Dissertations / Theses on the topic 'Duffing Oscillator'

This thesis is a study of bifurcation trees of periodic motions in a parametric Duffing oscillator. The bifurcation trees from period-1 to period-4 motions are investigated by a semi-analytic method. For the semi-analytic method, the discretization of differential equations of nonlinear dynamical systems is obtained to attain the implicit mapping structure. Following the development of implicit mapping structure, the periodic nodes of periodic motions are computed. The stability and bifurcation conditions are carried out by the eigenvalue analysis. For a better understanding of nonlinear behaviors of periodic motions, the harmonic frequency-amplitude characteristics are presented by the finite Fourier series. Numerical simulations are illustrated to verify the analytical predictions. Based on the comparison of numerical and analytical result, the trajectory, time history, harmonic amplitude and harmonic phase plots of period-1 to period-4 motions are completed.

La thèse concerne l'étude des cycles limites d'une équation différentielle sur le plan (la deuxième partie du 16ème problème de Hilbert). La notion de "cycle limite" a une grande importance dans la théorie de la stabilité, elle est introduite par Poincaré vers la fin du 19ème siècle et désigne une orbite périodique isolée. Le but de cette thèse est : d'établir l'existence d'une borne supérieure finie, pour le nombre de cycle limites d'une équation quadratique dans le plan. Ce problème est aussi appelé 16ème problème d' Hilbert infinitésimal. Probablement, l'outil le plus fondamental pour l'étude de la stabilité et les bifurcations des orbites périodiques est l'application de Poincaré, défini par Henri Poincaré en 1881. Cependant, la méthode de Melnikov nous donne une excellente procédure pour déterminer le nombre de cycles limites dans une bande continue de cycles qui sont préservés sous perturbation. En effet, le nombre, les positions et les multiplicités des équations différentielles planes perturbées avec une petite perturbation non nulle sont déterminées par le nombre, les positions et les multiplicités des zéros des fonctions génératrices. La fonction de Melnikov est plus précisément, appelé la fonction de Melnikov de premier- ordre. Si cette fonction est identiquement nulle à travers la bande continue de cycles, on calcule ce qu'on appelle " la fonction de Melnikov d'ordre supérieure ". Ensuite, une analyse d'ordre supérieure est nécessaire, ce qui peut être fait par " l'algorithme de Françoise. Les discussions et les calculs présentés dans notre travail sont limités non seulement à la fonction de Melnikov de premier ordre, mais aussi pour les fonctions de Melnikov de deuxième -ordre. Ces outils seront utiles pour résoudre notre problématique. Les activités de recherche menées dans le cadre de cette recherche sont divisées en quatre parties : La première partie de cette thèse, traite les systèmes dynamiques plans et l'existence de cycles limites. Nous souhaitons après résoudre le problème suivant: Calculer la cyclicité de l'oscillateur asymétrique perturbé de Duffing. Dans la deuxième partie, nous sommes intéressés de la cyclicité à l'extérieur de l'anneau périodique de l'oscillateur de Duffing pour une perturbation particulière, puis, nous fournissons un diagramme de bifurcation complet pour le nombre de zéros de la fonction de Melnikov associée dans un domaine complexe approprié en se basant sur le principe de l'argument. Le nombre de cette cyclicité est égal à trois. Dans la troisième partie, nous étudions la cyclicité à l'intérieur ainsi que à l'extérieur de double boucle homocline pour une perturbation cubique arbitraire de l'oscillateur de Duffing en utilisant les mêmes techniques de Iliev et Gavrilov dans le cas d'un Hamiltonien asymétrique de degré quatre. Notre principal résultat est que deux au plus cycle limite peuvent bifurquer de la double homocline. D'autre part, il est représenté, qu'après bifurcation de eight-loop un cycle limite étranger est née, qui ne soit pas contrôlée par un zéro lié par les intégrales Abéliennes, ce cycle supplémentaire est appelé " Alien "
This thesis concerns the study of limit cycles of a differential equation in the plane (The second part of the 16th Hilbert problem). The concept of "limit cycle" has a great importance in the theory of stability; Poincaré introduces this notion at the end of the 19th century and denotes an isolated periodic orbit. The purpose of this thesis: Find an upper bound finite to the number of limit cycles of a quadratic equation in the plane. This problem is so- called the infinitesimal Hilbert 16th problem. Probably, the most basic tool for studying the stability and bifurcations of periodic orbits is the Poincaré, defined by Henri Poincaré in 1881. However, Melnikov's method gives us an excellent method for determining the number of limit Cycles in a continuous band of cycles that are preserved under perturbation. In fact, the number, positions and multiplicities of perturbed planar differential equations for a small nonzero parameters, are determined by the number, positions and multiplicities of the zeros of the generating functions. The Melnikov function is more precisely, called the first-order Melnikov function. If this function is identically equal zero across the continuous band of cycles, one computes the so-called "Higher order Melnikov function". Then, a higher order analysis is necessary which can be done by making use of the so called "the algorithm of Françoise". The discussions and computation presented in this thesis are restricted not only to the first order Melnikov function, but also to the second-order Melnikov functions. These tools will be useful to resolve the question problem. The research activities in the framework of this thesis are divided into four parts: The first part of this thesis, discusses planar dynamical systems and the existence of limit cycles. We wish to solve the following problem: Calculate the cyclicity of the perturbed asymmetric oscillator Duffing. In the second part, we are interested of the cyclicity of the exterior period annulus of the asymmetrically perturbed Duffing oscillator for a particular perturbation, then, we provide a complete bifurcation diagram for the number of zeros of the associated Melnikov function in a suitable complex domain based on the argument principle. The number of this cyclicity is equal to three. In the third part, we study the cyclicity of the interior and exterior eight-loop especially for arbitrary cubic perturbations by using the same techniques of Iliev and Gavrilov in the case of an asymmetric Hamiltonian of degree four. Our main result is that at most two limit cycles can bifurcate from double homoclinic loop. On the other hand, it is appears after bifurcation of eight-loop an "Alien" limit was born, which is not covered by a zero of the related Abelian integrals

