A parametric identification procedure is proposed that combines the method of multiple scales and higher-order statistics to efficiently and accurately model nonlinear systems. A theoretical background for the method of multiple scales and higher-order statistics is given. Validation of the procedure is performed through applying it to numerical simulations of two nonlinear systems. The results show how the procedure can successfully characterize the system damping and nonlinearities and determine the corresponding parameters. The procedure is then applied to experimental measurements from two structural systems, a cantilevered beam and a three-beam frame. The results show that quadratic damping should be accounted for in both systems. Moreover, for the three-beam frame, the parametric excitation is much more important than the direct excitation. To show the flexibility of the procedure, numerical simulations of ship motion under parametric excitation are used to determine nonlinear parameters govening the relation between pitch, heave, and roll motions. The results show a high level of agreement between the numerical simulation and the mathematical model with the identified parameters. / Master of Science

Made available in DSpace on 2017-07-21T19:25:59Z (GMT). No. of bitstreams: 1 CarlosEduardoKrassinskiSoares.pdf: 1234525 bytes, checksum: 758cda42769aa4bdf65a24c3f1cca4ed (MD5) Previous issue date: 2010-12-17 / In this dissertation, we study the physical properties of an open system, composed of two interacting bosonic modes (quantum harmonic oscillators) of different frequencies, with a explicitly time dependent coupling. It is assumed that each of the bosonic modes is under the effect of a thermal reservoir and the dynamics of their quantum properties is studied. In the case of quadratic systems and Gaussian initial states, the quantum properties of the coupled modes evolve in time according to the time evolution of the non-symmetrized second order moment which is described in the framework of the non-unitary evolution of quantum quadratic systems. The dynamical properties are determined by solving systems of differential equations for the second moments. These systems of differential equations have time-dependent coefficients and in the limit of weak coupling, the Method of Multiple Scales for constructing and solving differential equations is applied. This approach determines the differential equations solutions in a perturbative series of an appropriate parameter associated with orders of magnitude of the couplings. In this way it is obtained a description of the temporal behavior of the squeezing and the purity for each mode in the particular cases of parametric amplification and conversion. In this context, it is also considered the problem of the quantitative measure of entanglement, analyzing its dynamical behavior for different values of the system parameters and Gaussian initial states configurations. / Nesta dissertação, estudam-se propriedades físicas de um sistema aberto, composto por dois modos bosˆonicos interagentes (osciladores harmônicos quânticos) de frequências distintas, com o acoplamento explicitamente dependente do tempo. Assume-se cada um dos modos bosônicos sob efeito de reservatórios téermicos e analisa-se a dinâmica de suas propriedades quânticas. Em se tratando de sistemas quadraticos e estados iniciais Gaussianos, determina-se a evolução ao temporal das propriedades dos modos acoplados a partir da evolução ao temporal dos momentos de segunda ordem não o simetrizados na formulação de sistemas quadraticos de evolucção não unitária. As propriedades dinamicas são o determinadas solucionando-se sistemas de equacões diferenciais para os segundos momentos não simetrizados. Sendo sistemas de equções diferenciais com coeficientes dependentes do tempo, no limite do acoplamento fraco, aplica-se o Método das Escalas Múltiplas para a construção e a resolução das equações diferenciais, determinando-se suas soluções em séries perturbativas em um parâmetro adequado, associado as ordens de grandeza dos acoplamentos. Dessa maneira, analisa-se a dinâmica da medida de compressão e da pureza para os casos da amplificação e conversão paramétrica para diferentes valores de parâmetros de estados iniciais. Neste contexto, considera-se também o problema da medida quantitativa do emaranhamento, analisando seu comportamento para diferentes conjuntos de parˆametros do sistema e configurações de estados iniciais Gaussianos.

