A pendulum with its supporting point vibrating in both the x and the y direction is analyzed. Numerical simulation by computer is used to analyze the motion of the pendulum. Chaotic motion of the system is observed. Threshold values for chaos are obtained by simulation. The Lyapunov exponent and the fast Fourier transform ( FFT ) are used as the criteria to determine if the system is chaotic. Two predictive theoretical criteria, the Melnikov criterion and a period-doubling criterion, are then applied to the system. The results obtained by simulation and by theoretical criteria are shown to be in good agreement. A brute-force approach is used to supplement the results. It is found that the motion of this simple driven pendulum will have very complicated behavior. Multiple attractors can be shown to coexist. / Master of Science

Two prominent continuum mechanics based incremental nonlinear finite element formulations are reviewed. An introduction to different material response measures suitable for nonlinear analysis, in addition to an overview of the Total and Updated Lagrangian reference frames, serve as the starting point for this review. The two nonlinear formulations are specialized for use with a geometrically nonlinear plane truss finite element. The truss formulations are then implemented into separate geometrically nonlinear finite element codes. Numerical comparisons of five test structures are carried out using ABAQUS and both programs. ABAQUS serves as the bench-mark by which the solution accuracy of the two programs is judged. / Master of Science / incomplete_metadata

Microelectromechanical Systems are ubiquitous in modern technology, with applications ranging from accelerometers in smartphones to ultra-high precision motion stages used for atomically-precise positioning. With the appropriate selection of materials and device design, MEMS resonators with ultra-high quality factors can be fabricated at minimal cost. As the sizes of such resonators decrease, however, their mechanical, electrical, and material properties can no longer be treated as linear, as can be done for larger-scale devices. Unfortunately, adding nonlinear effects to a system changes its dynamics from exactly-solvable to only solvable in specific cases, if at all. Despite (and because of) these added complications, nonlinear effects open up an entirely new world of behaviors that can be measured or taken advantage of to create even more advanced technologies. In our resonators, oscillations are induced and measured using aluminum nitride transducers. I used this mechanism for several separate highly-sensitive experiments. In the first, I demonstrate the incredible sensitivity of these resonators by actuating a mechanical resonant mode using only the force generated by the radiation pressure of a laser at room temperature. In the following three experiments, which use similar mechanisms, I demonstrate information transfer and force measurements by taking advantage of the nonlinear behavior of the resonators. When nonlinear resonators are strongly driven, they exhibit sum and difference frequency generation, in which a large carrier signal can be mixed with a much smaller modulation to produce signals at sum and difference frequencies of the two signals. These sum and difference signals are used to detect information encoded in the modulation signal using optical radiation pressure and acoustic pressure waves. Finally, in my experiments, I probe the nonlinear nature of the piezoelectric material rather than take advantage of the nonlinear resonator behavior. The relative sizes of the linear and nonlinear portions of the piezoelectric constant can be determined because the force applied to the resonator by a transducer is independent of the dielectric constant. This method allowed me to quantify the nonlinear constants.

