This thesis presents an approximate solution of second order relative motion equations. The equations of motion for a Keplerian orbit in spherical coordinates are expanded in Taylor series form using reference conditions consistent with that of a circular orbit. Only terms that are linear or quadratic in state variables are kept in the expansion. A perturbation method is employed to obtain an approximate solution of the resulting nonlinear differential equations. This new solution is compared with the previously known solution of the linear case to show improvement, and with numerical integration of the quadratic differential equation to understand the error incurred by the approximation. In all cases, the comparison is made by computing the difference of the approximate state (analytical or numerical) from numerical integration of the full nonlinear Keplerian equations of motion. / Master of Science

[pt] O presente trabalho, de natureza teórico-experimental, objetiva estudar a influencia de perturbações acústicas num escoamento turbulento e subsônico de ar, em desenvolvimento num duto de seção circular de 3000 mm de comprimento e 50,8 mm de diâmetro interno. Perturbações acústicas senoidais são introduzidas no escoamento nas frequências naturais de ressonâncias do duto (56 e 112 Hz), acoplado ao túnel de vento propulsor do escoamento. O trabalho experimental compreende medições dos campos de velocidade média, turbulência e pressão e de características da onda acústica utilizada como fonte de perturbação do escoamento. O campo de pressão induzido pelo escoamento não-perturbado foi medido por técnicas convencionais. Já na situação de escoamento perturbado acusticamente, uma análise da técnica de medição fez-se necessária devido às dificuldades intrínsecas à medição de pressão em escoamentos periódicos. Com base na resposta de diferentes sistemas de medição, desenvolveu-se uma modelagem via técnica de parâmetros concentrados, o que permitiu o estudo da resposta do transdutor de pressão sob o ponto de vista de sistemas não lineares. Para identificar os diversos fatores que distorcem o valor da pressão indicada pelos transdutores na situação de escoamento perturbado, uma intercomparação dos resultados obtidos via diferentes sistemas (micromanometro, sensor capacitivo de diafragma e microfone capacitivo) foi realizada, permitindo validar o modelo proposto. Conceitos de linhas de transmissão, fundamentados à luz da analogia elétrica, foram utilizados para explicar a origem dos erros sistemáticos, também reportados em outros trabalhos, e estabelecer uma base para modelagem via parâmetros distribuídos, permitindo estabelecer restrição para seleção do comprimento do tubo de conexão, que interliga o transdutor ao sistema de medição. Medições de velocidade média e de turbulência foram realizadas nos escoamentos perturbadas e não-perturbado pela técnica da anemometria de fio quente. Para algumas das situações investigadas, a excitação acústica induziu estímulos da ordem de 100 por cento nas flutuações de velocidade, decorrentes do acoplamento da turbulência e da onda perturbadora, relativamente ao escoamento não-perturbado, resultado que apresenta interesse em aplicações tecnológicas. Na busca dos efeitos introduzidos pela perturbação acústica no escoamento turbulento, foi estudada teórica e experimentalmente a mecânica da propagação da onda e os efeitos da dissipação viscosa na sua propagação. / [en] A theoretical and experimental study of acoustic pertubations interacting with a turbulent subsonic air flow inside a circular cross-section duct (length 3000 mm and diameter 50.8 mm) was performed. Sinusoidal acoustic perturbations are introduced in the flow at resonante frequencies of the pipe, namely 56 and 112 Hz. For the undisturbed flow, static pressure field was measured by means of standard techniques. Measurement of pressure field in the disturbed flow condition is a difficult task and requires more sophisticated methods. A concentrated parameter model based on system response was developed in order to study the response of a the pressure trasnducer under the non-linear point of view. As to estimate the influence of some parameters involved in the measurement, different types of transducers were used (micromanometer, capacitive sensor and a condenser michophone) to promote the intercomparison between available techniques and the esxperimental pressure a data obtained. The transmission line concepts, based upon the electrical analogy, were also used to explain the source of systematic errors present in the measuring technique. Technical recommendations, based on a distributed parameters modelling, are provided leading to restrictions of the length of the tube which connects the transducer to the wall pressure taps. Measurements of mean velocity and intensity of turbulence were made by means of a hot-wire anemometer. In some of the conditions investigated, in comparison with the undisturbed turbulent flow, acoustic perturbations indeced and increase of velocity fluctuations of the order of 100 per cent, due to the coupling of the turbulence and the excitation acoustic wave.

