Nowadays, three-phase induction motors are widely used on industrial and other types of processes. Therefore, accurate knowledge of an induction motor performance is very essential to have an idea of its operation conditions. This study is a sequel of a previous one, where Direct and Soft starting methods of three-phase motors has been simulated and compared. As in the previous study, the theory behind this one is based on representing the real motor by aset of equations and values in Matlab, forming a corresponding idealistic motor in a way where all the physical effects are similar. The motor is started under three different frequencies in the VVVF method using supporting simulation of the current, torque, speed,efficiency and power factor curves. The results of the three starting methods are then discussed and compared. / Numera är tre-fas asynkronmotorer i stor utsträckning på industriella och andra typer av processer. Därför är det mycket viktigt att ha exakt kunskap om en induktionsmotorprestanda för att ha en uppfattning om dess driftsförhållanden . Denna studie är en fortsättning av en tidigare, där direkt och mjukstart metoder för trefasmotorer har simulerats och jämförts. Såsom i den tidigare studien, är teorin bakom denna en baserat på representerar den verkligamotorn av en uppsättning ekvationer och värden i Matlab, som bildar en motsvarande ideell motor på ett sätt där alla de fysiska effekterna är likartade . Motorn startas under tre olika frekvenser i VVVF metod med stöd simulering av ström, vridmoment, hastighet, effektivitetoch effektfaktorn kurvor. Därefter, resultaten av de tre startmetoder diskuteras och jämföras. / في الوقت الحالي تستخدم المحركات التحريضية ثلاثية الطور بشكل واسع في التطبيقات الصناعية و غيرها. و لهذا فإن المعرفة الدقيقة بأداء المحرك التحريضي أساسية لإعطاء فكرة عن ظروف تشغيله. إن هذه الدراسة هي تتمة لدراسة سابقة حيث تمت محاكاة و مقارنة طريقتي الإقلاع المباشر و الناعم للمحرك التحريضي ثلاثي الطور. كما في الدراسة السابقة, فإن هذه الدراسة مبنية على تمثيل المحرك الحقيقي بمجموعة من المعادلات و القيم الاسمية في برنامج ماتلاب لتكوين محرك مثالي مطابق, بحيث تكون جميع الآثار الفيزيائية مماثلة للمحرك الحقيقي. يتم إقلاع المحرك عند ثلاث ترددات مختلفة بطريقة تغيير التردد و التوتر و يتم محاكاة هذا الإقلاع عبر منحنيات التيار، العزم، السرعة، المردود و عامل الاستطاعة ثم تقارن نتائج طرق الإقلاع الثلاثة.

In this thesis, a simulation model developed in MATLAB® in consideration of system losses based on lithium ion-battery is presented. The purpose of the simulation model is to investigate peak shaving potential in the residential distribution network. In other word to determine an optimal threshold limit and battery capacity depending on if the battery system is placed at the transformer or household level. In the report there were economic calculations executed showing that profitability of investing in a battery system depends on the threshold limit and battery capacity. / I denna rapport presenteras analys av toppeffektreducering med ett simuleringsprogram baserad på litium-jon batteri med hänsyn till systemförlust. Simuleringsmodellen är uppbyggd i MATLAB® där metoder som Coulomb counting implementerades. Syftet med simuleringsprogrammet är att definiera en optimal tröskelgräns samt batterikapacitet på transformator- och hushållsnivån. I rapporten utfördes ekonomiska beräkningar som tyder på att lönsamheten för investering av ett batterisystem beror på tröskelgräns och batterikapacitet.