The recent demonstration of strong interactions between optical force and mechanical motion of an optomechanical structure has led to the triumphant result of mechanical ground-state cooling, where the quantum nature of a macroscopic object is revealed. Another intriguing demonstration of quantum physics on a macroscopic level is the measurement of the Casimir force which is a manifestation of the zero- point energy. An interesting aspect of the Casimir effect is that the anharmonicity of the Casimir potential becomes significant when the separation of microscale objects is in the sub-100nm regime. This regime is readily accessible by many of the realized gradient-force-based optomechanical structures. Hence, a new avenue of probing the Casimir effect on-chip all-optically has become available. We propose an integrated optomechanical platform, consisting of a suspended photonic crystal membrane evanescently coupled with a silicon-on-insulator substrate, for (i) measuring the Casimir force gradient and (ii) counteracting the attractive force by exerting a resonantly enhanced repulsive optical gradient force. This thesis first presents the full characterization of the optomechanical properties of the system in vacuo. The interplay of the optical gradient force (optomechanical coupling strength \(g_{om}/2\pi=- 66GHz/nm\)) and the photothermal force manifested in the optical spring effect and dynamic backaction is elucidated. Static displacement by the repulsive force of 1nm/mW is also demonstrated. In the second part of the thesis, the nonlinear mechanical signatures upon a strong coherent drive are reported. By resonantly driving the photonic crystal membrane with a piezo-actuator and an optical gradient force, we observed mechanical frequency mixing, mechanical bistability and non-trivial interactions of the Brownian peak with the driving signal. Finally we present our recent progress in establishing electro- static control of individual photonic crystal membranes to reduce and calibrate the electrostatic artifact which plagues Casimir measurements. The results discussed in this thesis point towards an auspicious future of a complete realization of a Casimir optomechanical structure and novel applications with nonlinearity afforded by the Casimir force and the optical gradient force. / Engineering and Applied Sciences

Orientadores: Janito Vaqueiro Ferreira, Alberto Luiz Serpa / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecânica / Made available in DSpace on 2018-08-16T11:31:00Z (GMT). No. of bitstreams: 1 Martins_PauloAugustodeOliveira_M.pdf: 56662402 bytes, checksum: d0d469664498e80bed0d21cef0f7f7ac (MD5) Previous issue date: 2005 / Resumo: O objetivo deste trabalho é estudar a aplicação do método "arc-length" em funções de resposta em frequência não-lineares (RFs), para se obter principalmente regiões instáveis do caminho de equilíbrio da curva de receptância ?(?). Para isto, é necessário conhecer os vários procedimentos de continuação existentes com o intuito de se escolher o melhor método que executasse este objetivo. O procedimento "arc-length" tem que ser usado em conjunto com algum algoritmo para solução de sistema de equações não-lineares. Logo, é utilizado o procedimento de Newton-Raphson, devido a sua razão de convergência e confiabilidade. A não-linearidade assumida neste trabalho é obtida através de funções descritivas, sendo que, no modelo matemático, a não-linearidade pode ser acrescentada tanto na matriz rigidez quanto na matriz de amortecimento, e sua posição irá depender do tipo de função descritiva que se está utilizando. Para verificar a funcionalidade do método, implementou-se um procedimento gráfico mostrando em cada passo de iteração até sua convergência. Na equação do resíduo para o caso de impedância ou receptância, o parâmetro de frequência (?) é modificado para (??), pois (?) é o fator que relaxa a frequência. Já, os resultados deste trabalho obtidos com o método "arc-length" são apresentados graficamente, utilizando vários modelos, os quais são mostrados os casos lineares e não-lineares. Estes gráficos são comentados e analisados, e são mostradas as dificuldades encontradas. Os resultados deste trabalho são de grande importância para a validação do método proposto e também para trabalhos futuros, no qual este método pode ser empregado junto a outros que determinam soluções periódicas, no qual o "arc-length" pode ser utilizado para obter a estabilidade da resposta sistema / Abstract: The objective of this work is to study the applicability of the arc-length method in nonlinear frequency response (RF), mainly to obtain unstable regions of the receptance curve ?