Asymmetric Gyrostat Research Articles

The Set of Relative Equilibria of a Stationary Orbital Asymmetric Gyrostat

Under consideration is the well-known problem of relative equilibria (an equilibrium position in the orbital coordinate system) of a gyrostat satellite and their dependence on the design parameters. A new geometric approach to the analysis of the set of relative equilibria is developed. It is proposed to determine the relative equilibria in the corresponding three-dimensional Euclidean space using special aggregated parameters of the systemby the coordinates of the intersection points of two pairs of corresponding hyperbolic cylinders with the sphere of the unit radius. It is shown that, for arbitrary values of the gyrostatic moment and other parameters of the system, there are at least eight different relative equilibria.

The set of relative equilibria of a stationary orbital asymmetric gyrostat

The set of relative equilibria of a stationary orbital asymmetric 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.

Stability of the permanent rotations of an asymmetric gyrostat in a uniform Newtonian field

The stability of the permanent rotations of a heavy gyrostat is analyzed by means of the Energy-Casimir method. Sufficient and necessary conditions are established for some of the permanent rotations. The geometry of the gyrostat and the value of the gyrostatic moment are relevant in order to get stable permanent rotations. Moreover, the necessary conditions are also sufficient, for some configurations of the gyrostat.

Chaotic motion in a class of asymmetrical Kelvin type gyrostat satellite

This work is an investigation on the roots of chaotic attitudinal motion in a class of asymmetrical gyrostat satellites. The result shows that for a class of Kelvin type gyrostat satellite, there is an equivalent rigid spinning satellite with the same attitude dynamics. Finding some constants of motion and eliminating the cyclic coordinates, the rotational kinetic energy is changed to a quadratic form and using Jordan canonical form of the associated inertia tensor and transforming the coordinate system, the Hamiltonian has been changed to those of a rigid satellite. The Hamiltonian has been split into integrable and non-integrable parts. Using Deprit canonical transformation and Andoyer variables the integrable part has been reduced to a one-dimensional form. The reduced Hamiltonian shows that the regular dynamics of the satellite can be chaotic, under the influence of gravitational effects. To demonstrate various attitudinal dynamics of the satellite, a second-order Poincaré map is employed. This research shows firstly, that the attitudinal dynamics of Kelvin type gyrostat satellites and rigid satellites follow the same dynamical patterns, secondly, for non-linear analysis of dynamics of gyrostat satellite based on the perturbation methods, there is a preferable form for Hamiltonian of the system in the near-integrable fashion and thirdly the chaotic motion is originated from the gravitational field effects that can be suppressed by increasing the attitudinal energy of the satellite in comparison with the translational energy.

Chaotic Attitude Tumbling of an Asymmetric Gyrostat in a Gravitational Field

The chaotic attitude tumbling of an asymmetric gyrostat is investigated in detail. The gyrostat has three sym- metrical wheels along the principal axes rotating about a e xed point under the action of either the gravity torques or the gravity-gradient torques. With the use of the Deprit canonical variables, the Euler attitude equations are transformed into Hamiltonian form. This makes the Poincare-Arnold-Melnikov (PAM) function developed by Holmes and Marsden applicable. The physical parameters triggering the chaotic attitude are established. The analytical results are checked by using the fourth-orderRunge -Kutta simulation in terms of the Euler parameters (quaternions ). The relationships of the following physical parameters are established: moments of inertia of carri- ers and wheels, positions of the mass center, kinetic energy and moment of momentum of the torque-free gyrostat, and initial attitude leading to chaotic motion. The results show that the PAM function is a powerful analytical tool for the treatment of the dynamics of nonlinear gyrostat orientations.

Near-Integrability in a Class of Asymmetrical Gyrostat Satellites

Article Near-Integrability in a Class of Asymmetrical Gyrostat Satellites was published on June 1, 2002 in the journal International Journal of Nonlinear Sciences and Numerical Simulation (volume 3, issue 2).

Chaotic dynamics of an asymmetrical gyrostat

The chaotic motions of an asymmetrical gyrostat, composed of an asymmetrical carrier and three wheels installed along its principal axes and rotating about the mass center of the entire system under the action of both damping torques and periodic disturbance torques, are investigated in detail in this paper. By introducing the Deprit's variables, one can derive the attitude dynamical equations that are well suited for the utilization of the Melnikov's integral developed by Wiggins and Shaw. By using the elliptic function theory, the homoclinic solutions of the attitude motion of a torque-free asymmetrical gyrostat are obtained analytically, based upon the Wangerin's method developed by Wittenburg. Transversal intersections of the stable and unstable manifolds (typically a necessary condition for chaotic motions to exist) are detected by the techniques of Melnikov's functions. The bifurcation curve between the compound parameters is depicted and discussed. By using a fourth-order Runge–Kutta integration algorithm as a tool of the numerical simulation, the long-term dynamical behavior of the system shows that the technique of the Melnikov's function could successfully be employed to predict the compound physical parameters that correspond to the chaotic dynamical motions of an asymmetrical gyrostat.