L'objectif de ce travail de thèse est d'étudier analytiquement et numériquement le comportementdynamique non-linéaire des rotors, en prenant en compte des effets significatifs comme les grandesdéformations en flexion, les non-linéarités géométriques et le cisaillement. Le manuscrit est diviséen trois parties principales. Dans la première partie, le principe de Hamilton est utilisé pour formulerles équations du mouvement qui prennent en compte un ensemble d’effets non-linéaires comme desdéformations d'ordre supérieur en flexion et le cisaillement. De plus, si les supports du rotor nepermettent pas à l'arbre de se déplacer dans la direction axiale, il y a alors une force dynamiqueharmonique agissant axialement sur le rotor en fonctionnement. Ces modèles se composentd’équations différentielles non-linéaires du deuxième et du quatrième ordre.Les deux parties suivantes sont consacrées à la résolution des différents modèles non-linéairesdéveloppés dans la première partie. Des méthodes analytiques et numériques sont appliquées afin detraiter les équations non-linéaires du mouvement. Une méthode basée sur des développementsasymptotiques, la méthode des échelles multiples (MEM) est utilisée. Les courbes de réponse sonttracées pour différentes résonances possibles et l'effet de la non-linéarité est discuté par rapport àl'analyse linéaire. La réponse forcée du système provoquée par un balourd est également présentéepour plusieurs configurations du rotor. Lorsque les déformations de cisaillement sont prises encompte, l'analyse est effectuée pour différents élancements afin de mettre en évidence cet effet sur ladynamique d’un système arbre-disque / The objective of the present work is to investigate the nonlinear dynamic behavior of the rotorsystems analytically and numerically, taking into account the significant effects, for example, higherorder large deformations in bending, geometric nonlinearity and shear effects.This thesis is dividedinto two major parts. In the first part, Hamilton’s principle is used to derive the equations of motionwhich take into account various effects, for example, nonlinearity due to higher order largedeformations in bending and shear effects. In addition, if the supports of the rotor do not allow theshaft to move in the axial direction, then there will be a dynamical force acting axially on the rotoras it operates. The mathematical models are composed of coupled nonlinear differential equations ofthe 2nd and the 4th order.In the second part, the resolution of various nonlinear models developed in the first part isaddressed. Analytical and numerical methods are applied for treating the nonlinear equations ofmotion. A method based on asymptotic developments, the method of multiple scales (MMS) is used.The response curves are plotted for different possible resonance conditions and the effect ofnonlinearity is discussed with respect to the linear analysis. The forced response of the system due toa mass unbalance is also presented for various configurations of the rotor. When shear deformationsare taken into account, the analysis is performed for various slenderness ratios to highlight sheareffects on the dynamics of the shaft-disk rotor systems

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

No description available.

Bifurcations in Huang's chaotic chemical reactor system leading from simple dynamics into chaotic regimes are considered. Following the linear stability analysis, the periodic orbit resulting from a Hopf bifurcation of any of the six fixed points is constructed analytically by the method of multiple scales across successively slower time scales, and its stability is then determined by the resulting final secularity condition. Furthermore, we run numerical simulations of our chemical reactor at a particular fixed point of interest, alongside a set of parameter values that forces our system to undergo Hopf bifurcation. These numerical simulations then verify our analysis of the normal form.

Energy harvesting of ambient and aeroelastic vibrations is important for reducing the dependence of wireless sensing and networks on batteries. We develop a configuration for a piezoelectric energy harvester with the capability to wirelessly communicate vibration measurements while using those vibrations to power the sensing and communication devices. Particularly, we perform experiments that aim at identifying challenges to overcome in the development of such a configuration. Towards that objective, we successfully tested a self-powered real-time point-to-point wireless communication system between a vibration sensor and transmission and receiving modules. The sensing device and transmission module are powered by the vibrating object using a piezoelectric energy harvester. The communication is established by using two XBee modules. In the second part of this dissertation, we address the optimization of the output power of piezoelectric energy harvesters of aeroelastic vibrations. Given the complexity of high-fidelity simulations of the coupling between the fluid flow, structural response and piezoelectric transduction, we develop and experimentally validate a phenomelogical reduced-order model for energy harvesting from wake galloping. We also develop a high-fidelity simulation for the same phenomena. The modeling and high-fidelity simulations can be a part of a multi-disciplinary optimization framework to be used in the design and operation of galloping-based energy harvesters. / Doctor of Philosophy / Energy harvesting of ambient or flow-induced vibrations is important for reducing the dependence on batteries in wireless sensing and networks to monitor deterioration conditions, environmental pollution or wildlife conservation. Balancing the benefits and shortcomings of a specific approach, namely piezoelctric transduction, for energy harvesting from vibrations, we address a specific challenge related to the development of a configuration that allows for communicating measured vibrations using their power. Furthermore, given the low levels of output power from piezoelectric transduction, we address the need to optimize power output levels through the development of predictive models that depend on geometry and speed of the fluid flow.

