55 research outputs found

    Fourth SIAM Conference on Applications of Dynamical Systems

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    Rotating potential of a stochastic parametric pendulum

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    The parametric pendulum is a fruitful dynamical system manifesting some of the most interesting phenomena of nonlinear dynamics, well-known to exhibit rather complex motion including period doubling, fold and pitchfork bifurcations, let alone the global bifurcations leading to chaotic or rotational motion. In this thesis, the potential of establishing rotational motion is studied considering the bobbing motion of ocean waves as the source of excitation of a oating pendulum. The challenge within this investigation lies on the fact that waves are random, as well as their observed low frequency, characteristics which pose a broader signi cance within the study of vibrating systems. Thus, a generic study is conducted with the parametric pendulum being excited by a narrow-band stochastic process and particularly, the random phase modulation is utilized. In order to explore the dynamics of the stochastic system, Markov-chain Monte-Calro simulations are performed to acquire a view on the in uence of randomness onto the parameter regions leading to rotational response. Furthermore, the Probability Density Function of the response is calculated, applying a numerical iterative scheme to solve the total probability law, exploiting the Chapman-Kolmogorov equation inherent to Markov processes. A special case of the studied structure undergoing impacts is considered to account for extreme weather conditions and nally, a novel design is investigated experimentally, aiming to set the ground for future development

    5th EUROMECH nonlinear dynamics conference, August 7-12, 2005 Eindhoven : book of abstracts

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    5th EUROMECH nonlinear dynamics conference, August 7-12, 2005 Eindhoven : book of abstracts

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    Nonlinear dynamics and chaos: Their relevance to safe engineering design

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    As many engineering systems are neither linear nor nearly linear, they are normally modelled by nonlinear equations for which closed-form analytical solutions are unobtainable. However with the advent of powerful computers, equations can be readily integrated numerically, so that the response from a given set of starting conditions is easily established. Unlike linear systems where all initial conditions lead to one type of motion, be it to an equilibrium point or to a harmonic oscillation, nonlinear systems can exhibit chaotic transients which can setlle down to a rich and complex variety of competing steady state solutions. Associated with each steady state solution is its basin of attraction. Under the variation of a control parameter, as the attractors move and bifurcate, the basins also undergo corresponding changes and metamorphoses. Associated with the homoclinic tangling of the invariant manifolds of the saddle solution, basin boundaries can change in nature from smooth to fractal, resulting in regions of chaotic transients. The aim of the thesis is to investigate how the size and nature of the basin of attraction changes with a control parameter. We show that there can exist a rapid loss of engineering integrity accompanying the rapid erosion and stratification of the basin. We explore the engineering significance of the basin erosions that occur under increased forcing. Various measures of engineering integrity are introduced: a global measure assesses the overall basin area; a local measure assesses the distance from the attractor to the basin boundary; a velocity measure is related to the size of impulse that could be sustained without failure; and a stochastic integrity measure assesses the stability of an attractor subjected to an external noise excitation. Since engineering systems may be subjected to pulse loads of finite duration, attention is given to both the absolute and transient basins of attraction. The significant erosion of these at homoclinic tangencies is particularly highlighted in the present study, the fractal basins having a severely reduced integrity under all four criteria. We also apply the basin erosion phenomena to the problem of ship capsize. We make a numerical analysis of the steady state and transient motions of the semi-empirical nonlinear differential equations, which have been used to model the resonant rolling motions of real ships. Examinadon of the safe basin in the space of the starting conditions shows that transient capsizes can occur at a wave height that is a small fraction of that at which the final steady state motions lose their stability. It is seen that the basin is eroded quite suddenly throughout its central region by gross striations, implying that transient capsize might be a reasonably repeatable phenomenon, offering a new approach to the quantification of ship stability in waves. We conclude from this thesis that the stability of nonlinear engineering systems may, in the future, be based on the basin erosion phenomenon relating to chaotic transients and incursive fractals

    A New Method to Predict Vessel Capsizing in a Realistic Seaway

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    A recently developed approach, in the area of nonlinear oscillations, is used to analyze the single degree of freedom equation of motion of a oating unit (such as a ship) about a critical axis (such as roll). This method makes use of a closed form analytic solution, exact upto the rst order, and takes into account the the complete unperturbed (no damping or forcing) dynamics. Using this method very-large-amplitude nonlinear vessel motion in a random seaway can be analysed with techniques similar to those used to analyse nonlinear vessel motions in a regular (periodic) or random seaway. The practical result being that dynamic capsizing studies can be undertaken considering the shortterm irregularity of the design seaway. The capsize risk associated with operation in a given sea state can be evaluated during the design stage or when an operating area change is being considered. Moreover, this technique can also be used to guide physical model tests or computer simulation studies to focus on critical vessel and environmental conditions which may result in dangerously large motion amplitudes. Extensive comparitive results are included to demonstrate the practical usefulness of this approach. The results are in the form of solution orbits which lie in the stable or unstable manifolds and are then projected onto the phase plane
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