Analytical solutions of periodic motions in a periodically excited, Duffing oscillator with a time-delayed displacement are developed through the Fourier series, and the stability and bifurcation of such periodic motions are discussed through eigenvalue analysis. The analytical bifurcation trees of period-1 motions to chaos in the time-delayed Duffing oscillator is presented through asymmetric period-1 to period-4 motions. Four independent symmetric period-3 motions were obtained. Two independent symmetric period-3 motions are not relative to chaos, while the other two includes bifurcation trees of period-3 motion to chaos, which are presented through period-3 to period-6 motions. Stable periodic motions are illustrated from numerical and analytical solutions. The appropriate initial history functions for periodic motions are analytically computed from the analytical solutions of periodic motions. Without the appropriate initial history functions, such a time-delayed system cannot yield periodic motions directly.

In this paper, bifurcation trees of periodic motions in a periodically forced, time-delayed, hardening Duffing oscillator are analytically predicted by a semi-analytical method. Such a semi-analytical method is based on the differential equation discretization of the time-delayed, non-linear dynamical system. Bifurcation trees for the stable and unstable solutions of periodic motions to chaos in such a time-delayed, Duffing oscillator are achieved analytically. From the finite discrete Fourier series, harmonic frequency-amplitude curves for stable and unstable solutions of period-1 to period-4 motions are developed for a better understanding of quantity levels, singularity and catastrophes of harmonic amplitudes in the frequency domain. From the analytical prediction, numerical results of periodic motions in the time-delayed, hardening Duffing oscillator are completed. Through the numerical illustrations, the complexity and asymmetry of period-1 motions to chaos in nonlinear dynamical systems are strongly dependent on the distributions and quantity levels of harmonic amplitudes. With the quantity level increases of specific harmonic amplitudes, effects of the corresponding harmonics on the periodic motions become strong, and the certain complexity and asymmetry of periodic motion and chaos can be identified through harmonic amplitudes with higher quantity levels.

A study is presented of the instability mechanisms of a damped axisymmetric circular disk of uniform thickness rotating about its axis with constant angular velocity and subjected to various transverse space-fixed loading systems. The natural frequencies of spinning floppy disks are obtained for various nodal diameters and nodal circles with a numerical and an approximate method. Exploiting the fact that in most physical applications the thickness of the disk is small compared with its outer radius, we use their ratio to define a small parameter. Because the nonlinearities appearing in the governing partial-differential equations are cubic, we use the Galerkin procedure to reduce the problem into a finite number of coupled weakly nonlinear second-order equations. The coefficients of the nonlinear terms in the reduced equations are calculated for a wide range of the lowest modes and for different rotational speeds. We have studied the primary resonance of a pair of orthogonal modes under a space-fixed constant loading, the principal parametric resonance of a pair of orthogonal modes when the disk is subject to a massive loading system, and the combination parametric resonance of two pairs of orthogonal modes when the excitation is a linear spring. Considering the case of a spring moving periodically along the radius of the disk, we show how its frequency can be coupled to the rotational speed of the disk and lead to a principal parametric resonance. In each of these cases, we have used the method of multiple scales to determine the equations governing the modulation of the amplitudes and phases of the interacting modes. The equilibrium solutions of the modulation equations are determined and their stability is studied.
Master of Science