[pt] Sistemas dinâmicos não-lineares são comuns em engenharia. Este tipo de problema é resolvido por integração numérica das equações de movimento ou por métodos analíticos aproximados (métodos de perturbação) ou semi- analíticos como o método do Balanço Harmônico. A integração numérica é um processo lento e oneroso em análises paramétricas. Já os outros métodos aproximados são extremamente rápidos, mas são menos precisos e em pro- blemas com certos tipos de não-linearidade, tais como expoentes fracionários, são de difícil, ou impossível, aplicação. Neste trabalho, são apresentados dois métodos alternativos, baseados nas séries de Taylor, para a análise de sistemas não-lineares. No primeiro método, a resposta é escrita em série de Taylor e propriedades de simetria do sistema no espaço de fase são utilizadas para se determinar a relação freqüência-amplitude ou pontos fixos da resposta. No segundo método a solução é escrita em série de Fourier e as amplitudes dos harmônicos são determinadas da mesma forma que os coeficientes da série de Taylor. A simetria do sistema agora fica implícita na solução em série de Fourier, e a relação freqüência- amplitude ou os pontos fixos da resposta são obtidos utilizando equações suplementares. Através de comparações com outros métodos, mostra-se que os métodos desenvolvidos são de fácil implementação e precisos. Estes possuem as vantagens de serem aplicados a problemas com diversos tipos de não-linearidade e de fornecerem uma resposta em série de Fourier onde as amplitudes são determinadas analiticamente resolvendo-se um sistema de equações algébricas lineares. / [en] Nonlinear dynamical systems are rather common in engineering. This class of problems is usually solved by numerical integration or through the use of ap- proximate analytical methods (perturbation methods) or semi-analytical meth- ods such as the harmonic balance method. The numerical integration is a slow and cumbersome process in parametric analyses. The other methods are usu- ally extremely fast but they are less precise and their application to problems involving certain types of non-linearity, such as fractional power non-linearities, are difficult or even impossible. In this work two alternative methodologies for the analysis of non-linear dynamical systems, based on Taylor series expan- sions, are proposed. In the first method, the solution of the initial value prob- lem is obtained by expanding the response in Taylor series and the symmetries of the response in phase space are used to obtain the frequency-amplitude rela- tion or the fixed points of the steady-state response. In the second method the response is written as a Fourier series and the modal amplitudes are obtained using the same methodology used in the previous method for the determina- tion of the coefficients of the Taylor expansion. The symmetries of the response are implicit in the Fourier series, and supplementary equations are proposed for the determination of the frequency-amplitude relation and the fixed points of the response. Comparisons with other existing methods show that the two proposed methods are precise and can be easily applied to the analysis of sev- eral dynamical systems. The main advantages of the proposed methods are that they can be applied to several types of non-linearities and that analytic expression for the Fourier coefficients can be obtained by the solution of a system of linear algebraic equations.

A procedure is presented for using a primary resonance excitation in experimentally identifying the nonlinear parameters of a model approximating the response of a cantilevered beam by a single mode. The model accounts for cubic inertia and stiffness nonlinearities and quadratic damping. The method of multiple scales is used to determine the frequency-response function for the system. Experimental frequency- and amplitude-sweep data are compared with the prediction of the frequency-response function in a least-squares curve-fitting algorithm. The algorithm is improved by making use of experimentally known information about the location of the bifurcation points. The method is validated by using the parameters extracted to predict the force-response curves at other nearby frequencies. We then compare this technique with two other techniques that have been presented in the literature. In addition to the amplitude- and frequency-sweep technique presented, we apply a second frequency-domain technique and a time-domain technique to the second mode of a cantilevered beam. We apply the restoring-force surface method assuming no a priori knowledge of the system and use the shape of the surface to guide us in assuming a form for the equation of motion. This equation is used in applying the frequency-domain techniques: a backbone curve-fitting technique based on the describing-function method and the amplitude- and frequency-sweep technique based on the method of multiple scales. We derive the equation of motion from a Lagrangian and discover that the form assumed based on the restoring-force surface is incorrect. All of the methods are reapplied with the new form for the equation of motion. Differences in the parameter estimates are discussed. We conclude by discussing the limitations encountered for each technique. These include the inability to separate the nonlinear curvature and inertia effects and problems in estimating the coefficients of small terms with the time-domain technique. / Master of Science

