Les pieds d’aubes de soufflantes de turboréacteurs font face à des sollicitations de type fretting. Il en résulte deux types d'endommagements: (i) l'amorçage et la propagation de fissures, (ii) l'usure des surfaces en contact. OBJECTIF: Afin de fournir les outils permettant de répondre à la problématique industrielle, une méthode semi-analytique permettant de traiter des problèmes de contacts élasto-plastiques revêtus et/ou hétérogènes est développé à partir d'éléments existants et de solutions analytiques et numériques novatrices. METHODE: La structure est simplifiée en supposant un contact entre deux massifs élastiques semi-infinis. Des solutions analytiques donnant pour: + les contributions élémentaires de chargements normaux et tangentiels constants sur une surface rectangulaire + les contributions élémentaires de déformations plastiques supposées constantes sur un volume parallélépipédique + les contributions élémentaires de déformations d’incompatibilité liées à un problème hétérogène (inclusions, revêtements, endommagement,...) supposées constantes sur un volume parallélépipédique. >> les déplacements en surface ou les contraintes dans le volume. Les déplacements en surface ou les contraintes dans le volume sont alors exprimés en utilisant des produits de convolution discrets entre des coefficients d'influence et la source surfacique (chargements en surface) ou volumique (déformations plastiques ou d'incompatibilité). Le problème normal et le problème tangentiel en glissement total ou en glissement partiel peuvent alors être résolus, en prenant en compte les effets plastiques et hétérogènes. L'algorithme d'optimisation sous contrainte utilisé (contact elastique) est celui développé par L.Gallego tandis que la base du solver plastique utilisé a été développé par C. Jacq (contact elasto-plastique sans frottement) RESULTATS: De nouvelles solutions analytiques sont obtenues pour le calcul des déplacements résiduels tangentiels. / Fan blade roots of jet engines are operating under fretting sollicitations. This results in two types of damage: (i) initiation and crack propagation, and (ii) wear of contact surfaces. OBJECTIVE: A semi-analytical method has been developed in order to solve the industrial problem. This method is fast-computing and able to deal with elastic-plastic coated contacts and / or heterogeneous, and has bee developed from existing items and innovative analytical and numerical solutions. METHOD: The structure is simplified by assuming two semi-infinite elastic solid in contact, and analytical solutions for: + Contributions of elementary normal and tangential loads constant over a rectangular area + Contributions of elementary plastic strains constant on a parallelepiped + Contributions basic incompatibility of deformations related to a heterogeneous (inclusions, coatings, damage ,...) constant on a parallelepiped. >> on surface displacements and stresses in the volume. Surface displacements and stresses in the volume are then expressed using discrete convolution products (2D or 3D-FFT) between the coefficients of influence and the surfacic (loading surface) or volumic (plastic deformation or inconsistency) source. The normal and tangential problems in both the gross-slip and stick-slip regime can be solved, taking into account the effect of plasticity and heterogeneities. The contact algorithm used (elastic contact) has been developed by L. Gallego while the base of the plastic solver has been developed by C. Jacq (frictionless elastic-plastic contact) RESULTS: New analytical solutions are given for calculating the residual tangential displacements.

