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.     

Dynamics of an axisymmetric gyrostat satellite. Equilibrium positions and their stability

The dynamics of an axisymmetric gyrostat satellite in a circular orbit in the central Newtonian force field is investigated. All the equilibrium positions of the gyrostat satellite in the orbital system of coordinates are determined, and the conditions for their existence are analysed. All the bifurcation values of the system parameters at which the number of equilibrium positions changes are found. It is shown that, depending on the values of the parameters of the problem, the number of equilibrium positions of a gyrostat satellite can be 8, 12 or 16. The evolution of regions where the sufficient conditions for stability of the equilibrium positions hold is investigated.     

Analysis of the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame

A method that enables a conclusion to be drawn regarding the nature of the Lyapunov stability of the equilibria positions of autonomous systems with partial dissipation from the nature of the secular stability and truncated equations of the linear approximation is developed based on the Barbashin–Krasovskii theorem. The method is used to investigate the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame in a circular orbit. For the case when the axis of the gyroscope frame is oriented parallel to one of the principal axes of inertia of the satellite, the nature of the Lyapunov stability of all the equilibria, with the exception of a few branching points, is determined. It is established that gyroscopic stabilization, which is possible when there is no dissipation in the axis of the frame for certain equilibria that are unstable in the secular sense, breaks down when dissipation occurs.     

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.     

The properties of the steady motions of a body carrying a system of two-degree-of-freedom powered gyroscopes

The steady motions of a rigid body carrying several two-degree-of-freedom powered gyroscopes in a uniform external field are investigated. It is shown that when the installation scheme of the gyroscopes in the carrying body is collinear, the problem of determining the steady motions of the system and analysing their secular stability reduces for the most part to the previously solved, similar problem for a system with one gyroscope. It is established that when there is dissipation in the axes of the gyroscope frames, the system tends asymptotically to a state of rest if the absolute value of the total angular momentum of the system lies in the segment of possible absolute values of the angular momentum of the gyroscope rotors. The results of an analysis of the steady motions of a system carrying two gyroscopes with a non-collinear installation scheme are presented.     

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.     

Asymptotic properties of the motions of a heavy gyroscope with internal dissipation

The limiting motions of a heavy gyroscope, simulated by a system of rigid bodies, are considered when there is internal friction. The whole set of limiting motions is determined and the nature of their stability is studied in detail for cases when the carried body of the gyroscope has a) three degrees and b) one degree of freedom with respect to the supporting body. The results of an analysis of case a are extended to the motion of a gyroscope with a fluid filling. For case b, the values of the parameters are determined for which the gyroscope, apart from steady rotations, has unsteady limiting motions that are integrable motions in the special Bobylev-Steklov case.     

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 critical case of a pair of zero roots in a two-degree-of-freedom hamiltonian system

The problem of the motion of an autonomous two-degree-of-freedom Hamiltonian system in the neighbourhood of its equilibrium position is considered. It is assumed that the characteristic equation of the linearized system has a pair of pure imaginary roots. The roots of the other pair are assumed to be close to or equal to zero, and in the latter case non-simple elementary dividers correspond to these roots. The problem of the existence, bifurcations and orbital stability of families of periodic motions, generated from the equilibrium position, is solved. Conditionally periodic motions are analysed. The problem of the boundedness of the trajectories of the system in the neighbourhood of the equilibrium position in the case when it is Lyapunov unstable, is considered. Non-linear oscillations of an artificial satellite in the region of its steady rotation around the normal to the orbit plane are investigated as an application.     

Eine genäherte Behandlung des schweren symmetrischen Kreisels in nicht-Eulerschen Koordinaten

Summary The gyroscope has many technical applications as an essential member in a system with several degrees of freedom, the (dynamic) stability of which with respect to a certain equilibrium position may be of particular interest. To investigate the stability of a system by the well-known method of small oscillations, the movements of the different members must be described by convenient coordinates. As to the gyroscope, it is necessary for this purpose to replaceEuler's angles by other coordinates, with which the rigorous equations of its movement are deduced. Assuming only small displacements of the axis of symmetry from its undisturbed equilibrium position, and by making the problem dimensionless, a simple solution can be obtained for the nutation and precession of a heavy gyroscope both as a symmetrical top and as a gyroscopic pendulum. Further, the damping of the gyroscope by its movement around the equilibrium position has also been considered in order to be able to estimate its influence on the stability.     

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.     

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 mixing of a viscous fluid in a layer between rotating eccentric cylinders

The motion of an incompressible viscous fluid in a thin layer between two circular cylinders, inserted into one another, with parallel axes is investigated. The cylinders rotate relative to one another about an axis parallel to the axes of the cylinders. The stream function of the unsteady plane-parallel flow that occurs is found by solving the boundary-value problem for the equations of hydrodynamic lubrication theory. The motion of the fluid particles is found from the solution of a non-autonomous time-periodic Hamiltonian system with a Hamiltonian equal to the stream function. The positions of fluid particles over time intervals that are a multiple of the period of rotation (Poincaré points) are calculated. The set of points is investigated using a Poincaré mapping on the phase flow. The observed transition to chaotic motion is related to the mixing of the fluid particles and is investigated both numerically and using a mapping, calculated with an accuracy up to the third power of the small eccentricity. The optimum mode of motion is observed when the area of the mixing (chaos) region reaches its highest value.     

