in solid dynamics, a finite element discretization in space and time-integration schemes. For the contact problem, we begin by comparing the main methods proposed in the literature. We then focus on the so-called modified mass method recently introduced by H. Khenous, P. Laborde et Y. Renard, for which we propose a semi-explicit variant. In addition, we prove a convergence result of the space semi-discrete solutions to a continuous solution in the frictionless viscoelastic case. We also analyze the space semi-discrete and fully discrete problems in the friction Coulomb case. For the dynamic fracture problem, using a fully explicit scheme is impossible or not robust enough. Therefore, we propose time-integration schemes where the boundary condition is treated in an implicit way. Finally, we present and analyze augmented Lagrangian methods for static fracture problems

Les contacts entre une corde vibrante et un obstacle unilatéral interviennent dans de nombreux instruments de musique tels que la tampoura, le sitar ou encore la basse électrique. Ils introduisent une forte non-linéarité dans le mouvement de la corde, qui se traduit notamment par d'importants transferts d'énergie. Dans ce manuscrit, une étude à la fois numérique et expérimentale du contact corde/obstacle est proposée. Trois types de schémas conservatifs sont développés. Ils s'appuient sur une description modale de la corde, ce qui permet un ajustement des fréquences propres et amortissements pour chaque mode. De plus, deux modèles de contact et frottement sont considérés, l'un régularisant et l'autre non. Des comparaisons à la solution analytique dans le cas d'un obstacle ponctuel sont menées et permettent une première validation des modèles. Des protocoles expérimentaux sont ensuite mis en place pour l'étude d'une corde isolée en présence d'un obstacle ponctuel, puis dans le cas d'une corde installée sur une basse électrique, munie ou non de frettes. Une forte concordance des signaux expérimentaux et numériques est observée sur des temps longs, montrant la capacité des modèles à rendre compte des éléments principaux observés expérimentalement. Enfin, une étude paramétrique est conduite dans le cas d'une basse munie de frettes, montrant l'influence de paramètres tels que la position de pincement de la corde et le réglage des frettes sur le son produit. Ces ajustements sont directement liés à des problématiques récurrentes et centrales rencontrées par les musiciens et les luthiers. / Collisions between a vibrating string and a unilateral obstacle arise in a number of instruments such as tanpuras, sitars and electric basses. Contacts introduce a strong nonlinearity in the string motion, resulting in energy transfers. In this document, a numerical and experimental study of string/obstacle contacts is investigated. Three conservative numerical schemes are developed. They rely on a modal description of the string, so that a fine tuning of eigenfrequencies and damping parameters is possible for each mode. Moreover, contact and friction models are numerically treated either with a penalty approach or with a nonsmooth contact dynamics method. In the case of a point obstacle, comparisons to an analytical solution constitute a first validation of models. Experiments are then carried out in the case of an isolated string vibrating against a point obstacle, as well as in the case of a string vibrating against the neck of a fretted or fretless electric bass on which the string is installed. A good agreement is observed between numerical and experimental signals over long durations, showing the ability of models to take into account the essential physical features experimentally observed. Finally, a parametric study is led on a fretted electric bass, showing the influence of parameters such as the plucking position and the fretboard profile on the resulting sound. These adjustements are directly related to common and central issues met by guitar makers and players.

La motivation industrielle et mécanique du problème sera présentée pour un problème continu: élasticité linéaire avec une contrainte unilatérale. Un système masse-ressort avec un contact unilatéral en découle par discrétisation. Le but de cette thèse est d'étudier ces systèmes à vibro-impact de N degrés de liberté avec un contact unilatéral. Le système résultant est linéaire en l'absence de contact; Il est régi par une loi d'impact autrement. L'auteur identifie les modes non linéaires qui présentent une phase de contact collant pour un modèle à deux degrés de liberté en présence d'un obstacle rigide. L'application de premier retour de Poincaré est un outil fondamental pour étudier la dynamique près de solutions périodiques. Étant donné que la section de Poincaré est un sous-ensemble de l'interface de contact dans l'espace des phases, elle peut être tangente aux orbites pour les contacts rasants et conduire à une singularité en « racine carrée » déjà connue en Mécanique. Cette singularité est revisitée dans un cadre mathématique rigoureux. Elle implique la discontinuité du temps de premier retour. Enfin, l’instabilité des modes linéaire rasants est abordée. / The mechanical motivation is presented for a PDE with a constraint. The purpose of this thesis is to study N degree-of-freedom vibro-impact systems with an unilateral contact. The resulting system is linear in the absence of contact; it is governed by an impact law otherwise. The author identifies some nonlinear modes that display a sticking phase. The First Return Map is a fundamental tool to explore periodic solutions. Since the Poincaré section is a subset of the contact interface in the phase-space, it can be tangent to orbits which yields the well-known square-root singularity. This singularity is here revisited in a rigorous mathematical framework. Moreover, the study of this singularity implies a more important singularity: the discontinuity of the first return time. Finally, the square-root dynamics near the linear grazing modes which may lead to the instability of these linear grazing modes is studied.