The diploma thesis is focused on the use of heuristic and metaheuristic methods to stabilization and controlling the selected systems distinguished by the deterministic chaos behavior. There are discussed parameterization of chosen optimization methods, which are the genetic algorithm, simulated annealing and pattern search. The thesis also introduced the suitable controlling methods and the definition of the objective function. In the theoretical part of the thesis there is a brief introduction to the deterministic chaos theory. The next chapters describes the most common and deployed methods in~the~control theory, especially OGY and Pyragas methods. The practical part of the thesis is divided into two chapters. The first one describes the~stabilization of the artifical chaotic systems with the time delayed Pyragas method - TDAS and its modification ETDAS. The second chapter shows the real chaotic system control. The Duffing oscillator system was chosen to serve this purpose.

La détection de signaux magnétiques faibles a suscité une attention considérable en raison de ses applications potentielles dans des domaines tels que la médecine et la nanotechnologie. Diverses méthodes ont été employées pour améliorer la détection de signaux faibles, notamment les dispositifs SQUID, les capteurs diamant et les résonateurs magnétoélectriques (ME). Le choix de la méthode dépend de facteurs tels que le contexte d'application, le coût et les exigences de sensibilité. Parmi ces méthodes, les résonateurs MEMS-ME ont retenu l'attention en raison de leur flexibilité de conception, de leur compacité et compatibilité avec les circuits intégrés. Dans ces résonateurs à microéchelle, l'interaction entre films magnétostrictifs et piézoélectriques permet d'obtenir des effets de contrainte à l'échelle-micrométrique, offrant ainsi une grande précision et une résolution spatiotemporelle élevée. Cette thèse explore le régime nonlinéaire du fonctionnement des résonateurs, caractérisé par des formes asymétriques, des bifurcations et des résonances nonlinéaires. Le régime nonlinéaire permettre des modes de fonctionnement analogiques, qui sont obtenus en balayant la fréquence d'excitation jusqu'au point de bifurcation. Malgré les défis liés au comportement nonmonotonique, le régime nonlinéaire se révèle être une méthode précieuse pour détecter les signaux faibles. La bistabilité, courante dans les résonateurs fonctionnant de manière nonlinéaire, n'est pas largement utilisée dans les configurations piézoélectriques. Cette thèse explore le potentiel du régime de bifurcation dans les résonateurs actionnés piézoélectriquement et présente dispositif de preuve-de-concept pour quantifier les changements du signal en mesurant la fréquence. Mathématiquement, les équations différentielles sont transformées en équations de Duffing normalisées à l'aide de la méthode de Galerkin, permettant ainsi aux comportements dynamiques de se manifester à travers les coefficients Duffing. Différents modèles ont été développés pour répondre à différentes conditions et hypothèses, révélant des liens entre les paramètres mécaniques. Combler l'écart entre les modèles basés sur l'amplitude des vibrations et les données d'impédance s'est avéré complexe mais réalisable. Grâce à des expériences et à l'affinement itératif du modèle, le modèle basé sur l'amplitude des vibrations a fourni des informations sur les réponses en fréquence, bien qu'il ne prédise pas directement l'ampleur et la phase de l'impédance. La recherche a reconnu les limitations liées à l'emplacement de l'axe neutre dans les films minces monocouches, suggérant de réévaluer les hypothèses, de tenir compte des effets multicouches et d'effectuer des simulations numériques pour obtenir des représentations plus précises. Le concept d'axe neutre relatif a été introduit, reconnaissant les écarts par rapport aux prédictions du modèle monocouche. Cette approche a été justifiée de manière transparente et alignée sur le comportement expérimental observé. Parallèlement, la fabrication de microcantilevers à base de PZT, composants essentiels du capteur résonant, a subi de multiples itérations pour relever les défis. Ces améliorations itératives ont abouti à un processus de fabrication plus robuste et fiable. En conclusion, cette étude a fait progresser la compréhension des résonateurs actionnés piézoélectriquement et de leur potentiel dans la détection de signaux faibles. Les améliorations itératives de la fabrication et les modèles mathématiques ont contribué au développement de dispositifs de détection multifonctionnels. Elle a mis en lumière l'interaction complexe entre la nonlinéarité et la résonance dans les systèmes de résonateurs, fournissant informations pour des recherches futures et des applications pratiques
