A new concept, extracting energy from sea waves by parametric pendulor, has been explored in this project. It is based on the conversion of vertical oscillations to rotational motion by means of a parametrically-excited pendulor, i.e. a pendulum operating in rotational mode. The main advantage of this concept lies in a direct conversion from vertical oscillations to rotations of the pendulum pivot. This thesis, firstly, reviewed a number of well established linear and nonlinear theories of sea waves and Airy’s sea wave model has been used in the modelling of the sea waves and a parametric pendulum excited by sea waves. The third or fifth order Stokes’s models can be potentially implemented in the future studies. The equation of motion obtained for a parametric pendulum excited by sea waves has the same form as for a simple parametrically-excited pendulum. Then, to deepen the fundamental understanding, an extensive theoretical analysis has been conducted on a parametrically-excited pendulum by using both numerical and analytical methods. The numerical investigations focused on the bifurcation scenarios and resonance structures, particularly, for the rotational motions. Analytical analysis of the system has been performed by applying the perturbation techniques. The approximate solutions, resonance boundary and existing boundary of rotations have been obtained with a good correspondence to numerical results. The experimental study has been carried out by exploring oscillations, rotations and chaotic motions of the pendulum.

This thesis develops methods to exploit static and dynamic instabilities in electrostatic MEMS to develop new MEMS devices, namely dynamically actuated micro switches and binary micro gas sensors. Models are developed for the devices under consideration where the structures are treated as elastic continua. The electrostatic force is treated as a nonlinear function of displacement derived under the assumption of parallel-plate theorem. The Galerkin method is used to discretize the distributed-parameter models, thus reducing the governing partial differential equations into sets of nonlinear ordinary-differential equations. The shooting method is used to numerically solve those equations to obtain the frequency-response curves of those devices and the Floquet theory is used to investigate their stability. To develop the dynamically actuated micro switches, we investigate the response of microswitches to a combination of DC and AC excitations. We find that dynamically actuated micro switches can realize significant energy savings, up to 60 %, over comparable switches traditionally actuated by pure DC voltage. We devise two dynamic actuation methods: a fixed-frequency method and a shifted-frequency method. While the fixed-frequency method is simpler to implement, the shifted-frequency method can minimize the switching time to the same order as that realized using traditional DC actuation. We also introduce a parameter identification technique to estimate the switch geometrical and material properties, namely thickness, modulus of elasticity, and residual stress. We also develop a new detection technique for micro mass sensors that does not require any readout electronics. We use this method to develop static and dynamic binary mass sensors. The sensors are composed of a cantilever beam connected to a rigid plate at its free end and electrostatically coupled to an electrode underneath it. Two versions of micro mass sensors are presented: static binary mass sensor and dynamic binary mass sensor. Sensitivity analysis shows that the sensitivity of our static mass sensor represents an upper bound for the sensitivity of comparable statically detected inertial mass sensors. It also shows that the dynamic binary mass sensors is three orders of magnitude more sensitive than the static binary mass sensor. We equip our mass sensor with a polymer detector, doped Polyaniline, to realize a formaldehyde vapor sensor and demonstrate its functionality experimentally. We find that while the static binary gas sensor is simpler to realize than the dynamic binary gas sensor, it is more susceptible to external disturbances.

Response and stability of vessel rolling motion with strongly nonlinear softening stiffness will be studied in this dissertation using the methods of stochastic dynamical systems. As one of the most classic stability failure modes of vessel dynamics, large amplitude rolling motion in random beam waves has been studied in the past decades by many different research groups. Due to the strongly nonlinear softening stiffness and the stochastic excitation, there is still no general approach to predict the large amplitude rolling response and capsizing phenomena. We studied the rolling problem respectively using the shaping filter technique, stochastic averaging of the energy envelope and the stochastic Melnikov function. The shaping filter technique introduces some additional Gaussian filter variables to transform Gaussian white noise to colored noise in order to satisfy the Markov properties. In addition, we developed an automatic cumulant neglect tool to predict the response of the high dimensional dynamical system with higher order neglect. However, if the system has any jump phenomena, the cumulant neglect method may fail to predict the true response. The stochastic averaging of the energy envelope and the Melnikov function both have been applied to the rolling problem before, it is our first attempt to apply both approaches to the same vessel and compare their efficiency and capability. The inverse of the mean first passage time based on Markov theory and rate of phase space flux based on the stochastic Melnikov function are defined as two different, but analogous capsizing criteria. The effects of linear and nonlinear damping and wave characteristic frequency are studied to compare these two criteria. Further investigation of the relationship between the Markov and Melnikov based method is needed to explain the difference and similarity between the two capsizing criteria.

