Safety criteria for aperiodic dynamical systems

The use of dynamical system models is commonplace in many areas of science and engineering. One is often interested in whether the attracting solutions in these models are robust to perturbations of the equations of motion. This question is extremely important in situations where it is undesirable to have a large response to perturbations for reasons of safety. An especially interesting case occurs when the perturbations are aperiodic and their exact form is unknown. Unfortunately, there is a lack of theory in the literature that deals with this situation. It would be extremely useful to have a practical technique that provides an upper bound on the size of the response for an arbitrary perturbation of given size. Estimates of this form would allow the simple determination of safety criteria that guarantee the response falls within some pre-specified safety limits. An excellent area of application for this technique would be engineering systems. Here one is frequently faced with the problem of obtaining safety criteria for systems that in operational use are subject to unknown, aperiodic perturbations. In this thesis I show that such safety criteria are easy to obtain by using the concept of persistence of hyperbolicity. This persistence result is well known in the theory of dynamical systems. The formulation I give is functional analytic in nature and this has the advantage that it is easy to generalise and is especially suited to the problem of unknown, aperiodic perturbations. The proof I give of the persistence theorem provides a technique for obtaining the safety estimates we want and the main part of this thesis is an investigation into how this can be practically done. The usefulness of the technique is illustrated through two example systems, both of which are forced oscillators. Firstly, I consider the case where the unforced oscillator has an asymptotically stable equilibrium. A good application of this is the problem of ship stability. The model is called the escape equation and has been argued to capture the relevant dynamics of a ship at sea. The problem is to find practical criteria that guarantee the ship does not capsize or go through large motions when there are external influences like wind and waves. I show how to provide good criteria which ensure a safe response when the external forcing is an arbitrary, bounded function of time. I also consider in some detail the phased-locked loop. This is a periodically forced oscillator which has an attracting periodic solution that is synchronised (or phase-locked) with the external forcing. It is interesting to consider the effect of small aperiodic variations in the external forcing. For hyperbolic solutions I show that the phase-locking persists and I give a method by which one can find an upperbound on the maximum size of the response

Dynamics of a class of vortex rings

The contour dynamics method is extended to vortex rings with vorticity varying linearly from the symmetry axis. An elliptic core model is also developed to explain some of the basic physics. Passage and collisions of two identical rings are studied focusing on core deformation, sound generation and stirring of fluid elements. With respect to core deformation, not only the strain rate but how rapidly it varies is important and accounts for greater susceptibility to vortex tearing than in two dimensions. For slow strain, as a passage interaction is completed and the strain relaxes, the cores return to their original shape while permanent deformations remain for rapidly varying strain. For collisions, if the strain changes slowly the core shapes migrate through a known family of two-dimensional steady vortex pairs up to the limiting member of the family. Thereafter energy conservation does not allow the cores to maintain a constant shape. For rapidly varying strain, core deformation is severe and a head-tail structure in good agreement with experiments is formed. With respect to sound generation, good agreement with the measured acoustic signal for colliding rings is obtained and a feature previously thought to be due to viscous effects is shown to be an effect of inviscid core deformation alone. For passage interactions, a component of high frequency is present. Evidence for the importance of this noise source in jet noise spectra is provided. Finally, processes of fluid engulfment and rejection for an unsteady vortex ring are studied using the stable and unstable manifolds. The unstable manifold shows excellent agreement with flow visualization experiments for leapfrogging rings suggesting that it may be a good tool for numerical flow visualization in other time periodic flows

Statistical energy analysis of engineering structures

This thesis was submitted for the degree of Doctor of Philosophy and awarded by Brunel University.This thesis examines the fundamental equations of the branch of linear oscillatory dynamics known as Statistical Energy Analysis (SEA). The investigation described is limited to the study of two, point coupled multi-modal sub-systems which form the basis for most of the accepted theory in this field. Particular attention is paid to the development of exact classical solutions against which simplified approaches can be compared. These comparisons reveal deficiencies in the usual formulations of SEA in three areas, viz., for heavy damping, strong coupling between sub-systems and for systems with non-uniform natural frequency distributions. These areas are studied using axially vibrating rod models which clarify much of the analysis without significant loss of generality. The principal example studied is based on part of the structure of a modem warship. It illustrates the simplifications inherent in the models adopted here but also reveals the improvements that can be made over traditional SEA techniques. The problem of heavy damping is partially overcome by adopting revised equations for the various loss factors used in SEA. These are shown to be valid provided that the damping remains proportional so that inter-modal coupling is not induced by the damping mechanism. Strong coupling is catered for by the use of a correction factor based on the limiting case of infinite coupling strength, for which classical solutions may be obtained. This correction factor is used in conjunction with a new, theoretically based measure of the transition between weakly and strongly coupled behaviour. Finally, to explore the effects of non-uniform natural frequency distributions, systems with geometrically periodic and near-periodic parameters are studied. This important class of structures are common in engineering design and do not posses the uniform modal statistics commonly assumed in SEA. The theory of periodic structures is used in this area to derive more sophisticated statistical models that overcome some of these limitations

ftp ejde.math.swt.edu ftp ejde.math.unt.edu (login: ftp) Nonclassical Sturm-Liouville problems and

Schrödinger operators on radial trees

A numerical study of viscous vortex rings using a spectral method

Viscous, axisymmetric vortex rings are investigated numerically by solving the incompressible Navier-Stokes equations using a spectral method designed for this type of flow. The results presented are axisymmetric, but the method is developed to be naturally extended to three dimensions. The spectral method relies on divergence-free basis functions. The basis functions are formed in spherical coordinates using Vector Spherical Harmonics in the angular directions, and Jacobi polynomials together with a mapping in the radial direction. Simulations are performed of a single ring over a wide range of Reynolds numbers (Re approximately equal gamma/nu), 0.001 less than or equal to 1000, and of two interacting rings. At large times, regardless of the early history of the vortex ring, it is observed that the flow approaches a Stokes solution that depends only on the total hydrodynamic impulse, which is conserved for all time. At small times, from an infinitely thin ring, the propagation speeds of vortex rings of varying Re are computed and comparisons are made with the asymptotic theory by Saffman. The results are in agreement with the theory; furthermore, the error is found to be smaller than Saffman's own estimate by a factor square root ((nu x t)/R squared) (at least for Re=0). The error also decreases with increasing Re at fixed core-to-ring radius ratio, and appears to be independent of Re as Re approaches infinity). Following a single ring, with Re=500, the vorticity contours indicate shedding of vorticity into the wake and a settling of an initially circular core to a more elliptical shape, similar to Norbury's steady inviscid vortices. Finally, we consider the case of leapfrogging vortex rings with Re=1000. The results show severe straining of the inner vortex core in the first pass and merging of the two cores during the second pass