Bifurcations of the relative equilibria of a gyrostat satellite for a special case of the alignment of its gyrostatic moment

The bifurcations of the equilibria of a gyrostat satellite with a centre of mass moving uniformly in a circular Kepler orbit around an attracting centre are investigated. It is assumed that the axis of rotation of a statically and dynamically balanced flywheel rotating at a constant relative angular velocity is fixed in the principal central plane of inertia of the gyrostat containing the axis of its mean moment of inertia and that it is not collinear with any principal central axis of inertia of the system. The problem is solved in a direct formulation, that is, the whole set of equilibria with respect to the orbital system of coordinates of the gyrostat satellite is determined using the given moments of inertia, the value of the gyroscopic moment and the direction cosines of the axis of rotation of the flywheel and the changes in this set are investigated as a function of the bifurcation parameter, that is, the magnitude of the gyrostatic moment of the system. A parametric analysis of the relative equilibria of the three possible classes of equilibria for a system in a circular orbit in a central Newtonian force field is carried out using computer algebra facilities.     

Non-linear oscillations of a satellite at 1:1:1 resonance

The motion of a satellite about its centre of mass in a central Newtonian gravitational field is investigated. The satellite is considered to be a dynamically symmetrical rigid body. It is assumed that the ratio of the polar and equatorial principal central moments of inertia of the satellite is 4/3, or close to this. The orbit of the centre of mass is elliptic, and the orbit eccentricity is assumed to be small. In the limit case, when the orbit of the centre of mass is circular, a steady motion exists (corresponding to relative equilibrium of the satellite in the orbital system of coordinates) in which the axis of dynamic symmetry is directed along the velocity vector of the centre of mass of the satellite; here, the frequencies of the small linear oscillations of the axis of symmetry are equal or close to one another. But in an elliptic orbit of small eccentricity, multiple 1:1:1 resonance occurs in this case, as the oscillation frequencies mentioned are equal or close to the frequency of motion of the centre of mass of the satellite in orbit. The non-linear problem of the existence, bifurcations and stability of periodic motions of the satellite with a period equal to the rotation period of its centre of mass in orbit is investigated.     

The oscillations of a satellite about a direction fixed in absolute space

The stability of the plane oscillations of a satellite about the centre of mass in a central Newtonian gravitational field is investigated. The orbit of the centre of mass is circular and the principal central moments of inertia of the satellite are different. In unperturbed motion, one of the axes of inertia is perpendicular to the plane of the orbit, while the satellite performs periodic oscillations about a direction fixed in absolute space. The problem of the stability of these oscillations with respect to plane and spatial perturbations is investigated.     

Rotations of a near-symmetrical satellite in an elliptical orbit with Mercury-type resonance

The motion of a satellite, i.e., a rigid body, about to the centre of mass under the action of the gravitational moments of a central Newtonian gravitational field in an elliptical orbit of arbitrary eccentricity is investigated. It is assumed that the satellite is almost dynamically symmetrical. Plane periodic motions for which the ratio of the average value of the absolute angular velocity of the satellite to the average motion of its centre of mass is equal to 3/2 (Mercury-type resonance) are examined. An analytic solution of the non-linear problem of the existence of such motions and their stability to plane perturbations is given. In the special case in which the central ellipsoid of inertia of the satellite is almost spherical, the stability to spatial perturbations is also examined, but only in a linear approximation. ©2008.     

Chaos and optimal control of steady-state rotation of a satellite-gyrostat on a circular orbit

The present paper is devoted to discuss both the chaos and optimal control of the steady rotations of a satellite-gyrostat on a circular orbit. In this the satellite is controlled with the help of three independent control moments that are developed by three rotors attached to the satellite principal axes of inertia and rotate with the help of motors rigidly mounted on the satellite body. The optimal controllers that asymptotically stabilize these chaotic rotations and minimize the required like-energy cost are derived as a function of the phase coordinates of the system. The asymptotic stability of the resulting nonlinear system is proved using the Liapunov technique. Numerical study and examples are introduced.     