(?) following the equilibrium way. To do this, it is necessary to know the several path-following procedures with the intuit of choosing the best method to do this task. The arc-length procedure must be used together with some optimization algorithm, so that this solves the nonlinear system of equations. Therefore, the procedure used is Newton-Raphson, due to its convergence ratio and reliability, although others Newton methods can be used with this purpose. The nonlinearity assumed in this work is obtained through describing functions, this nonlinearity is increased in the stiffness matrix and damping matrix of the mathematical model. To verify the functionality of the method, a graphic procedure was implemented to show the convergence at each iteration step. In the equation of residue, for the case of impedance or receptance, the parameter of frequency (?) is modified for (??), because (?) is the factor that relaxes the frequency. The results obtained with the method are represented graphically using several models, which are treated for linear and nonlinear cases. These graphs are commented properly, analyzed and the difficulties are showed presented. The results of this work are of great importance to validate the proposed method and also for future works, in which this method can be used in conjunction to others that determine periodic solutions, in which the arc-length can be used to obtain the stability of the system / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

Orientador: Douglas Eduardo Zampieri / Dissertação (mestrado) - Universidade Estadual de Campinas, Faculdade de Engenharia Mecanica / Made available in DSpace on 2018-08-05T06:44:34Z (GMT). No. of bitstreams: 1 Motta_DanieldaSilva_M.pdf: 1219534 bytes, checksum: 63a8c2a0d7734b3fc5ba39e3e5386e99 (MD5) Previous issue date: 2005 / Resumo: Este trabalho analisa e compara sistemas semi-ativos de controle de suspensões veiculares com um sistema passivo não-linear. O modelo matemático do sistema de suspensão, detalhado nesta dissertação, tem sete graus de liberdade, de modo a representar um veículo completo com suas quatro rodas. As duas leis de controle semi-ativas utilizadas foram baseadas na teoria do skyhook, sendo que em uma destas leis, o coeficiente de amortecimento varia continuamente (semi-ativa CVD) e, na outra, este coeficiente pode assumir um valor máximo ou um valor mínimo (semi-ativa ON-OFF). Para a análise dos sistemas foi desenvolvido um programa usando a ferramenta computacional SIMULINK, programa que permite a avaliação de diversas situações de sistemas veiculares. Os resultados evidenciam uma alternância de melhor desempenho entre os sistemas passivo não linear, semi-ativo ON-OFF e semi-ativo CVD, sendo que para uma avaliação mais detalhada do desempenho dos sistemas semi-ativos se faz necessária uma otimização dos parâmetros utilizados nas leis de controle. Neste trabalho fica bastante evidente a importância de se considerar o comportamento não linear do conjunto amortecedor e mola no sistema passivo / Abstract: This work describes the analysis and comparison of a vehicle suspension semi-active controlled to a non linear passive system. The mathematic model of a suspension system, detailed in this work, has seven degrees of freedom in order to represent a full vehicle system. The two semi-active control laws used in this work are based on the skyhook theory. In the first one the damping coefficient is continuously variable (semi-active CVD) and, in the second one the damping coefficient can assume a maximum or a minimum value (semi-active ON-OFF). To analyze the systems, it was developed a program using the SIMULINK computational tool. This program can evaluate different situations of vehicle suspension systems. The results show that the non linear passive system, semi-active ON-OFF and semi-active CVD alternate the better performance. To have a better understanding of semi-active performance, an optimization of the parameters used in the control laws is needed. This work also explains the importance of considering the non linear behavior of passive systems elements / Mestrado / Mecanica dos Sólidos e Projeto Mecanico / Mestre em Engenharia Mecânica

Orientador: Lúcio Tunes dos Santos / Dissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica / Made available in DSpace on 2018-08-18T19:22:30Z (GMT). No. of bitstreams: 1 Gower_ArturLewis_M.pdf: 2962671 bytes, checksum: 60eacb0b1c81be752469204420ce44d3 (MD5) Previous issue date: 2011 / Resumo: Nesta tese expomos uma investigação qualitativa e quantitativa sobre a propagação de ondas com simetria radial em materiais elásticos pré-esticados, isotrópicos e homogêneos. Esse tipo de modelo se aproxima a uma explosão em um sólido como, por exemplo, uma fonte em sísmica. Além disso, o comportamento qualitativo da dinâmica radial ajuda o melhor entendimento do caso tridimensional. O trabalho tem um enfoque mais específico na modelagem, condições de choques, a dinâmica próximo do centro do corpo e análise da solução analítica do material linearizado / Abstract: This thesis is an exploration into the qualitative and quantitative behaviour of wave propagation with radial symmetry in materials which are prestressed elastic, isotropic and homogeneous. This type of model approximates an explosion in a solid, such as a seismic source. Also, an understanding of the radial dynamics can bring insight and contribute to our comprehension of the general 3D case. This thesis focuses on modelling, shock conditions, the dynamics close to the centre of the body and examining the analytical solution of an linearised material / Mestrado / Matematica Aplicada / Mestre em Matemática Aplicada