The detection of weak magnetic field signals has gained significant attention for its potential applications in fields such as medicine, geophysics, and nanotechnology. Various methods, including Superconducting Quantum Interference Devices (SQUIDs), optically pumped magnetometers (diamond sensors), and magnetoelectric (ME) resonators, have been used to enhance the detection of these weak signals. The choice of detection method depends on factors such as application context, available resources, cost, and sensitivity requirements. Among these methods, MEMS ME resonator-based sensors have garnered attention due to their design flexibility, compactness, and compatibility with integrated circuits. In these microscale resonators, the interaction between magnetostrictive and piezoelectric thin films enables a strain-mediated effect at micro- and nano-scales, resulting in high precision and spatial-temporal resolution. The thesis delves into the nonlinear regime in resonator operation, characterized by nonlinearity in vibrational responses, including asymmetrical peak shapes, multivalued responses, bifurcations, and nonlinear resonances. The nonlinear regime, particularly bifurcation, promises enhanced sensing capabilities and analog operation modes by sweeping the excitation frequency. Despite challenges like noise-activated stochastic switching, the nonlinear regime is valuable for detecting weak signals. Bistability in resonators within the nonlinear regime, underutilized in piezoelectric configurations, is explored. A proof-of-concept device quantifies signal changes through jumping frequency. Mathematically, differential equations are transformed into normalized Duffing equations using Galerkin's method, enabling dynamic behaviors to manifest through coefficients. Distinct models accommodate various conditions and assumptions, revealing connections between mechanical parameters and normalized coefficients in linear and nonlinear regimes. Bridging the gap between vibrational amplitude-based models and impedance data is complex but achievable. Experiments and iterative model refinement provide insights into frequency responses. Limitations regarding the neutral axis in monolayer thin films are acknowledged, with suggestions to reevaluate assumptions, consider multilayer effects, and employ numerical simulations. A relative neutral axis concept is introduced, transparently justified, and aligned with observed experimental behavior. The nonlinear regime widens resonance peaks, enhancing sensitivity in magnetic field detection. Parameters like piezoelectric and dielectric coefficients influence the transition to the nonlinear regime. The research extends beyond ideal scenarios, requiring further investigation to replicate the bifurcation regime under different conditions. In parallel, the fabrication of PZT-based microcantilevers, vital components of the resonant sensor, underwent multiple iterations to address challenges. These iterative improvements resulted in a more robust and reliable fabrication process. In conclusion, this study advanced the understanding of piezoelectrically actuated resonators and their potential applications in weak signal detection. The iterative fabrication enhancements and mathematical models contributed to the development of multifunctional sensing devices. The research also emphasized the importance of bridging the gap between vibrational amplitude-based models and impedance data. Finally, it shed light on the intricate interplay of nonlinearity and resonance in resonator systems, providing insights for future investigations and practical applications

