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With the extreme miniaturization of NEMS, the role of nonlinear dynamics has become increasingly important -- even when the dynamics are driven by the Brownian force. This nonlinearity has imposed a mechanical noise floor on the linear frequency measurements made in sensing applications. Given that NEMS also become more sensitive as they become smaller, this floor has resulted in a complex interplay between the nonlinear regime and the linear sensitivity required to make continued advancements in exercising ultra-sensitive measurements. Recently, this has led to efforts to more accurately characterize the edge of the linear regime. Inside of the nonlinear regime, there are also ongoing fundamental studies in theory and experiment to partially characterize the nonlinear behavior of NEMS. Theoretically, these systems are frequently studied by decomposing the nonlinear continuous system into one or more nonlinear oscillators. However, in many of these works, the nonlinear spring constants are estimated at the lowest order. As such, there is a clear need to more accurately characterize and scale the nonlinear coefficients for NEMS. This work considers a long and slender NEMS resonator in the form of a doubly-clamped beam in tension. Using nonlinear Euler-Bernoulli beam theory, the geometric nonlinearity due to the stretching of the neutral axis is considered. We extensively explore simplifications in using a Galerkin discretization of the continuous system, where a single mode's dynamics are described as a damped, Duffing oscillator. We examine limitations of current approaches and find that using a tensioned and doubly-clamped mode shape for the trial function more accurately predicts experiment. Additionally, we find that doubly-clamped beams of finite tension may have their boundary conditions modified to that of a hinged-hinged beam in tension with little to no loss of generality. This modification allows for closed-form scaling of the critical amplitude and dynamic range at arbitrary mode number and tension. We extend this approach to scale the relative influence of bending, tension, and nonlinearity with increasing mode number n, finding that bending and nonlinear influences quickly outgrow the contributions of the intrinsic tension. Where applicable, these results are compared with experiment, and we obtain good agreement. To validate the approximate Galerkin formulations, we develop a finite element method to calculate the nonlinear coefficients of symmetric NEMS resonators. Unlike previous works, the present formulation may include all nonlinearity due to geometry, and the nonlinear amplitude-frequency backbone described by the Duffing oscillator is found as an excellent approximation for large amplitude beams. For beams of zero intrinsic tension, the finite element method obtains excellent agreement with the literature. For beams of finite tension and varying mode number, we find the error from the Galerkin discretization is small (≈5%). In addition, we theoretically explore the stochastic dynamics of a Duffing oscillator driven nonlinearly by the Brownian force. To access this regime experimentally with current nanomechanical systems, we motivate an experimental "synthetic noise" to approximate the Brownian force in the proximity of a single mode. As a measure of drive magnitude, we vary an effective temperature to explore the linear and nonlinear stochastic dynamics of a doubly-clamped nanoresonator. Using similitude and the Fluctuation-Dissipation theorem, we show that varying the effective temperature of the synthetic noise offers a window into the fundamental limits of thermally-driven nonlinearities. We compare theory, numerics, and experiment where applicable, obtaining good agreement for both limits of frequency shifts in the weakly nonlinear case. This research was supported by the National Science Foundation, grant number CMMI-2001559, and portions of the computations were conducted using the resources of Virginia Tech's Advanced Research Computing (ARC) center. / Master of Science / Nanoelectromechanical systems (NEMS) are nanoscale mechanical structures that convert physical stimuli (force, mass, acceleration, charge, etc.) to measurable electrical signals. Due to the extremely small size of NEMS, they offer an unprecedented level of sensitivity in a variety of measurement applications. However, as NEMS become smaller, the response of these mechanical structures begin to exhibit nonlinear behaviors. Said otherwise, proportional inputs (such as drive strength) do not result in proportional outputs. These nonlinear behaviors include a variety of undesired effects, such as multi-valued unstable/stable solutions and a noisy resonant frequency. In this work, we study a NEMS resonator in the form of a doubly-clamped beam, and we consider the stretching of the mid-plane as the nonlinearity. Here, the stretching of the mid-plane (an axial strain) is nonlinearly dependent on the amplitude of vibration, inducing a nonlinear tension force. With this model, it is typical to represent a singular nonlinear vibrational mode as a simple harmonic oscillator with an additional cubic term. In order to better characterize the edge of the nonlinear regime, the relative strength of the cubic term must be known. We thoroughly explore existing and new simplifications to obtain the nonlinear coefficient for the cubic term, demonstrating two possible approaches for better accuracy in beams of varying tension and mode number. These simplifications are validated by comparing with the present finite element method to determine the nonlinearity in symmetric NEMS resonators. Using these new insights, theory and numerics are used to explore the behavior of a doubly-clamped beam in a stochastic (random) force field. This force field is tailored to represent the collisions of surrounding molecules at the nanoscale, allowing exploration of nonlinear behavior at its fundamental limits. Where applicable, theory and numerics are compared to experiment, and we obtain good agreement. This research was supported by the National Science Foundation, grant number CMMI-2001559, and portions of the computations were conducted using the resources of Virginia Tech's Advanced Research Computing (ARC) center.