La fatigue à nombre de cycles élevé (HCF) est un mode de défaillance courant et dangereux pour les aubages de turbomachines. Elle est induite par les efforts dynamiques élevés générés lors de résonances présentes dans la plage de fonctionnement de ces machines. Les dispositifs amortisseurs basés sur l’utilisation du frottement sec, tels que les nageoires ou les frotteurs sous-plateformes, permettent de réduire les amplitudes vibratoires, voire de repousser les fréquences de résonance hors des zones de fonctionnement. Cependant la conception de ces dispositifs reste encore largement basée sur l’empirisme et ils peuvent être la source d’un effet de désaccordage potentiellement nuisible. L’objectif ici est de développer des modélisations adaptées au traitement du problème de vibration des aubages en présence de frottement sec, ceci afin de mieux maîtriser les comportements physiques mis en jeu et donc, de mieux maîtriser leur processus de conception. Plusieurs modélisations numériques sont testées et confrontées à des résultats de référence. Une comparaison entre procédures de résolution temporelle et fréquentielle est menée et montre l’efficacité des méthodes fréquentielles. La méthode fréquentielle de la balance harmonique à plusieurs harmoniques est adaptée au problème et exploitée dans le cadre d’une étude énergétique. Cette étude conduit à une meilleure compréhension des phénomènes mis en jeu lors de l’aplatissement des pics en fonctionnement ; elle permet de démontrer que l’alternance des états de contact glissant et bloqué est à l’origine de cet aplatissement et non la dissipation d’énergie comme souvent avancé dans la littérature. Enfin, la méthode est exploitée pour décrire le comportement des disques aubés désaccordés. La méthode de Monte Carlo est utilisée pour obtenir les caractéristiques statistiques de la réponse forcée d’un système discret, en tenant compte des variations stochastiques des paramètres du contact notamment, la charge normale, la raideur du contact et le coefficient de frottement. Les résultats obtenus permettent de mieux comprendre les effets de la nature variable de ces paramètres fondamentaux sur la dynamique d’ensemble du système non linéaire. / Fatigue with high number of cycles (HCF) is a current and dangerous mode of failure for the blades of turbo shaft engines. It is induced by the high dynamic stresses generated at resonance in the operating range of these machines. The rubbing devices based on use of dry friction, such as shrouds or under-platform dampers, make it possible to reduce the vibratory amplitudes, to even push back the resonance frequencies out of the operation zones. However the design of these devices remains still largely based on empiricism and they can be the source of a potentially harmful effect of mistuning. The goal is to develop adapted modelling for the treatment of blades vibration problem in the presence of dry friction, this in order to better control the concerned physical behaviours and thus, to better control their process of design. Several numerical modelling are tested and confronted with reference results. A comparison between procedures of time and frequency domains resolution is carried out and shows the effectiveness of frequential methods. The frequential method of the Harmonic Balance including several harmonics is adapted to the problem and is used within an energy study. This study leads to a better comprehension of the phenomena of peaks flattening and it shows that the alternation of the states of slipping and sticking contact is the real cause of this flattening and not energy dissipation like often advanced in the literature. Lastly, the method is exploited to describe the behaviour of mistuned bladed disks. The Monte Carlo method is used to obtain the statistical characteristics of the forced response for a lumped system, by taking account of the stochastic variations of parameters in the contact, specifically the normal load, the stiffness of the contact and the coefficient of friction. Results obtained make it possible to better include/understand the effects of the variable nature of these fundamental parameters on dynamics of nonlinear system.

