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.     

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.     

On motion of a symmetric gyrostat in a Newtonian force field

A spherical cavity in a sphere-shaped gyrostat contains a spherical rotor, which is rotating at a constant angular velocity relative to the outer sphere. The centres of the outer sphere, the cavity and the rotor coincide. Attached to the outer sphere are identical point masses, placed at the vertices of an octahedron. A study is presented of the influence of the rotation of the rotor on the existence and stability of steady motions of the gyrostat about its mass centre in the Newtonian field of a fixed attracting centre. Interest centres on motions in which the radius vector of the gyrostat centre and the gyrostatic moment are collinear. It is shown that the existence of a gyrostatic moment may essentially modify the stability properties of the steady motions discovered.     

The steady motions of a satellite with a two-degree-of-freedom powered gyroscope in a central gravitational field and their stability

The positions of relative equilibrium of a satellite carrying a two-degree-of-freedom powered gyroscope, in the axes of the framework of which only dissipative forces can act, are investigated within the limits of a restricted circular problem. For the case when the “satellite - gyroscope” system possesses the property of a gyrostat and the axis of the gyroscope frame is directed parallel to one of the principal central axes of inertia of the satellite, all the equilibrium positions are found as a function of the magnitude of the angular momentum of the rotor. It is established that the minimum number of equilibrium positions is equal to 32 and, in certain ranges of values of the system parameters, it can reach 80. All the positions satisfying the sufficient conditions for stability are also determined. The number of them is either equal to 4 or 8 depending on the values of the system parameters.     

Stability of steady rotations of a heavy gyrostat about its principal axis

Stability of steady rotations of a gyrostat about its principal axis is investigated with the use of the Arnol'd —Moser theorem /1, 2/ extended to stationary motions /3, 4/. It is shown that steady rotations are stable for all parameter values that belong to the region where the necessary stability conditions are satisfied, except for some manifold of lesser dimension.     

Single-axis equilibrium orientations to the attracting center of a symmetrical prolate orbital gyrostat with an elastic rod

Under study in the restricted formulation is the motion of a symmetrical prolate stationary gyrostat along a Keplerian circular orbit in a central Newtonian field of forces. An elastic homogeneous rod, rectilinear in the undeformed state, is rigidly clamped by one end in the body of gyrostat along its axis of symmetry. There is a point mass at the free end of the rod. The inextensible elastic rod, for simplicity of constant circular cross-section, performs infinitesimal space oscillations in the process of system motion. In this case, we neglect the terms in the system’s tensor of inertia which are nonlinear with respect to displacements of the points of the rod.We consider the following (so-called semi-inverse) problem: Under what kinetic momentumof the flywheel, among the relative equilibriums of the system (the states of rest relative to the orbital coordinate system) does there exist an equilibrium such that the axis, arbitrarily chosen in the coordinate system associated with the gyrostat, is collinear with the local vertical? In the discretization of the problem, we present the values of the Lagrange coordinates that define the deformation of the rod for these equilibria and the value of gyrostatic moment providing the presence of the equilibrium in question.     

Integrable cases of the problem of the free motion of a gyrostat

The free spatial motion of a gyrostat in the form of a carrier body with a triaxial ellipsoid of inertia and an axisymmetric rotor is considered. The bodies have a common axis of rotation, which coincides with one of the principal axes of inertia of the carrier. In the Andoyer–Deprit variables the equations of motion reduce to a system with one degree of freedom. Stationary solutions of this system are found, and their stability is analysed. Cases in which the longitudinal moment of inertia of the carrier is greater than the largest of the transverse moments of inertia of the system of bodies, is smaller than the smallest or belongs to a range composed of the moments of inertia indicated, are investigated. General analytical solutions that describe the motion on separatrices and in regions corresponding to oscillations and rotation on the phase portrait are obtained for each case. The results can be interpreted as a development of the Euler case of the motion of a rigid body about a fixed point when one degree of freedom, namely, relative rotation of the bodies, is added.     

