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.     

Optimal configurations for dual-spin satellites subject to gravitational torques

Dual-spin or gyrostat satellites subject to gravitational torques can adopt an infinite number of possible equilibria obtained by adjusting the magnitude and direction of the rotor angular momentum within the satellite. This paper seeks to answer the question, which of these equilibria is best — and best is chosen here to mean most stable in the sense that the energy required to perturb the orientation by any prescribed amount is maximized, i.e. the smallest eigenvalue of the Hessian matrix of the dynamic potential energy is maximized. Using this criterion, it is shown that the conventional configuration for dual-spin satellites with the angular momentum of the rotor, the spacecraft principal axis of maximum moment of inertia, and the perpendicular to the orbital plane coincident is not always the best orientation. The optimal configuration is shown to have the minimum moment of inertia always aligned with the local vertical, but the principal axis of maximum moment of inertia, shifts from the perpendicular to the orbital plane to lying in-plane as the angular momentum of the rotor is increased from zero (corresponding to a rigid gravity gradient satellite) to some sufficiently large value which is determined as a function of parameters. For angular momentum greater than this value, global optimality is established analytically, and otherwise local optimality is proved analytically with global optimality demonstrated numerically.     

Short-axis-mode rotation of a free rigid body by perturbation series

A simple rearrangement of the torque free motion Hamiltonian shapes it as a perturbation problem for bodies rotating close to the principal axis of maximum inertia, independently of their triaxiality. The complete reduction of the main part of this Hamiltonian via the Hamilton–Jacobi equation provides the action-angle variables that ease the construction of a perturbation solution by Lie transforms. The lowest orders of the transformation equations of the perturbation solution are checked to agree with Kinoshita’s corresponding expansions for the exact solution of the free rigid body problem. For approximately axisymmetric bodies rotating close to the principal axis of maximum inertia, the common case of major solar system bodies, the new approach is advantageous over classical expansions based on a small triaxiality parameter.     

Equivalent problems in rigid body dynamics — I

The general problem of motion of a rigid body about a fixed point under the action of stationary non-symmetric potential and gyroscopic forces is considered. The equations of motion in the Euler-Poisson form are derived. An interpretation is given in terms of charged, magnetized gyrostat moving in a superposition of three classical fields. As an example, the problem of motion of a satellite — gyrostat on a circular orbit with respect to its orbital system is reduced to that of its motion in an inertial system under additional magnetic and Lorentz forces.When the body is completely symmetric about one of its axes passing through the fixed point, the above problem is found to be equivalent to another one, in which the body has three equal moments of inertia and the forces are symmetric around a space axis. The last problem is well-studied and the given analogy reveals a number of integrable cases of the original problem. A transformation is found, which gives from each of these cases a class of integrable cases depending on an arbitrary function. The equations of motion are also reduced to a single equation of the second order.     

The eccentric case of a fast-rotating,gravity-gradient-perturbed satellite attitude solution

A closed-form first-order perturbation solution for the attitude evolution of a triaxial space object in an elliptical orbit is presented. The solution, derived using the Lie–Deprit method, takes into account gravity-gradient torque and is facilitated by an assumption of fast rotation of the object. The formulation builds on the earlier implementation of Lara and Ferrer, which assumes a circular orbit. The previously presented work—which assumes spin about an object’s axis of maximum inertia—is further extended by the explicit presentation of the transformations required to apply the solution to an object spinning about its axis of minimum inertia. Additionally, several numerical analyses are presented to more completely assess the utility of the solution. These studies (1) validate the elliptical solution, (2) assess the impact of varying the small parameter of the perturbation procedure, (3) analyze the assumption of fast rotation, and (4) apply the solution to the common and important scenario of a tumbling rocket body.     

Stabilization due to gyroscopic coupling in dual-spin satellites subject to gravitational torques

The stability of attitude equilibria relative to gravitational torques for a rigid satellite in a circular orbit has been divided into three inertia regions, the Lagrange region of assured Liapunov stability, the Beletskii-Delp region which is often described as stabilized due to gyroscopic coupling, and an assured instability region. The generalization of these regions to the case of dual-spin or gyrostat satellites whose internal spin momentum is along a principal axis is treated here. The stability boundaries are obtained for all possible equilibrium orientations for such vehicles, and the variations of these boundaries corresponding to changes in the internal momentum magnitude, or to aligning the momentum with a different principal axis, are determined.Alexander von Humboldt Research Fellow at the Institut für Mechanik; on sabbatical leave from Columbia University, New York, U.S.A.     