[pt] Esta tese apresenta o estudo dinâmico de três sistemas pendulares e de um sistema de corpo livre no espaço com 3 graus de liberdade. O primeiro sistema pendular consiste de um pêndulo acoplado ao centro de uma roda, que rola sem escorregar na direção horizontal, enquanto o segundo, se baseia num pêndulo simples, porém com comprimento variável, que ao mudar seu tamanho consegue ganhar/perder energia para aumentar/diminuir a amplitude de seu movimento e finalmente o terceiro está baseado num pêndulo duplo que, a despeito de restrições impostas ao movimento consegue inserir/retirar energia do sistema de forma similar ao segundo. O modelo de corpo livre no espaço está baseado na suspensão cardânica de um giroscópio e se utiliza de um modelo didático real de um giroscópio para observação das características dinâmicas. A partir destes exemplos estudou-se formas de controle não-linear para movimentar os sistemas de maneira a utilizar-se da mudança de posição interna do centro de massa para injetar e retirar energia dos sistemas. Foram gerados modelos matemáticos simulados no Simulink valendo-se do Matlab para análise, e geradas animações também com o Matlab para melhor observação dos efeitos. Em paralelo, para dois destes sistemas foram construídos em bancada experimentos para comprovação dos resultados numéricos, e os resultados são comparados em cada caso, analisando as diferenças. Ao final, todas as observações sobre os estudos foram analisadas, e comentários feitos baseados nos resultados, além de sugerir trabalhos futuros. / [en] This thesis presents the study of the dynamics of three pendulum systems and a 3DoF free body in space. The first pendular system is based on a simple pendulum coupled to the center of a wheel that translates horizontally without slip; while the second system is based on a simple pendulum, with variable length, which is able to acquire/lose energy to grow/diminish the amplitude of its movement; and finally the third one is based on a double pendulum that, in spite of movement restrictions, can as well inject/drop energy like the second system. The free body in space is based on a real gyroscope for didactical use, which is helpful for the observation of the dynamic characteristics of the motion. Using these examples a non-linear control was designed to drive the system by using the property that changing the internal position of the center of mass it is possible to inject or to subtract energy from the systems. Mathematical models were simulated with Simulik software, Matlab was used for the analysis, and animations were created also with Matlab for a better sight of the effects. In parallel, there were developed 2 test rig systems for verification of the numerical results. In the conclusions all the considerations about the study were analyzed, and comments made on the results, as well also future developments are suggested.

Dans cette thèse, nous étudions différents aspects mathématiques et numériques des équations de Gross-Pitaevskii et de Schrödinger non linéaire. Nous commençons (chapitre 1) par introduire différents modèles à partir des systèmes physiques que sont les condensats de Bose-Einstein et les impulsions lumineuses dans les fibres optiques. Cette modélisation conduit aux équations aux dérivées partielles stochastiques suivantes : l'équation de Gross-Pitaevskii stochastique et l'équation de Schrödinger non linéaire avec dispersion aléatoire. Ensuite, dans le second chapitre, nous nous intéressons au problème de l'existence et l'unicité d'une solution de ces équations. On montre notamment que le problème de Cauchy a une solution pour l'équation de Gross-Pitaevskii stochastique avec rotation grâce à la construction de la solution associée au problème. Nous abordons ensuite dans le troisième chapitre le problème du calcul des états stationnaires pour l'équation de Gross-Pitaevskii. Nous développons une méthode pseudo-spectrale de discrétisation du Continuous Normalized Gradient Flow, associée à une résolution itérative préconditionnée des sous-espaces de Krylov. Le quatrième chapitre concerne l'étude de schémas pseudo-spectraux pour la dynamique de l'équation de Gross-Pitaevskii et de Schrödinger non linéaire. On procède à une étude numérique de ces schémas (schéma de splitting de Lie et de Strang, ainsi qu'un schéma de relaxation). De plus, on analyse le schéma de Lie dans le cadre de l'équation de Schrödinger non linéaire avec dispersion aléatoire. Finalement, nous présentons, dans le cinquième chapitre, une boîte à outils Matlab (GPELab) développée dans le but de fournir les méthodes numériques que nous avons étudiées / The aim of this Thesis is to study various mathematical and numerical aspects related to the Gross-Pitaevskii and nonlinear Schrödinger equations. We begin (chapter 1) by introducing a few models starting from the physics of Bose-Einstein condensates and optical fibers. This naturally leads to introducing a stochastic Gross-Pitaevskii equation and a nonlinear Schrödinger equation with random dispersion. Next, in the second chapter, we analyze the existence and uniqueness problem for these two equations. We prove that the Cauchy problem admits a solution for the stochastic