L'objectif de cette thèse est d'étudier le contrôle passif des vibrations avec un absorbeur non linéaire de type Nonlinear Energy Sink (NES) à Vibro-Impact (VI). Plusieurs aspects ont été étudiés : l’influence des paramètres sur les régimes vibratoires observés, l’optimisation du design sous différentes excitations, l'application sur des systèmes vibratoires non linéaires et enfin l'étude de deux NES-VI en parallèle.Tout d'abord, l’influence des paramètres de design sur les régimes vibratoires et les bifurcations est étudiée de façon analytique, numérique et expérimentale. Ainsi différentes bifurcations et des réponses fortement modulées de type chaotique sont présentées.Ensuite, l’efficacité des régimes vibratoires est comparée, le mécanisme ainsi décrit constitue la base de l’optimisation du design de l’absorbeur face à différents types d’excitation.Le mécanisme d’activation du NES-VI est étudié analytiquement puis validé expérimentalement. Un critère d’optimisation du design est proposé, puis appliqué sur différents systèmes au comportement vibratoire non linéaire.Finalement, dans le but d’améliorer l’efficacité et la robustesse, le montage de deux NES-VI en parallèle est étudié expérimentalement. Le principe d’activation séparé est alors observé. / The objective of this thesis is to study the passive control of vibration by a Vibro-Impact (VI) Nonlinear Energy Sink (NES). Several aspects have been developed: the influence of parameters on response regimes, the optimization design mechanism under different excitations, the application to vibration control of nonlinear systems and the study of two VI NES in parallel.Firstly, the influence of parameters on response regimes and bifurcation is analytically, numerically and experimentally studied, respectively. Different bifurcation routes and chaotic strongly modulated response are presented.Then, the efficiency of typical response regimes is compared, and its mechanism lays the foundation for the optimization design of different parameters under different types of excitation.Thirdly, the activation characteristic of VI NES is analytically studied and experimentally validated. An optimization design criterion is proposed for the vibration control of nonlinear system.Finally, the vibration control by two VI NES in parallel is experimentally studied with the purpose of efficiency and robustness improvement. The principle of separate activation of VI NES is observed

Ce travail de recherche a été réalisé au sein du Laboratoire de Tribologie et Dynamique des Systèmes de l’École Centrale de Lyon (FRANCE) en cotutelle avec l’Unité de Mécanique, Modélisation et Productique (U2MP) à l’École Nationale d’Ingénieurs de Sfax (TUNISIE) dans le cadre du projet européen "Mid-Frequency". La prédiction du comportement dynamique des structures est une tâche importante dans la phase de conception de tout produit mécanique. Le choix de l’outil ou de la méthode utilisée dépend de plusieurs facteurs. Pour un système dynamique, la bande de fréquence d’étude est l’un des paramètres essentiels étant donné qu’il existe des approches appropriées pour chaque domaine fréquentiel. Ces derniers seront rapidement inapplicables en changeant le domaine d’application. Par exemple, les méthodes dites hautes fréquences ou globales sont très limitées dans la partie basse du spectre. De même, les méthodes dites basses fréquences deviennent, numériquement, très lourdes et peu performantes si l’on monte en fréquence. Les moyennes fréquences représentent alors les hautes fréquences pour les méthodes globales et les basses fréquences pour les méthodes locales. Comme les incertitudes jouent un rôle important dans les comportements vibratoires en moyennes fréquences, le travail présenté de ce mémoire est une contribution à la recherche d’une approche peu coûteuse en temps de calcul permettant l’extension d’une méthode locale : la méthode des éléments finis ondulatoires, à cette bande de fréquence pour les systèmes à caractère non déterministe. Cette contribution consiste à tenir compte des incertitudes présentes dans le système étudié pour évaluer la dispersion des différents paramètres (spectraux, de diffusion, dynamiques, etc.) et leurs effets sur la réponse globale (cinématique et énergétique) de la structure. Le travail présenté peut être partagé en deux parties. La première concerne le développement des formulations explicites et directes des dispersions des différents paramètres. Cette partie se base sur l’utilisation de la méthode de perturbation à l’ordre un. La deuxième partie est une généralisation de la première. En effet, l’utilisant de la projection des variables aléatoires sur la base des polynômes de chaos permet une évaluation plus générale des effets des incertitudes sur la dynamique des structures périodiques en moyennes fréquences. / The prediction of dynamic behavior of structures is an important task in the design step of any mechanical product. There are many factors affecting the choice of the used methods. For a dynamic system, the frequency band under study is one of the important parameters since for each frequency range exists its appropriate approach which can be quickly inapplicable in other domains. For example, the high frequency methods are very limited in the lower part of the spectrum. Similarly, the so-called low-frequency methods become numerically inefficient if it goes up in frequency range. The mid-frequencies then represent the high-frequencies for global and low frequencies for local methods. Knowing that uncertainties play an important role on the vibro-acoustics behavior in mid-frequencies, the presented work is a contribution to the research approach, with inexpensive computing time, allowing the extension of a local method, called ’the wave finite element method’, in this frequency band. These contributions consist in taking into account uncertainties in the studied system to evaluate the dispersion of all parameters (spectral, diffusion, dynamics, etc.) and their effects on the global response (kinematic and energetic) of the structure. The presented work can be divided into two main parts. The first one involves the development of an explicit and direct formulation describing the dispersion of different parameters; this part is based on the first-order perturbation method. The second part is a generalization of the first one; indeed, using the chaos polynomial projection of all random variables allows a more general assessment of the effects of uncertainties on the dynamics of periodic structure in mid-frequency range.