We model the nonlinear dynamics of a cantilever beam with tip mass system subjected to different excitation and exploit the nonlinear behavior to perform sensitivity analysis and propose a parameter identification scheme for nonlinear piezoelectric coefficients. First, the distributed parameter governing equations taking into consideration the nonlinear boundary conditions of a cantilever beam with a tip mass subjected to principal parametric excitation are developed using generalized Hamilton's principle. Using a Galerkin's discretization scheme, the discretized equation for the first mode is developed for simpler representation assuming linear and nonlinear boundary conditions. We solve the distributed parameter and discretized equations separately using the method of multiple scales. We determine that the cantilever beam tip mass system subjected to parametric excitation is highly sensitive to the detuning. Finally, we show that assuming linearized boundary conditions yields the wrong type of bifurcation. Noting the highly sensitive nature of a cantilever beam with tip mass system subjected to parametric excitation to detuning, we perform sensitivity of the response to small variations in elasticity (stiffness), and the tip mass. The governing equation of the first mode is derived, and the method of multiple scales is used to determine the approximate solution based on the order of the expected variations. We demonstrate that the system can be designed so that small variations in either stiffness or tip mass can alter the type of bifurcation. Notably, we show that the response of a system designed for a supercritical bifurcation can change to yield a subcritical bifurcation with small variations in the parameters. Although such a trend is usually undesired, we argue that it can be used to detect small variations induced by fatigue or small mass depositions in sensing applications. Finally, we consider a cantilever beam with tip mass and piezoelectric layer and propose a parameter identification scheme that exploits the vibration response to estimate the nonlinear piezoelectric coefficients. We develop the governing equations of a cantilever beam with tip mass and piezoelectric layer by considering an enthalpy that accounts for quadratic and cubic material nonlinearities. We then use the method of multiple scales to determine the approximate solution of the response to direct excitation. We show that approximate solution and amplitude and phase modulation equations obtained from the method of multiple scales analysis can be matched with numerical simulation of the response to estimate the nonlinear piezoelectric coefficients. / Master of Science