Chaotic motion of an asymmetric gyrostat in the magnetic field of the earth

The attitude motion of an asymmetric magnetic gyrostat satellite in a circular orbit subjected to both gravitational and magnetic forces of the earth is investigated by using the version of Melnikov's method developed for a two-degree of freedom Hamiltonian system withS1 symmetry. For this purpose Deptrit's canonical variables are introduced to establish the Hamiltonian structure for this problem. It is found that the motion is chaotic in the sense of Smale's horseshoe under certain conditions. The effects of the magnetic moment of the gyrostat and the speed of the rotor in the gyrostat on the global motion are also studied by numerical computation. It is shown that as the magnetic moment of the gyrostat increases, the chaotic area in the Poincare map will enlarge; as the rotor speed increases, a chaotic motion will turn into a regular motion.

Chaotic motion of a gyrostat with asymmetric rotor

In this paper, we discuss an asymmetric gyrostat with a rotor to which a mass point is attached, this breaks the symmetry and perturbs the integrable system periodically. We take this system as a Euler–Poinsot motion perturbed by a small periodic excitation, and apply the Melnikov method to determine the intersection of the stable and unstable manifold of the system's hyperbolic point, since, this is the cause of chaos. We also manifest the chaotic motion in angular momentum space by the Poincaré surface of section. The stability about the rotating axis is also showed in the portrait of the Poincaré surfaces of section.

Complete analysis of bifurcations in the axial gyrostat problem

We analyse the phase flow evolution of the torque free asymmetric gyrostat motion. The gyrostat consists of a triaxial rigid body and a symmetric rotor spinning around one of the principal axis of inertia of the gyrostat. The problem is converted into a two parametric quadratic Hamiltonian with the phase space on the sphere. As the parameters evolve, the appearance - disappearance of centres and saddle points is originated by a sequence of pitchfork bifurcations. When the gyrostat is axial symmetric, there are motions of the rotor that break the degeneracy through an oyster bifurcation while other motions simply shift the degeneracy along a minor circle.

Chaotic motion of an asymmetric gyrostat in the gravitational field

The motion of an asymmetrical gyrostat, consisting of an asymmetrical carrier and an axisymmetric rotor, rotating about a fixed point under the action of the gravitational force, is studied in this paper. Deprit's canonical variables are introduced to describe the attitude motion of the gyrostat. We show that the motion is chaotic in the sense of Smale's horseshoe by using Melnikov's method developed by Holmes and Marsden. The effect of the rotor on the global motion of the gyrostat is also studied. It is found that, as the rotor speed increases, a chaotic motion will turn into a regular motion.

Global motion of a dissipative asymmetric gyrostat

The motion of an asymmetrical gyrostat with external or internal energy dissipation is discussed by use of the qualitative method. The attitude motion of the gyrostat is described by two angular coordinates of the total angular momentum H relative to a body-fixed reference frame. The dynamical equation is written for an autonomous system whose singularity corresponds to the permanent rotation of the gyrostat. The global motion of the gyrostat is determined by the number and distribution of singularities. The type of each singularity varies with the variation of mass geometry and rotor speed, and a bifurcation phenomenon can be observed.

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.

On regular precessions of a heavy gyrostat: PMM vol. 37, n≗4, 1973, pp. 746–753

The regular precessions of a heavy asymmetric gyrostat are found by direct integration of the system of Zhukovskii equations written in the principal axes of inertia. The properties of these motions are investigated; the possibility of controlling them is revealed. Forces capable of causing a regular precession in the gyrostats are investigated. An idea was developed in [1] on the preferability of investigating the motion of a heavy gyrostat fixed at one point before the investigation of the motions of the classical rigid body ( ∗ ∗ Kharlamova E.I., Algebraic invariant relations of the differential equations of rigid body dynamics. Doctorate Dissertation, 1971. ) (see footnote on the next page). Still earlier [2] an attempt at such an investigation was made for an asymmetric gyrostat by Grioli's method and the dynamic possibility of regular precessions in such gyrostats was proved. Later on in [4] a solution of this problem was obtained in special nonprincipal axes of inertia, which was not written as a function of time.