Poincaré and the Three Body Problem

The purpose of the thesis is to present an account of Henri Poincare's famous memoir on the three body problem, the final version of which was published in Acta Mathematica in 1890 as the prize-winning entry in King Oscar II's 60th birthday competition. The memoir is reknowned both for its role in providing the foundations for Poincare's celebrated three volume Méthodes Nouvelles de la Mécanique Céleste, and for containing the first mathematical description of chaotic behaviour in a dynamical system. A historical context is provided both through consideration of the problem itself and through a discussion of Poincaré's earlier work which relates to the mathematics developed in the memoir. The organisation of the Oscar competition, which was undertaken by Gösta Mittag-Leffler, is also described. This not only provides an insight into the late 19th century European mathematical community but also reveals that after the prize had been awarded Poincare found an important error in his work and substantially revised the memoir prior to its publication in Acta. The discovery of a printed version of the original memoir personally annotated by Poincaré has allowed for a detailed comparative study of the mathematics contained in both versions of the memoir. The error is explained and it is shown that it was only as a result of its correction that Poincaré discovered the chaotic behaviour now associated with the memoir. The contemporary reception of the memoir is discussed and Poincaré's subsequent work in celestial mechanics and related topics is examined. Through the consideration of sources up to 1920 the influence and impact of the memoir on the progress of the three body problem and on dynamics in general is assessed

Control of colocated geostationary satellites

Control of the inter-satellite distances within a cluster of colocated satellites located in the same GEO window is examined with regards to the close approaches between pairs of satellites. Firstly, the orbital evolution and station keeping control of a single GEO satellite is examined and a new IBM PC based software program capable of performing both these functions autonomously from initial values of the orbital position and date is detailed and validated. Cluster design ideas are then examined in detail and the propagation software is used to generate data for a cluster of four satellites. Two test cases are examined to quantify the frequency of close approaches between individual satellite pairs, each test case using a different orbital element separation strategy but the same station keeping control scheme. The results of the study are then compared with previous research and discussions are presented on the advantages of each method. Finally, a cluster geometry correction manoeuvre, based on Hill's equations of relative motion, is presented which requires only those thrusters used by typical station keeping. This manoeuvre is integrated into the computer software and the two test cases noted previously are again propagated and the close approach results analysed to demonstrate the reduction in the number of close approaches below 5 km

The influence of gyroscopic forces on the dynamic behavior and flutter of rotating blades

The structural dynamics of a cantilever turbomachine blade mounted on a spinning and precessing rotor are investigated. Both stability and forced vibration are considered with a blade model that increases in complexity (and verisimilitude) from a spring-restrained point mass, to a uniform cantilever, to a twisted uniform cantilever turbomachine blade mounted on a spinning and precessing rotor are investigated. Both stability and forced vibration are considered with a blade model that increases in complexity (and verisimilitude) from a spring-restrained point mass, to a uniform cantilever, to a twisted uniform cantilever, to a tapered twisted cantilever of arbitrary cross-section. In every instance the formulation is from first principles using a finite element based on beam theory. Both ramp-type and periodic-type precessional angular displacements are considered. In concluding, forced vibrating and flutter are studied using the final and most sophisticated structural model. The analysis of stability is presented and a number of numerical examples are worked out

An Efficient Implementation of the GMC Micromechanics Model for Multi-Phased Materials with Complex Microstructures

An efficient implementation of the generalized method of cells micromechanics model is presented that allows analysis of periodic unidirectional composites characterized by repeating unit cells containing thousands of subcells. The original formulation, given in terms of Hill's strain concentration matrices that relate average subcell strains to the macroscopic strains, is reformulated in terms of the interfacial subcell tractions as the basic unknowns. This is accomplished by expressing the displacement continuity equations in terms of the stresses and then imposing the traction continuity conditions directly. The result is a mixed formulation wherein the unknown interfacial subcell traction components are related to the macroscopic strain components. Because the stress field throughout the repeating unit cell is piece-wise uniform, the imposition of traction continuity conditions directly in the displacement continuity equations, expressed in terms of stresses, substantially reduces the number of unknown subcell traction (and stress) components, and thus the size of the system of equations that must be solved. Further reduction in the size of the system of continuity equations is obtained by separating the normal and shear traction equations in those instances where the individual subcells are, at most, orthotropic. The reformulated version facilitates detailed analysis of the impact of the fiber cross-section geometry and arrangement on the response of multi-phased unidirectional composites with and without evolving damage. Comparison of execution times obtained with the original and reformulated versions of the generalized method of cells demonstrates the new version's efficiency

Development and analysis of a high fidelity linearized J₂ model for satellite formation flying

Thesis (S.M.)--Massachusetts Institute of Technology, Dept. of Aeronautics and Astronautics, 2001.Includes bibliographical references (p. 99-100).by Samuel A. Schweighart.S.M

Novel piecewise trajectory shaping in Hill's canonical variables

Shape-based methods have been proven to be computationally efficient techniques to quickly estimate the cost of low-thrust interplanetary trajectories. However, in some cases the solution is far from optimal, like in the case of the exponential sinusoid, or requires a special treatment when the motion is not completely planar. More recent developments allows for a full three-dimensional representation of the trajectory but either constraints need to be imposed on the thrust direction or approximations need to be introduced on the trajectory time-evolution, causing the domain of representable trajectories to shrink. As a consequence, trajectories transferring to highly inclined or highly eccentric orbits can lead to infeasible control laws. This paper presents a new analytical framework for the quick estimation of the $\Delta v$ and peak thrust of two-point boundary value low-thrust transfers. The novelty of this method is that it solves an inverse optimal control problem in Hill's canonical variables. The parameterisation in Hill's variables was selected so that the shaping of the in-plane and out-of-plane motions can be treated separately and the boundary conditions can be analytically satisfied. This choice leads to a computationally efficient extraction of the control profile and allows for the integration of known analytical solutions for the in-plane motion. The computation of the value of the objective function (usually the total $\Delta v$ or the spacecraft final mass) and path constraints is reduced to computationally inexpensive quadratures. The shaping proposed in this paper is piecewise continuous and allows for a flexible full three-dimensional representation of the trajectory. In particular, the out-of-plane motion is represented by piecewise continuous functions so that one can independently maximise both the change of inclination and the variation of the longitude of the ascending node. The method is applied to some well-known test cases, a rendezvous with Mars, asteroid 1989ML and comet Tempel-1, and the results compared to the solutions obtained with exponential sinusoid, pseudoequinoctial elements and spherical shaping

Large-Scale periodic solar velocities: An observational study

Observations of large-scale solar velocities were made using the mean field telescope and Babcock magnetograph of the Stanford Solar Observatory. Observations were made in the magnetically insensitive ion line at 5124 A, with light from the center (limb) of the disk right (left) circularly polarized, so that the magnetograph measures the difference in wavelength between center and limb. Computer calculations are made of the wavelength difference produced by global pulsations for spherical harmonics up to second order and of the signal produced by displacing the solar image relative to polarizing optics or diffraction grating