Orientador: Francisco de Assis Magalhães Gomes Neto / Tese (doutorado )- Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica / Made available in DSpace on 2018-08-22T00:37:54Z (GMT). No. of bitstreams: 1 Senne_ThadeuAlves_D.pdf: 4979548 bytes, checksum: 0317347b212e7649dd09e5d5388e618e (MD5) Previous issue date: 2013 / Resumo: Nos últimos anos, a otimização topológica vem sendo amplamente adotada nas indústrias automotiva e aeroespacial, e no projeto de um tipo especial de estruturas, denominado mecanismo flexível. Grande parte dos trabalhos na área de otimização topológica considera que a estrutura possui uma relação linear entre deformações e deslocamentos, ou seja, supõe-se que os deslocamentos sofridos pela estrutura sejam pequenos. Todavia, para algumas estruturas, essa hipótese não é válida, sendo necessário supor que os deslocamentos são grandes, o que implica numa relação não linear entre deformações e deslocamentos. Nesse caso, dizemos que a estrutura está sob não linearidade geométrica. O objetivo desta tese de doutorado é a obtenção da topologia ótima de estruturas e de mecanismos flexíveis sob não linearidade geométrica através um novo algoritmo de otimização, denominado Programação Linear por Partes Sequencial (PLPS). Este método consiste na resolução de subproblemas de programação linear por partes convexas, onde são introduzidas informações sobre a diagonal da matriz Hessiana da função objetivo. Para acelerar o algoritmo, tais subproblemas são convertidos em problemas de programação linear. Provamos que a PLPS é globalmente convergente a pontos estacionários. Além disso, nossos experimentos numéricos realizados com estruturas e mecanismos flexíveis sujeitos a grandes deslocamentos mostram que a PLPS é eficiente e robusta / Abstract: In the last years, topology optimization has been broadly applied in the automotive and aerospatial industries, and to a special kind of structure, named compliant mechanism. Most papers on topology optimization consider that the structure has a linear relation between strains and displacements, meaning that the displacements of the structure are small. However, for some structures this assumption is not valid, and it is necessary to suppose that the displacements are large, implying in a nonlinear relation between strains and displacements. In this case, we say that the structure is under geometrical nonlinearity. The objective of this doctoral thesis is to obtain the optimum topology of structures and compliant mechanisms under geometrical nonlinearity through a new optimization algorithm, named Sequential Piecewise Linear Programming (SPLP). This method consists in the solution of convex piecewise linear programming subproblems that contain information about the diagonal of the Hessian matrix of the objective function. To speed up the algorithm, these subproblems are converted into linear programming ones. We prove that the SPLP is globally convergent to stationary points. Besides, our numerical experiments with structures and compliant mechanisms under large displacements also show that the SPLP is efficient and robust / Doutorado / Matematica Aplicada - Otimização / Doutor em Matemática Aplicada

Les nonlinéarités dans les systèmes nanomécaniques peuvent provenir d’effets dispersif ou dissipatif et ce dans divers systèmes (résistifs, inductifs et capacitifs). Au-delà de l’intérêt fondamental pour tester la réponse dynamique d’un système non-linéaire à plusieurs dégrées de libertés, les nonlinéarités de tels systèmes ouvre la voie vers des capteurs nanomécanique et le traitement du signal. Le résonateur nanomécanique dont la réponse nonlinéaire est étudié, est une membrane suspendue à cristal photonique bidimensionnel utilisée comme miroir déformable. Sa faible masse et sa haute réflectivité en font un candidat idéal pour l’électro-opto-mécanique. L’actuation d’une telle membrane dans le domaine fréquentiel du MHz est rendu possible par des électrodes inter-digitées en dessous de la membrane assurant ainsi l’uniformité de la force d’actuation sur cette dernière. La fabrication de telles structures est basée sur l’intégration hétérogène 3D.La force électrostatique qui s’applique sur la membrane induit des non-linéarités mécaniques avec notamment un effet bistable, des résonances superharmoniques et des résonances stochastiques.La membrane est mise en mouvement par un potentiel électrique V(t) = Vdc + Vac cos(w.t), où Vdc est l’amplitude du courant continu, Vac l’amplitude du courant alternatif à la fréquence d’excitation w;. Le système