On s'intéresse à la résolution numérique de problèmes de contact et de fissuration en dynamique. Le problème de contact envisagé est le problème de Signorini avec ou sans frottement de Coulomb. Quant au problème de fissuration, il s'agit d'un modèle de zone cohésive avec trajet de fissuration pré-défini. Ces problèmes se caractérisent par la présence d'une condition aux limites non-régulière et se formulent comme des inéquations variationnelles d'évolution ou des inclusions différentielles. Pour les résoudre numériquement, nous combinons, comme il est courant en dynamique des solides, une discrétisation en espace par éléments finis et des schémas d'intégration en temps (de types différences finies). Pour le problème de contact, nous commençons par comparer les principales méthodes proposées dans la littérature. Nous étudions ensuite plus particulièrement la méthode dite de masse modifiée récemment introduite par H. Khenous, P. Laborde et Y. Renard. Nous en proposons une variante semi-explicite. Par ailleurs, nous prouvons un résultat de convergence des solutions semi-discrètes en espace vers une solution continue dans le cas d'un problème de Signorini sans frottement et d'un matériau viscoélastique. Nous analysons également les methodes semi-discrètes en espace et totalement discrètes dans le cas d'un problème de Signorini avec frottement de Coulomb. Pour le problème de fissuration dynamique, la non-régularité de la condition aux limites rend impossible ou peu robuste l'utilisation de schémas totalement explicites. Nous proposons donc des schémas où cette condition aux limites est traitée de façon implicite. Enfin, nous présentons et analysons des méthodes de lagrangien augmenté pour la résolution numérique du problème de fissuration en statique / The present work deals with the numerical solution of dynamic contact and fracture problems. The contact problem is a Signorini problem with or without Coulomb friction. The fracture problem uses a cohesive zone model with a prescribed crack path. These problems are characterized by a non-regular boundary condition and can be formulated with evolutionary variational inequations or differential inclusions. For the numerical solution, we combine, as usual in solid dynamics, a finite element discretization in space and time-integration schemes. For the contact problem, we begin by comparing the main methods proposed in the literature. We then focus on the so-called modified mass method recently introduced by H. Khenous, P. Laborde et Y. Renard, for which we propose a semi-explicit variant. In addition, we prove a convergence result of the space semi-discrete solutions to a continuous solution in the frictionless viscoelastic case. We also analyze the space semi-discrete and fully discrete problems in the friction Coulomb case. For the dynamic fracture problem, using a fully explicit scheme is impossible or not robust enough. Therefore, we propose time-integration schemes where the boundary condition is treated in an implicit way. Finally, we present and analyze augmented Lagrangian methods for static fracture problems

Cette thèse est consacrée à l’étude de méthodes de domaines fictifs pour les éléments finis. Ces méthodes, initialement conçues pour l’approximation de problèmes d’interactions fluide/structure, consistent à prolonger un domaine réel par un domaine de géométrie simple appelé domaine fictif. On applique ces méthodes à un problème de contact unilatéral sans frottement en petite déformation entre deux corps élastiques séparés par une distance initiale non nulle et possédant par ailleurs des conditions aux bords de type Dirichlet et Neumann. Les deux premiers chapitres sont consacrés à l’introduction des méthodes de domaines fictifs et du problème unilatéral de contact de deux corps élastiques. Le chapitre 3 est consacré à l’analyse a priori et à l’étude numérique de ce problème de contact en domaine fictif avec les conditions aux bords de Dirichlet et de contact qui sont prises en compte à l’aide d’une méthode de type Nitsche. Des résultats théoriques de consistance de la méthode discrète, d’existence et d’unicité sont présentés. Afin d’obtenir une estimation d’erreur a priori optimale, une stabilisation de la méthode de domaine fictif est nécessaire. Ces résultats sont validés numériquement sur des cas tests en dimensions deux et trois. Le chapitre 4 est consacré à l’étude d’un estimateur d’erreur de type résidu d’un problème de contact sans domaine fictif entre un corps élastique et un corps rigide. Les résultats théoriques sont également validés sur deux cas tests numériques : un domaine rectangulaire avec seulement une partie de la zone de contact en contact effectif ainsi qu’un contact de type Hertz en dimensions deux et trois. Le chapitre 5 est une généralisation du chapitre 4 à l’approche domaine fictif et au cas de deux corps élastiques. / This thesis is dedicated to the study of the fictitious domain methods for the finite element methods. These methods, initially designed for the fluid-structure interaction, consist in immersing the real domain in a simply-shaped and a geometrically bigger domain called the fictitious domain. We apply these methods to a unilateral frictionless contact problem in small deformation of two deformable elastics bodies separated by an initial gap and satisfying boundary Dirichlet and Neumann conditions. The first two chapters are devoted to the introduction of these methods and to the unilateral contact problem. The chapter 3 is dedicated to a theoretical study for Dirichlet and contact boundary conditions taken into account with a Nitsche type method. Some theoretical results are presented: the consistency of the discrete method, existence and uniqueness results. To obtain an optimal a priori error estimate, a stabilized fictitious domain method is necessary. These results are numerically validated using Hertz contact in two and three dimensions. The chapter 4 is devoted to the study of a residual-based a posteriori error estimator, without the fictitious domain approach, between an elastic body and rigid obstacle. The numerical study of two tests cases will be performed: a rectangular domain with only a part of the potential zone of contact in effective contact as well as a Hertz contact in two and three dimensions. The chapter 5 is a generalization of the chapter 4 to the fictitious domain approach and the care of to two elastics bodies.