The motion of a symmetric gyrostat around the center of mass in a circular orbit in the presence of viscoelastic bars and disc

In the present work we consider asymmetric gyrostat which has a homogeneous viscoelastic disc and two bars attached to it. Furthermore, the gyrostat has a rotor oriented inside it such that the rotor is statically and dynamically balanced. This sytem has a rotational motion around its center of mass in a circular orbit under a central gravitational field. Bending vibrations of the bars and the disc are accompanied by dissipation of energy, which is the cause of the evolution of the system's rotational motion. Using the method of separation of motion and averaging, the approximate equations describing the evolution of rotational motion in terms of Andoyer canonical variables are obtained. The stationary motions for the system are deduced, together with the conditions of its stability.     

Tidal friction in close-in satellites and exoplanets: The Darwin theory re-visited

This report is a review of Darwin’s classical theory of bodily tides in which we present the analytical expressions for the orbital and rotational evolution of the bodies and for the energy dissipation rates due to their tidal interaction. General formulas are given which do not depend on any assumption linking the tidal lags to the frequencies of the corresponding tidal waves (except that equal frequency harmonics are assumed to span equal lags). Emphasis is given to the cases of companions having reached one of the two possible final states: (1) the super-synchronous stationary rotation resulting from the vanishing of the average tidal torque; (2) capture into the 1:1 spin-orbit resonance (true synchronization). In these cases, the energy dissipation is controlled by the tidal harmonic with period equal to the orbital period (instead of the semi-diurnal tide) and the singularity due to the vanishing of the geometric phase lag does not exist. It is also shown that the true synchronization with non-zero eccentricity is only possible if an extra torque exists opposite to the tidal torque. The theory is developed assuming that this additional torque is produced by an equatorial permanent asymmetry in the companion. The results are model-dependent and the theory is developed only to the second degree in eccentricity and inclination (obliquity). It can easily be extended to higher orders, but formal accuracy will not be a real improvement as long as the physics of the processes leading to tidal lags is not better known.     

Synchronous spin-orbital resonance locking of large planetary satellites

A numerical investigation of the chaotic rotation of large planetary satellites before their synchronous spin-orbital resonance locking with regard to tidal friction is carried out. The rotational dynamics of seven large satellites greater than 1000 km in diameter and with known inertial parameters (Io, Europa, Ganymede, Callisto (J1–J4), Tethys (S3), Iapetus (S8), and Ariel (U1)) in the epoch of synchronous resonance locking is modeled. All of these satellites have a small dynamic asymmetry. The planar case is considered in which the satellite’s axis of rotation is orthogonal to the plane of orbit. The satellites possessing an initial rapid rotation pass through various resonant states during the tidal evolution. Here, the probability of their locking into these states exists. The numerical experiments presented in this paper have shown that, with a rather high arbitrariness in the choice of initial states, the satellites during the course of the tidal evolution of their rotational motion have passed without interruption through the regions of the 5: 2, 2: 1, and 3: 2 resonances in the phase space and are locked into the 1: 1 resonance. The estimate for the tidal deceleration time is obtained both theoretically and on the numerical experimental basis.     

Exact solution of a triaxial gyrostat with one rotor

The problem of the attitude dynamics of a triaxial gyrostat under no external torques and one constant internal rotor, is a three degrees-of-freedom system, although thanks to the existence of integrals of motion it can be reduced to only one degree-of-freedom problem. We introduce coordinates to represent the orbits of constant angular momentum as a flow on a sphere. This representation shows that the problem is equivalent to a quadratic Hamiltonian depending on two parameters. We find the exact solution of the orbits in terms of elliptic functions. By making use of properties of elliptic functions we find the solution at each region of the parametric partition from the solution of one region. We also prove that heteroclinic orbits are planar curves.     