se comporte alors comme une capacité de sorte que la force qui s’applique sur la membrane varie de manière quadratique avec la tension appliquée. Selon la tension DC ou AC, le comportement de la structure est différent. L’augmentation de la tension DC induit une augmentation de la tension de polarisation sur le matériau qui par conséquent modifie la fréquence propre de la membrane. Tandis que l’augmentation de la tension AC cause l’augmentation de l’amplitude des oscillations de la membrane pouvant aller jusqu’à atteindre le régime non-linéaire.Dans une première série de mesure, la membrane est excitée à la résonance avec une fréquence w; égale à la fréquence du mode mécanique fondamental wm. A partir de la réponse fréquentielle du système, il est possible d’identifier différents modes mécaniques de la membrane sondé optiquement. Pour une excitation plus importante, il est possible d’observer des effets de bistabilité mécanique. Ces non-linéarités sont dues à l’élongation au niveau des points d’ancrage de la membrane.La méthode la plus commune pour agir sur la membrane est l’excitation proche de la résonance fondamentale. Cependant la technique de la résonance superharmonique peut également être utilisée. Cela consiste à appliquer la fréquence d’excitation w; à une fréquence égale à wm/n où n est un entier. La possibilité d’utiliser cette technique est fortement dépendante des nonlinéarités présentes dans le système. Ainsi, l’existence d’une résonance super harmonique à wm/n résulte de la présence d’une nonlinéarité d’ordre n. Dans une seconde série de mesure, un balayage des résonances superharmoniques en fonction de la fréquence et de la puissance a été réalisé en modulant la tension à la fréquence wm/n et en enregistrant la réponse de la membrane autour de wm. Il a été ainsi possible d’observer des résonances superharmoniques allant de n=2 jusqu’à 8. Il a également été possible d’obtenir l’évolution de la phase le long des résonances et ce pour toutes celles observées.Dans une dernière série de mesure, nous utilisons la nonlinéarité présente pour observer des effets de résonance stochastique. L’idée est d’amplifier un signal de faible amplitude (basse fréquence) en injectant du bruit (haute fréquence) dans le système nonlinéaire. Dans le cas de notre système, nous avons été capables d’observer des résonances stochastiques à la fois en amplitude et en phase. Une étude comparative de ces deux régimes est détaillée. Le fait de pouvoir observer la résonance stochastique en phase peut permettre d’envisager la réalisation de communications codées en phase. / Nonlinearities in nanomechanical systems can arise from various sources such as spring and damping mechanisms and resistive, inductive, and capacitive circuit elements. Beyond fundamental interests for testing the dynamical response of discrete nonlinear systems with many degrees of freedom, non-linearities in nanomechanical devices, open new routes for nanomechanical sensing, and signal processing.The nonlinear response of a nanomechanical resonator consisting in a suspended photonic crystal membrane acting as a deformable mirror has been investigated. The low-mass and high reflectivity of suspended membranes pierced by a two-dimensional photonic crystal, makes them good candidates as electro-optomechanical resonator. Actuation of the membrane motion in the MHz frequency range is achieved via interdigitated electrodes placed underneath the membrane. The choice of these electrodes is due to the fact they are able to uniformly actuate these membranes. The processing of such platforms relies on 3D-heterogenous integration process.The applied electrostatic force induces mechanical non-linearities, in particular bistability, superharmonic resonances and stochastic resonance.The membrane is actuated by an electric load V(t) = Vdc + Vac cos(w.t), where Vdc is the DC polarization voltage, Vac the amplitude of the applied AC voltage, and w; the excitation frequency. The system acts as a capacitive system and thus the force applied on the membrane varies as a quadratic function of the applied voltage. Application of either DC or AC voltages can have different implications. Increasing the DC voltage increases the polarizing voltage on the material which in turn causes modulation of the eigenfrequency of the membranes. While an increase in the periodic AC voltage causes the membrane to oscillate more, pushing the system towards non-linear regime.In a first series of experiments, the membrane is actuated resonantly, with an excitation frequency w; equal to the fundamental mechanical modes frequency wm. From the frequency response spectra of the system it was possible to identify different mechanical modes of these membranes via optical measurements. For increased actuation voltages, bistability effects are observed with two different behaviors (spring hardening or softening). The mechanical nonlinearities due to stretching at the clamping point dominate