http://hdl.handle.net/1721.1/7841 Search the Channel search http://dspace.mit.edu/simple-search http://hdl.handle.net/1721.1/34616 Title: Rocking and rolling down an incline : the dynamics of nested cylinders on a ramp <br/> <br/>Authors: Vener, David Paul <br/> <br/>Abstract: In this thesis, I report the results of a combined experimental and theoretical investigation of a journal bearing, specifically, a cylinder suspended in a viscous fluid housed within a cylindrical shell, rolling down an incline under the influence of gravity. Particular attention is given to rationalizing the distinct modes of motion observed. We performed a series of experiments in which the inner cylinder density and the fluid viscosity were varied. Three distinct types of behavior were observed. First, in what we shall call the "rocking" mode, after an initial settling period, the shell rocks back and forth without moving down the ramp. Second, we observed "slow, quasi-steady rolling"; this mode is characterized by the system proceeding down the hill at essentially a constant velocity. Finally, the cylinders roll down the incline with constant acceleration; we shall call this mode "unbounded acceleration." An accompanying theoretical model is developed and enables us to rationalize the rocking and accelerating modes. In the rocking solutions, potential and kinetic energy are dissipated in the fluid as the inner cylinder approaches the bottom of the outer cylinder.; (cont.) In the accelerating solutions, the whole system moves as a solid body so that no dissipation occurs and potential energy is continually converted into kinetic energy. In order to understand the quasi-steady motion, we analyze the motion of a similar system: a metal cylinder is placed inside a larger plastic cylinder filled with fluid and attached to a motor which fixes the larger cylinder's rotation rate. Our observations of this system, specifically, the differences between experiments and theory lead us to consider the effect of internal friction due to surface roughness. The resulting model's predictions are well supported by our observations. Finally, to rationalize the slow, quasi-steady rolling motion of the system, we incorporate surface roughness and cavitation into the theoretical model. These effects provide a restoring force on the inner cylinder; however, we find that surface roughness is the dominant effect. <br/> <br/>Description: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; Includes bibliographical references (p. 109-111). Sat, 29 Oct 2005 22:58:59 GMT http://hdl.handle.net/1721.1/34543 Title: Three-dimensional solitary waves in dispersive wave systems <br/> <br/>Authors: Kim, Boguk, Ph. D. Massachusetts Institute of Technology <br/> <br/>Abstract: Fully localized three-dimensional solitary waves, commonly referred to as 'lumps', have received far less attention than two-dimensional solitary waves in dispersive wave systems. Prior studies have focused in the long-wave limit, where lumps exist if the long-wave speed is a minimum of the phase speed and are described by the Kadomtsev-Petviashvili (KP) equation. In the water-wave problem, in particular, lumps of the KP type are possible only in the strong-surface-tension regime (Bond number, B > 1/3), a condition that limits the water depth to a few mm. In the present thesis, a new class of lumps is found that is possible under less restrictive physical conditions. Rather than long waves, these lumps bifurcate from infinitesimal sinusoidal waves of finite wavenumber at an extremum of the phase speed. As the group and phase velocities are equal there, small-amplitude lumps resemble fully localized wavepackets with envelope and crests moving at the same speed, and the wave envelope along with the induced mean-flow component are governed by a coupled Davey-Stewartson equation system of elliptic-elliptic type. The lump profiles feature algebraically decaying tails at infinity owing to this mean flow. In the case of water waves, lumps of the wavepacket type are possible when both gravity and surface tension are present on water of finite or infinite depth for B < 1/3.; (cont.) The asymptotic analysis of these lumps in the vicinity of their bifurcation point at the minimum gravity-capillary phase speed, is in agreement with recent fully numerical computations by Parau, Cooker & Vanden-Broeck (2005) as well as a formal existence proof by Groves & Sun (2005). A linear stability analysis of the gravity-capillary solitary waves that also bifurcate at the minimum gravity-capillary phase speed, reveals that they are always unstable to transverse perturbations, suggesting a mechanism for the generation of lumps. This generation mechanism is explored in the context of the two-dimensional Benjamin (2-DB) equation, a generalization to two horizontal spatial dimensions of the model equation derived by Benjamin (1992) for uni-directional, small-amplitude, long interfacial waves in a two-fluid system with strong interfacial tension. The 2-DB equation admits solitary waves and lumps of the wavepacket type analogous to those bifurcating at the minimum gravity-capillary phase speed in the water-wave problem. Based on unsteady numerical simulations, it is demonstrated that the transverse instability of solitary waves of the 2-DB equation results in the formation of lumps, which propagate stably and are thus expected to be the asymptotic states of the initial-value problem for fully localized initial conditions. <br/> <br/>Description: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2006.; Includes bibliographical references (p. 119-122). Sat, 29 Oct 2005 22:58:59 GMT http://hdl.handle.net/1721.1/27866 Title: Mesh generation for implicit geometries <br/> <br/>Authors: Persson, Per-Olof, 1973- <br/> <br/>Abstract: We present new techniques for generation of unstructured meshes for geometries specified by implicit functions. An initial mesh is iteratively improved by solving for a force equilibrium in the element edges, and the boundary nodes are projected using the implicit geometry definition. Our algorithm generalizes to any dimension and it typically produces meshes of very high quality. We show a simplified version of the method in just one page of MATLAB code, and we describe how to improve and extend our implementation. Prior to generating the mesh we compute a mesh size function to specify the desired size of the elements. We have developed algorithms for automatic generation of size functions, adapted to the curvature and the feature size of the geometry. We propose a new method for limiting the gradients in the size function by solving a non-linear partial differential equation. We show that the solution to our gradient limiting equation is optimal for convex geometries, and we discuss efficient methods to solve it numerically. The iterative nature of the algorithm makes it particularly useful for moving meshes, and we show how to combine it with the level set method for applications in fluid dynamics, shape optimization, and structural deformations. It is also appropriate for numerical adaptation, where the previous mesh is used to represent the size function and as the initial mesh for the refinements. Finally, we show how to generate meshes for regions in images by using implicit representations. <br/> <br/>Description: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.; Includes bibliographical references (p. 119-126). Fri, 29 Oct 2004 22:58:59 GMT http://hdl.handle.net/1721.1/40014 Title: Generalized straightening laws for products of determinants <br/> <br/>Authors: Taylor, Brian David <br/> <br/>Description: Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 1997.; Includes bibliographical references (p. 157-159). Tue, 29 Oct 1996 22:58:59 GMT