Gross-Pitaevskii equation with a rotational term by constructing the solution associated with the linear. The third chapter is concerned with the computation of stationary states for the Gross-Pitaevskii equation. We develop a pseudo-spectral approximation scheme for the Continuous Normalized Gradient Flow formulation, combined with preconditioned Krylov subspace methods. This original approach leads to the robust and efficient computation of ground states for fast rotations and strong nonlinearities. In the fourth chapter, we consider some pseudo-spectral schemes for computing the dynamics of the Gross-Pitaevskii and nonlinear Schrödinger equations. These schemes (the Lie's and Strang's splitting schemes and the relaxation scheme) are numerically studied. Moreover, we proceed to a rigorous numerical analysis of the Lie scheme for the associated stochastic PDEs. Finally, we present in the fifth chapter a Matlab toolbox (called GPELab) that provides computational solutions based on the schemes previously introduced in the Thesis

En växande global befolkning och stor efterfrågan på grödor driver den teknologiska utvecklingen framåt inom det moderna lantbruket. I sin utveckling av nya styr- och reglerfunktioner för frontlastare behövde Ålö AB en pålitlig och effektiv plantmodell av deras frontlastare Quicke Q5M. Plantmodellen skulle inkludera det tillhörande mekaniska och hydrauliska systemet och kunna rendera simulering och 3D-animering i realtid. En automatisk nedställningsfunktion för material skulle också utvecklas för att automatisera den vanligt förekommande sysslan att ställa ned material från en upphöjd position. En plantmodell för Quicke Q5M med ett lastkännande hydraulsystem utvecklades i MATLAB Simulink med Simscape Multibody och Simscape Fluids. Prestandan av den simulerade plantmodellen utvärderades. Plantmodellen visade grundläggande överensstämmelse när den jämfördes med sensordata från en frontlastare i verkligheten. En reglerfunktion för automatisk nedställning av material utvecklades i Simulink. En algoritm för att detektera kollision med mark konstruerades genom att analysera sensordata från en frontlastare i verkligheten som ställt ned material på marken. För att detektera materialets kontakt med mark utfördes hastighets- och accelerationsestimeringar på sensordata från vinkelgivare och trycksensorer. Vid kontakt med mark avslutades nedställningsrörelsen automatiskt. Prestandan och noggrannheten av reglersystemet under olika scenarion analyserades och diskuterades. Trots att simuleringar av plantmodellen och den föreslagna nedställningsfunktionen såg realistiska ut var det svårt att kvantifiera överensstämmelsen mellan plantmodell och frontlastare då inga utförligare fysiska tester hanns med. Framtida utveckling av plantmodell och nedställningsfunktion rekommenderas därför att börja med ett implementeringstest av reglerfunktionen i en traktor i verkligheten. Det skulle ge ett mått på hur pass bra plantmodellen representerar frontlastaren samt möjliggöra utvärdering och vidare justering av reglerfunktionens parametrar i en verklig fysisk miljö. / A growing global population and large demands for crops worldwide is driving agricultural technological advances forward. In developing new automatic control functions for front end loader applications, Ålö AB needed a reliable and effective plant model of their front end loader Quicke Q5M. The plant model should include the associated mechanical and hydraulic system and be capable of rendering simulation and 3D animation in real time. An automated put down control function would also be of benefit for farming operators, speeding up the commonly performed farming task of putting down material from an elevated position. A plant model for the Quicke Q5M with a Load Sensing hydraulic system was developed in MATLAB Simulink with Simscape Multibody and Simscape Fluids. The performance of the simulated plant model was evaluated. The plant model showed some consistent results when compared to sensor data logs from a real-world front end loader. An automatic put down control function for material was also developed and tested in Simulink. A collision detection algorithm was constructed by analyzing sensor data logs from a real-world tractor putting down material on the ground. Velocity and acceleration estimations were made from angular position and pressure sensors to detect the contact between material and ground. The control function automatically stopped the lowering movement when contact was detected. The performance and accuracy of the control system under different scenarios were analyzed and discussed. Although the simulations of the plant model and proposed control function seemed realistic, the consistency of the plant model was hard to quantify since there was no time left for any thourough physical tests. It is recommended that future development of both the plant model and control system begin with an implementation test of the control system in a real-world tractor. This would help determine how accurately the plant model represents the front end loader. It would also allow evaluation and further tuning of the control function in a real physical environment.