<p>A screened GPM approach in an EMTO-CPA framework was investigated in order to study its ability of describing transition temperatures in phase separating systems, i. e. systems giving either a random or a cluster structure depending on the temperature and the relative concentration of the ingoing atoms of the binary alloy used for the study. A motivation for the study is that the method works well for ordering systems, i. e. systems giving either a random or ordered structure dependent on the temperature and the relative concentration of the components in the binary alloy. Thereby is it of interest to find out the methods capacity in phase separating systems. The so called GPM potentials derived in the approach were applied in statistical Monte Carlo simulations for this purpose. The systems chosen for the investigation were the RhPd and the AlZn binary alloy systems. For both systems the method showed acceptable accuracy when properties as lattice parameter and mixing enthalpy were calculated. The quality of the derived GPM potentials has also been checked by calculating ordering energy for different ordered structures; directly from first principles calculations and from the GPM approach. The results were in acceptable agreement and thereby indicating that the GPM potentials were reliable. The transition temperatures in the RhPd phase diagram, derived by the statistical Monte Carlo simulations showed anyway deviation from experimental results. The error in the predictions might be due to the existing concentration dependencies in the GPM potentials.The conclusion from this study is that the Monte Carlo scheme might be inconvenient in order to handle the concentration dependencies seen in the GPM potentials.</p>

Part I: The present work is to calculate dimensionless three-dimensional universal phase and lift force diagrams for a microbubble or pendant drop in a static liquid on a solid surface or orifice. Studying microbubble dynamics is important due to its controlling mass, momentum, energy and concentration transfer rates encountered in micro- and nano-sciences and technologies. In this work, dimensionless phase and force diagrams are presented by applying an equation for microbubble shape to accuracy of the second order of small Bond number provided by O¡¦Brien (1991). Two dimensionless independent parameters, Bond number and contact angle (or base radius), are required to determine dimensionless phase and force diagrams governing static and dynamic states of a microbubble. The phase diagram divides the microbubble surface into three regions, the apex to inflection, inflection to neck, and neck to the edge of microbubble. The growth, collapse, departure and entrapment of a microbubble on a surface thus can be described. The lift forces include hydrostatic buoyancy, difference in gas and hydrostatic pressures at the microbubble base, capillary pressure and surface tension resulted from variation of circumference. The force to attach the microbubble to solid surface is the surface tension resulted from variation of circumference, which is not accounted for in literature. Adjusting the base radius to control static and dynamic behaviors of a microbubble is more effective than Bond number. Part II: Controlling states and growth of a microscale bubble (or pendant drop) in a static liquid on a surface by introducing general phase diagrams is proposed. Microbubbles are often used to affect transport phenomena in micro- and nano-technologies. In this work, a general phase diagram is provided by applying a perturbation solution of Young-Laplace equation for bubble shape with truncation errors of the second power of small Bond number. The three-dimensional phase diagram for a given Bond number is uniquely described by the dimensionless radius of curvature at the apex, contact angle and base radius of the microbubble. Provided that initial and end states are chosen, adjusting two of them gives the desired states and growth, decay and departure of the bubble described by path lines in the phase diagram. A universal three-dimensional phase diagram for a microbubble is also introduced.

A screened GPM approach in an EMTO-CPA framework was investigated in order to study its ability of describing transition temperatures in phase separating systems, i. e. systems giving either a random or a cluster structure depending on the temperature and the relative concentration of the ingoing atoms of the binary alloy used for the study. A motivation for the study is that the method works well for ordering systems, i. e. systems giving either a random or ordered structure dependent on the temperature and the relative concentration of the components in the binary alloy. Thereby is it of interest to find out the methods capacity in phase separating systems. The so called GPM potentials derived in the approach were applied in statistical Monte Carlo simulations for this purpose. The systems chosen for the investigation were the RhPd and the AlZn binary alloy systems. For both systems the method showed acceptable accuracy when properties as lattice parameter and mixing enthalpy were calculated. The quality of the derived GPM potentials has also been checked by calculating ordering energy for different ordered structures; directly from first principles calculations and from the GPM approach. The results were in acceptable agreement and thereby indicating that the GPM potentials were reliable. The transition temperatures in the RhPd phase diagram, derived by the statistical Monte Carlo simulations showed anyway deviation from experimental results. The error in the predictions might be due to the existing concentration dependencies in the GPM potentials.The conclusion from this study is that the Monte Carlo scheme might be inconvenient in order to handle the concentration dependencies seen in the GPM potentials.