The steady motions of gyrostats with equal moments of inertia in a central force field

The problem of the motion of a gyroscope in a central force field is considered. It is assumed that the principal central moments of inertia of the gyrostat are equal to one another, while the centre of mass moves in a circular orbit in a plane passing through the attracting centre. The steady motions of the gyrostat and their stability are investigated. The case when the mass distribution allows of the symmetry group of a tetrahedron is considered as an example.     

The global stability of the steady rotations of a solid

The dynamical Euler equations describing the motion of a non-symmetrical solid about the centre of mass in the field of a constant external moment and a dissipative one are considered. It is assumed that the external moment specified with respect to axes attached to the body acts about the intermediate central axis of inertia of the body. The conditions for global asymptotic stability as well as the stability in total of steady rotations of the solid are obtained.     

Non-linear oscillations of a 1:1 resonance Hamiltonian system

The non-linear oscillations of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its stable equilibrium position are considered. It is assumed that the Hamilton function is sign-definite in the neighbourhood of the equilibrium position and that the values of the frequencies of its linear oscillations are equal or close to one another (1:1 resonance). The investigation is carried out using the example of the problem of the motion of a dynamically symmetrical rigid body (satellite) about its centre of mass in a circular orbit in a central Newtonian gravitational field. In this problem there is relative equilibrium of the rigid body in the orbital system of coordinates, for which its axis of dynamic symmetry is directed along the velocity vector of the centre of mass. Resonance occurs when the ratio of the polar and equatorial principal central moments of inertia is equal to 4/3 or is close to it. The problem of the existence, bifurcation and orbital stability of the periodic motions of a rigid body generated from its relative equilibrium is solved. Some aspects of the existence of quasiperiodic motions are also considered.     

The motion of a body with a plane of symmetry over a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation

The problem of the motion of a rigid body possessing a plane of symmetry over the surface of a three-dimensional sphere under the action of a spherical analogue of Newtonian gravitation forces is considered. Approaches to introducing spherical analogues of the concepts of centre of mass and centre of gravity are discussed. The spherical analogue of “satellite approach” in the problem of the motion of a rigid body in a central field, which arises on the assumption that the dimensions of the body are small compared with the distance to the gravitating centre, is studied. Within the framework of satellite approach, assuming plane motion of the body, the question of the existence and stability of steady motions is investigated. A spherical analogue of the equation of the plane oscillations of a body in an elliptic orbit is derived.     

The steady motions of a system of two elastically coupled bodies

The problem of the existence, branching and stability of the steady motions of a system of two elastically coupled bodies in a central gravitational field is considered. Each body is simulated by a weightless rod with point masses at opposite ends. It is assumed that the rods are essentially attached at their mass centres, and the composite body is moving in a plane containing the attracting centre. Both trivial and non-trivial steady motions are studied, on the assumption that none of the principal axes of inertia of the body coincides with the radius vector of the centre of mass or with a tangent to the orbit; it is also assumed that the rods are not orthogonal to one another. The stability of all steady motions is fully investigated and an atlas of bifurcation diagrams presented.     

On the duration of the total eclipse of a satellite of a body

This paper solves the problem of the duration of the total eclipseof a satellite of a body, provided that (a) the body is sphericalin shape and the mass distribution inside the body is sphericallysymmetrical; (b) the satellite is a very small object; (c) thesatellite's orbit around a body is circular and the satellitecan pass through the centre of its shadow; (d) a source of lightis the source of the conical body's shadow (umbra) and it doesnot change its apparent position in the sky; (e) there are noperturbations on the satellite's orbit. The solution presentedrepresents the general discussion of the problem based on utilizationof some parts of the theory of contraction mappings.     

