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.     

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.     

BIFURCATION ANALYSIS AND PHASE PORTRAITS OF AN ASYMMETRIC TRIAXIAL GYROSTAT WITH TWO ROTORS

This paper deals with the bifurcations and phase portraits of an asymmetric triaxial gyrostat with two rotors, which is a 3-dimensional generalized Hamiltonian system with a quadratic Hamiltonian depending on three independent parameters. The number and stability of equilibria are analyzed, and corresponding bifurcation conditions of parameters are obtained. Moreover, by Maple software, all possible phase portraits are plotted out. Except for some planar orbits under particular parametric conditions, genera...     

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.     

The equilibrium positions of a satellite carrying a three-degree-of-freedom powered gyroscope in a central gravitational field

For a satellite, carrying an arbitrary number of three-degree-of-freedom powered gyroscopes, the whole set of equilibrium positions in a central gravitational field in a circular orbit is determined and a detailed analysis of their secular stability is presented. The asymptotic properties of the satellite motions when there is dissipation in the axes of the gyroscope frames are investigated.     

The orbital motion of a tetrahedral gyrostat

The orbital motion of a gyrostat whose mass distribution admits of the symmetry group of a regular tetrahedron is examined. The equations of motion and their first integrals are presented. The order of the equations of motion is reduced using a Routh–Lyapunov approach. The reduced potential and the equations for its critical points are presented. Some solutions of these equations are indicated, and a mechanical interpretation of the steady motions corresponding to them is given. Equations of motion similar to the well known equations of relative motion of a gyrostat in an elliptical orbit in the satellite approximation are derived assuming that the dimensions of the body are small compared with its distance from the attracting centre. A three-dimensional analogue of Beletskii's equation that relies on the use of the true anomaly as the independent variable is presented. Three classes of steady configurations are determined by Routh's method in the case of a circular orbit, and the conditions for their stability are investigated.     

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.     

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.     

Separation of Variables in the Kovalevskaya–Goryachev–Chaplygin Gyrostat

We construct separation variables for the Kovalevskaya–Goryachev–Chaplygin gyrostat for arbitrary values of the parameters. We show that different separation variables can be constructed for the same integrable system if different integrals of motion are chosen.     

Evolution of the precessional motion of unbalanced gyrostats of variable structure

The precessional motion of an unbalanced gyrostat of variable structure when acted upon by dissipative and accelerating external and internal moments, which depend on the angular velocities of the bodies (the carrier and the rotor) is considered. A qualitative method of analysing the phase space of non-autonomous dynamical systems is developed, based on the determination of the curvature of the phase trajectory. The motion is analysed and the conditions for obtaining the required modes of nutational-precessional motion of unbalanced gyrostats of variable structure are synthesized using this method. A number of cases of the motion of a gyrostat of variable structure, including free motion, motion when there are constant internal and reactive moments and, also, under the action of the moments of resistance forces, proportional to the angular velocities, is investigated. The possible evolutions in the above-mentioned cases of motion and the causes of these evolutions are determined. The conditions for evolution with a decreasing amplitude of the nutational oscillations are obtained.     

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 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.     

On stability in the presence of multiple resonance of odd order

The stability of the trivial solution of an autonomous system of ordinary differential equations is investigated in the critical case of n pairs of pure imaginary roots when odd-order multiple resonance is present. All possible cases of the presence of a third-order double resonance are examined for a canonic system. The stability problem for the relative equilibrium of a satellite on a circular orbit is analyzed as an example.     

Bifurcation of periodic solutions of a system close to a Lyapunov system

Bifurcation of 2π-periodic solutions (2π-ps) of a system of second-order differential equations close to a Lyapunov system is investigated. The case of principal resonance, when an eigenfrequency of the linear oscillations of the unperturbed system is close to the frequency of the perturbing impulse, is considered. It is shown that, at certain values of the problem parameters, bifurcation of the 2π-ps that are generated from an equilibrium position, occurs. A constructive method is proposed for finding the bifurcation curve, as well as 2π-ps on it. The examples considered are bifurcation of 2π-ps in the problem of the oscillations of a mathematical pendulum with a horizontally vibrating suspension point, and in the problem of the planar oscillations of an artificial satellite in a weakly elliptical orbit. The bifurcation curves for these examples are constructed and the corresponding 2π-ps are found.     

The equilibrium positions of an ellipsoid with a cavity containing a liquid on a horizontal plane

The equilibrium positions of an ellipsoid with an ellipsoidal cavity, partially filled with an ideal incompressible liquid, on a horizontal plane in a uniform gravitational field are considered. All trivial and non-trivial equilibrium positions are found and the conditions for their stability are obtained. The results are presented in the form of bifurcation diagrams.     

On dynamics of an aeroelastic system with two degrees of freedom

An aeroelastic system with one translational and one rotational degrees of freedom is considered. Evolution of the set of equilibrium positions of the system (including both trivial and oblique ones) is examined depending on parameters of the system. Stability criteria are obtained for these equilibria. Numerical simulation of behavior of the system is performed for different values of parameters, including in the area of large angles of attack. Limit cycles existing in the system and their domains of attraction are studied depending on the flow speed.     

Stabilization of a quasi-conservative system subjected to high frequency excitation

The existence and stability conditions for the steady motions and equilibrium positions of non-linear quasi-conservative systems with fast external perturbations having quasi-periodic and random components are investigated. A change of variables is proposed which reduces Lagrange's equations of the system to standard form. It is shown the averaged system of the first approximation has a canonical form and the effect of fast perturbations (not necessarily potential) is equivalent to a change in the system's potential. This leads to stabilization of unstable equilibrium positions and to the appearance of additional stationary points different from the equilibrium positions of the unperturbed system. The approach used is illustrated by examples; the stability of a pendulum on an elastic suspension when there is suspension point, and the steady motion of a sphere subjected to a high-frequency load. The critical loading of a double pendulum loaded by a pulsating tracking force is estimated. A form of wide-band random perturbations capable of stabilizing the system is considered.     

The relative equilibria of a tethered gyrostat in a central Newtonian field

The motion of an orbital tether system comprising a massive body and a gyrostat of small mass attached to it by a non-extensible weightless tether is examined. The body performs unperturbed motion in a Kepler orbit. There are several different equilibria of the system relative to a uniformly rotating system of coordinates. These equilibria are interpreted geometrically using Mohr circles. Despite being the simplest example of an orbital tether system with a gyrostat, it exhibits a wealth of dynamic properties. There are also more complex orbital tether systems which contain more than one gyrostat [1].     

The self-induced rotational and oscillatory motions of an aerodynamic pendulum

The behaviour of an oscillatory mechanical system with alternating dissipation is considered taking an aerodynamic pendulum as an example. The phase portraits are investigated, their rearrangements are studied and the critical values of a parameter are determined. The equilibrium positions of the pendulum and the self induced rotational and oscillatory states of the motion are determined and their stability is investigated.