Obtaining the modal parameters of a deflected and rolling tire represents a challenge due to the complex vibration characteristics that cause the tire's symmetry distortion and the natural frequencies' bifurcation phenomena. The modal parameters are usually extracted using a detailed finite element model. The main issue with full modal models (FEA, for example) is the inability to integrate the tire modal model with the vehicle models to tune the suspension system for optimal ride comfort. An in-plane rigid–elastic-coupled tire model was used to examine the 200 DOF finite difference method (FDM) modal analysis accuracy under non-ground contact and non-rotating conditions. The discrete in-plane rigid–elastic-coupled tire model was modified to include the contact patch restriction, centrifugal force, Doppler, and Coriolis effects, covering a range of 0-300 Hz. As a result, the influence of the contact patch and the rotating tire conditions on the natural frequencies and modes were obtained through modal analysis. The in-plane rigid–elastic-coupled modal model with varying conditions was created that connects any two DOFs around the tire's tread or sidewall as inputs or outputs. The vertical movement of the wheel was incorporated into the in-plane rigid–elastic-coupled tire modal model to extract the transfer function (TF) that connects road irregularities as an input to the wheel's vertical movement as an output. The TF was utilized in a quasi-static manner to obtain the tire's enveloping characteristics rolling over short wavelength obstacles as a direct function of vertical wheel displacement under varying contact patch length constraints. The tire modal model was implemented with the quarter car model to obtain the vehicle response rolling over short wavelength obstacles. Finally, a sensitivity analysis was performed to examine the influence of tire parameters and pretension forces on natural frequencies. / Doctor of Philosophy / The goal of vehicle manufacturers is to predict the vehicle's behavior under various driving conditions using mathematical models and simulation. Automotive companies rely heavily on computational simulation tools instead of real-time tests to shorten the product development cycle and reduce costs. However, the interaction between the tire and the road is one of the most critical aspects to consider when evaluating automobile stability and performance. The tires are responsible for generating the forces and moments that drive and maneuver the vehicle. Tires are complex products due to their intricate design, and their characteristics are affected by many factors such as vertical load, inflation pressure, speed, and a road with an uneven surface profile. Consequently, this project aims to describe the influence of various driving circumstances and load conditions on tire properties, as well as to develop a model that can represent the vertical tire and vehicle behavior while traveling over a cleat under different vehicle loads.

The thesis presents a novel computational method for analysing the static and dynamic behaviour of a multi-damaged beam using local and non-local elasticity theories. Most of the lumped damage beam models proposed to date are based on slender beam theory in classical (local) elasticity and are limited by inaccuracies caused by the implicit assumption of the Euler-Bernoulli beam model and by the spring model itself, which simplifies the real beam behaviour around the crack. In addition, size effects and material heterogeneity cannot be taken into account using the classical elasticity theory due to the absence of any microstructural parameter. The proposed work is based on the inhomogeneous Euler-Bernoulli beam theory in which a Dirac's delta function is added to the bending flexibility at the position of each crack: that is, the severer the damage, the larger is the resulting impulsive term. The crack is assumed to be always open, resulting in a linear system (i.e. nonlinear phenomena associated with breathing cracks are not considered). In order to provide an accurate representation of the structure's behaviour, a new multi-cracked beam element including shear effects and rotatory inertia is developed using the flexibility approach for the concentrated damage. The resulting stiffness matrix and load vector terms are evaluated by the unit-displacement method, employing the closed-form solutions for the multi-cracked beam problem. The same deformed shapes are used to derive the consistent mass matrix, also including the rotatory inertia terms. The two-node multi-damaged beam model has been validated through comparison of the results of static and dynamic analyses for two numerical examples against those provided by a commercial finite element code. The proposed model is shown to improve the computational efficiency as well as the accuracy, thanks to the inclusion of both shear deformations and rotatory inertia. The inaccuracy of the spring model, where for example for a rotational spring a finite jump appears on the rotations' profile, has been tackled by the enrichment of the elastic constitutive law with higher order stress and strain gradients. In particular, a new phenomenological approach based upon a convenient form of non-local elasticity beam theory has been presented. This hybrid non-local beam model is able to take into account the distortion on the stress/strain field around the crack as well as to include the microstructure of the material, without introducing any additional crack related parameters. The Laplace's transform method applied to the differential equation of the problem allowed deriving the static closed-form solution for the multi-cracked Euler-Bernoulli beams with hybrid non-local elasticity. The dynamic analysis has been performed using a new computational meshless method, where the equation of motions are discretised by a Galerkin-type approximation, with convenient shape functions able to ensure the same grade of approximation as the beam element for the classical elasticity. The importance of the inclusion of microstructural parameters is addressed and their effects are quantified also in comparison with those obtained using the classical elasticity theory.

