In this dissertation, the evolution of a pressure wave driven by a harmonic signal on the boundary during gas combustion is studied. The problem is modeled by a nonlinear, hyperbolic partial differential equation. Steady-state behavior is investigated using the perturbation method to ensure that enough time has passed for any transient effects to have dissipated. The zeroth, first and second-order perturbation solutions are obtained and their moduli are plotted against frequency. It is seen that the first and second-order corrections have unique maxima that shift to the right as the frequency decreases and to the left as the frequency increases. Dispersion relations are determined and their limiting behavior investigated in the low and high frequency regimes. It is seen that for low frequencies, the medium assumes a diffusive-like nature. However, for high frequencies the medium behaves similarly to one exhibiting relaxation. The phase speed is determined and its limiting behavior examined. For low frequencies, the phase speed is approximately equal to sqrt[ω/(n+1)] and for high frequencies, it behaves as 1/(n+1), where n is the mode number. Additionally, a maximum allowable value of the perturbation parameter, ε = 0.8, is determined that ensures boundedness of the solution. The location of the peak of the first-order correction, xmax, as a function of frequency is determined and is seen to approach the limiting value of 0.828/sqrt(ω) as the frequency tends to zero and the constant value of 2 ln 2 as the frequency tends to infinity. Analytic expressions are obtained for the approximate general perturbation solution in the low and high-frequency regimes and are plotted together with the perturbation solution in the corresponding frequency regimes, where the agreement is seen to be excellent. Finally, the solution obtained from the perturbation method is compared with the long-time solution obtained by the finite-difference scheme; again, ensuring that the transient effects have dissipated. Since the finite-difference scheme requires a right boundary, its location is chosen so that the wave dissipates in amplitude enough so that any reflections from the boundary will be negligible. The perturbation solution and the finite-difference solution are found to be in excellent agreement. Thus, the validity of the perturbation method is established.

Esse trabalho consiste na determinação das velocidades críticas do vento e das amplitudes das vibrações numa estrutura composta por uma viga engastada suspensa por um estai (cabo), submetida aos efeitos de vento e chuva. Foi considerada a deformação no cabo devido ao carregamento do peso próprio e o acoplamento não-linear das vibrações do cabo e da viga. Três modos de vibração são de especial interesse, chamados de primeiro modo global (flexão da viga e vibração no cabo), primeiro modo local (vibração no cabo, com flexão na viga desprezável) e primeiro modo à torção. O modelo foi reduzido a três graus de liberdade. A modelagem dos carregamentos aerodinâmicos aplicados na viga foi feita segundo procedimentos tradicionais. O carregamento aerodinâmico aplicado ao cabo sob efeito de chuva e vento também foi levado em consideração. Para a redução do modelo matemático, os coeficientes de rigidez e de amortecimento equivalente são definidos e dependem parametricamente da velocidade do vento. Os termos não-lineares são devidos ao acoplamento das vibrações do cabo e da viga à flexão (no plano do cabo) e também aos efeitos aeroelásticos no cabo. Os seguintes regimes instáveis são avaliados: o efeito de galope (galloping) no cabo, o drapejamento (flutter) unimodal na torção e o drapejamento (flutter) do modo de flexão da viga em conjunto com vibrações transversais do cabo. / This paper is concerned with determining wind critical velocities and post-critical vibration amplitudes in a cable-stayed beam, under wind-rain condition. It is considered the cable sag due to the dead load plus the non-linear coupling between the vibration of both the cable and the beam. Three modes are of special interest, namely the first global mode (beam bending & cable vibration), the first local mode (cable vibration & negligible beam bending) and the first torsion mode. A reduced mathematical model, with three degrees of freedom, is also developed. With regard to the modelling of the aerodynamic loads applied to the beam, it can be performed after extension of classical guidelines. The aerodynamic loads applied to the cable under rain are also taken into account. For the reduced mathematical model, equivalent damping and stiffness coefficients will be defined, which depend parametrically on the wind velocity. Non-linear terms appear due to the coupling between the cable and the beam bending vibrations, and also to the aero-elastic non-linear effects on the cable. Different unstable regimes are surveyed such as the cable galloping, the unimodal flutter in torsion and the unimodal flutter with beam bending and cable vibrations coupled.