Le comportement local au contact et son interaction avec la dynamique globale du système sont à l'origine d’innombrables problèmes de contact concernant plusieurs disciplines telles que la tribologie, la géophysique, la mécanique de vibration ou la mécanique de la rupture. Lorsque deux corps élastiques sont en mouvement relatif avec une interface de frottement, des vibrations induites se produisent dans le système. Dans un point de vue macroscopique, le scénario macroscopique de stick-slip survenant pendant le mouvement relatif est caractérisé par la chute soudaine de la force de frottement (état de glissement), séparées par des périodes d'accumulation d'énergie élastique (état d’adhérence). Autrement, une instabilité dynamique se produit quand un mode de vibration du système mécanique devient instable en raison des forces de frottement. Ces types d'instabilités, générées par des forces de frottement, ont été principalement objet de publies traitant de problèmes spécifiques dans différents domaines. Dans ce contexte, des analyses expérimentales et numériques ont été ici mis en place pour comprendre comme le comportement de l'interface locale affecte la réponse macroscopique du système et vice-versa, au cours de scénarios d'instabilité. Les scénarios macroscopiques (instabilité de « stick-slip macroscopique », instabilité modale, glissement continu stable), survenant entre deux milieux élastiques simples en mouvement relatif, ont été étudiés numériquement et expérimentalement. Un dispositif expérimental dédié (TRIBOWAVE) a été développé et a permis de reproduire et examiner les différents scénarios de frottement dans des conditions aux limites bien contrôlées. Les mêmes scénarios de frottement ont été reproduits par des simulations numériques transitoires. Une loi de frottement en fonction du temps d’adhérence (stick) a été définie à partir des essais expérimentaux. La loi de frottement obtenue a été mise en œuvre dans le modèle numérique, conduisant à une validation quantitative des scénarios de frottement par les expériences. Les simulations transitoires non linéaires, l’analyse aux valeurs propres complexes et les tests expérimentaux ont permis de dessiner des cartes de scénarios d'instabilité en fonction des paramètres clés du système. Validé par la comparaison avec les mesures des signaux expérimentaux globaux (forces, accélérations / vitesse), le modèle numérique a permis d'étudier le couplage entre le comportement du contact local (distribution de l'état du contact, propagation des ondes et des ruptures, précurseurs) et la réponse dynamique du système au cours du « stick-slip macroscopique », de l’instabilité due au couplage modale et du glissement continu stable. La compréhension du couplage entre le contact et la dynamique des systèmes apportera de nouvelles améliorations sur le contrôle des instabilités de contact et les problèmes d'usure connexes. / Local contact behavior and its interaction with the global dynamics of the system are at the origin of innumerable contact issues concerning several different disciplines like tribology, geophysics, vibration mechanics or fracture mechanics. When two elastic media are in relative motion with a frictional interface, friction induced vibrations arise into the system. By a macroscopic point of view, the “macroscopic stick-slip” scenario occurring during relative motion is characterized by sudden friction force drops (sliding state) along the time, separated by periods of elastic energy accumulation (stick state). Instead, the mode dynamic instability occurs when a vibration mode of the mechanical system becomes unstable, due to frictional contact forces. This kind of instabilities, generated by frictional forces, have been mainly object of papers dealing with specific issues in different domains. In this context, experimental and numerical analyses have been focused here on understanding how the local interface behavior affects the macroscopic frictional response of the system, and, conversely, during instability scenarios. The macroscopic frictional scenarios (macroscopic stick-slip instability, mode coupling instability, stable continuous sliding) arising between two simple elastic media in relative motion have been investigated numerically and experimentally. A newer experimental setup (TRIBOWAVE) has been developed and it allowed to reproduce and to investigate the different scenarios under well-controlled boundary conditions. The same frictional scenarios have been reproduced by transient numerical simulations. A dedicated friction law as a function of adherence (sticking) time has been recovered by means of experimental tests. The obtained friction law has been implemented in the numerical model, leading to a quantitative validation of the simulated scenarios by the experiments. Nonlinear transient simulations, complex eigenvalue analyses and experimental tests allowed for drawing instability maps as a function of system key parameters. The numerical model, validated by the comparison with the experimental global measurements (forces, accelerations/velocity), allowed for investigating the coupling between the local contact behavior (contact status distribution, wave and rupture propagation, precursors) and the system dynamic response during macroscopic stick-slip instability, mode coupling instability and stable continuous sliding. The understanding of the coupling between contact and system dynamics will bring to further improvements on the control of contact instabilities and related wear issues.