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.     

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.     

Linear problems of the stability of a type of rotation of a satellite about the centre of mass

The stability in the first approximation of the rotation of a satellite about a centre of mass is investigated. In the unperturbed motion the satellite performs, in absolute space, three rotations around the normal to the orbital plane in a time equal to two periods of rotation of its centre of mass in the orbit (Mercury-type rotation). Three cases of such rotations are considered: the rotations of a dynamically symmetrical satellite and a satellite, the central ellipsoid of inertia of which is close to a sphere, in an elliptic orbit of arbitrary eccentricity, and the rotation of a satellite with three different principal central moments of inertia in a circular orbit.     

Stability of stationary rotations of multidimensional rigid body

It is well known that rotations of a free three-dimensional rigid body around the long and short axes of inertia are stable, while the rotation around the intermediate axis is unstable. We generalize this result to the case of a rigid body in a space of arbitrary dimension.     

On the relative equilibria of a satellite-gyrostat, their branchings and stability

The set of relative equilibria of a satellite-gyrostat in a Newtonian gravitational field is studied. The simple geometrical form of this set is described. The branching and stability of the equilibria of a symmetric gyrostat are considered. The results are represented by bifurcation diagrams, on which the degree of stability of the equilibria is distributed in accordance with a law whereby the stability changes at a fixed value of the gyrostatic moment.     

The asymptotic properties of the motions of satellites in a central field due to internal dissipation

A satellite in the form of a system of bodies that does not have the property of a gyrostat in the general case is considered. An algorithm for determining all the equilibrium configurations of the system that correspond to steady motions in a central gravitational field and an algorithm for analysing their stability are given. A method based on Routh's first theorem is used to investigate the asymptotic stability of the steady motions in the unconstrained problem. Three effects caused by internal dissipation are established in a model example: stabilization of the satellites in a neighbourhood of rotations about a normal to the orbital plane, which is codirectional with the axis of the largest moment of inertia, evolution of elliptic orbits into circular orbits, and capture of the satellites in resonant oscillatory modes of motion.     

Zerlegung der Darboux-Drehung in Zwei Ebene Drehungen

The Darboux rotation for space curves in Euclidean space E³ is decomposed into two simultaneous rotations. The axes of these simultaneous rotations are joined by a simple mechanism. One of these axes is a parallel of the principal normal of the curve, the direction of the other is the direction of the Darboux vector of the curve. This decomposition of the Darboux rotation yields a necessary condition for the curve to be closed.

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Herrn Prof. Dr.Oswald Giering zum 60. Geburtstag gewidmet     

A class of three invariant relations for the equations of motion of a gyrostat with a variable gyrostatic moment

The motion of a dynamically symmetrical gyrostat under the action of potential and gyroscopic forces with a variable gyrostatic moment, which can be described by generalized equations of the Kirchhoff–Poisson class, is considered. The conditions for the existence of three linear invariant relations of a special type are obtained, and new solutions of the equations of motion, expressed either in the form of elementary functions or elliptic functions of time, are obtained. ©2013     

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.     

Modeling of chaotic motion of gyrostats in resistant environment on the base of dynamical systems with strange attractors

A chaotic motion of gyrostats in resistant environment is considered with the help of well known dynamical systems with strange attractors: Lorenz, Rössler, Newton–Leipnik and Sprott systems. Links between mathematical models of gyrostats and dynamical systems with strange attractors are established. Power spectrum of fast Fourier transformation, gyrostat longitudinal axis vector hodograph and Lyapunov exponents are find. These numerical techniques show chaotic behavior of motion corresponding to strange attractor in angular velocities phase space. Cases for perturbed gyrostat motion with variable periodical inertia moments and with periodical internal rotor relative angular moment are considered; for some cases Poincaré sections areobtained.     

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.     

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.