<p>The primary focus of this dissertation is the characterization of snap-through buckling of discrete and continuous systems. Snap-through buckling occurs as the consequence of two factors, first the destabilization, or more often the disappearance of, an equilibrium position under the change of a system parameter, and second the existence of another stable equilibrium configuration at a remote location in state space. In this sense snap-through buckling is a global dynamic transition as the result of a local static instability.</p><p> </p><p>In order to better understand the static instabilities that lead to snap-through buckling, the behavior of mechanical systems in the vicinity of various local bifurcations is first investigated. Oscillators with saddle-node, pitchfork, and transcritical bifurcations are shown analytically to exhibit several interesting characteristics, particularly in relation to the system damping ratio. A simple mechanical oscillator with a transcritical bifurcation is used to experimentally verify the analytical results. The transcritical bifurcation was selected since it may be used to represent generic bifurcation behavior. It is shown that the damping ratio may be used to predict changes in stability with respect to changing system parameters.</p><p>Another useful indicator of snap-through is the presence of chaos in the dynamic response of a system. Chaos is usually associated snap-through, as in many systems large amplitude responses are typically necessary to sufficiently engage the nonlinearities that induce chaos. Thus, a pragmatic approach for identifying chaos in experimental (and hence noisy) systems is also developed. The method is applied to multiple experimental systems showing good agreement with identification via Lyapunov exponents.</p><p>Under dynamic loading, systems with the requisite condition for snap-through buckling, that is co-existing equilibria, typically exhibit either small amplitude response about a single equilibrium configuration, or large amplitude response that transits between the static equilibria. Dynamic snap-through is the name given to the large amplitude response, which, in the context of structural systems, is obviously undesirable. This phenomenon is investigated using experimental, numerical, and analytical means and the boundaries separating safe (non-snap-through) from unsafe (snap-through) dynamic response in forcing parameter space are obtained for both a discrete and a continuous arch. Arches present an ideal avenue for the investigation of snap-through as they typically have multiple, often tunable, stable and unstable equilibria. They also have many direct applications in both civil engineering, where arches are a canonical structural element, and mechanical engineering, where arches may be used to approximate the behavior of curved plates and panels such as those used on aircraft.</p> / Dissertation

This thesis describes the comparison of time-lapse field observations of suncups on alpine snow with numerical simulations. The simulations consist of solutions to a nonlinear partial differential equation which exhibits spontaneous pattern formation from a low amplitude, random initial surface. Both the field observations and the numerical solutions are found to saturate at a characteristic height and fluctuate chaotically with time. The timescale of these fluctuations is found to be instrumental in determining the full set of parameters for the numerical model such that it mimics the nonlinear dynamics of suncups. These parameters in turn are related to the change in albedo of the snow surface caused by the presence of suncups. This suggests the more general importance of dynamical behaviour in gaining an understanding of pattern formation phenomena.