L'objectif de nos études est d’appliquer la technique de radiométrie photothermique modulée, technique non intrusive et applicable à distance, pour d’une part, mesurer l'épaisseur et la diffusivité thermique d'une plaque, et d’autre part, caractériser une couche sur un substrat. Un modèle thermique du chauffage 3D a été développé avec prise en compte de l’échange thermique par convection dans le cas d'une plaque, et de la résistance thermique de l'interface dans le cas d'une couche sur un substrat. Une analyse de sensibilité des paramètres sur le déphasage et des études multiparamétriques ont été réalisées à l'aide d'un code de calcul développé sous Matlab. Des formules simples ont ainsi été déterminées pour mesurer l'épaisseur et la diffusivité thermique d'une plaque ainsi que le rapport des effusivités thermiques dans le cas d'une couche sur un substrat. Les formules établies pour les plaques ont été validées expérimentalement sur des plaques d’épaisseur variant de 100μm à 500μm pour différents métaux : inox 304L, nickel, titane, tungstène, molybdène, zinc et fer. L’incertitude de ces déterminations est inférieure à 10% pour l'épaisseur et inférieure à 15% pour la diffusivité thermique. La technique a ensuite été appliquée à des gaines de Zircaloy-4, qui représentent une application très intéressante dans le domaine du nucléaire : les résultats montrent que la présence de la couche d'oxyde, d’épaisseur quelques μm, n'a que très peu d’influence sur les déterminations de l'épaisseur et de la diffusivité thermique du Zircaloy-4. Le comportement du déphasage à hautes fréquences (> 1 kHz) ouvre de plus de nouvelles perspectives, avec la possibilité d’étendre le domaine d’application de la méthode aux couches semi-transparentes et aux couches très minces (inférieures au μm). / Modulated photothermal radiometry, a remote non-intrusive technique, was used to measure the thickness and the thermal diffusivity of a metal plate and to characterize a layer on a substrate. A thermal model of 3D heating was developed with considering the thermal exchange by convection for a plate and the thermal resistance of the interface for a layer on a substrate. The sensibility analysis and the multi-parameter studies on the phase shift were performed by the code developed with the Matlab software. Simple formulas were obtained to determine the thickness and the thermal diffusivity of a plate and the ratio of the thermal effusivities for a layer on a substrate. The obtained formulas were experimentally validated for 100 μm - 500 μm plate thickness of various metals (stainless steel 304L, nickel, titanium, tungsten, molybdenum, zinc and iron). The uncertainty of the measurements was lower than 10 % for thickness and lower than 15 % for thermal diffusivity determination. The same technique was applied in the study on Zircaloy-4 cladding that may be of particular interest for the nuclear industry. It was found that the presence of the oxide layer of some μm thickness had practically no effect on the thickness and the thermal diffusivity measurements of Zircaloy-4 cladding. However, the observed effect of a phase shift on high frequency (> 1kHz) may open new perspectives and widen the field of the method application for semi-transparent layers and for very thin layers (of less than μm thickness).