Although dynamical systems have a multitude of classical uses in physics and applied mathematics, new research in theoretical computer science shows that dynamical systems can also be used as a highly secure method of encrypting data. Properties of Lorenz and similar systems of equations yield chaotic outputs that are good at masking the underlying data both physically and mathematically. This paper aims to show how Lorenz systems may be used to encrypt text and image data, as well as provide a framework for how physical mechanisms may be built using these properties to transmit encrypted wave signals.

<p>This thesis focuses on chaotic stationary waves, both quantum mechanical and classical. In particular we study different statistical properties regarding thesewaves, such as energy transport, intensity (or density) and stress tensor components. Also, the methods used to model these waves are investigated, and somelimitations and specialities are pointed out.</p>

Transition to turbulence and flow control are studied by means of numerical simulations for different simple shear flows. Linear and non-linear optimisation methods using the Lagrange multiplier technique are employed. In the linear framework as objective function the standard disturbance kinetic energy is chosen and the constraints involve the linearised Navier–Stokes equations. We consider both the optimal initial condition leading to the largest disturbance energy growth at finite times and the optimal time-periodic forcing leading to the largest asymptotic response for the case of the flat plate boundary layer excluding the leading edge. The optimal disturbances for spanwise wavelengths of the order of the boundary layer thickness are streamwise vortices exploiting the lift-up mechanism to create streaks. For long spanwise wavelengths it is the Orr mechanism combined with the amplification of oblique wave packets that is responsible for the disturbance growth. Also linear optimal disturbances are computed around a leading edge and the effect of the geometry is considered. It is found that two-dimentional disturbances originating upstream, relative to the leading edge of the plate are inefficient at generating a viable disturbance, while three dimentional disturbances are more amplified. In the non-linear framework a new approach using ideas from non-equilibrium thermodynamics is developed. We determine the initial condition on the laminar/turbulent boundary closest to the laminar state. Starting from the general evolution criterion of non-equilibrium systems we propose a method to optimise the route to the statistically steady turbulent state, i.e. the state characterised by the largest entropy production. This is the first time information from the fully turbulent state is included in the optimisation procedure. The method is applied to plane Couette flow. We show that the optimal initial condition is localised in space for realistic flow domains, while the disturbance visits bent streaks before breakdown. Feedback control is applied to the bypass-transition scenario with high levels of free-stream turbulence. The flow is the flat-plate boundary layer. In this scenario low frequency perturbations enter the boundary layer and streamwise elongated disturbances emerge due to non-modal growth. The so-called streaky structures are growing in amplitude until they reach high enough energy levels and break down into turbulent spots via their secondary instability. When control is applied in the form of wall blowing and suction, the growth of the streaks is delayed, which implies a delay of the whole transition process. Additionally, a comparison with experimental work is performed demonstrating a remarkable agreement in the disturbance attenuation once the differences between the numerical and experimental setup are reduced. Open-loop control with wall travelling waves by means of blowing and suction is applied to a separating boundary layer. For downstream travelling waves we obtain a mitigation of the separation of the boundary layer while for upstream travelling waves a significant delay in the transition location accompanied by a modest reduction of the separated region. / QC 20110518

This thesis focuses on chaotic stationary waves, both quantum mechanical and classical. In particular we study different statistical properties regarding thesewaves, such as energy transport, intensity (or density) and stress tensor components. Also, the methods used to model these waves are investigated, and somelimitations and specialities are pointed out.

The task of finding the beat in music is simple for most people, but surprisingly difficult to replicate in a robot. Progress in this problem has been made using various preprocessing techniques (Hitz 2008; Tomic and Janata 2008). However, a real-time method is not yet available. Methods using a class of oscillators called relay relaxation oscillators are promising. In particular, systems of forced Hopf oscillators (Large 2000; Righetti et al. 2006) have been used with relative success. This work describes current methods of beat tracking and develops a new method that incorporates the best ideas from each existing method and removes the necessity for preprocessing.

Two methods for control of swarms are described. The first of these methods, the Virtual Attractive-Repulsive (VARP) method, is based on potentials defined between swarm elements. The second control method, or the abstraction method, is based on controlling the macroscopic characteristics of a swarm. The derivation of a new control law based on the second method is described. Numerical simulation and analytical interpretation of the result is also presented.