Dans la première partie de la thèse, les schémas d’intégration conservatifs sont appliqués aux poutres co-rotationnelles 2D. Les cinématiques d'Euler-Bernoulli et de Timoshenko sont abordées. Ces formulations produisent des expressions de l'énergie interne et l'énergie cinétique complexe et fortement non-linéaires. L’idée centrale de l’algorithme consiste à définir, par intégration, le champ des déformations en fin de pas à partir du champ de vitesses de déformations et non à partir du champ des déplacements au travers de la relation déplacement-déformation. La même technique est appliquée aux termes d’inerties. Ensuite, une poutre co-rotationnelle plane avec rotules généralisées élasto-(visco)-plastiques aux extrémités est développée et comparée au modèle fibre avec le même comportement pour des problèmes d'impact. Des exemples numériques montrent que les effets de la vitesse de déformation influencent sensiblement la réponse de la structure. Dans la seconde partie de cette thèse, une théorie de poutre spatiale d’Euler-Bernoulli géométriquement exacte est développée. Le principal défi dans la construction d’une telle théorie réside dans le fait qu’il n’existe aucun moyen naturel de définir un trièdre orthonormé dans la configuration déformée. Une nouvelle méthodologie permettant de définir ce trièdre et par conséquent de développer une théorie de poutre spatiale en incorporant l'hypothèse d'Euler- Bernoulli est fournie. Cette approche utilise le processus d'orthogonalisation de Gram-Schmidt couplé avec un paramètre rotation qui complète la description cinématique et décrit la rotation associée à la torsion. Ce processus permet de surmonter le caractère non-unique de la procédure de Gram-Schmidt. La formulation est étendue au cas dynamique et un schéma intégration temporelle conservant l'énergie est également développé. De nombreux exemples démontrent l’efficacité de cette formulation. / In the first part of the thesis, energymomentum conserving algorithms are designed for planar co-rotational beams. Both Euler-Bernoulli and Timoshenko kinematics are addressed. These formulations provide us with highly complex nonlinear expressions for the internal energy as well as for the kinetic energy which involve second derivatives of the displacement field. The main idea of the algorithm is to circumvent the complexities of the geometric non-linearities by resorting to strain velocities to provide, by means of integration, the expressions for the strain measures themselves. Similarly, the same strategy is applied to the highly nonlinear inertia terms. Next, 2D elasto-(visco)-plastic fiber co-rotational beams element and a planar co-rotational beam with generalized elasto-(visco)-plastic hinges at beam ends have been developed and compared against each other for impact problems. In the second part of this thesis, a geometrically exact 3D Euler-Bernoulli beam theory is developed.The main challenge in defining a three-dimensional Euler-Bernoulli beam theory lies in the fact that there is no natural way of defining a base system at the deformed configuration. A novel methodology to do so leading to the development of a spatial rod formulation which incorporates the Euler-Bernoulli assumption is provided. The approach makes use of Gram-Schmidt orthogonalisation process coupled to a one-parametric rotation to complete the description of the torsional cross sectional rotation and overcomes the non-uniqueness of the Gram-Schmidt procedure. Furthermore, the formulation is extended to the dynamical case and a stable, energy conserving time-stepping algorithm is developed as well. Many examples confirm the power of the formulation and the integration method presented.