Rotational evolution of a symmetric gyrostat with visco-elastic bars around the center of mass in a circular orbit

The motion of a gyrostat in a circular orbit in a Newtonian field of force is considered. The gyrostat has four homogeneous viscoelastic bars attached to it. Rotation of the symmetric rotor inside the rigid body is statically and dynamically balanced. Bending deformations of the bars, accompanied by dissipation of energy, are the cause of the evolution of the system's rotational motion. Approximate equations describing this evolution are derived, together with averaged equations in Andoyer variables.     

How fast do Galilean satellites spin?

Each of the Galilean satellites, as well as most other satellites whose initial rotations have been substantially altered by tidal dissipation, has been widely assumed to rotate synchronously with its orbital mean motion. Such rotation would require a small permanent asymmetry in the mass distribution in order to overcome the small mean tidal torque. Since Io and Europa may be substantially fluid, they may not have the strenght to support the required permanent asymmetry. Thus, each may rotate at the unknown but slightly nonsynchronous rate that corresponds to zero mean tidal torque. This behaviour may be observable by Galileo spacecraft imaging. It may help explain the longitudinal variation of volcanism on Io and the cracking of Europa's crust.     

Exact analytic solutions for the rotation of an axially symmetric rigid body subjected to a constant torque

New exact analytic solutions are introduced for the rotational motion of a rigid body having two equal principal moments of inertia and subjected to an external torque which is constant in magnitude. In particular, the solutions are obtained for the following cases: (1) Torque parallel to the symmetry axis and arbitrary initial angular velocity; (2) Torque perpendicular to the symmetry axis and such that the torque is rotating at a constant rate about the symmetry axis, and arbitrary initial angular velocity; (3) Torque and initial angular velocity perpendicular to the symmetry axis, with the torque being fixed with the body. In addition to the solutions for these three forced cases, an original solution is introduced for the case of torque-free motion, which is simpler than the classical solution as regards its derivation and uses the rotation matrix in order to describe the body orientation. This paper builds upon the recently discovered exact solution for the motion of a rigid body with a spherical ellipsoid of inertia. In particular, by following Hestenes’ theory, the rotational motion of an axially symmetric rigid body is seen at any instant in time as the combination of the motion of a “virtual” spherical body with respect to the inertial frame and the motion of the axially symmetric body with respect to this “virtual” body. The kinematic solutions are presented in terms of the rotation matrix. The newly found exact analytic solutions are valid for any motion time length and rotation amplitude. The present paper adds further elements to the small set of special cases for which an exact solution of the rotational motion of a rigid body exists.     

Empiric models of the Earth’s free core nutation

Free core nutation (FCN) is the main factor that limits the accuracy of the modeling of the motion of Earth’s rotational axis in the celestial coordinate system. Several FCN models have been proposed. A comparative analysis is made of the known models including the model proposed by the author. The use of the FCN model is shown to substantially increase the accuracy of the modeling of Earth’s rotation. Furthermore, the FCN component extracted from the observed motion of Earth’s rotational axis is an important source for the study of the shape and rotation of the Earth’s core. A comparison of different FCN models has shown that the proposed model is better than other models if used to extract the geophysical signal (the amplitude and phase of FCN) from observational data.     

Attitude stability of a gravity-stabilized gyrostat satellite

This paper treats analytically the problem of the stability of the attitude motions of a gravity-stabilized gyrostat satellite that is in a circular orbit around a spherical planet. The vehicle considered consists of a body with no special symmetries that has any number of rotors attached to it. The internal angular momentum vector due to these rotors is parallel to one of the principal axes of the entire satellite; this axis is aligned with (or close to) the normal to the orbit plane. Both the cases in which each rotor is driven by a motor at a constant spin rate relative to the main body of the vehicle and the one in which each rotor is rotating freely, without any friction, are treated. Stability (both infinitesimal and in the sense of Liapunov) of the attitude motions of the vehicle can be quickly predicted by using the results derived here, which are summarized in the form of a continuous, three-dimensional, stability diagram.National Research Council Post-Doctoral Research Associate at NASA-Ames Research Center, Moffett Field, Calif. Parts of this research were included in [12].     

