Gravity-gradient stabilization of gyrostat satellites with rotor axes in principal planes

Three special classes of equilibrium orientations of gyrostat satellites subject to gravitational torques have been treated in the literature. Here we find the set of all equilibria for a restricted class of gyrostat configurations. Those configurations for which the internal angular momentum vector (or the rotor axis) is aligned with a principal axis have been treated in a separate work, where it is shown that at one, and only one, rotor speed there exists a continuum of equilibrium orientations. When the rotor axis is moved away from a principal axis in such a way that it is contained in a plane formed by two principal axes, it is shown that the continuum disappears, and we have a new set of eight equilibrium orientations which have not previously been described. The stability of these orientations is then investigated using the Hamiltonian as a Liapunov testing function. For properly chosen satellite inertia ratios some of these orientations are stable, and might be used in future gravitygradient stabilized satellites.This research is sponsored by the United States Air Force under Project RAND-Contract No. F44620-67-C-0045-monitored by the Directorate of Operational Requirements and Development Plans, Deputy Chief of Staff, Research and Development, Hq. USAF. Views or conclusions contained in this study should not be interpreted as representing the official opinion or policy of the United States Air Force. The material presented here was originally published in RAND Corporation Memorandum RM-5921-PR. The author wishes to acknowledge his indebtedness to Dr. R. E. Roberson for helpful discussions, and for suggesting a research area, part of which is treated here.     

An algorithm for finding the position of a point relative to a fixed polygonal boundary

This article describes a method that determines the position of a point relative to a fixed boundary. Its advantage over the other existing algorithms described below is that in most cases it is necessary that only part of the algorithm be executed to determine the position of the point. A possible inconvenience of using this algorithm is that the contour boundary has to be entered as data. For problems where the boundary often changes the algorithm is not suitable in its present form. The applications envisaged here are to cases where the position of a very large number of points have to be determined relative to one fixed polygonal boundary whose shape is known in advance, for example, blanking out of contours in automatic contouring, or selecting drill holes which lie in the particular part of a mineral deposit.     

Topological properties of disjoint channel networks within enclosed regions

Enclosure of some portion of one or more natural stream-drainage basins by superposition of a rectangle on a map of drainage network results in fragmentation of the natural basins into a set of disjoint channel networks. Each of these may have some channel links and forks of the natural network plus truncated links intersected by the enclosure boundary. The topological properties of the network elements in the enclosure are used to set up a model of random network patterns, in which the number of disjoint channel networks is expressed as a function of the number of links and forks in the enclosures. This function is shown to be a multiplicative constant times the square root of the number of links or forks. Empirical data on square and rectangular enclosures of several sizes from the Inez (Kentucky)Quadrangle map showed that the predicted multiplicative constants do not agree with observation, but that the square-root relation seems to hold at least to a first approximation. The models thus can be used as a norm against which deviations of real-world enclosures from network pattern randomness can be studied.     

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