Les Structures atomiques Quasi-périodiques (QP) possèdent des propriétés particulières, notamment dans le domaine vibrationnel. Il pourrait être intéressant de pouvoir transférer ces propriétés à des méta-matériaux macroscopiques. Des réseaux de poutres quasi-périodiques 2D sont étudiés dans cette thèse dans le cadre du modèle élément finis (EF) poutre Euler Bernoulli. Ces réseaux de poutres peuvent facilement être produits par fabrication additive ou par découpe laser. Il est possible de faire varier l'élancement des poutres (le ratio hauteur sur longueur) qui est un paramètre intéressant pour modifier la réponse mécanique du réseau. En utilisant la méthode EF, l'influence de l'élancement des poutres sur la réponse vibratoire des réseaux de poutres QP est étudiée. La méthode numérique Kernel Polynomial (KPM) est adaptée avec succès de la dynamique moléculaire aux réseaux de poutres pour étudier leurs modes de vibration sans avoir à diagonaliser complètement la matrice dynamique. Les réseaux de poutres QP présentent des propriétés similaires à leur compère atomique: en particulier la localisation de modes sur des sous-structures et une relation de dispersion hiérarchisée. Le comportement à la fracture est aussi étudié étant donné que les symétries présentes dans les QP pourraient permettre des réseaux de poutres ne présentant pas de plans faibles pour la propagation de fissures. Cela a été démontré d'après des calculs EF statiques avec un critère de fracture fragile sur l'énergie de déformation. Les simulations statiques ne suffisent pas car elles ne peuvent pas capturer les phénomènes dynamiques complexes qui apparaissent lors de la fissuration fragile. Les propriétés de vibration du QP pourraient aussi avoir un impact sur la propagation dynamique de fissure. Un modèle dynamique de fissuration est développé afin d'étudier l'impact de l'élancement sur la capacité des réseaux de poutres QP à dissiper de l'énergie par fissuration. Finalement une méthode Coarse Graining est développée pour identifier un milieu Cosserat continu équivalant au réseau de poutres QP pour différentes échelles. Cette méthode permet d'identifier la densité, les déformations, les contraintes et donc les modules d'élasticité du milieu Cosserat équivalent, permettant ainsi une meilleure compréhension du rôle des sous structures précédemment identifiées. / Quasi periodic (QP) structures have shown peculiar properties in the atomistic domain, especially the vibrational one. It could be interesting to be able to transpose these properties in macroscopic meta-materials. Quasi periodic 2D beam lattices are studied in this thesis due to the simplicity of the Euler Bernoulli finite element (FE) model. These beam lattices can easily be produced by additive manufacturing or by laser cutting. It is possible to vary the beam slenderness (i.e the ratio of height over length) that is a interesting parameter to modify the mechanical response of the lattice. Using finite element method, the influence of the beam slenderness over the vibration behavior of the QP beam lattices will be studied. The Kernel Polynomial numerical Method (KPM) is successfully adapted from molecular dynamics simulations in order to study vibrational modes of FE beam lattices without having to fully diagonalize the dynamical matrix. The QP lattices show similar properties as their atomic counterpart e.g mode localization over sub-stuctures and hierarchical dispersion relation. The fracture behavior is also studied, as the special symmetries allowed by the quasi periodicity could result in beam lattices without weak planes for crack propagation. It was proved to be true from static FE simulations with a brittle strain energy breaking criterion. Static simulations were not enough and do not grasp the complex dynamical phenomena taking place in brittle fracture. A dynamic crack propagation model was thus developed. The vibrational properties of quasi periodic structures could also have an impact on the dynamic crack propagation. Several simulations are run in order to study the impact of the slenderness on the energy dissipated by fracture of QP lattices. Finally, a coarse graining method (CG) was developed to identify a continuous Cosserat medium at different scales from the FE beam model. This CG method allows to identify, density, strain, stress and elastic moduli of an equivalent continuous Cosserat. This allows a better understanding of the role of previously identified characteristic sub structures.

This thesis presents an approach to modelling dynamic mass capture which is applied to a number of system models. The models range from a simple 2D Euler-Bernoulli beam with point masses for the end-effector and target to a 3D Timoshenko beam model (including torsion) with rigid bodies for the end-effector and target. In addition, new models for torsion, as well as software to derive the finite element equations from first principles were developed to support the modelling. Results of the models are compared to a simple experiment as done by Ben Rhody. Investigations of offset capture are done by simulation to show why one would consider using a 3D model that includes torsion. These problems have relevance to both terrestrial robots and to space based robotic systems such as the manipulators on the International Space Station capturing payloads such as the SpaceX Dragon capsule. One could increase production in an industrial environment if industrial robots could pick up items without having to establish a zero relative velocity between the end effector and the item. To have a robot acquire its payload in this way would introduce system dynamics that could lead to the necessity of modelling a previously ‘rigid’ robot as flexible.