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Capes
Nesta dissertação, analisamos a dinâmica de osciladores parametricamente forçados, com enfoque na amplificação de pequenos sinais. Iniciamos por uma revisão da ressonância paramétrica e da amplificação paramétrica em um oscilador linear parametricamente excitado. Em seguida, estudamos dois tipos de osciladores não-lineares parametricamente forçados e concluímos a dissertação com a análise de um dímero parametricamente excitado. Basicamente, analisamos os fenômenos de ressonância paramétrica e de amplificação paramétrica, comparando os resultados obtidos analiticamente (via métodos da média ou do balanço harmônico) com os obtidos via integração numérica das equações do movimento. Em todos os casos, obtivemos a linha de transição para a instabilidade paramétrica do oscilador paramétrico. Nós excitamos os amplificador paramétrico com e sem dessintonia entre entre o bombeamento e o sinal externo ac. Verificamos que o ganho da amplificação paramétrica depende da sensitivamente na fase do sinal externo ac e na amplitude do bombeamento. Mostramos que tais sistemas podem ser facilmente utilizados para recepção e decodificação de sinais com modulação de fase. Além disso, obtivemos séries temporais, envelopes e transformadas de Fourier para a resposta da amplificação paramétrica de pequenos sinais ac. Especificamente nos casos dos osciladores de Duffing parametricamente forçados, obtivemos e analisamos linhas de bifurcação e a amplitude dos ciclos limites como função da frequência e da amplitude de bombeamento. Adicionalmente, conseguimos obter uma relação analítica para os ganhos do sinal e do idler dos osciladores não-lineares parametricamente forçados pelo método do balanço harmônico. Os resultados obtidos implicam que os amplificadores paramétricos não-lineares podem ser excelentes detectores, especialmente em pontos próximos a bifurcações para instabilidade, em que apresentam altos ganhos e largura de banda bem estreitas. Por último, investigamos também o comportamento de dois osciladores lineares acoplados e parametricamente estimulados, com e sem força externa ac. Tais sistemas são muito sensíveis à fase do sinal a ser amplificado e podem ser utilizados para criar amplificadores sintonizáveis em função do parâmetro de acoplamento.
In this dissertation, we studied the dynamics of parametrically-driven oscillators, with a focus on the amplification of small signals. We begin with a revision of parametric resonance and parametric amplification in a linear oscillator parametrically excited. Next, we studied two types of nonlinear parametrically-driven oscillators and finished the dissertation with an analysis of a parametric dimer. Basically, we analyzed the phenomena of parametric resonance and parametric amplification by comparing the results obtained analytically (via the averaging or harmonic balance methods) with those of numerical integration of the equations of motion. In all cases, we obtained the transition line to parametric instability of the parametric oscillator. We excited the parametric amplifier with and without detuning between the pump and the external signal. We found that the parametric amplification depends sensitively on the phase of the external ac signal and on the internal pump amplitude. We showed that such amplifiers can be easily used for the reception and decoding of signals with phase modulation. Furthermore, we obtained time series, envelopes, and Fourier transforms of the response of the parametric amplifier to small external ac signals. Specifically in the cases of the parametrically-driven Duffing oscillators, we obtained and analysed the bifurcation lines and the amplitude of limit cycles as function of the pump amplitude and frequency. In addition, we derived an expression for the signal and idler gains of the nonlinear parametrically-driven oscillators with the harmonic balance method. The results imply that the nonlinear parametric amplifiers can be excellent detectors, specially near bifurcations to instability, due to their high gains and narrow bandwidths. Finally, we studied the dynamics of two linear oscillators coupled and parametrically excited, with and without external ac driving. We found that such systems have a wealth of dynamical responses. They present parametric amplification that is dependent on the coupling parameter and on the phases of the external ac signals. Such systems may be used as tunable amplifiers.

In support of society's technological evolution, the study of nonlinear systems in engineering and sciences has become a vital research area. Aiming to contribute in this field, this thesis investigates the behaviour of nonlinear systems using the 'Wiener theories'. As a useful example the Duffing oscillator is investigated in this work. In many real-life applications, nonlinear systems are excited randomly so this work examines systems under white-noise excitation using the Wiener series. Equivalent Linearisation (EL) is a well-known and simple method that approximates a nonlinear system by an equivalent linear system. However, it has deficiencies which this thesis attempts to improve. Initially, the performance of EL for different types of nonlinearities will be assessed and an alternative method to enhance it is suggested. This requires the calculation of the first Wiener kernel of various system defined quantities. The first Wiener kernel, as it will be shown, is the foundation of this research and a central element of the Wiener theory. In this thesis, an analytical proof to explain the interesting behaviour of the first Wiener kernel for a system with nonlinear stiffness is included using an energy transfer approach. Furthermore, the method mentioned above to enhance EL known as the Single-Pole Fit method (SPF) is to be tested for different kinds of systems to prove its robustness and validity. Its direct application to systems with nonlinear stiffness and nonlinear damping is shown as well as its ability to perform for systems with two degrees of freedom where an extension of the SPF method is required to achieve the desired solution. Finally, an investigation to understand and replicate the complex behaviour observed by the first Wiener kernel in the early chapters is carried out. The groundwork for this investigation is done by modelling an isolated nonlinear spring with a series of linear filters and certain nonlinear operations. Subsequently, an attempt is made to relate the principles governing the successful spring model presented to the original nonlinear system. An iterative procedure is used to demonstrate the application of this method, which also enables this new modelling approach to be related to the SPF method.