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

A rapid growth of installed wind power capacity is expectedin the next few years. However, the siting of wind turbines ona large scale raises concerns about their environmental impact,notably with respect to noise. To this end, variable speed windturbines offer a promising solution for applications in denselypopulated areas like the European countries, as this designwould enable an efficient utilisation of the masking effect dueto ambient noise. In rural and recreational areas where windturbines are sited, the ambient noise originates from theaction of wind on the vegetation and about the listener's ear(pseudo-noise). It shows a wind speed dependence similar tothat of the noise from a variable speed wind turbine and cantherefore mask the latter for a wide range of conditions.However, a problem inherent to the design of these machines istheir proclivity to pure tone generation, because of theenhanced difficulty of avoiding structural resonances in themechanical parts. Pure tones are deemed highly annoying and areseverely regulated by most noise policies. In relation to thisproblem, the vibration transmission of structure-borne sound tothe tower of the turbine is investigated, in particular whenthe tower is stiffened at its upper end. Furthermore, sincenoise annoyance due to wind turbine is mostly a masking issue,the wind-related sources of ambient noise are studied and theirmasking potentials assessed. With this aim, prediction modelsfor wind-induced vegetation noise and pseudo-noise have beendeveloped. Finally, closely related to the effect of masking,is the difficulty, regularly encountered by local authoritiesand wind farm developers, to measure noise immission from windturbines. A new measurement technique has thus been developedin the course of this work. Through improving thesignal-to-noise ratio between wind turbine noise and ambientnoise, the new technique yields more accurate measurementresults. Keywords: Masking, vibration transmission, diffraction,ambient noise, pseudo-noise, cylindrical shell, perturbationmethods, structural mobility, acoustic outdoor measurement. / QC 20100616

Philosophiae Doctor - PhD / In this thesis, we consider some nonlinear differential models that govern unsteady magneto-hydrodynamic convective flow and mass transfer of viscous, incompressible,electrically conducting fluid past a porous plate with/without heat sources. The study focusses on the effect of a combination of a number of physical parameters (e.g., chemical reaction, suction, radiation, soret effect,thermophoresis and radiation absorption) which play vital role in these models.Non dimensionalization of these models gives us sets of differential equations. Reliable solutions of such differential equations can-not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a fitted operator finite difference method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame- ters, we present extensive numerical simulations for each of these models. Finally, we confirm theoretical results through a set of specificc numerical experiments.

Ce travail concerne l'étude de monocouches auto-assemblées (SAMs) sur l'or par simulation moléculaire. Des SAMs électroactives formées de chaines ferrocenylalcanethiols et alcanethiols et des SAMs constituées de β-cyclodextrines immobilisées sur des surfaces pouvant donner lieu à la formation de complexes d'inclusion à l'interface ont été étudiées. L'objectif était d'obtenir des grandeurs macroscopiques qui soient directement comparables aux grandeurs expérimentales. Pour cela, des simulations de dynamique moléculaire ont été couplées à des calculs de perturbation thermodynamique afin d'obtenir des grandeurs rédox et des propriétés thermodynamiques d'association. La reproduction de grandeurs expérimentales a dans un premier temps permis de valider les méthodologies de simulation et les champs de forces utilisés. Ceci a ensuite conduit à envisager la simulation moléculaire comme une technique prédictive pour l'étude de nouveaux systèmes. Les grandeurs macroscopiques obtenues ont pu être interprétées grâce à une caractérisation structurale et énergétique des processus mis en jeu. / This work concerns the study of self-assembled monolayers (SAMs) on gold surfaces by molecular simulation. Electroactive SAMs formed by both ferrocenylalkanethiol and alkanethiol chains and SAMs of immobilized β-cyclodextrins that can form inclusion complexes at the interface were investigated. The objective of this study was to use molecular simulation to reproduce macroscopic properties that can be directly compared with experimental results. Molecular dynamics simulations are coupled to perturbation methods in order to calculate redox properties and thermodynamic properties of association. The comparison with experimental data allows us to validate simulation methodologies and forcefields and to consider simulation as a predictive tool for the study of new systems. Molecular dynamic also provides a rationalization of the macroscopic properties at the atomic level by a structural and energetic analysis of the processes involved in the reactions.