A busca de sistemas de comunicação eficientes e econômicos é cada vez maior, principalmente com o aumento da demanda por tráfego e banda de transmissão. Conhecidas pela utilização eficiente de largura de banda, as redes ópticas de segmentação espectral flexível têm sido intensamente estudadas e são consideradas boas candidatas para a próxima geração de redes. Tais redes flexíveis utilizam formatos de modulação multinível e multiplexação multiportadora. Sistemas com alta capacidade de transmissão devem ser extremamente confiáveis, pois a falha em um dispositivo ou enlace da rede acarreta enorme perda de dados. Por isso, as técnicas de proteção e restauração do tráfego devem ser eficientes, de modo a garantir a sobrevivência da rede. Neste trabalho, foi desenvolvido um algoritmo capaz de lidar com cada requisição de conexão, encontrar um caminho óptico para transmissão e reservar outro caminho para recuperação do tráfego em caso de falha na rede. A seleção dos caminhos primário e de proteção utiliza a técnica smart-fit, que escolhe a solução com menor custo final, definido pela distância da rota percorrida somada ao custo do índice inicial da faixa espectral alocada, buscando equilibrar a escolha entre a posição no espectro e rota escolhida. Além disso, são definidos custos diferentes para slots de frequência livres e compartilhados, a fim de incentivar o compartilhamento espectral em caminhos de proteção. Dentre as técnicas de alocação espectral, a varredura com janela espectral varre o espectro em todos os enlaces da rota, slot por slot, em busca de uma faixa livre com tamanho suficiente para atender uma demanda. Neste trabalho, foi desenvolvida uma técnica chamada busca lógica, que lida simultaneamente com todos os enlaces da rota ao realizar operações lógicas com seus espectros. Em seguida, é realizada uma convolução entre a janela espectral, com tamanho da demanda, e o espectro resultante. A partir desse resultado, é possível saber as posições no espectro onde a demanda poderá ser alocada. Tal técnica, como será demonstrado, é mais veloz que a varredura com janela espectral. Para garantir a eficácia e confiabilidade do algoritmo, utilizando o software MATLAB, avaliou-se a probabilidade de bloqueio e probabilidade de bloqueio de banda, a fragmentação espectral média na rede, o grau de compartilhamento, a taxa de utilização espectral e a taxa de capacidade reservada. O algoritmo desenvolvido foi capaz de organizar o tráfego na rede de modo ordenado e pouco fragmentado, além de utilizar de maneira eficiente os recursos destinados à proteção. / The search for efficient and economical communication systems is increasing, especially with the increased demand for traffic and bandwidth transmission. The elastic optical networks, well known for their efficient using of bandwidth, have been thoroughly studied and are considered good candidates for the next generation networks. Such flexible networks use multilevel modulation formats and multicarrier multiplexing. Systems with high transmission capacity should be extremely reliable, as a failure in a device or link brings huge data loss. Therefore, traffic protection and restoration techniques should be efficient, to ensure network survivability. In this work, we propose an algorithm capable of managing each connection request, finding an optical path for transmission and a protection path to recover traffic in case of network failure. The selection of the primary and protection path uses the smart-fit technique, which chooses the solution with the lower final cost, defined by the route\'s distance plus the index\'s cost of the first slot of the allocated spectrum, in order to balance the choice between the position in the spectrum and the chosen route. In addition, different costs for free or shared frequency slots are defined, so as to encourage spectral sharing in protection paths. Among the spectrum allocation techniques, scanning with spectrum window scans the spectrum in all links of the route, slot by slot, searching for a free band large enough to meet the demand. In this work, we developed a new technique, called logical search, that simultaneously deals with all the links in the route, by performing logical operations with them. Then, a convolution is performed between a spectrum window, with the same size of the demand, and the resulting spectrum. From this result, it is possible to know all the positions in the spectrum where the demand may fit. This technique, as will be shown, is faster than the scan with spectrum window. To guarantee the efficiency and reliability of the algorithm, using the MATLAB software, we evaluated the blocking probability and bandwidth blocking probability, the average spectral fragmentation in the network, the shareability ratio, the spectral utilization ratio and the spare capacity ratio. The developed algorithm was able to organize the traffic in an orderly and less fragmented way, and use the protection resources efficiently.