On the Stability of Periodic Mercury-type Rotations

We consider the stability of planar periodic Mercury-type rotations of a rigid body around its center of mass in an elliptical orbit in a central Newtonian field of forces. Mercurytype rotations mean that the body makes 3 turns around its center of mass during 2 revolutions of the center of mass in its orbit (resonance 3:2). These rotations can be 1) symmetrical 2π- periodic, 2) symmetrical 4π-periodic and 3) asymmetrical 4π-periodic. The stability of rotations of type 1) was investigated by A.P.Markeev. In our paper we present a nonlinear stability analysis for some rotations of types 2) and 3) in 3rd- and 4th-order resonant cases, in the nonresonant case and at the boundaries of regions of linear stability.     

Conservative methods of controlling the rotation of a gyrostat

A method for shaping the control of the rotation of a gyrostat consisting of a rigid body, within which there are three rotors rotating about non-coplanar axes rigidly connected to the body, is discussed. The state of the system is defined by the position and angular velocity of rotation of the body, as well as by the angular velocities of the rotors. Control is achieved by torques applied to the rotors. The idea behind the proposed control method is to choose the controlling torques so that the angular velocities of rotation of the rotors are linear functions of the components of the angular velocity vector of the body. The linear dependence thus specified defines a 3 × 3 matrix, that is, a “controlled inertia tensor.” This matrix, which is specified by the parameters of the control selected, does not necessarily have the properties of an inertia tensor. As a result of such a choice of controls, the equations that define the variation of the angular velocity of the body are written in a form similar to Euler's dynamical equations. The system of equations obtained is used to formulate and solve problems of controlling the angular motion of a satellite in a circular orbit. The proposed method for constructing controlling actions enables both the Lagrangian structure of the equations of motion and the fundamental symmetries of the problem to be maintained. Expressions for the torques acting on the rotors and realizing the motion of the required classes are written in explicit form.     

The permanent rotations of a balanced non-autonomous gyrostat

The permanent rotations of a gyrostat about its fixed centre of gravity are investigated. It is assumed that the lines of action of the time-dependent gyrostatic momentum vector maintain a constant position in a reference system attached to the carrier body. It is shown that, if the total angular momentum of the gyrostat is non-zero, permanent rotations can only occur about its principal axes of inertia. In that case the gyrostatic momentum vector must be collinear with one of the principal axes of inertia of the gyrostat.     

On the motion of chaplygin''s sphere on a horizontal plane

The motion of a heavy sphere on a fixed horizontal plane is considered. It is assumed that the centre of mass of the sphere is at its geometric centre, while the principal central moments are different (Chaplygin's sphere). Using the method of averaging, the motion of the sphere is investigated under slip conditions when there is low viscous and also low dry friction. It is shown that when the sphere moves with viscous friction it tends, for the majority of initial data, to rotate about the longest of the axes of the principal central moments of inertia. The motion of the sphere centre tends to become uniform so that the slip velocity approaches zero exponentially. A system of averaged equations, which is fully integrable, is obtained in the case of almost equal moments of inertia, when the friction is dry. The solutions are analyzed.     

On the stability of a satellite cylindrical precession in an elliptic orbit : PMM vol. 40, n 6, 1976, pp. 1040–1047

The stability of motion of a dynamically symmetric satellite with respect to its center of mass in a central Newtonian gravitational field is investigated. The satellite is a solid body whose center of mass moves on an elliptic orbit. The particular case in which the satellite axis of symmetry is normal to the orbit plane (the so-called cylindrical precession [1, 2]) and its absolute angular velocity projection on the axis of symmetry is zero, is examined. Analytical and numerical methods are used. Regions of Liapunov instability and of stability in the first approximation are. obtained in the parameter space of the problem (the inertial parameter and the orbit eccentricity). Detailed nonlinear analysis is carried out in the latter, and the formal stability of the satellite cylindrical precession is proved. The question of stability for the majority of intial conditions is also considered [4].     