This thesis involves the study of four problems in the area of nonlinear vibrations, using the asymptotic method of multiple scales(MMS). Accordingly, it consists of four sequentially arranged parts. In the first part of this thesis we study some nonlinear dynamics related to the amplitude control of a lightly damped, resonantly forced, harmonic oscillator. The slow flow equations governing the evolution of amplitude and phase of the controlled system are derived using the MMS. Upon choice of a suitable control law, the dynamics is represented by three coupled ,nonlinear ordinary differential equations involving a scalar free parameter. Preliminary study of this system using the bifurcation analysis package MATCONT reveals the presence of Hopf bifurcations, pitchfork bifurcations, and limit cycles which seem to approach a homoclinic orbit. However, close approach to homoclinic orbit is not attained using MATCONT due to an inherent limitation of time domain-based continuation algorithms. To continue the limit cycles closer to the homoclinic point, a new algorithm is proposed. The proposed algorithm works in phase space with an ordered set of points on the limit cycle, along with spline interpolation. The algorithm incorporates variable stretching of arclength based on local curvature, through the use of an auxiliary index-based variable. Several numerical examples are presented showing favorable comparisons with MATCONT near saddle homoclinic points. The algorithm is also formulated with infinitesimal parameter increments resulting in ordinary differential equations, which gives some advantages like the ability to handle fold points of periodic solution branches upon suitable re-parametrization. Extensions to higher dimensions are outlined as well. With the new algorithm, we revisit the amplitude control system and continue the limit cycles much closer to the homoclinic point. We also provide some independent semi-analytical estimates of the homoclinic point, and mention an a typical property of the homoclinic orbit. In the second part of this thesis we analytically study the classical van der Pol oscillator, but with an added fractional damping term. We use the MMS near the Hopf bifurcation point. Systems with (1)fractional terms, such as the one studied here, have hitherto been largely treated numerically after suitable approximations of the fractional order operator in the frequency domain. Analytical progress has been restricted to systems with small fractional terms. Here, the fractional term is approximated by a recently pro-posed Galerkin-based discretization scheme resulting in a set of ODEs. These ODEs are then treated by the MMS, at parameter values close to the Hopf bifurcation. The resulting slow flow provides good approximations to the full numerical solutions. The system is also studied under weak resonant forcing. Quasiperiodicity, weak phase locking, and entrainment are observed. An interesting observation in this work is that although the Galerkin approximation nominally leaves several long time scales in the dynamics, useful MMS approximations of the fractional damping term are nevertheless obtained for relatively large deviations from the nominal bifurcation point. In the third part of this thesis, we study a well known tool vibration model in the large delay regime using the MMS. Systems with small delayed terms have been studied extensively as perturbations of harmonic oscillators. Systems with (1) delayed terms, but near Hopf points, have also been studied by the method of multiple scales. However, studies on systems with large delays are few in number. By “large” we mean here that the delay is much larger than the time scale of typical cutting tool oscillations. The MMS up to second order, recently developed for such large-delay systems, is applied. The second order analysis is shown to be more accurate than first order. Numerical integration of the MMS slow flow is much faster than for the original equation, yet shows excellent accuracy. A key point is that although certain parameters are treated as small(or, reciprocally, large), the analysis is not restricted to infinitesimal distances from the Hopf bifurcation. In the present analysis, infinite dimensional dynamics is retained in the slow flow, while the more usual center manifold reduction gives a planar phase space. Lower-dimensional dynamical features, such as Hopf bifurcations and families of periodic solutions, are also captured by the MMS. The strong sensitivity of the slow modulation dynamics to small changes in parameter values, peculiar to such systems with large delays, is seen clearly. In the last part of this thesis, we study the weakly nonlinear whirl of an asymmetric, overhung rotor near its gravity critical speed using a well known two-degree of freedom model. Gravity critical speeds of rotors have hitherto been studied using linear analysis, and ascribed to rotor stiffness asymmetry. Here we present a weakly nonlinear study of this phenomenon. Nonlinearities arise from finite displacements, and the rotor’s static lateral deflection under gravity is taken as small. Assuming small asymmetry and damping, slow flow equations for modulations of whirl amplitudes are developed using the MMS. Inertia asymmetry appears only at second order. More interestingly, even without stiffness asymmetry, the gravity-induced resonance survives through geometric nonlinearities. The gravity resonant forcing does not influence the resonant mode at leading order, unlike typical resonant oscillations. Nevertheless, the usual phenomena of resonances, namely saddle-node bifurcations, jump phenomena and hysteresis, are all observed. An unanticipated periodic solution branch is found. In the three dimensional space of two modal coefficients and a detuning parameter, the full set of periodic solutions is found to be an imperfect version of three mutually intersecting curves: a straight line, a parabola, and an ellipse. To summarize, the first and fourth problems, while involving routine MMS involve new applications with rich dynamics. The second problem demonstrated a semi-analytical approach via the MMS to study a fractional order system. Finally, the third problem studied a known application in a hitherto less-explored parameter regime through an atypical MMS procedure. In this way, a variety of problems that showcase the utility of the MMS have been studied in this thesis.