The local boundedness of the perturbed motions of an imperfect gyroscope in gimbals with dissipative and accelerating forces

An unbalanced dynamically symmetrical gyroscope in gimbals with constructive imperfections is considered in a central Newtonian field of forces. It is assumed that there is a moment of forces of viscous friction acting on the axis of rotation of one of the rings of the suspension and an accelerating (electromagnetic) moment applied to the axis of rotation of another ring. The equations of motion have a partial solution for which the basic plane of the frame is perpendicular to the direction from the specified fixed point of the frame to the centre of gravitation, the basic plane of the mantle is parallel to this direction and the rotor rotates with an arbitrary constant angular velocity.

The equations of perturbed motions of the reduced system with two degrees of freedom are obtained to within third-order terms at the corresponding position of equilibrium. In the domain of admissible values of the parameters F_o the characteristic equation of the system is considered and its coefficients are written down. A domain in F_o is specified in which complex conjugate pairs of the eigenvalues have small moduli of the real parts but the absolute values of the second- to fourth-order off-resonance mistuning between the imaginary parts are not small. For an imperfect gyroscope in gimbals with dissipative and accelerating forces the sufficient conditions of the local uniform boundedness of motions perturbed with respect to the specified partial solution are obtained in this domain. The conditions found provide the local uniform boundedness of solutions irrespective of the forms of higher than the third order in the equations of perturbed motions. These conditions are obtained in the form of constraints for the coefficients of the normal form and, finally, for the original parameters of the system and the real and imaginary parts of the eigenvalues. To provide a clear interpretation of the results, special cases when all but two parameters are fixed are analysed. The domains of local uniform boundedness are constructed in the two-dimensional domains F_o using a personal computer.     

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.     

Optimal stabilization of the equilibrium positions of a rigid body using rotors

In this work, the problem of optimal stabilization of the equilibrium positions of a rigid body using internal rotors is studied. The conditions for the optimal stabilization of the equilibrium positions are used to deduce a feedback control law as functions of the phase coordinates of the body and the parameters describing the equilibrium positions. The Lyapunov function is used to prove the asymptotic stability of these positions. Special cases and analysis of the obtained results are presented to assess the present method. Moreover, some of the results are compared with those obtained in the literature using other methods. In contrast to the usual methods in the literature, which stabilize some of the equilibrium positions of the rigid body, the present one has the advantage of stabilizing all the equilibrium positions with optimal control law.     

Algorithms for computing the equilibrium of a compact torus with several magnetic axes

We discuss some issues connected with the computation of equilibrium states of plasma configurations in the shape of a compact torus with several magnetic axes. The computation method proposed for such systems employs solution control, establishing a feedback between the discharge parameters and the equilibrium-producing control currents so that the position and the shape of the plasma system satisfy given requirements. The article also investigates the proposed numerical method based on an iterative process. A method of determining the iteration parameter is described and the question of the number of iterations is considered.Translated from Vychislitel'naya Matematika i Matematicheskoe Obespechenie EVM, pp. 138–142, 1985.     

The orbital stability of periodic motions of a Hamiltonian system with two degrees of freedom in the case of 3:1 resonance

The problem of the orbital stability of periodic motions, produced from an equilibrium position of an autonomous Hamiltonian system with two degrees of freedom is considered. The Hamiltonian function is assumed to be analytic and alternating in a certain neighbourhood of the equilibrium position, the eigenvalues of the matrix of the linearized system are pure imaginary, and the frequencies of the linear oscillations satisfy a 3:1 ratio. The problem of the orbital stability of periodic motions is solved in a rigorous non-linear formulation. It is shown that short-period motions are orbitally stable with the sole exception of the case corresponding to bifurcation of short-period and long-period motions. In this particular case there is an unstable short-period orbit. It is established that, if the equilibrium position is stable, then, depending on the values of the system parameters, there is only one family of orbitally stable long-period motions, or two families of orbitally stable and one family of unstable long-period motions. If the equilibrium position is unstable, there is only one family of unstable long-period motions or one family of orbitally stable and two families of unstable long-period motions. Special cases, corresponding to bifurcation of long-period motions or degeneration in the problem of stability, when an additional analysis is necessary, may be exceptions. The problem of the orbital stability of the periodic motions of a dynamically symmetrical satellite close to its steady rotation is considered as an application.     

Dipole and Multipole Flows with Point Vortices and Vortex Sheets

An exact method is presented for obtaining uniformly translating distributions of vorticity in a two-dimensional ideal fluid, or equivalently, stationary distributions in the presence of a uniform background flow. These distributions are generalizations of the well-known vortex dipole and consist of a collection of point vortices and an equal number of bounded vortex sheets. Both the vorticity density of the vortex sheets and the velocity field of the fluid are expressed in terms of a simple rational function in which the point vortex positions and strengths appear as parameters. The vortex sheets lie on heteroclinic streamlines of the flow. Dipoles and multipoles that move parallel to a straight fluid boundary are also obtained. By setting the translation velocity to zero, equilibrium configurations of point vortices and vortex sheets are found.