La dynamique granulaire amenant à l'état critique présente un intérêt dans la compréhension de la déstabilisation menant à l'avalanche. Son étude permet d'avoir des pistes de compréhension sur des mécanismes plus complexes telles les catastrophes géophysiques (séismes, glissements de terrain, éboulements). Ainsi, lorsqu'un milieu granulaire tridimensionnel sous gravité est quasi-statiquement incliné, des précurseurs sont observés à partir d'une dizaine de degrés avant l'avalanche. Ces précurseurs correspondent à des réarrangements collectifs de grains observés à la surface qui apparaissent pseudo-périodiquement avec l'angle d'inclinaison. Cette thèse fournit une caractérisation expérimentale des précurseurs détectés à la surface par méthode optique et dans le volume par méthodes acoustiques (linéaire et non linéaire). Tout d'abord, de bonnes corrélations sont trouvées entre les réarrangements à la surface et dans le volume. Dans un second temps, l'étude est poursuivie avec une liste non exhaustive de paramètres influant sur les propriétés des précurseurs. L'état de surface des grains est crucial pour la dynamique des précurseurs. Puis, une tentative de description de la déstabilisation est réalisée avec notamment la mesure de la variation des paramètres élastiques. Les précurseurs d'avalanches correspondent à des pertes successives de rigidité du système, suivies du renforcement de ce dernier. / The granular dynamics leading to the avalanche is of interest in understanding the destabilization conducting to one. Its study implies a good train of throught in the understanding of more complex mechanisms such as geophysical disasters (earthquakes, landslides, rockslides). Thus, when a three-dimensional granular medium under gravity is quasi-statically tilted, precursors are observed from the tilt of ten degrees before the avalanche. These precursors correspond to collective rearrangements of grains observed on the free surface which appear pseudo-periodically with the angle of inclination. In order to understand this phenomenon, the thesis presents an experimental characterisation of the precursors detected on the surface by optical method and in the bulk by acoustic methods (linear and nonlinear). Firstly, good correlations were found between the surface and bulk rearrangements which led to extending the study with a non-exhaustive list of parameters affecting the precursors properties. The surface features of the grains have appeared to be particularly crucial in the precursors dynamics. Finally, an attempt at a description of the destabilization is approached with the measurement of the variation of the elastic parameters revealing that precursors of avalanches correspond to successive loss of rigidity of the system, followed by hardening of the latter.

[pt] Durante a perfuração de poços de petróleo, a coluna de perfuração apresenta um comportamento dinâmico complexo, esta tese foca no estudo experimental e na modelagem matemática deste comportamento. Neste trabalho, destaca-se as vibrações autoexcitadas axiais, laterais e torcionais, que podem levar a efeitos como o bit bouncing, o whirling e stick-slip torcional. A primeira contribuição desta tese é a análise experimental de um bancada de testes, que fornece informações sobre a dinâmica de sistemas torcionais. A influência dos parâmetros de controle não lineares na resposta do sistema é investigada, identificando as condições sob as quais o fenômeno stick-slip ocorre. Em segundo lugar, a tese propõe estratégias de identificação de sistemas para sistemas não lineares, utilizando a mesma bancada de testes supracitada. Uma abordagem híbrida para a identificação é proposta, onde técnicas de modelagem de caixa cinza e caixa preta são combinadas para calibrar os parâmetros do sistema, particularmente aqueles associados ao atrito. Essa abordagem aumenta a precisão das estimativas em comparação com os métodos tradicionais de caixa cinza, mantendo a interpretabilidade. Além disso, a pesquisa emprega physics-informed deep learning para estimar os parâmetros mecânicos e de atrito do modelo de dois graus de liberdade. A calibração usando dados experimentais obtidos de uma bancada de testes fornece informações sobre o comportamento de sistemas de perfuração. Finalmente, a tese apresenta investigações experimentais sobre o acoplamento entre oscilações torcionais e axiais utilizando uma bancada experimental de perfuração em escala de laboratório modificada e adaptada equipada com brocas e amostras de rocha reais. Em resumo, esta tese aumenta a compreensão da dinâmica de colunas de perfuração e apresenta aplicações úteis para técnicas de identificação de sistemas na análise de oscilações torcionais e axiais. / [en] During drilling for oil extraction purposes, the drill string experiences complex dynamic behavior, and this work delves into the experimental study and the mathematical modeling of such behavior. Self-excited vibrations, such as axial, lateral, and torsional vibrations, which can lead to detrimental effects such as bit bouncing, whirling, and torsional stick-slip are highlighted in this thesis. Distinct aspects of drilling dynamics are considered in this investigation to enhance the understanding of various phenomena. Initially, an experimental analysis of a lab-scale rig is conducted, providing valuable insights into the dynamics of such systems. And the influence of control parameters on the system’s response is examined, particularly in identifying the conditions under which the stick-slip phenomenon is likely to occur. Secondly, the thesis proposes system identification strategies for nonlinear systems, specifically focusing on the same laboratory test rig. An innovative ensemble approach is proposed, which combines gray and black-box modeling techniques to effectively calibrate the parameters of a dynamical system, particularly those associated with friction. This approach improves prediction accuracy compared to traditional gray-box methods while maintaining interpretability in the dynamic responses. Furthermore, the research employs physics-informed deep learning to estimate the low-dimensional model mechanical and friction parameters. Calibration using experimental data obtained from a specialized setup provides insights into the drill-string system s behavior. Finally, the thesis involves experimental investigations on the coupling between torsional and axial oscillations using a modified and adapted lab-scale drilling rig equipped with real drill bits and rock samples. In summary, this thesis advances our understanding of drill-string dynamics and presents helpful applications for system identification techniques.