This thesis is about using structural-dynamics based methods to address the existing challenges in the field of Structural Health Monitoring (SHM). Particularly, new structural-dynamics based methods are presented, to model areas of damage, to do damage diagnosis and to estimate and predict the sensitivity of structural vibration properties like natural frequencies to the presence of damage. Towards these objectives, a general analytical procedure, which yields nth-order expressions governing mode shapes and natural frequencies and for damaged elastic structures such as rods, beams, plates and shells of any shape is presented. Features of the procedure include the following: 1. Rather than modeling the damage as a fictitious elastic element or localized or global change in constitutive properties, it is modeled in a mathematically rigorous manner as a geometric discontinuity. 2. The inertia effect (kinetic energy), which, unlike the stiffness effect (strain energy), of the damage has been neglected by researchers, is included in it. 3. The framework is generic and is applicable to wide variety of engineering structures of different shapes with arbitrary boundary conditions which constitute self adjoint systems and also to a wide variety of damage profiles and even multiple areas of damage. To illustrate the ability of the procedure to effectively model the damage, it is applied to beams using Euler-Bernoulli and Timoshenko theories and to plates using Kirchhoff's theory, supported on different types of boundary conditions. Analytical results are compared with experiments using piezoelectric actuators and non-contact Laser-Doppler Vibrometer sensors. Next, the step of damage diagnosis is approached. Damage diagnosis is done using two methodologies. One, the modes and natural frequencies that are determined are used to formulate analytical expressions for a strain energy based damage index. Two, a new damage detection parameter are identified. Assuming the damaged structure to be a linear system, the response is expressed as the summation of the responses of the corresponding undamaged structure and the response (negative response) of the damage alone. If the second part of the response is isolated, it forms what can be regarded as the damage signature. The damage signature gives a clear indication of the damage. In this thesis, the existence of the damage signature is investigated when the damaged structure is excited at one of its natural frequencies and therefore it is called ``partial mode contribution". The second damage detection method is based on this new physical parameter as determined using the partial mode contribution. The physical reasoning is verified analytically, thereupon it is verified using finite element models and experiments. The limits of damage size that can be determined using the method are also investigated. There is no requirement of having a baseline data with this damage detection method. Since the partial mode contribution is a local parameter, it is thus very sensitive to the presence of damage. The parameter is also shown to be not affected by noise in the detection ambience.

This thesis presents an approach to modelling dynamic mass capture which is applied to a number of system models. The models range from a simple 2D Euler-Bernoulli beam with point masses for the end-effector and target to a 3D Timoshenko beam model (including torsion) with rigid bodies for the end-effector and target. In addition, new models for torsion, as well as software to derive the finite element equations from first principles were developed to support the modelling. Results of the models are compared to a simple experiment as done by Ben Rhody. Investigations of offset capture are done by simulation to show why one would consider using a 3D model that includes torsion. These problems have relevance to both terrestrial robots and to space based robotic systems such as the manipulators on the International Space Station capturing payloads such as the SpaceX Dragon capsule. One could increase production in an industrial environment if industrial robots could pick up items without having to establish a zero relative velocity between the end effector and the item. To have a robot acquire its payload in this way would introduce system dynamics that could lead to the necessity of modelling a previously ‘rigid’ robot as flexible.

L'objet de cette thèse est l'étude de la stabilisation de modèles d'interaction fluide-structure par des contrôles de dimension finie agissant sur la frontière du domaine fluide. L'écoulement du fluide est décrit par les équations de Navier-Stokes incompressibles tandis que l'évolution de la structure, située à la frontière du domaine fluide, satisfait une équation d'Euler-Bernoulli avec amortissement. Dans le chapitre 1, nous étudions le cas où le contrôle est une condition aux limites de Dirichlet sur les équations du fluide (contrôle par soufflage/aspiration). Nous obtenons des résultats de stabilisation locale du système non-linéaire autour d'une solution stationnaire instable de ce système. Dans les chapitres 2 et 3, nous nous intéressons au cas où le contrôle est une force appliquée sur la structure (contrôle par déformation de paroi). Dans le chapitre 2, nous considérons un modèle simplifié, où l'équation d'Euler-Bernoulli pour la structure est remplacée par un système de dimension finie. Nous construisons des lois de contrôle pour les systèmes de dimension infinie, ou pour leurs approximations semi-discrètes, capables de stabiliser les systèmes linéarisés avec un taux de décroissance exponentielle prescrit, et localement les systèmes non-linéaires. Nous présenterons des résultats numériques permettant de vérifier l'efficacité de ces lois de contrôles. / The aim of this thesis is to study the stabilization of fluid-structure interaction models by finite dimensional controls acting at the boundary of the fluid domain. The fluid flow is described by the incompressible Navier-Stokes equations while the displacement of the structure, localized at the boundary of the fluid domain, satisfies a damped Euler-Bernoulli beam equation. First, we study the case where the control is a Dirichlet boundary condition in the fluid equations (control by suction/blowing). We obtain local feedback stabilization results around an unstable stationary solution of this system. Chapters 2 and 3 are devoted to the case where control is a force applied to the structure (control by boundary deformation). We consider, in chapter 2, a simplified model where the Euler-Bernoulli equation for the structure is replaced by a system of finite dimension. We construct feedback control laws for the infinite dimensional systems, or for their semi-discrete approximations, able to stabilize the linearized systems with a prescribed exponential decay rate, and locally the nonlinear systems. We present some numerical results showing the efficiency of the feedback laws.