This dissertation is concerned with identification of nonlinear instabilities in rotating multibody systems and subsequent control to eliminate the vibrations. Three nonlinear mechanical systems of this type are investigated and instabilities arising from their inherent nonlinearities are shown to exist for a range of system parameters and conditions. Subsequently, various nonlinear methods of vibration control have been employed to eliminate or suppress the instabilities. Analytical and numerical models have been designed to demonstrate various unstable dynamical behaviour with consistent results. The motion is studied by means of time history, phase space, frequency spectrum, Poincare map, Lyapunov characteristic exponents and Correlation Dimension. Numerical simulations have also shown the effectiveness and robustness of the control techniques over a range of instability conditions for each model. The dynamics of a rotating body with internal energy dissipation is first investigated. Such a model may be considered to be representative of a simplified spinning spacecraft. A comprehensive stability analysis is performed and regions of highly nonlinear behaviour are identified for more rigorous investigation. Numerical simulations using typical satellite parameter values are performed and the system is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft for a range of forcing amplitude and frequency. Analysis of this model using Melnikov’s method is performed and a sufficient criterion for chaotic instabilities is obtained in terms of system parameters. Evidence is also presented, indicating that the onset of chaotic motion is characterised by period doubling as well as intermittency. Subsequently, Control of chaotic vibrations in this model is achieved using three techniques. The control methods are implemented on the model under instability conditions. The first two control techniques, recursive proportional feedback (RPF) and continuous delayed feedback are recently developed model independent methods for control of chaotic motion in dynamical systems. As such these methods are employed on all three rotating multibody systems in this dissertation. Control of chaotic instability in this model is also achieved using an algorithm derived using Lyapunov’s second method. Each technique is outlined and the effectiveness of the three strategies in controlling chaotic motion exhibited by the present system is compared and contrasted. The dynamics of a dual-spin spacecraft with internal energy dissipation in the form of an axial nutational damper is also investigated for non-linear phenomena. The problem involves the study of a body with internal moving parts that is characterised by a coupling of the motions of the damper mass and the angular rotations of the platform and rotor of the spacecraft. Two realistic spacecraft parameter configurations are investigated and each is found to exhibit chaotic motion when a sinusoidally varying torque is applied to the spacecraft rotor for a range of forcing amplitude and frequency. Onset of chaotic motion was shown to be characterised by period doubling and Hopf bifurcations. An investigation of the effects of damping upon the configuration is also performed. Predicted instabilities indicate the range of rotor speeds, perturbation amplitudes and damping coefficients to be avoided in the design of dual-spin spacecraft. Control of chaotic vibrations in this model is also achieved using recursive proportional feedback (RPF) and continuous delayed feedback. Subsequently a more effective model dependent method based on energy considerations is derived and implemented. The effectiveness and robustness of each technique is shown using numerical simulations. Another rotating multibody system that is physically distinct from the previously described models is also investigated for nonlinear instabilities and control. The model is in the form of a driveline which incorporates a commonly used coupling called a Hooke’s joint. In particular, torsional instabilities due to fluctuating angular velocity ratio across the joint are examined. Linearised equations are used for the prediction of critical speed ranges where parametric instabilities characterised by exponential build up of torsional response amplitudes occur. Predicted instabilities indicate the range of driveshaft speeds to be avoided during the design of a driveline which employs a Hooke's joint. Numerical simulations further demonstrate the existence of parametric, quasi-periodic and chaotic instabilities. Subsequently, suppression of these vibrations is achieved using the previously described model independent techniques. Chaotic vibrations have also been observed in a range of simple mechanical systems such as a periodically kicked rotor, forced pendulum, synchronous rotor, aeroelastic panel flutter and impact print hammer to name but a few. It is thus becoming of increasing importance to engineers to be aware of chaotic phenomena and be able to recognise, quantify and eliminate these undesirable vibrations. The analytical and numerical methods described in this dissertation may be usefully employed by engineers for detecting as well as controlling chaotic vibrations in an extensive range of physical systems.

