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(searched for: doi:10.2514/8.6410)
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Computational Mathematics and Mathematical Physics, Volume 60, pp 74-81; https://doi.org/10.1134/s0965542520010091

Abstract:
Computer algebra and numerical methods were used to investigate the properties of a nonlinear algebraic system determining the equilibrium orientations of a system of two bodies connected by a spherical hinge that move in a circular orbit under the action of a gravitational torque. Primary attention was given to equilibrium orientations of the two-body system in the special cases when one of the principal axes of inertia of both the first and second body coincides with the normal to the orbital plane, the radius vector, or the tangent to the orbit. To determine the equilibrium orientations of the two-body system, the set of stationary algebraic equations of motion was decomposed into nine subsystems. The system of algebraic equations was solved by applying algorithms for constructing Gröbner bases. The equilibrium positions were determined by numerically analyzing the roots of the algebraic equations from the constructed Gröbner basis.
, S.A. Mirer, V.V. Sazonov
Published: 31 October 1976
Acta Astronautica, Volume 3, pp 651-669; https://doi.org/10.1016/0094-5765(76)90104-1

Abstract:
The paper studies the dynamics of a satellite with gravitational attitude control. It discusses small oscillations of the satellite-stabilizer system in the plane of a circular orbit in the vicinity of the two equilibrium positions most interesting from the practical standpoint. The necessary and sufficient conditions are obtained for asymptotic stability of these equilibrium positions. The values of the parameters at which the maximal speed of response of the system is achieved are analytically determined.
R. Rangarajan, R. C. Flanagan
Celestial Mechanics and Dynamical Astronomy, Volume 12, pp 231-249; https://doi.org/10.1007/bf01230215

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R. L. Borrelli, I. P. Leliakov
Journal of Optimization Theory and Applications, Volume 10, pp 344-361; https://doi.org/10.1007/bf00935399

Abstract:
The principal result of this paper is an analytical technique for optimizing the transient response characteristics for a class of passively stabilized satellites in the presence of practical design parameter constraints. The technique developed is directed at linearized models and is treated in detail only for quartic characteristic equations. The results, even for the special case treated, are sufficiently broad to provide a generalization of some techniques which have appeared in the literature. Briefly, it is shown that certain coalescent root configuration represent a hierarchy of transient optima, and conditions are derived which give the optimum transient characteristics as functions of the available variable system parameters.
V. A. Sarychev
Published: 1 January 1970
Gas Accretion onto Galaxies pp 574-583; https://doi.org/10.1007/978-94-011-7551-7_40

Abstract:
The dynamics of a satellite with a gravitational stabilization system is investigated. The parameters providing minimum duration of the natural oscillation damping of the system are determined. Main attention is given to the analysis of oscillation of the system in the orbit plane.
Hermann Stümke
Grundzüge der Flugmechanik und Ballistik pp 441-486; https://doi.org/10.1007/978-3-663-07250-8_6

Abstract:
Solange sich drehsymmetrische Flugkörper in tieferen Luftschichten bewegen und damit unter dem Einfluß von merklichen Luftkräften stehen, sind ihre Flugleistungen ebenso wie die der spiegelsymmetrischen Flugzeuge wesentlich von der richtigen Orientierung ihrer Symmetrieachse relativ zur Bewegungsrichtung abhängig. Eine Analyse der möglichen Drehbewegungen ist daher wieder unerläßlich. Als neues Element gegenüber spiegelsymmetrischen Flugzeugen erscheint hier die Möglichkeit der Stabilisierung mittels schneller Drehbewegungen um die Längsachse. Die damit verbundenen Fragen der sekundären Ballistik stehen im Mittelpunkt der folgenden Ausführungen.
V. A. Sarychev
Published: 1 January 1966
Applied Mechanics pp 429-435; https://doi.org/10.1007/978-3-662-29364-5_58

Abstract:
One of the important lines in the development of the technique of space flights is the creation of stabilized Earth satellites. The solution of this problem allows one to recover to Earth a satellite or a container with the results of observations, to create satellites-reflectors used for purposes of radiocommunication, television, etc. Depending on the tasks settled, orientation of an artificial satellite can be achieved by active (with energy consumption for orientation) or passive (not requiring energy) means. The use of active stabilization systems in projects with long life and high requirements for the accuracy of stabilization is connected with considerable consumption of energy or the working medium, with the weight and complexity of these systems. In many cases the use of satellite passive stabilization systems is preferable. The advantage of passive stabilization systems, as compared to active systems, is displayed especially distinctly in projects with long life if the stabilization accuracy of the order 1–5° is required. These requirements are satisfied, for instance, by meteorological satellites, by communication satellites and by many satellites launched for the purposes of scientific experimentation.
J. A. Lewis, E. E. Zajac
Published: 1 November 1964
Bell System Technical Journal, Volume 43, pp 2705-2765; https://doi.org/10.1002/j.1538-7305.1964.tb01025.x

Abstract:
This article gives the results of an analytical and numerical study of a two-gyro, gravity-oriented communications satellite. The principal purpose of the study was to uncover and solve the analytical problems arising in the design of passive gravity-gradient altitude control systems. Although the study was directed at satellite orientation, it is felt that many of the techniques developed have general use in the investigation of dynamical systems. We consider both small and large motions about the desired earth-pointing orientation. In the small-motion study, the goal is simultaneous optimization of the transient response and the forced response to perturbations caused by orbital eccentricity, magnetic torques, solar torques, thermal rod bending, and micrometeorite impact. In the large-motion study, we enumerate all possible equilibrium positions of the satellite and then consider initial despin after injection into orbit, inversion of the satellite from one stable equilibrium position to another by switching of gyro bias torques, and the decay of transient motions resulting from large initial angular rates. As a specific numerical example, we have treated a 300-lb satellite in a 6000-nm orbit, stabilized by a 60-ft extensible rod with a 20-lb tip mass, and by two single degree-of-freedom gyros, each with an angular momentum of 106 cgs units. Without a detailed discussion of hardware, it is concluded that such a system, having a total weight of 50 to 75 pounds including power supply, will provide a settling time for small disturbances of less than one orbit and will hold the antenna pointing error within a few degrees.
B. Paul, J. W. West, E. Y. Yu
Published: 1 September 1963
Bell System Technical Journal, Volume 42, pp 2195-2238; https://doi.org/10.1002/j.1538-7305.1963.tb00964.x

Abstract:
It is shown how the gravity-gradient effect may be utilized to design a long-lived, earth-pointing satellite attitude control system which requires no fuel supplies, attitude sensors or active control equipment. This two-body system is provided with a magnetic hysteresis damper which effectively damps out oscillations (librations) about the local vertical. The long rods, which must be extended in space from coiled up metal tapes, provide the required large moments of inertia and possess adequate rigidity and sufficient strength to endure the rigors of the extension process. The system is compatible with the requirements of multiple satellite launchings from a single last-stage vehicle. Analysis indicates that the gravitational torques are sufficient to keep the disturbing effects of solar radiation pressure, residual magnetic dipole moments, orbit eccentricity, rod curvature, eddy currents, and meteorite impacts within tolerable limits. It is believed that the high-performance, earth-pointing system described and analyzed in this paper represents an essential step in the development of high-capacity communications satellites requiring long life
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