Studium chování opticky zachycených částic nám umožňuje porozumět základním fyzikálním jevům plynoucím z interakce světla a hmoty. Předkládaná práce podává vysvětlení zesílení tažné síly působící na opticky svázané částice ve strukturovaném světelném poli, tzv. tažném svazku. Ukazujeme, že pohyb dvou opticky svázaných objektů v tažném svazku je silně závislý na jejich vzájemné vzdálenosti a prostorové orientaci, což rozšiřuje možnosti manipulace hmoty pomocí světla. Následně se práce zaměřuje na levitaci opticky zachycených částic ve vakuu. Představujeme novou metodologii na charakterizaci vlastností slabě nelinearního Duffingova oscilátoru reprezentovaného opticky levitující částicí. Metoda je založena na průměrování trajektorií s určitou počáteční pozicí ve fázovém prostoru sestávajícím z polohy a rychlosti částice a poskytuje informaci o parametrech oscilátoru přímo ze zaznamenaného pohybu. Náš inovativní postup je srovnán s běžně užívanou metodou založenou na analýze spektrální hustoty polohy částice a za využití numerických simulací ukazujeme její použitelnost i v nízkých tlacích, kde nelinearita hraje významnou roli.

Orientador: Masayoshi Tsuchida
Banca: José Manoel Balthalzar
Banca: Adalberto Spezamiglio
Resumo: Osciladores eletromecânicos podem ser modelados matematicamente através da equação de Duffing ou equação de Van der Pol, mesmo que sejam sistemas de escala nanomética. Nesta dissertação analisamos um oscilador forçado sujeito a um amortecimento não-linear, que é representado pela equação de Duffing - Van der Pol. Em geral, não é fácil obter solução analítica exata para esta equação, então a análise é feita utilizando a teoria de perturbações para obter uma solução analítica aproximada. Para isso consideramos certos parâmetros do problema como sendo pequenos parâmetros, e obtemos a solução na forma de expansão direta. Devido o fato da frequência natural do sistema dinâmico depender do pequeno parâmetro, essa expansão é não uniforme, ou seja, apresenta termos seculares mistos (termos de Poisson), e além disso possui pequenos divisores. Essas inconveniências são eliminadas aplicando o método das múltiplas escalas e o método da média. Inicialmente os pequenos parâmetros são escolhidos de modo que o problema não perturbado se reduz a um oscilador harmônico forçado, e na escolha posterior o problema não perturbado é um oscilador linear amortecido e forçado.
Abstract: Electromechanical oscillators can be mathematically modeled by a Du±ng equation or a Van der Pol equation, even if they are nanometric systems. In this work we studied a forced oscillator having nonlinear damping, that is represented by a Du±ng - Van der Pol equation. In general, it is not easy to get the exact analytical solution for this equation, then the analysis is done using the perturbation theory to get an approximate analytical solution. For this reason we considered that certain parameters of the problem are small parameters and we obtain the solution in the form of straightforward expansion. Due to the fact that natural frequency of the dynamic system depends on the small parameter, this expansion is not uniform, i.e. presents secular terms (Poisson terms) and also small-divisors. These inconveniences are eliminated using the method of multiple scales and the aver- aging method. Initially the small parameters are chosen so that the unperturbed problem is reduced to a forced harmonic oscillator, and in the subsequent choice the unperturbed is a forced oscillator having linear damping.
Mestre

On considere un systeme dynamique non lineaire de type duffing, avec une composante periodique parametrique dans la force de rappel et une force exterieure periodique de pulsation differente de celle de la force parametrique