Nearfield Acoustical Holography (NAH) is an acoustic field visualization technique that can be used to reconstruct three-dimensional (3-D) acoustic fields by projecting two-dimensional (2-D) data measured on a hologram surface. However, linear NAH algorithms developed and improved by many researchers can result in significant reconstruction errors when they are applied to reconstruct 3-D acoustic fields that are radiated from a high-level noise source and include significant nonlinear components. Here, planar, nonlinear acoustical holography procedures are developed that can be used to reconstruct 3-D, nonlinear acoustic fields radiated from a high-level noise source based on 2-D acoustic pressure data measured on a hologram surface. The first nonlinear acoustic holography procedure is derived for reconstructing steady-state acoustic pressure fields by applying perturbation and renormalization methods to nonlinear, dissipative, pressure-based Westervelt Wave Equation (WWE). The nonlinear acoustic pressure fields radiated from a high-level pulsating sphere and an infinite-size, vibrating panel are used to validate this procedure. Although the WWE-based algorithm is successfully validated by those two numerical simulations, it still has several limitations: (1) Only the fundamental frequency and its second harmonic nonlinear components can be reconstructed; (2) the application of this algorithm is limited to mono-frequency source cases; (3) the effects of bent wave rays caused by transverse particle velocities are not included; (4) only acoustic pressure fields can be reconstructed. In order to address the limitations of the steady-state, WWE-based procedure, a transient, planar, nonlinear acoustic holography algorithm is developed that can be used to reconstruct 3-D nonlinear acoustic pressure and particle velocity fields. This procedure is based on Kuznetsov Wave Equation (KWE) that is directly solved by using temporal and spatial Fourier Transforms. When compared to the WWE-based procedure, the KWE-based procedure can be applied to multi-frequency source cases where each frequency component can contain both linear and nonlinear components. The effects of nonlinear bent wave rays can be also considered by using this algorithm. The KWE-based procedure is validated by conducting an experiment with a compression driver and four numerical simulations. The numerical and experimental results show that holographically-projected acoustic fields match well with directly-calculated and directly-measured fields.

Les travaux présentés dans cette thèse s’intéressent au contrôle passif des systèmes soumis à une excitation harmonique à l’aide d’absorbeurs non-linéaires de type Nonlinear Energy Sink (NES). Plusieurs voies de recherches ont été dégagées visant : l’étude expérimentale du pompage énergétique, l’étude d’un type de NES alternatif puis la proposition d’une application pour le contrôle passif de l’instabilité de broutement en usinage.L’étude théorique d’un oscillateur linéaire couplé à un NES à raideur purement cubique permet de mettre en évidence les régimes de relaxation. Une procédure de dimensionnement est développée et appliquée à la réalisation d’un dispositif expérimental.La partie suivante est consacrée à l’étude d’un NES à vibro-impact (NES-VI). L’analyse théorique permet de révéler une stratégie de contrôle analogue aux cycles de relaxation observés dans le cas du NES à raideur cubique. Un dispositif expérimental est ensuite élaboré et permet de retrouver ces cycles de relaxation.La dernière partie est dédiée au contrôle passif de l’instabilité de broutement en tournage. La faisabilité à l’aide d’un NES à raideur cubique et d’un NES-VI est établie par l’étude théorique. Le contrôle passif à l’aide d’un NES-VI est réalisé expérimentalement / The work presented in this thesis deal with the passive control of dynamic systems subjected to an harmonic forcing using a Nonlinear Energy Sink (NES). Several research ways have been identified : the experimental study of an harmonic oscillator with an embedded NES, the study of an alternative type of NES and an application for the passive control of chatter instability during machining operations.The theoretical analysis of an harmonically oscillator with an embedded NES with cubic stiffness coupling allow us to reveal the relaxation cycles. A design procedure is presented and applied to the realization of an experimental setup.The next section is devoted to the study of a vibro-impact NES (VI-NES). The theoretical analysis gives an insight on analog relaxation cycle similar to those observed with a NES with cubic stiffness. An experimental setup is presented and these relaxation cycles are observed.In the last section, the idea of passively controling the chatter instability during turning operation using a NES is studied. This possibility is demonstrated theoretically as well for a classic cubic NES as for a VI-NES. This passive control is performed experimentally and a reduction of the vibration amplitude is observed

The paper considers systems of linear first-order ODEs with a randomly perturbed system matrix and stationary additive noise. For the description of the long-term behavior of such systems it is necessary to study their stationary solutions. We deal with conditions for the existence of stationary solutions as well as with their representations and the computation of their moment functions. Assuming small perturbations of the system matrix we apply perturbation techniques to find series representations of the stationary solutions and give asymptotic expansions for their first- and second-order moment functions. We illustrate the findings with a numerical example of a scalar ODE, for which the moment functions of the stationary solution still can be computed explicitly. This allows the assessment of the goodness of the approximations found from the derived asymptotic expansions.