This thesis presents results of investigations that have been carried out to understand dynamics in nonlinear Paul trap mass spectrometers. Of the three problems that have been taken up for study in this thesis, the first concerns understanding early/delayed ejection of ions in mass selective boundary ejection experiments. The second looks at the differential resolution observed in forward and reverse scan resonance ejection experiments. The third study explores a coupled nonlinear resonance within the nominally stable region of trap operation. The method of multiple scales has been to elucidate dynamics associated with early and delayed ejection of ions in mass selective ejection experiments in Paul traps. We develop a slow flow equation to approximate the solution of a weakly nonlinear Mathieu equation to describe ion dynamics in the neighborhood of the stability boundary of ideal traps (where the Mathieu parameter qz = qz* = 0.908046). For positive even multipoles in the ion trapping field, in the stable region of trap operation, the phase portrait obtained from the slow flow consists of three fixed points, two of which are saddles and the third is a center. As the qz value of an ion approaches qz*, the saddles approach each other, and a point is reached where all nonzero solutions are unbounded, leading to an observation of early ejection. The phase portraits for negative even multipoles and odd multipoles of either sign are qualitatively similar to each other and display bounded solutions even for qz > qz*, resulting in the observation of delayed ejection associated with a more gentle increase in ion motion amplitudes, a mechanism different from the case of the positive even multipoles. The second study investigates constraints on pre-ejection dynamical states which cause differential resolution in resonance ejection experiments using Paul traps with stretched geometry. Both analytical and numerical computations are carried out to elucidate the role of damping and scan rate in influencing coherence in ion motion associated with the forward and reverse scan. It has been shown that in the forward scan experiments, for a given damping, low scan rates result in coherent motion of ions oof a given mass at the jump point. At this point, the amplitude and phase of ions of a given mass, starting at different initial conditions, become effectively identical. As the scan rate is increased, coherence is destroyed. For a given scan rate, increasing damping introduces coherence in ion motion, while decreasing damping destroys this coherence. In reverse scan experiments, for a given damping, very low scan rates will cause coherent ion motion. Increasing the scan rate destroys this coherence. The effect of damping in reverse scan experiments is qualitatively similar to that in the forward scan experiments, but settling times in the forward scan are shorter, leading to improved coherence and resolution. For mass spectrometrically relevant scan rates and damping values, significantly greater coherence is obtained in the forward scan. In the third study we investigate the weakly coupled and nonlinear Mathieu equations governing ion motion in axial and radial directions in a Paul trap in the neighborhood of a nonlinear resonance point at az* = -0.2313850427 and qz* = 0.9193009931$. Using harmonic balance based approximate averaging up to second order; we obtain a slow flow that, we numerically demonstrate, approximates the actual ion dynamics. We find that the slow flow is Hamiltonian. We study the slow flow numerically with the objective of exploring and displaying some of the possible types of interesting ion motions. In particular, we choose specific but arbitrary parameter values; study the stability of the individual radial and axial motion invariant manifolds; examine the rather large times associated with escape of ions; notice regions in the averaged phase space wherein trajectories do not, in fact, escape; observe apparently chaotic dynamics preceding escape for ions that do escape; and note that trajectories that do not escape appear to be confined to 4-tori. We conclude with some comments on the implications for practical operation of the Paul trap near this resonant point.

Cracks often exist in structural members that are exposed to repeated loading, which will certainly lower the structural integrity. A crack on a structural member introduces a local flexibility which is a function of the crack depth and location. This may cause nonlinear dynamic response of the structure. In this thesis, a new method is suggested to detect and locate a crack in a structural component. The method is based on the fact that nonlinear response of a structure with a crack will be a function of the crack location and crack magnitude. The method suggested is the extension of a recently developed technique for identification of non-linearity in vibrating multi degree of freedom system. In this method, experimentally measured receptances at different forcing levels are used as input, and the existence and location of a nonlinearity are sought. In order to validate the method, simulated experimental data is used. Characteristics of a cracked beam are simulated by using experimentally obtained analytical expressions, given in the literature. The structure itself is modelled by using finite element method. Several case studies are performed to test and demonstrate the applicability, efficiency and sensitivity of the method suggested. The effect of crack depth on nonlinear system response is also studied in numerical examples.