La zircone 3Y-TZP présente un grand intérêt pour les applications dentaires, en implantologie orale, elle semble être un matériau extrêmement prometteur: elle allie une biocompatibilité à un aspect esthétique satisfaisant et présente aussi des propriétés mécaniques très supérieures aux autres céramiques. Ces bonnes propriétés mécaniques sont intimement liées à la microstructure du matériau, elle-même directement liée aux procédés d’élaboration comme nous l’a rappelé l’alarmante série de ruptures de plus de 800 têtes de prothèses de hanche en zircone au début des années 2000, due au phénomène de vieillissement de ce matériau. Cependant, les études cliniques menées à ce jour ne font toujours pas état des interrelations existant entre la microstructure, les propriétés mécaniques et la sensibilité au vieillissement. Il était donc primordial de valider et d’appliquer les connaissances acquises dans le domaine des sciences des matériaux pour l’application de la zircone 3Y-TZP en implantologie oral. C’est l’objectif de notre première publication. Nous avons également développé, dans notre deuxième publication, un protocole d’évaluation permettant dans un premier temps, d’évaluer l’effet de l’état de surface et de la microstructure sur la résistance mécanique d’implants neufs, puis de suivre leurs cinétiques de vieillissement tout en analysant l’évolution de la microstructure et son influence sur la résistance mécanique au cours du vieillissement. Tout cela permettant in fine, de prédire la durabilité d’un type d’implant. Nous avons par la suite, développé un programme informatique permettant la détection et la quantification du vieillissement pour un volume donné. Cette détection de la zone vieillie ou transformée est basée sur des modifications microstructurales caractéristiques engendrées lors du vieillissement. Enfin, nous avons pu mettre en évidence l’occurrence du phénomène de vieillissement in vivo, par l’analyse d’explants issus d’une étude clinique et démontrer leur probable implication dans ces cas d’échec. Le logiciel informatique de traitement d’image développé a été également, appliqué aux explants dans le but de mettre en lumière et d’expliciter l’occurrence du phénomène de vieillissement in vivo, afin de sensibiliser les acteurs de ce marcher aux problématiques rencontrées et d’optimiser de ce dispositif médical, à la lumière des connaissances actuelles. / 3Y-TZP zirconia is gaining interest in oral implantology, it seems to be a promising material with good biocompatibility, esthetic appearance and also the highest mechanical properties for a ceramic. These mechanical properties are closely related to the microstructure of the material itself directly related to production processes as we recalled the alarming series of breaks of more than 800 heads of zirconia hip replacements in the early 2000, due to the aging phenomenon of the material. However, clinical studies to date are still not state of the interrelationships between microstructure, mechanical properties and sensitivity to aging. It was therefore important to validate and apply the knowledge gained in the field of materials science for the application of 3Y-TZP zirconia oral implantology. This is the goal of our first publication. We also have developed in our second publication, a protocol to assess the effect of the surface modification and microstructure on the mechanical strength of new implants and follow their kinetics of aging and also, the evolution of the microstructure and its influence on the mechanical strength during aging, to predict the durability of a type of implant. We have subsequently developed a computer program for the detection and quantification of the aging for a given volume. This detection of the aged or transformed area is based on microstructural modifications produced during aging. Finally, we have been able to demonstrate the occurrence of the phenomenon of aging in vivo, by analyzing explants from a clinical study and demonstrate their involvement in the case of dental implant failure. The image processing developed was also applied to the explants in order to highlight and explain the occurrence of in vivo aging phenomenon. The objective is to optimize this medical device, in the light of current knowledge.

Η παρούσα διπλωματική εργασία, πραγματεύεται την εύρεση της κυματομορφής της μεταβολής της φωτεινότητας φωτονίων που διέρχονται από το χέρι ανθρώπου και δίνουν πληροφορία για την αρτηριακή πίεση και κατ’ επέκταση τον υπολογισμό του καρδιακού ρυθμού ενός ατόμου με τη χρήση τεχνικών επεξεργασίας εικόνας. Χρησιμοποιώντας μια σειρά από διόδους εκπομπής, στο ορατό και υπέρυθρο φάσμα, κατευθύνουμε το φως προς ένα δίκτυο ιστών όπου αυτό είναι λεπτό και το διαπερνά (δάκτυλο, λοβίο αυτιού κλπ). Στη συνέχεια, μέσω μιας βιντεοκάμερας παίρνουμε τα υπό εξέταση δεδομένα. Συγκρίνοντας την απορρόφηση του φωτός στις διαδοχικές εικόνες και ύστερα από κατάλληλη επεξεργασία των εικόνων με τη βοήθεια του Matlab οδηγούμαστε στην εύρεση του καρδιακού ρυθμού. / The present thesis deals with the determination of the waveform that depicts the fluctuation of the brightness of photons which pass through the hand of a person and provides information for the arterial pressure. Exploiting the results through digital image processing techniques, subject’s cardiac rhythm can be conclusively calculated. Using a series of diodes emitting in the visible spectrum along with a second series of diodes emitting in the infrared spectrum, we direct their light to a part of the human tissue which is thin (finger, earlobe etc) and can be easily penetrated. Afterwards via a CCD video camera we capture picture data of the light that is not absorbed. Cardiac rhythm can be calculated by comparing the absorption of light in successive pictures processed by digital imaging processing tools of Matlab.