[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.

[pt] Este trabalho apresenta um estudo do comportamento dinâmico de uma bancada de testes representativa do sistema real de perfuração, composta por um motor CC acoplado a um sistema físico torcional, sujeita a fontes de atrito que induzem um regime de stick-slip no sistema em duas regiões distintas. O estudo incluiu a identificação de parâmetros da bancada de testes por meio de uma série de ensaios experimentais; e a caracterização do atrito, por meio do levantamento experimental da curva do coeficiente de atrito, em função da velocidade angular dos rotores principais. O intuito inicial foi a obtenção de um modelo numérico que fosse o mais simples possível e que representasse bem a bancada de testes. Uma vez obtido o modelo numérico, prosseguiu-se com uma série de simulações que permitissem uma caracterização indireta do regime de atrito ao qual estivessem submetidos os rotores principais, partindo-se apenas de medições de parâmetros no motor. Esse estudo é de grande relevância para a compreensão qualitativa da dinâmica do sistema real de perfuração, uma vez que ainda hoje não há técnicas totalmente confiáveis para caracterização do comportamento da coluna no fundo do poço a partir de dados da superfície somente. / [en] This paper presents a study of the dynamic behavior of a representative test bench of a real rotary drilling system, comprising a DC motor coupled to a very exible torsional system subjected to sources of friction which can induce self-excitation into two distinct regions of the system. The study includes the identification of parameter settings from the test bench by means of a series of experimental tests and characterization of friction, by obtaining the experimental curve of the friction coefficient as a function of the angular speed of the main rotor. The initial aim was to obtain a numerical model as simple as possible, capable of representing the test bench. Once obtained the numerical model, a series of numerical simulations were done, which allow an indirect characterization of the friction condition to which main rotors were subjected, starting only with the parameters measured at the drive. This study is of great importance for a qualitative understanding of the dynamics of the real drilling system, since today there is no fully reliable techniques to characterize the behavior of the column in the deep from surface data only.

The principal objective of this investigation is to develop a constitutive model to simulate the hysteresis behaviour of unbonded flexible risers. A new constitutive model for flexible risers is proposed and a procedure for the identification of the related input parameters is developed using a multi-scale approach. The constitutive model is formulated in the framework of an Euler-Bernoulli beam model, with the addition of suitable pressure terms to the generalised stresses to account for the internal and external pressures, and therefore can be efficiently used for large-scale analyses. The developed non-linear relationship between generalised stresses and strains in the beam is based on the analogy between frictional slipping between different layers of a flexible riser and frictional slipping between micro-planes of a continuum medium in nonassociative elasto-plasticity. Hence, a linear elastic relationship is used for the initial response in which no-slip occurs; an onset-slip function is introduced to define the ‘noslip’ domain, i.e. the set of generalised stresses for which no slip occurs; a nonassociative rule with linear kinematic hardening is used to model the full-slip phase. The results of several numerical simulations for a riser of small-length, obtained with a very detailed (small-scale) non-linear finite-element model, are used to identify the parameters of the constitutive law, bridging in this way the small scale of the detailed finite-element simulations with the large scale of the beam model. The effectiveness of the proposed method is validated by the satisfactory agreement between the results of various detailed finite-element simulations for a short riser, subject to internal and external uniform pressures and cyclic bending and tensile loadings, with those given by the proposed constitutive law. The merit of the present constitutive law lies in the capturing of many important aspects of risers structural response, including the energy dissipation due to frictional slip between layers and the hysteretic response. This privilege allows one to accurately study the cyclic behavior of unbonded flexible risers subject to axial tension, bending moment, internal and external pressures.