[pt] Problemas dinâmicos governados por problemas de valor inicial (PVI) não lineares, em geral, despertam grande interesse da comunidade científica. O conhecimento da solução desses PVI facilita o entendimento das características dinâmicas do problema. Porém, infelizmente, muitos dos PVI de interesse não têm solução conhecida. Nesse caso, uma alternativa é o cálculo de aproximações para a solução. Métodos numéricos e analíticos são eficientes nessa tarefa e podem fornecer aproximações com a precisão desejada. Os métodos numéricos foram muito desenvolvidos nos últimos anos e amplamente aplicados em problemas de diversas áreas da engenharia. Pacotes computacionais de fácil utilização foram criados e hoje fazem parte dos mais tradicionais programas de simulação numérica. Entretanto, as aproximações numéricas têm uma desvantagem em relação às aproximações analíticas. Elas não permitem o entendimento de como a solução depende dos parâmetros do problema. Visto isso, esta dissertação foca na análise e implementação de técnicas analíticas denominadas métodos de perturbação. Foram estudados os métodos de Lindstedt-Poincaré e de múltiplas escalas de tempo. As metodologias foram aplicadas em um PVI envolvendo a equação de Duffing não amortecida. Programas em álgebra simbólica foram desenvolvidos com objetivo de calcular aproximações analíticas hierárquicas para a solução desse problema. Foi feita uma análise paramétrica, ou seja, estudo de como as condições iniciais e os valores de parâmetros influem nas aproximações. Além disso, as aproximações analíticas obtidas foram comparadas com aproximações numéricas calculadas através do método do Runge- Kutta. O método de múltiplas escalas de tempo também foi aplicado em um PVI que representa a dinâmica de um sistema massa-mola-amortecedor com atrito seco. Devido ao atrito, a resposta do sistema pode ser caracterizada em duas fases alternadas, a fase de stick e a fase de slip, compondo um fenômeno chamado stick-slip. Verificou-se que as aproximações obtidas para resposta do sistema pelo método de múltiplas escalas de tempo têm boa acurácia na representação da dinâmica do stick-slip. / [en] Dynamical problems governed by non-linear initial value problems (IVP), in general, are of great interest of the scientific community. The knowledge of the solution of these IVPs facilitates the understanding of the dynamic characteristics of the problem. However, unfortunately, many of the IVPs of interest does not present a known solution. In this case, an alternative is to calculate approximations for the solution. Numerical and analytical methods are efficient in this assignment and can provide approximations with the desired precision. Numerical methods have been developed over the last years and have been widely applied to dynamical problems in various engineering areas. Computational packages, easy to use, were created and today are part of the most traditional numerical simulation programs. However, numerical approximations have a disadvantage in relation to analytical approaches. They do not allow the understanding of how the solution depends on the problem parameters. Given this, this dissertation focuses on the analysis and implementation of analytical techniques called perturbation methods. The Lindstedt-Poincaré method and multiple time scales method were studied. The methodologies were applied in an IVP involving the non-damped Duffing equation. Symbolic algebra programs were developed with the purpose of calculating hierarchical analytical approximations to the solution of this problem. A parametric analysis was performed, in other words, a study of how the approximations are influenced by initial conditions and parameter values. In addition, the analytical approximations obtained were compared with numerical approximations calculated using the Runge-Kutta method. The multiple scales method was also applied in a IVP that represents the dynamics of a mass-spring-damper oscillator with dry friction. Due to friction, the system response can be characterized in two alternating phases, the stick phase and the slip phase, composing a phenomenon called stick-slip. It was verified that the approximations obtained for system response by the multiple scales method represent the stick-slip dynamics with good accuracy.

In this dissertation, we use the Natural transform decomposition method to obtain approximate analytical solution of fractional differential equations. This technique is a combination of decomposition methods and natural transform method. We use the Adomian decomposition, the homotopy perturbation and the Daftardar-Jafari methods as our decomposition methods. The fractional derivatives are considered in the Caputo and Caputo- Fabrizio sense. / Mathematical Sciences / M. Sc. (Applied Mathematics)