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Osciladores eletromecânicos podem ser modelados matematicamente através da equação de Duffing ou equação de Van der Pol, mesmo que sejam sistemas de escala nanomética. Nesta dissertação analisamos um oscilador forçado sujeito a um amortecimento não-linear, que é representado pela equação de Duffing - Van der Pol. Em geral, não é fácil obter solução analítica exata para esta equação, então a análise é feita utilizando a teoria de perturbações para obter uma solução analítica aproximada. Para isso consideramos certos parâmetros do problema como sendo pequenos parâmetros, e obtemos a solução na forma de expansão direta. Devido o fato da frequência natural do sistema dinâmico depender do pequeno parâmetro, essa expansão é não uniforme, ou seja, apresenta termos seculares mistos (termos de Poisson), e além disso possui pequenos divisores. Essas inconveniências são eliminadas aplicando o método das múltiplas escalas e o método da média. Inicialmente os pequenos parâmetros são escolhidos de modo que o problema não perturbado se reduz a um oscilador harmônico forçado, e na escolha posterior o problema não perturbado é um oscilador linear amortecido e forçado.
Electromechanical oscillators can be mathematically modeled by a Du±ng equation or a Van der Pol equation, even if they are nanometric systems. In this work we studied a forced oscillator having nonlinear damping, that is represented by a Du±ng - Van der Pol equation. In general, it is not easy to get the exact analytical solution for this equation, then the analysis is done using the perturbation theory to get an approximate analytical solution. For this reason we considered that certain parameters of the problem are small parameters and we obtain the solution in the form of straightforward expansion. Due to the fact that natural frequency of the dynamic system depends on the small parameter, this expansion is not uniform, i.e. presents secular terms (Poisson terms) and also small-divisors. These inconveniences are eliminated using the method of multiple scales and the aver- aging method. Initially the small parameters are chosen so that the unperturbed problem is reduced to a forced harmonic oscillator, and in the subsequent choice the unperturbed is a forced oscillator having linear damping.

Gyroscopes are inertial sensors that measure the rate or angle of rotation. One of the most promising technologies for reaching a high-performance MEMS gyroscope has been development of the micro-hemispherical shell resonator. (μHSR) This thesis presents the electronic control and read-out interface that has been developed to turn the μHSR into a fully functional micro-hemispherical resonating gyroscope (μHRG) capable of measuring the rate of rotation. First, the μHSR was characterized, which both enabled the design of the interface and led to new insights into the linearity and feed-through characteristics of the μHSR. Then a detailed analysis of the rate mode interface including calculations and simulations was performed. This interface was then implemented on custom printed circuit boards for both the analog front-end and analog back-end, along with a custom on-board vacuum chamber and chassis to house the μHSR and interface electronics. Finally the performance of the rate mode gyroscope interface was characterized, showing a linear scale factor of 8.57 mv/deg/s, an angle random walk (ARW) of 34 deg/sqrt(hr) and a bias instability of 330 deg/hr.

L'objet de ce memoire est l'etude d'un systeme dynamique non lineaire de type duffing, avec une excitation parametrique, periodique dans le temps, sur la force de rappel, et une force exterieure periodique de pulsation differente de celle de la force parametrique. Des solutions periodiques (synchronisees quand ces solutions sont asymptotiquement stables), dans le cas d'un rapport frequentiel rationnel (pulsation exterieure/pulsation parametrique), ont d'abord ete construites analytiquement. Le meme probleme a ete ensuite aborde dans le cas d'un rapport frequentiel irrationnel, avec determination des solutions quasi-periodiques par des methodes analytiques et numeriques. La seconde partie met en evidence une structure feuilletee dans un plan parametrique de coordonnees (pulsation, amplitude de la force parametrique) dans le cas du rapport frequentiel un. Ceci a permis de faire apparaitre de nouveaux types de communications entre feuillets du plan parametrique, par rapport a ceux decrits dans chaotic dynamic (c. Mira world scientific ed. )