[Formulae and special characters can only be approximated here. Please see the pdf version of the abstract for an accurate reproduction.] Wave run-up is the vertical uprush of water when an incident wave impinges on a free- surface penetrating body. For large volume offshore structures the wave run-up on the weather side of the supporting columns is particularly important for air-gap design and ultimately the avoidance of pressure impulse loads on the underside of the deck structure. This investigation focuses on the limitations of conventional wave diffraction theory, where the free-surface boundary condition is treated by a Stokes expansion, in predicting the harmonic components of the wave run-up, and the presentation of a simplified procedure for the prediction of wave run-up. The wave run-up is studied on fixed vertical cylinders in plane progressive waves. These progressive waves are of a form suitable for description by Stokes' wave theory whereby the typical energy content of a wave train consists of one fundamental harmonic and corresponding phase locked Fourier components. The choice of monochromatic waves is indicative of ocean environments for large volume structures in the diffraction regime where the assumption of potential flow theory is applicable, or more formally A/a < Ο(1) (A and a being the wave amplitude and cylinder radius respectively). One of the unique aspects of this work is the investigation of column geometry effects - in terms of square cylinders with rounded edges - on the wave run-up. The rounded edges of each cylinder are described by the dimensionless parameter rc/a which denotes the ratio of edge corner radius to half-width of a typical column with longitudinal axis perpendicular to the quiescent free-surface. An experimental campaign was undertaken where the wave run-up on a fixed column in plane progressive waves was measured with wire probes located close to the cylinder. Based on an appropriate dimensional analysis, the wave environment was represented by a parametric variation of the scattering parameter ka and wave steepness kA (where k denotes the wave number). The effect of column geometry was investigated by varying the edge corner radius ratio within the domain 0 <=rc/a <= 1, where the upper and lower bounds correspond to a circular and square shaped cylinder respectively. The water depth is assumed infinite so that the wave run-up caused purely by wave-structure interaction is examined without the additional influence of a non-decaying horizontal fluid velocity and finite depth effects on wave dispersion. The zero-, first-, second- and third-harmonics of the wave run-up are examined to determine the importance of each with regard to local wave diffraction and incident wave non-linearities. The modulus and phase of these harmonics are compared to corresponding theoretical predictions from conventional diffraction theory to second-order in wave steepness. As a result, a basis is formed for the applicability of a Stokes expansion to the free-surface boundary condition of the diffraction problem, and its limitations in terms of local wave scattering and incident wave non-linearities. An analytical approach is pursued and solved in the long wavelength regime for the interaction of a plane progressive wave with a circular cylinder in an ideal fluid. The classical Stokesian assumption of infinitesimal wave amplitude is invoked to treat the free-surface boundary condition along with an unconventional requirement that the cylinder width is assumed much smaller than the incident wavelength. This additional assumption is justified because critical wavelengths for wave run-up on a fixed cylinder are typically much larger in magnitude than the cylinder's width. In the solution, two coupled perturbation schemes, incorporating a classical Stokes expansion and cylinder slenderness expansion, are invoked and the boundary value problem solved to third-order. The formulation of the diffraction problem in this manner allows for third-harmonic diffraction effects and higher-order effects operating at the first-harmonic to be found. In general, the complete wave run-up is not well accounted for by a second-order Stokes expansion of the free-surface boundary condition and wave elevation. This is however, dependent upon the coupling of ka and kA. In particular, whilst the modulus and phase of the second-harmonic are moderately predicted, the mean set-up is not well predicted by a second-order Stokes expansion scheme. This is thought to be caused by higher than second-order non-linear effects since experimental evidence has revealed higher-order diffraction effects operating at the first-harmonic in waves of moderate to large steepness when k < < 1. These higher-order effects, operating at the first-harmonic, can be partially accounted for by the proposed long wavelength formulation. For small ka and large kA, subsequent comparisons with measured results do indeed provide a better agreement than the classical linear diffraction solution of Havelock (1940). To account for the complete wave run-up, a unique approach has been adopted where a correction is applied to a first-harmonic analytical solution. The remaining non-linear portion is accounted for by two methods. The first method is based on regression analysis in terms of ka and kA and provides an additive correction to the first-harmonic solution. The second method involves an amplification correction of the first-harmonic. This utilises Bernoulli's equation applied at the mean free-surface position where the constant of proportionality is empirically determined and is inversely proportional to ka. The experimental and numerical results suggest that the wave run-up increases as rc/a--› 0, however this is most significant for short waves and long waves of large steepness. Of the harmonic components, experimental evidence suggests that the effect of a variation in rc/a on the wave run-up is particularly significant for the first-harmonic only. Furthermore, the corner radius effect on the first-harmonic wave run-up is well predicted by numerical calculations using the boundary element method. Given this, the proposed simplified wave run-up model includes an additional geometry correction which accounts for rc/a to first-order in local wave diffraction. From a practical view point, it is the simplified model that is most useful for platform designers to predict the wave run-up on a surface piercing column. It is computationally inexpensive and the comparison of this model with measured results has proved more promising than previously proposed schemes.

Philosophiae Doctor - PhD / In this thesis, we consider some nonlinear di erential models that govern unsteady magneto-hydrodynamic convective ow and mass transfer of viscous, incompressible, electrically conducting uid past a porous plate with/without heat sources. The study focusses on the e ect of a combination of a number of physical parameters (e.g., chem- ical reaction, suction, radiation, soret e ect, thermophoresis and radiation absorption) which play vital role in these models. Non-dimensionalization of these models gives us sets of di erential equations. Reliable solutions of such di erential equations can- not be obtained by standard numerical techniques. We therefore resorted to the use of the singular perturbation approaches. To proceed, each of these model problems is discretized in time by using a suitable time-stepping method and then by using a tted operator nite di erence method in spatial direction. The combined methods are then analyzed for stability and convergence. Aiming to study the robustness of the proposed numerical schemes with respect to change in the values of the key parame-ters, we present extensive numerical simulations for each of these models. Finally, we con rm theoretical results through a set of speci c numerical experiments.