Orientador: Antonio Roberto Balbo / Coorientadora: Célia Aparecida dos Reis / Banca: Hassan Costa Arbex / Banca: Tatiana Miguel Rodrigues de Souza / Resumo: Atualmente a produção e o consumo de energia têm suma importância para a realização de diversas atividades humanas, o que implica em uma grande quantidade a ser produzida para atender a demanda, que, com o passar do tempo, aumenta cada vez mais. Esse aumento do consumo faz com que pesquisas sejam desenvolvidas com relação a sistemas de captação ou colheita de energia, que são chamados de "energy harvesting". Estas possibilitam sua conversão em energia elétrica, que pode ser diretamente utilizada ou armazenada para uso posterior. Um sistema de colheita de energia captada através de material piezoelétrico é aquele que colhe a energia gerada por vibração mecânica. Neste trabalho fez-se um estudo, em termos do plano de fase e estabilidade assintótica, de um modelo de vibração massa-mola-amortecedor associada a sistemas de colheita de energia, o qual é baseado em uma equação de Duffing. O modelo é formulado através de um sistema de equações diferenciais ordinárias não lineares e, para análise de suas soluções de equilíbrio, foi utilizado o método de Cardano - Tartaglia. Testes com um problema real são realizados apresentando as soluções obtidas pelo método, bem como uma simulação numérica do retrato de fase de dois pontos críticos destes, utilizando o software MatLab. / Abstract: Currently the production and consumption of energy is very important for the performance of various human activities, which entails a great demand to be met and, with the passage of time, increases more and more. This increase in energy consumption makes researches are developed in relation to energy harvesting systems called energy harvesting. These enable their conversion into electrical energy, which can be directly used or stored for later use. A system of energy harvesting captured by piezoelectric material is one that harvests the energy generated by mechanical vibration. In this work is investigated a mass-spring-damper vibration model associated with an energy harvesting system, which is based in a Duffing's equation. This model is formulated through a system of ordinary nonlinear differential equations and, to analyze of their equilibrium solutions, the Cardano - Tartaglia's method was used. Tests with an actual problem are done, presenting the solutions obtained by the method, as well as a numerical simulation a phase portrait analysis of two critical points of this, using the MatLab software. / Mestre

Thesis (M.S.)--Rochester Institute of Technology, 2005. / Typescript. Includes bibliographical references (leaves 144-146).