Wave propagation in elastic continuum is a subject of interest in many fields of engineering and has been explored for several decades from theoretical, computational and experimental perspective. In this context, a one/two-dimensional elastic continuum is more colloquially called a wave guide since they minimize energy loss by restricting the wave propagation in a specific direction/plane. These one-dimensional elastic wave guides can be dispersion free, for example, an elastic bar propagating longitudinal waves, or can exhibit dispersion, for example, flexural waves in Euler-Bernoulli or Timoshenko beams. However, in practical engineering structures, such wave guides invariably interact with structural components whose spatial scales are much smaller than the elastic continuum and the wavelength of the wave phenomena that they encounter. Such structural components can be modelled as discrete elements by considering point masses, stiffness and damping. The wave propagation characteristics when elastic wave guides interact with linear discrete elements are well known. The present work primarily dwells on the effect of weakly/strongly nonlinear discrete elements on the wave propagation characteristics in nondispersive elastic wave guides. Since, in general, closed form analytical solutions are seldom available for weakly/strongly/essentially nonlinear dynamical systems, in this thesis we propose a systematic approach based on classical perturbation techniques like Method of Multiple Scales (MMS) to find out the solutions. Firstly, we investigate an elastic continuum coupled to a Duffing oscillator with cubic stiffness nonlinearity through a linear spring. The elastic continuum is modelled as a simple, onedimensional bar. The propagation characteristics, the response of the bar and the oscillator motion are found out analytically by using a combination of Multiple Time Scales Analysis and Harmonic Balance method. The excitation pulse is considered to be a sinusoidal displacement function, imposed on the free end of the bar. The closed-form solutions for the responses are determined at different levels of approximation for the cases of primary, superharmonic and subharmonic resonance and also for a general non-resonant case. These closed-form solutions can be used as a reference to check solutions obtained by other standard procedures. Secondly, we study a system comprising of two simple, one-dimensional bars coupled to each other by means of a weakly damped, weakly nonlinear oscillator with cubic stiffness nonlinearity. The input condition is the same as in the previous case, i.e. a sinusoidal pulse applied to one of the bars as a displacement boundary condition. The closed-form solutions for the responses are found out for a general non-resonant case. The procedure for determining the responses analytically in the cases of primary and secondary resonance is also discussed. The same methodology is used to determine the responses. These closed-form solutions can be useful while analysing real-world systems like a two-stage rocket, where the stages can be modelled as simple, elastic bars and the connection between the two systems can be modelled as a nonlinear oscillator, such as the Duffing one or the Van der Pol one. Thirdly, a system comprising of a simple, one-dimensional bar attached with a snap-through truss is studied. The input condition is the same as for the previous systems, i.e. a sinusoidal pulse imposed on the free end of the bar as a displacement boundary condition. This system is essentially nonlinear and it is not possible to find a closed-form solution using standard procedures. However, solutions are provided for small amplitudes of the excitation pulse, taking some relevant assumptions. The same method based on Multiple Time Scales Analysis is used to determine the responses. Apart from providing analytical solutions, the problem is also solved numerically for the same system parameters using a Finite Difference algorithm. The close similarity between the analytical solutions and the numerical solutions is provided. A stability analysis is also done for the snap-through oscillator for sufficiently large amplitudes of the excitation pulse and the stable and unstable regions are shown. This stability diagram can be useful while determining important system input parameters like excitation amplitude and excitation frequency for which the oscillator will snap from one stable equilibrium position to another.

碩士
國立臺灣大學
土木工程學研究所
103
In the optimal control theory, the Hamiltonian formulation is a famous one which is convenient to find an optimally designed control force. However, when the performance index is a complicated function of control force, the Hamiltonian method is not easy to find the optimal closed-form solution, because one may encounter a two-point boundary value problem of nonlinear differential algebraic equations (DAEs). In this thesis, we address this issue via an novel approach, of which the optimal vibration control problem of Duffing oscillator is recast into a two-point nonlinear DAEs. We develop the corresponding and shooting methods, as well as a Lie-group differential algebraic equations (LGDAE) method to numerically solve the optimal control problems of nonlinear Duffing oscillators. Seven examples of a single Duffing oscillator and one coupled Duffing oscillators are used to test the performance of the present method.

碩士
國立臺灣大學
土木工程學研究所
100
In order to improve the dynamic characteristics, the stiffness and the damping of civil engineering structures to achieve a certain energy dissipation effect, and the active structural control is exerted additional force by the control elements to the traditional structural system. The active structural control system consists of three core components: the sensor, the control law and actuator. Sensors are used to measure the dynamic response which layout in structures and controllers of the civil engineering. According to the response of the structure and controllers, decide the timing, size and the direction of control being imposed on the structure. The actuator is developed the institutions by the control force and which is applied to structures of civil engineering through a series of dynamical systems. In the study of optimal control theory for nonlinear structures, one often encounters two-point boundary-value problems (TPBVPs). In this study, the numerical solution for two-point boundary value problem is the Lie group shooting method (LGSM), and then with the fourth-order Runge-Kutta method (RK4). The LGSM is a powerful technique to search the unknown initial conditions. These methods are gradually derived based on the closure property of the group, the Lie group property and the length preserving property in GPS, some simple mathematical derivation, the mid-point rule. And it will be used to this numerical solution of the linear optimal control, the single degree nonlinear Duffing osillator, as well as two degrees nonlinear Duffing osillator. In this thesis, we use programming language FORTRAN for the numerical analysis and plot the numerical results by the GRAPHER. Finally, we want to apply this method on the development of the civil engineering in the future.