Em uma abordagem realística de estruturas e componentes mecânicos admite-se a existência de trincas. A presença destas, geralmente, está associada ao fenômeno da fadiga. Existem diversos modelos matemáticos que descrevem a propagação de uma trinca. De forma geral, os modelos de propagação de trinca são classificados pelo tipo de carregamento, que pode ter amplitude de tensão constante (CATC) ou amplitude de tensão variável (CATV). Neste trabalho foi utilizado o modelo do tipo CATC proposto por Collipriest. Para muitas aplicações de engenharia, até um certo momento, não se faz necessário uma grande acurácia nas previsões do comportamento das estatísticas, a cerca da evolução de uma trinca, mas uma previsão confiável, dentro de certos limites, desse comportamento. Este trabalho apresenta resultados teóricos, que consistem em obter cotas, inferiores e superiores, que “envelopam” os estimadores dos momentos estatísticos de primeira e de segunda ordem da função “tamanho de trinca” baseadas no método “Fast Crack Bounds”. Essas cotas são polinômios, definidos na variável número de ciclos, que consideram as incertezas nos parâmetros que descrevem os modelos de propagação de trinca. O método de simulação de Monte Carlo foi utilizado para obter as realizações da função tamanho de trinca a partir de um conjunto de 10.000 amostras randômicas dos parâmetros característicos do modelo de Collipriest. Essas realizações foram utilizadas para obter os estimadores dos momentos estatísticos do tamanho de trinca. A eficiência das cotas para os estimadores dos momentos estatísticos do tamanho de trinca é avaliada através de funções “desvio relativo” entre as cotas e as soluções numéricas aproximadas do problema de valor inicial (PVI) que descreve o modelo de Collipriest. Em geral, a solução dos PVI que descrevem os modelos de propagação de trincas é obtida através do uso de métodos numéricos, como o método de Runge-Kutta de quarta ordem explícito (RK4). Neste trabalho foi utilizado o software MATLAB para obter as soluções do PVI que descreve o modelo de Collipriest, avaliar o tempo computacional da metodologia proposta, além dos desvios das cotas em relação às soluções aproximadas, confirmando sua eficiência. / In a realistic approach of structures and mechanical components, cracks are admitted. Their presence is usually associated with the fatigue phenomenon. There are several mathematical models that describe the propagation of a crack. In general, the crack propagation models are classified by the type of load, which can have constant stress amplitude (CSA) or variable stress amplitude (VSA). In this work, the CSA type model proposed by Collipriest was used. For many engineering applications, until a certain point, it is not necessary to have great accuracy in predictions of the behavior of statistics, about the evolution of a crack, but a reliable prediction, within certain limits, of this behavior. This work presents theoretical results, which consist of obtaining lower and upper bounds that "envelop" the estimators of the first and second order statistical moments of the crack size function based on the Fast Crack Bounds method. These bounds are polynomials defined in the variable “number of cycles” that consider the Metais - Fadiga uncertainties in the parameters that describe the crack propagation models. The efficiency of the bounds for the statistical moments of crack size is evaluated through the deviation between the bounds and the approximate numerical solutions of the initial value problems (IVP) that describes the Collipriest model. In general, the solution of the IVPs describing crack propagation models is obtained through the use of numerical methods, such as the explicit fourth order Runge-Kutta method (RK4). In this work, the MATLAB software was used to obtain the solutions of the IVP that describes the Collipriest model, to evaluate the computational time of the proposed methodology, besides the deviations of the bounds in relation to the approximated solutions, confirming its efficiency.