Olga Kholostova

The motion of a heavy rigid body with suspension point performing high-frequency periodic vibrations of small amplitude is considered. The study is carried out within the framework of an approximate autonomous system written in the form of the modified Euler – Poisson equations, to the right-hand sides of which the components of the vibration moment are added. The question of the existence of two particular motions of the body is resolved, they are permanent rotations and pendulum-type motions. It is shown that permanent rotations of the body can occur in the case of vibration symmetry relative to a vertically located axis. The search for pendulum-type motions is restricted to the case when the axis of these motions is one of the principal inertia axes of the body, as in the case of a heavy rigid body with a fixed point. Two basic variants of vibrations are considered, when the suspension point vibrates along a straight line and along an ellipse. To the latter variant any planar vibrations and a wide class of spatial vibrations of the suspension point are reduced. It is shown that for both basic cases of vibrations, pendulum-type motions are of two types. The motions of the first type are similar to the Mlodzeevsky’s pendulum-type motions of a heavy rigid body with a fixed point. For them, the body’s mass center lies in the principal plane of inertia, and the axis of the pendulum-type motions is perpendicular to this plane. Pendulum-type motions of the second type occur around the principal axis of inertia containing the body’s center of mass. Such motions are absent in the gravitational problem, they are caused by the presence of vibrations. To search for the pendulum-type motions, an approach is proposed that combines the results of the problem of gravitation (without vibration) and that of vibration (ignoring gravity). As an illustration, a number of examples of the interaction of the gravitational field and the vibration field corresponding to both basic variants of vibrations of the body’s suspension point are considered.

We consider the motion of a $ 2 \pi $-periodic in time two-degree-of-freedom Hamiltonian system in a neighborhood of the equilibrium position. It is assumed that the system depends on a small parameter $ e $ and other parameters and is autonomous at $ e = 0 $. It is also assumed that in the autonomous case there is a set of parameter values for which a 1:1 resonance occurs, and the matrix of the linearized equations of perturbed motion is reduced to a diagonal form. The study is carried out using an example of the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on an elliptical orbit with small eccentricity in the neighborhood of the cylindrical precession. The character of the motions of the reduced two-degree-of-freedom system in the vicinity of the resonance point in the three-dimensional  parameter space  is studied. Stability regions of the unperturbed motion (the cylindrical precession) and two types of parametric resonance regions corresponding to the case of zero frequency and the case of equal frequencies in the transformed approximate system of the linearized equations of perturbed motion are considered. The problem of the existence, number and stability of $ 2 \pi$-periodic motions of the satellite is solved, and conclusions on the existence of two- and three-frequency conditionally periodic motions are obtained.

We examine the motions of an autonomous Hamiltonian system with two degrees of freedom in a neighborhood of an equilibrium point at a 1:1 resonance. It is assumed that the matrix of linearized equations of perturbed motion is reduced to diagonal form and the equilibrium is linearly stable. As an illustration, we consider the problem of the motion of a dynamically symmetric rigid body (satellite) relative to its center of mass in a central Newtonian gravitational field on a circular orbit in a neighborhood of cylindrical precession. The abovementioned resonance case takes place for parameter values corresponding to the spherical symmetry of the body, for which the angular velocity of proper rotation has the same value and direction as the angular velocity of orbital motion of the radius vector of the center of mass. For parameter values close to the resonance point, the problem of the existence, bifurcations and orbital stability of periodic rigid body motions arising from a corresponding relative equilibrium of the reduced system is solved and issues concerning the existence of conditionally periodic motions are discussed.

The motion of the Lagrangian top, the fixed point of which performs small-amplitude vertical oscillations, is studied. The case, when values of constants $a$ and $b$ of cyclic integrals are related by $|a|=|b|$, is considered. It is known that for the classical Lagrangian top with a fixed point there exists either one regular precession of the top or none at $a=b$ depending on values of parameters of the problem (the parameter $a$ and the parameter, characterizing the position of the center of mass of the top on the axis of symmetry); at $a=-b$ there is no regular precessions.
The existence of periodic motions of the top close to regular precessions (with a period equal to the period of oscillations of the fixed point) is investigated in case of oscillations of the fixed point. In the first part of the present paper periodic motions of the top, which occur from regular precession, are found in case $a=b$ and stability of these motions is determined. In the second part it is additionally assumed that oscillations of the fixed point have a high frequency, and angular velocities of precession and self-rotation of the top are small. A qualitatively new result is obtained which does not have an analogue either in the classical problem or in a problem of motion of the top with small-amplitude oscillations of the fixed point: Motions of the top close to regular precessions are found, these motions exist both at $a=b$ and at $a=-b$, and their number is different (there are two, one or none at $a=b$, and there are two or none at $a=-b$) depending on values of parameters of the problem. The problem of stability of these motions is solved precisely with the help of the KAM-theory.