The stability of the plane periodic motions of a symmetrical rigid body with a fixed point

A rigorous non-linear analysis of the orbital stability of plane periodic motions (pendulum oscillations and rotations) of a dynamically symmetrical heavy rigid body with one fixed point is carried out. It is assumed that the principal moments of inertia of the rigid body, calculated for the fixed point, are related by the same equation as in the Kovalevskaya case, but here no limitations are imposed on the position of the mass centre of the body. In the case of oscillations of small amplitude and in the case of rotations with high angular velocities, when it is possible to introduce a small parameter, the orbital stability is investigated analytically. For arbitrary values of the parameters, the non-linear problem of orbital stability is reduced to an analysis of the stability of a fixed point of the simplectic mapping, generated by the system of equations of perturbed motion. The simplectic mapping coefficients are calculated numerically, and from their values, using well-known criteria, conclusions are drawn regarding the orbital stability or instability of the periodic motion. It is shown that, when the mass centre lies on the axis of dynamic symmetry (the case of Lagrange integrability), the well-known stability criteria are inapplicable. In this case, the orbital instability of the periodic motions is proved using Chetayev's theorem. The results of the analysis are presented in the form of stability diagrams in the parameter plane of the problem.     

Stabilization of the motions of mechanical systems with variable masses

A holonomic mechanical system with variable masses and cyclic coordinates is considered. Such a system can have generalized steady motions in which the positional coordinates are constant and the cyclic velocities under the action of reactive forces vary according to a given law. Sufficient Routh-Rumyantsev-type conditions for the stability of such motions are determined. The problem of stabilizing a given translational-rotational motion of a symmetric satellite in which its centre of mass moves in a circular orbit and the satellite executes rotational motion about its axis of symmetry is solved.     

On the Stability of a Class of Permanent Rotations of a Heavy Asymmetric Gyrostat

We consider the motion of an asymmetric gyrostat under the attraction of a uniform Newtonian field. It is supposed that the center of mass lies along one of the principal axes of inertia, while a rotor spins around a different axis of inertia. For this problem, we obtain the possible permanent rotations, that is, the equilibria of the system. The Lyapunov stability of these permanent rotations is analyzed by means of the Energy–Casimir method and necessary and sufficient conditions are derived, proving that there exist permanent stable rotations when the gyrostat is oriented in any direction of the space. The geometry of the gyrostat and the value of the gyrostatic momentum are relevant in order to get stable permanent rotations. Moreover, it seems that the necessary conditions are also sufficient, but this fact can only be proved partially.     

Quadratic integrals and the reducibility of the equations of motion of a complex mechanical system in a central field

A mechanical system, consisting of a non-variable rigid body (a carrier) and a subsystem, the configuration and composition of which may vary with time (the motion of its elements with respect to the carrier is specified), is considered. The system moves in a central force field at a distance from its centre which considerably exceeds the dimensions of the system. The effect of the system motion about the centre of mass on the motion of the centre of mass, which is assumed to be known, is ignored (the analogue of the limited problem [1] for a rigid body). The necessary and sufficient conditions for a quadratic integral of the motion around the centre of mass to exist are obtained in the case when there is no dynamic symmetry. It is shown that, for a quadratic integral to exist, it is necessary that the trajectory of the motion of the centre of mass should be on the surface of a certain circular cone, fixed in inertial space, with its vertex at the centre of the force field. If the trajectory does not lie on the generatrix of the cone, only one non-trivial quadratic integral can exist and the initial system, in the presence of this quadratic integral, reduces to autonomous form. For the motion of the centre of mass along the generatrix or the motion of the system around a fixed centre of mass, the necessary and sufficient conditions for a non-trivial quadratic integral to exist are obtained, which are generalizations of the energy integral, the de Brun integral [2] and the integral of the projection of the kinetic moment. When three non-trivial quadratic integrals exist, the condition for reduction to an autonomous system describing the rotation of the rigid body around the centre of mass and integrable in quadratures are indicated [3, 4].