Σε αυτή τη διπλωματική εργασία παρουσιάζονται μερικές μέθοδοι ομαλών διαταραχών και η εφαρμογή τους στις "διάσημες" μη γραμμικές συνήθεις διαφορικές εξισώσεις Duffing, Castor και van der Pol.Οι μέθοδοι διαταραχών μπορούν να χρησιμοποιηθούν για να βρούμε προσεγγιστικές λύσεις σε διαφορικές εξισώσεις οι οποίες είναι μη γραμμικές και μια ακριβή λύση δεν μπορεί να βρεθεί. Η μέθοδος της θεωρίας διαταραχών γίνεται με σεβασμό ως προς μια μικρή παράμετρο ε, 0<ε<<1. Οι προσεγγιστικές αυτές μέθοδοι προυποθέτουν ότι γνωρίζουμε πλήρως τη λύση του προβλήματος για την τιμή ε=0 μιας παραμέτρου και επιχειρούμε να εκφράσουμε τη γενική λύση, για 0<ε<<1, υπό μορφή σειράς όρων του ε,ε^2,...κ.τ.λ. / In this thesis we present some regular perturbation methods and their applications to the famous non-linear ordinary differential equations of Duffing, Castor and van der Pol. The method of perturbations can be used to develop approximate solutions to differential equations, which have nonlinearities or variable coefficients so that an exact solution cannot be constructed. The method of perturbation expansion is carried out with respect to a small parameter ε,0<ε<<1. These approximate methods require that we know the solution of the problem for ε=0 and try to expess the general solution, for 0<ε<<1, as a series of terms ε,ε^2,...e.t.c.

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.

Ordinary differential equations of various types appear in the mathematical modelling in mechanics. Differential equations obtained are usually rather complicated nonlinear equations. However, using suitable approximations of nonlinearities, one can derive simple equations that are either well known or can be studied analytically. An example of such "approximative" equation is the so-called Duffing equation. Hence, the question on the existence of a periodic solution to the Duffing equation is closely related to the existence of periodic vibrations of the corresponding nonlinear oscillator.

The objective of this dissertation is to develop a mechanical model of a nonlinear piezoelectric energy harvesting system induced by Duffing oscillator and predict the periodic motions of such a nonlinear dynamical system under different excitation frequency. In this dissertation, analytical distributed-parameter electromechanical modeling of a piezoelectric energy harvester will be presented. The electromechanically coupled circuit equation excited by infinitely many vibration modes is derived. The governing electromechanical equations are reduced to ordinary differential equations in modal coordinates and eventually an infinite set of algebraic equations is obtained for the complex modal vibration response and the complex voltage response of the energy harvester beam. One single vibration mode is chosen and discussed. The periodic motions are obtained through an implicit mapping method with high accuracy, stability and bifurcations of periodic motions are determined by the eigenvalue analysis. Frequency-amplitude characteristics of periodic motions are achieved by the Fourier transform

[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.

The constantly increasing volume of information in modern society demands a better understanding of the physics and modeling of optical phenomena, and in particular, optical waveguides which are the central component of information systems. Two ways of advancing this physics are to push current technologies into new regimes of operation, and to study novel materials which offer superior properties for practical applications. This dissertation considers two problems, each addressing the above-mentioned demands. The first relates to the influence of high-order nonlinear effects on pulse collisions in existing high-speed communication systems. The second part is a study of pulse dynamics in a novel nanocomposite medium which offers great potential for both optical waveguide physics and applications. The nanocomposite consists of metallic nanoparticles embedded in a host medium. Under resonance conditions, the optical field excites plasmonic oscillations in the nanoparticles, which induce a strong nonlinear response.Analytical and computational tools are used to study these problems. In the first case, a double perturbation method, in which the small parameters are the reciprocal of the relative frequency of the colliding solitons and the coefficient of quintic nonlinearity, reveals that the leading order effects on collisions are radiation emission and phase shift of the colliding solitons. The analytical results are shown to agree with numerics. For the case of pulse dynamics in nanocomposite waveguides, the resonant interaction of the optical field and material excitation is studied in a slowly-varying envelope approximation, resulting in a system of partial differential equations. A family of solitary wave solutions representing the phenomenon of self-induced transparency are derived. Stability analysis reveals the solitary waves are conditionally stable, depending on the sign of the perturbation parameter. A characterization of two-pulse interaction indicates high sensitivity to relative phase, and collision dynamics vary from highly elastic to the extreme case where one wave is immediately destroyed by the collision, depositing its energy into a localized hotspot of material excitation. This last scenario represents a novel mechanism for &quot;stopping light&quot;.