Rotating and non-rotating beams are widely used to model important engineering struc-tures. Hence, the vibration analyses of these beams are an important problem from a structural dynamics point of view. Depending on the beam dimensions, they are mod-eled using different beam theories. In most cases, the governing differential equations of these types of beams do not yield any simple closed-form solutions; hence we look for the inverse problem approach in determining the beam property variations given certain solutions. The long and slender beams are generally modeled using the Euler-Bernoulli beam theory. Under the premise of this theory, we study (i) the second mode tailoring of non-rotating beams having six different boundary conditions, (ii) closed-form solutions for free vibration analysis of free-free beams, (iii) closed-form solutions for free vibration analysis for gravity-loaded cantilever beams, (iv) closed-form solutions for free vibration analysis of rotating cantilever and pinned-free beams and (v) beams with shared eigen-pair. Short and thick beams are generally modeled using the Timoshenko beam theory. Here, we provide analytical closed-form solutions for the free vibration analysis of ro-tating non-homogeneous Timoshenko beams. The Rayleigh beam provides a marginal improvement over the Euler-Bernoulli beam theory without venturing into the math-ematical complexities of the Timoshenko beam theory. Under this theory, we provide closed-form solutions for the free vibration analysis of cantilever Rayleigh beams under three different axial loading conditions - uniform loading, gravity-loading and centrifu-gally loaded. We assume simple polynomial mode shapes which satisfy the different boundary conditions of a particular beam, and derive the corresponding beam property variations. In case of the shared eigenpair, we use the mode shape of a uniform beam which has a closed-form solution and use it to derive the stiffness distribution of a corresponding axially loaded beam having same length, mass variation and boundary condition. For the Timoshenko beam, we assume polynomial functions for the bending displacement and the rotation due to bending. The derived properties are demonstrated as benchmark analytical solutions for approximate and numerical methods used for the free vibration analysis of beams. They can also aid in designing actual beams for a pre-specified frequency or nodal locations in some cases. The effect of different parameters in the derived property variations and the bounds on the pre-specified frequencies and nodal locations are also studied for certain cases. The derived analytical solutions can also serve as a benchmark solution for different statistical simulation tools to find the probabilistic nature of the derived stiffness distri-bution for known probability distributions of the pre-specified frequencies. In presence of uncertainty, this flexural stiffness is treated as a spatial random field. For known probability distributions of the natural frequencies, the corresponding distribution of this field is determined analytically for the rotating cantilever Euler-Bernoulli beams. The derived analytical solutions are also used to derive the coefficient of variation of the stiffness distribution, which is further used to optimize the beam profile to maximize the allowable tolerances during manufacturing.

A partial differential equation in space and time represents the physics of rotating beams. Mostly, the numerical solution of such an equation is an available option as analytical solutions are not feasible even for a uniform rotating beam. Although the numerical solutions can be obtained with a number of combinations (in space and time), one tries to seek for a better alternative. In this work, various numerical techniques are applied to the rotating beam problems: finite element method, meshless methods, and B-spline finite element methods. These methods are applied to the governing differential equations of a rotating Euler-Bernoulli beam, rotating Timoshenko beam, rotating Rayleigh beam, and cracked Euler-Bernoulli beam. This work provides some elegant alternatives to the solutions available in the literature, which are more efficient than the existing methods: the p-version of finite element in time for obtaining the time response of periodic ordinary differential equations governing helicopter rotor blade dynamics, the symmetric matrix formulation for a rotating Euler-Bernoulli beam free vibration problem using the Galerkin method, and solution for the Timoshenko beam governing differential equation for free vibration using the meshless methods. Also, the cracked Euler-Bernoulli beam free vibration problem is solved where the importance of higher order polynomial approximation is shown. Finally, the overall response of rotating blades subjected to aerodynamic forcing is obtained in uncoupled trim where the response is independent of the overall helicopter configuration. Stability analysis for the rotor blade in hover and forward flight is also performed using Floquet theory for periodic differential equations.