the resonator dynamics.The most commonly used method to act upon the membrane is the primary-resonance excitation, in which the frequency of the excitation is tuned closed to the fundamental natural frequency of the nanostructure. Superharmonic resonance can also be implemented. It consists in applying an excitation frequency w; equal to wm/n, with n being integer. Existence of these superharmonic resonances is highly dependent on the non-linearity of the system. For example existence of n-th order non-linearity results in presence wm/n superharmonic resonance. In a second series of experiments, frequency-power sweep for superharmonic resonance has been performed, by modulating the electric load at a frequency wm/n and recording the response of the membrane at the fundamental frequency wm. High-order superharmonic resonances are observed with n=2 up to 8. Under superharmonic excitation, the control of the phase across the resonance has been shown for every observed resonance.In the next set of experiments, we used the nonlinearity existing in the system to perform stochastic resonance. The idea of stochastic resonance is amplification of a weak signal (with low frequency) by means of noise injected (higher frequency) in a nonlinear system. For our system we were able to achieve stochastic resonance with both amplitude and phase noise. A comparative study between these two schemes was also done in details. The idea of observing stochastic resonance in phase is very interesting as it opens doors to realize phase encoded communications.

The nonlinear dynamic characteristics and stability of floating vehicles are investigated theoretically and experimentally. Mathematical models of such floating bodies are used to investigate their complicated motions in regular waves. In particular, we address the phenomenon of indirectly exciting the roll motion of a vessel due to nonlinear couplings of the heave, pitch, and roll modes. In the analytical approach the method of multiple scales is used to determine first-order approximations to the solutions, yielding a system of nonlinear first-order equations governing the modulation of the amplitudes and phases of the system. The fixed-point solutions of these equations are determined and their bifurcations are investigated. Hopf bifurcations are found. Numerical simulations are used to investigate the bifurcations of the ensuing limit cycles and how they produce chaos. Experiments are conducted with tanker and destroyer models. They demonstrate some of the nonlinear effects, such as the jump phenomenon, the subcritical instability, and the coexistence of multiple solutions. The experimental results are qualitatively in good agreement with the results predicted by the theory. Coupling of the pitch and roll motions of a vessel when their frequencies are in the ratio of two-to-one is modeled by a two-degree-of-freedom system. The damping in the pitch mode is modeled by a linear viscous damping, whereas that of the roll mode is modeled by the sum of a linear viscous part and a quadratic viscous part. The effect of the quadratic damping is investigated when either mode is externally excited through a primary resonance. Force-response and frequency-response curves are generated. Coexistence of multiple solutions is found. The jump phenomenon continues to exist, whereas the saturation phenomenon ceases in the presence of quadratic damping. Hopf bifurcations are found. They indicate conditions for the nonexistence of steady-state periodic responses. Instead, the response is an amplitude- and phase-modulated motion consisting of both modes. Floquet theory is used to determine the stability of limit-cycle solutions. They undergo a pitchfork bifurcation followed by a cascade of period-doubling bifurcations, leading to chaos and hence chaotically modulated motions. When the roll mode is excited, the quadratic damping causes the region between the two Hopf bifurcation frequencies to shrink. However, the quadratic damping which may be introduced by attaching antirolling devices does not eliminate complicated motions completely in this region. The dynamic stability and excessive motion of the roll mode of a vessel in following or head regular waves is investigated theoretically and experimentally. The motion is modeled by a three-degree-of-freedom system with quadratic and cubic nonlinearities. The heave and pitch modes are linearized and their harmonic solutions are coupled into the roll mode. The resulting nonlinear ordinary-differential equation with time-varying coefficients is used to determine the stability of the roll mode for the case of principal parametric resonance. Experiments with a tanker model were conducted to validate the theory. They demonstrate the jump phenomenon and subcritical instability. They also reveal that the large-amplitude roll response depends not only on the encounter frequency but also on the position of the model relative to the waves. / Ph. D.