Soliman, Mohamed Sanaa; (1990) Nonlinear dynamics and chaos: Their relevance to safe engineering design. Doctoral thesis (Ph.D), UCL (University College London).

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Abstract

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.

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