Vladimir Ol'shanskii
Publications:
Ol'shanskii V. Y.
Nonregular Precession of a Gyrostat in Three Force Fields
2024, Vol. 20, no. 4, pp. 513-528
Abstract
In this paper, the conditions of nonregular precession with a constant ratio of the velocities
of precession and proper rotation for a gyrostat in the superposition of two homogeneous and
one axisymmetric field are obtained. The case where the gyrostat has axial dynamical symmetry
and the proper rotation axis coincides with the body’s symmetry axis is singled out. It is shown
that in the case where the gyrostatic momentum is collinear to the symmetry axis, nonregular
precession is possible with a precession velocity equal to, twice as large as, or twice as small
as the proper rotation velocity. In each of these cases, the condition expressing the ratio of
the axial and equatorial inertia moments of the body in terms of the nutation angle coincides
with the corresponding condition obtained earlier for the nonregular precession of a solid in
three homogeneous fields. In the particular case of the gyrostat’s spherical symmetry, when the
precession speed is half or twice as large as its proper rotation speed, the cosine of the nutation
angle is equal to one fourth; at equal speeds, the nutation angle should be equal to sixty degrees.
The sets of admissible positions of the forces’ centers for the general case of nonorthogonal
fields are found. The precession of a gyrostat whose gyrostatic momentum is deflected from the
symmetry axis is considered. The possibility of nonregular precession is shown for the case where
the precession velocity is twice as large as the proper rotation velocity. The solution is expressed
in terms of elementary functions. The rotation of the gyrostat is either periodic or the rotation
velocity tends to zero and the carrier body of the gyrostat approaches the equilibrium position.
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Ol'shanskii V. Y.
On quadratic integral Poincare–Zhukovsky’s equations
2012, Vol. 8, No. 3, pp. 523-540
Abstract
For Poincaré–Zhukovsky’s equations with non-diagonal matrices in the Hamiltonian, we obtain conditions for existence of the quadratic integral $({\bf YS},{\bf K}) = \rm{const}$ and the explisit form of it. It is shown that if the integral exists, then the equations reduce to the Schottky’s case.
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Ol'shanskii V. Y.
Abstract
The Poincaré – Zhukovsky – Hough model describing the motion of a rigid body with an
ellipsoidal cavity filled with an ideal vortex liquid is used. The possibility of regular precession
in a uniform force field of a system not possessing axial symmetry is shown. For the case where
the axis of proper rotation is one of the system principal inertia axes and the center of gravity
lies on this axis, two conditions of precession are obtained. One of the conditions coincides with
the condition of regular precession in the absence of external forces for the system without axial
symmetry found earlier by the author. This condition imposes one constraint on the system
configuration. The other condition relates the proper rotation and precession velocities to the
mechanical parameters of the system. A record is given of the conditions in the form of relations
between the inertia moments of the rigid shell and the semiaxes of the ellipsoidal cavity, as well
as between the distance to the center of gravity and the nutation angle, the precession velocity
and the proper rotation velocity. It is shown that in the case where the cavity differs little from
the sphere, the conditions obtained differ from the Lagrange conditions for an axisymmetric rigid
body with a fixed point in a uniform gravity field by small values of the second order.
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