Piezoelectric actuators (PEA) hold the most promise for precision positioning applications due to their capability of producing extremely small displacements down to 10 pm (1 pm = 10-12 m) as well as their high stiffness and force output. The piezoelectric-driven stick-slip (PDSS) actuator, working on the friction-inertia concept, has the capacity of accomplishing an unlimited range of motion. It also holds the promises of simple configuration and low cost. On the other hand, the PDSS actuator has a relatively low efficiency and low loading capability, which greatly limits its applications. The purpose of this research is to improve the performance of the PDSS actuators by employing specially-designed working surfaces.<p> The working surfaces, referred as anisotropic friction (AF) surfaces in this study, can provide different friction forces depending on the direction of relative motion of the two surfaces, and are used in this research to accomplish the aforementioned purpose. To fabricate such surfaces, two nanostructure technologies are employed: hot filament chemical vapour deposition (HFCVD) and ion beam etching (IBE). The HFCVD is used to deposit diamond on silicon substrates; and the IBE is used to etch the diamond crystalloid with a certain angle with respect to the coating surface to obtain an unsymmetrical-triangle microstructure. <p> A PDSS actuator prototype containing the AF surfaces was developed in this study to verify the function of the AF surfaces and characterize the performance of PDSS actuators. The designed surfaces were mounted on the prototype; and the improvement in performance was characterized by conducting a set of experiments with both the normal isotropic friction (IF) surfaces and the AF surfaces, respectively. The results illustrate that the PDSS actuator with the AF surface has a higher efficiency and improved loading capability compared to the one with the IF surfaces.<p> A model was also developed to represent the displacement of the novel PDSS actuator. The dynamics of the PEA and the platform were approximated by using a second order dynamic system. The pre-sliding friction behaviour involved was investigated by modifying the LuGre friction model, in which six parameters (Note that three parameters are used in the LuGre model) were employed to represent the anisotropic friction. By combining the PEA mechanism model, the modified friction model, and the dynamics of end-effector, a model for the PDSS actuator with the AF surface was developed. The model with the identified parameters was simulated in MATLAB Simulink and the simulation results obtained were compared to the experimental results to verify the model. The comparison suggests that the model developed in this study is promising to represent the displacement of the novel PDSS actuators with AF surfaces.

Previous studies show that the Piezoelectric-Driven Stick-Slip (PDSS) actuator is a promising device in many micropositioning and micromanipulation applications, where positioning with a long range and a high resolution is required. However, research in this area is still in its early stage and many issues remain to be addressed. One key issue is the representation of the dynamic displacement of the end-effector. It is known that such factors as the dynamics of piezoelectric actuator (PEA) and the presliding friction involved can significantly contribute to the displacement dynamics. Although this has been widely accepted, specific quantitative relationship between the aforementioned factors and the displacement dynamics has rarely been defined. The aim of this research is to develop a model to represent the displacement of the end-effecter of the PDSS actuators, in which both the presliding friction and the PEA dynamics are addressed. <p>In order to represent the presliding friction, the models reported in literatures, including Dahl model [Olsson, et al., 1998], Reset Integrator model [Haessig and Friedland 1991], LuGre model [Canudas de Wit et al., 1995] and Elastoplastic model [Dupont et al., 2002] were reviewed and examined; and the LuGre model was chosen to be used because of its efficiency and simple formulation. On the other hand, a linear second order dynamic system model was employed to represent the combination of a PEA and its driven mechanism. On the basis of the pre-sliding friction model and the linearized PEA dynamics model, a model representative of the end-effector displacement of the PDSS actuator model was developed. <p>In order to validate experimentally the developed PDSS model, a displacement measuring and data acquisition experiment system was established and a prototype was developed based on dSPACE and Simulink. On the prototyped actuator, two experiments were designed and conducted to identify the parameters involved in the model. One experiment is for the determination of the parameters of the second order system for the dynamics of the combination of a PEA and its driven mechanism; and other one is for the determination of the parameters of the chosen friction model. The identified parameters were then employed in the developed PDSS model to simulate the displacements and the results were compared with the experimental results that were obtained under the same operating conditions as the simulation. The comparison suggests that the model developed in this study is promising for the end-effector displacement of the PDSS actuator.