Gyrostatic Moment Research Articles

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.

New integrable cases in the dynamics of rigid bodies

This paper aims to investigate the rotatory dynamical motion of a charged symmetric gyrostat when its center of mass is slightly displaced from the axis of dynamic symmetry as a novel case. A uniform electromagnetic field, due to the presence of point charge located on the axis of symmetry, and a gyrostatic moment are acted on the gyrostatic motion. Equations of motion are derived and solved analytically using the small parameter method of Poincaré for irrational frequencies’ case of the examined motion of the gyrostat. The approximate formulas of the rotating Euler’s angles of the gyrostat are obtained. The time histories of the achieved results and the Euler’s ones are graphed to display the excellent influence of the chosen parameters of the gyrostat on the motion. Moreover, the phase plane diagrams of these results are graphed also to reveal the stability of the gyrostatic motion at any time. The attained results can be used to improve the dynamical behavior of several engineering applications, particularly those that employ the gyroscopic theory such as satellites, submarines, compasses, and automatic pilots on aircraft and ships.

On oscillations of a gyrostat around stable permanent rotations

The oscillations of a gyrostat with constant gyrostatic moment are investigated by the method of averaging for the Euler, Lagrange and Kovalevskaya cases, which are analogous to the oscillations studied earlier /1/ of a solid around its stable permanent rotations.

On periodic motions of a gyrostat in a newtonian force field

The problem of existence of periodic motions of a gyrostat is studied. The gyrostat consists of a rigid body and a rotor, the axis of which is stationary with respect to the rigid body in a central Newtonian force field. In /1,2/, Poincaré's method of small parameter was used to show the existence of periodic motions of a gyrostat with a single fixed point in a central Newtonian force field. It was assumed that the gyrostat differs little from a dynamically symmetric one, and the constant gyrostatic moment was assumed to be sufficiently small. Moreover, it was assumed that the center of gravity of the gyrostat is sufficiently close to the fixed point, and the center of attraction is sufficiently far removed from the gyrostat.

Periodic motions of a gyrostat with a fixed point in a newtonian field: PMM, vol. 43, no. 1, 1979, pp. 175–178

Poincaié's method of small parameter is used to investigate the existence of periodic motions of a gyrostat with a fixed point, consisting of a rigid body and a rotor. The friction at the rotor shaft and other dissipative effects are neglected. The gyrostatic moment is assumed constant. The free Euler -Poinsot motion is chosen to represent the generating solution.

Uniaxial stabilization of symmetric satellite gyrostat at libration points

i. Consider the dynamically symmetric gyrostat with a gyrostatic moment collinear with the dynamic-symmetry axis; the center of mass of the gyrostat resides at one of the libration points of the bounded circular three-body problem. The notation used is as in [5]: O~xyz denotes a coordinate system rotating with angular velocity ~ about the z axis, the origin of which coincides with the center of mass of m, and m2, and the x axis passes through the centers of mass M, and M2 of the masses m, and m2, while

On the Routh theorem and the Chetaev method for constructing the liapunov function from the integrals of the equations of motion: PMM vol. 33, n≗5, 1969, pp. 904–912

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 a motion of an equilibrated gyroscope in the newtonian force field

This article sheds light on the rotatory motion of a symmetric gyrostat around a fixed point under the effectiveness of both a magnetic field and a Newtonian one besides the action of a gyrostatic moment around the principal inertia’s axes. The gyrostat is assumed to have initially high angular velocity around the principal axis of dynamic symmetry. The controlling system of motion and its first integrals are significantly decreased to a new autonomous system and only one first integral. Poincaré’s method of a small parameter is utilized to achieve the asymptotic solutions of the controlling system. Euler’s angles are evaluated to perceive the motion at every blink. The achieved solutions are graphically portrayed through certain plots for various values of the charge causing the magnetic field and gyrostatic projection's values to estimate the output of these values on the gyrostatic motion. Moreover, the graphs of phase planes for these solutions are sketched to provide a complete explanation of the stability of considered motion. Many applications such as submarines, aircraft, and satellites, are of great interest to the topic concerned.

On the construction of periodic solutions of quasi-linear autonomous systems with several degrees of freedom

This paper attempts to explore the general rotatory 3D motion of a magnetic heavy solid body around one fixed point under the impact of a gyrostatic moment when the body awarded initially high angular velocity about one of its inertia principal axes. The asymptotic technique of Krylov-Bogoliubov-Mitropolski and its adjustment are applied to achieve new asymptotic periodic solutions of the controlling equations of motion. Discussion of these solutions is presented taking into consideration their graphs representations. The graphical depictions of such solutions and their corresponding phase planes are plotted to give an induction about its behaviour during the interval time of motion and to reveal the good effect of the different applied forces and moments on the body's motion. The application of the achieved results of the present work can be widely found in gyroscopic devices especially that used inertia reference systems like in satellites, airplanes, and missiles. Moreover, for devices that are responsible for the stability of motion for these applications.

Water-proofing concrete: Anon. ( Sci. Amer., cx, No. 21, 425.)

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.