An important part of the study of dynamical systems is the fitting of models to time-series data. That is, given the data, a series of observations taken from a (not fully understood) system of interest, we would like to specify a model, a mathematical system which generates a sequence of “simulated” observations. Our aim is to obtain a “good” model — one that is in agreement with the data. We would like this agreement to be quantitative — not merely qualitative. The major subject of this thesis is the question of what good quantitative agreement means. Most approaches to this question could be described as “predictionist”. In the predictionist approach one builds models by attempting to answer the question, “given that the system is now here, where will it be next?” The quality of the model is judged by the degree to which the states of the model and the original system agree in the near future, conditioned on the present state of the model agreeing with that of the original system. Equivalently, the model is judged on its ability to make good short-term predictions on the original system. The main claim of this thesis is that prediction is often not the most appropriate criterion to apply when fitting models. We show, for example, that one can have models that, while able to make good predictions, have long term (or free-running) behaviour bearing little resemblance to that exhibited in the original time-series. We would hope to be able to use our models for a wide range of purposes other than just prediction — certainly we would like our models to exhibit good free-running behaviour. This thesis advocates a “behaviourist” approach, in which the criterion for a good model is that its long-term behaviour matches that exhibited by the data. We suggest that the behaviourist approach enjoys a certain robustness over the predictionist approaches. We show that good predictors can often be very poorly behaved, and suggest that well behaved models cannot perform too badly at the task of prediction. The thesis begins by comparing the predictionist and behaviourist approaches in the context of a number of simplified model-building problems. It then presents a simple theory for the understanding of the differences between the two approaches. Effective methods for the construction of well-behaved models are presented. Finally, these methods are applied to two real-world problems — modelling of the response of a voltage-clamped squid “giant” axon, and modelling of the “yearly sunspot number”.

Thesis (Ph.D.)--Aberdeen University, 2009. / Title from web page (viewed on July, 1 2009). Includes bibliographical references.

Made available in DSpace on 2016-12-12T20:15:53Z (GMT). No. of bitstreams: 1 resumo.pdf: 30947 bytes, checksum: f3ad8c83ce0b37705512036923795354 (MD5) Previous issue date: 2009-02-27 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / In this work we investigate the dynamical behavior of two dynamical systems: (i) a symmetric linear coupling of three quadratic maps, and (ii) the generalized Lorenz equations obtained by Stenflo. For the discrete-time system represented by the coupling of three quadratic maps, we study the emergence of quasiperiodic states arising from Naimark-Sacker bifurcations of stable periodic orbits pertaining to 1×2n cascade, in particular period-1 and period-2 orbits. We also study the change in the structure of the basin of attraction of the chaotic attractors, in the neighborhood of chaos-hyperchaos transition. For the continuous-time system represented by the Lorenz-Stenflo equations, we analytically investigate, by using Routh-Hurwitz Test, the stability of three fixed points, although without explicit solution of the eigenvalue equation. We determine the precise location where pitchfork and Hopf bifurcations of the fixed points occur, as a function of the parameters of the system. Lyapunov exponents, parameter-space and phase-space portraits, and bifurcation diagrams were used to numerically characterize periodic and chaotic attractors in both systems. / Neste trabalho investigamos o comportamento de dois sistemas dinâmicos (i) Um acoplamento linear simetrico de três mapas quadraticos, e (ii) as equações eneralizadas de Lorenz, obtidas por Stenfio Para o sistema discreto, representado elo acoplamento linear dos três mapas quadraticos, estudamos a emergência de tados quase-periodicos, surgindo da bifurcação de Naimark-Sacker, a partir de uma bita estavel pertencendo a cascata 1 x 2n em particular orbitas de periodo um e eriodo dois Tambem estudamos a mudança na estrutura das bacias de atração do rator caotico, na vizinhança da transição caos-hipercaos Para o sistema de tempo ntínuo representado pelas equações de Lorenz-Stenflo, investigamos analiticamente elo método de Routh-Hurwitz, a estabilidade dos três pontos de equilíbrio, mas sem solução explicita da equação de autovalores. Determinamos a localização precisa de as bifurcações do tipo forquilha e Hopf acontecem, a partir dos pontos de uilíbrio, como uma função dos parâmetros do sistema. Expoentes de Lyapunov, agramas no espaço de parâmetros e espaço de fase e diagramas de bifurcação foram ilizados para caracterizar numericamente os atratores periódicos e caóticos em ibos os sistemas

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