First-order theory of satellite attitude motion application to HIPPARCOS

This work aims at finding an analytic solution corresponding to the attitude evolution in space of a satellite submitted to disturbing torques. This paper presents a basic frame applicable to any perturbed rotation satellite, and a method of resolution leading to a formal solution which is given here to the first order. Thus, the main problem is the slow rotation of a body with three unequal axes of inertia, essentially submitted to a dominant solar radiation pressure torque, with the axis pointing far away from a position of equilibrium. The comparison of the results with a numerical integration based upon a HIPPARCOS model is convincing.     

Hybrid fuzzy sliding mode control for motorised space tether spin-up when coupled with axial and torsional oscillation

A specialised hybrid controller is applied to the control of a motorised space tether spin-up space coupled with an axial and a torsional oscillation phenomenon. A seven-degree-of-freedom (7-DOF) dynamic model of a motorised momentum exchange tether is used as the basis for interplanetary payload exchange in the context of control. The tether comprises a symmetrical double payload configuration, with an outrigger counter inertia and massive central facility. It is shown that including axial and torsional elasticity permits an enhanced level of performance prediction accuracy and a useful departure from the usual rigid body representations, particularly for accurate payload positioning at strategic points. A simulation with given initial condition data has been devised in a connecting programme between control code written in MATLAB and dynamics simulation code constructed within MATHEMATICA. It is shown that there is an enhanced level of spin-up control for the 7-DOF motorised momentum exchange tether system using the specialised hybrid controller.     

Secular Motion in a 2nd Degree and Order-Gravity Field with no Rotation

The motion of a particle about a non-rotating 2nd degree and order-gravity field is investigated. Averaging conditions are applied to the particle motion and a qualitative analysis which reveals the general character of motion in this system is given. It is shown that the orbit plane will either be stationary or precess about the body's axis of minimum or maximum moment of inertia. It is also shown that the secular equations for this system can be integrated in terms of trigonometric, hyperbolic or elliptic functions. The explicit solutions are derived in all cases of interest.This revised version was published online in October 2005 with corrections to the Cover Date.     

Magnetospheric asymmetries associated with the y-component of the IMF

Observed magnetospheric asymmetries which occur in response to the y-component of the IMF are discussed in terms of the open model of the magnetosphere. The torque which the IMF exerts on the magnetosphere about the Earth-Sun axis results in asymmetric addition of open flux tubes to the tail lobes about the noon-midnight meridian. In response an IMF-associated B_y field appears across the tail lobes. The ratio between internal and external B_y fields will generally be same as the ratio between internal and external electric fields. If the tail flux asymmetry is related to an asymmetric distribution of the field normal to the tail magnetopause then an asymmetry in tail lobe electric field and plasma populations will immediately result, as observed. If the flux asymmetry is associated with a twist in the tail then the B_y field will appear but not necessary the electric field and plasma asymmetries. Generally both effects may occur together. Simple open tail lobe models are derived which demonstrate the asymmetry effects. These represent more physically satisfactory models of the tail and its plasma populations than available hitherto, but they remain somewhat unrealistic in a number of respects. Finally, it is shown that the observed asymmetry effects on closed (auroral zone) field lines may be at least qualitatively accounted for if the cross-magnetosphere IMF-associated B_y field pervades not only the open but also the closed field line regime, as may be generally expected.     

The secular variation of pulsar magnetic dipole moments

The time dependences of the inertia tensor and of a dissipative torque caused by the nonleptonic weak interaction have been investigated for a certain class of pulsars with no solid core. Early in the life of the pulsar, the angular velocity vector is predicted to move with respect to fixed body axes in such a way that it becomes perpendicular to the magnetic dipole moment. During this motion, the solid outer shell suffers plastic deformation so that the dipole moment becomes approximately collinear with a principal axis. After 10⁴ or 10⁵ yr, the dissipative torque is negligibly small compared with the electromagnetic torque, the Euler equations are those for a simple rigid body, and alignment of spin and dipole moment occurs. If the dipole moment discussed by Lyneet al. (1975) is interpreted as being equal to the component perpendicular to the spin, its secular decay is a natural property of this model and is not a consequence of field decay through electrical resistivity.