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No. of bitstreams: 2 Dissertação - Rosane Gonçalves Ferreira - 2016.pdf: 4272548 bytes, checksum: a5f44a1be60a4ace1d85167dc75c33c4 (MD5) license_rdf: 0 bytes, checksum: d41d8cd98f00b204e9800998ecf8427e (MD5) Previous issue date: 2016-03-04 / Vibrations of mechanical systems have a wide field of research, where many work have been dedicated. Such importance is due to the fact that most human activities involve vibrations. It is worth noting that many device can suffer or produce vibrations, such as, machines, structures, motors, turbines. Vibratory systems, generally can produce complex behavior, thus the analysis of such dynamics behavior needs to use sophisticated mathematical tools. The mathematical model assigns important features of real processes with respect to linear and non-linear differential equations. In this work we are interested in the analysis of behavior of delayed mechanical systems. Time delayed can compromise the performance of controls even adding instability in the systems. On the other hand, write choose of delays can improve its performance. Systems with time delay, similar to ordinary systems, are molded by ordinary and/or partial differential equations, but, unlikely ordinary differential equations, delayed differential equations, also known as functional differential equations, are molded on Banach spaces with infinite dimension, which introduce serious difficulty in analysis of stability, since that, the spectra of solution semi-group associated with the linear part of the model can presents infinite eigenvalues. Thus, our contribution of the study of dynamics behavior of such systems will be in two directions. In the first one, we apply the perturbation method of multiple scales in themodel of differential equations, since that the system shows nonlinear vibrations. It is worth noting that the differential analysis used in the stage regarding differential equations in Banach spaces, which has infinite dimension, this approach differ substantially from standards approaches. Then we obtain numerical solutions for the amplitude at steady state using the Newton Raphson method and then we made a numerical analysis of the model of stability with delay and without delay to different parameters, using the Runge-Kuttamethod. / As vibrações possuem um campo extenso de estudos, ao quais trabalhos inteiros têm sido dedicados. Tamanha importância deve-se ao fato de que a maioria das atividades humanas envolve vibrações. Muitos sistemas construídos sofrem ou produzem vibração, tais como máquinas, estruturas, motores, turbinas e sistemas de controle. Umsistema vibratório geralmente apresenta comportamento complexo, assim a análise do comportamento dinâmicos envolve o uso de ferramentas matemáticas sofisticadas. O modelo matemático incorpora os aspectos importantes do processo real, em termos de equações diferenciais lineares ou não lineares. Neste trabalho nosso objetivo é analisar o comportamento de um modelo de sistemas mecânicos. Os tempos de atrasos quando presentes em controladores e atuadores podem ser motivo de ineficiência ou mesmo causar a instabilidade do sistema. Porém, o controle adequado desses atrasos pode melhorar o desempenho de sistemas mecânicos. Os sistemas com tempo de atraso, assim como os sistemas ordinários, são modelados por equações diferenciais ordinárias ou parciais, mas diferentemente das equações ordinárias, equações com tempo de atraso, também conhecidas como equações funcionais, são modeladas em espaços de dimensão infinita, o que dificulta enormemente a análise de estabilidade, uma vez que, o espectro do semigrupo solução associado à parte linear do modelo pode apresentar infinitos autovalores. Assim, nossa contribuição ao estudo do comportamento dinâmico de tais sistemas foi feito em duas partes. Na primeira, aplicamos o método de perturbação das múltiplas escalas no sistema de equações diferenciais do modelo, uma vez que o sistema apresenta vibrações não lineares. Nesta parte, é importante ressaltar que a análise diferencial usada foi em um espaço de dimensão infinita, também conhecido como espaço de Banach; esta análise difere substancialmente daquela usada no caso ordinário. Em seguida obtemos soluções numéricas para a amplitude em estado estacionário usando o método de Newton Raphson e depois fizemos uma análise numérica da estabilidade do modelo com atraso e sem atraso para diferentes parâmetros, usando o método de Runge- Kutta.