Analysis of regular precession conditions for asymmetrical liquid-filled rigid bodies
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(2020) 132:46
ORIGINAL ARTICLE
Analysis of regular precession conditions for asymmetrical liquid-filled rigid bodies V. Yu. Ol’shanskii1 Received: 21 April 2020 / Revised: 17 July 2020 / Accepted: 25 August 2020 © Springer Nature B.V. 2020
Abstract To describe the rotation of a “rigid mantle + liquid core” system, the Poincaré–Hough– Zhukovsky equations are used. An analysis is made of the previously obtained (Ol’shanskii in Celest Mech Dyn Astron 131(12):Article number:57, 2019) conditions for regular precession of a system that does not have an axial symmetry. Upon receipt of the conditions, it is considered that the external torque can be neglected as, for example, for free-floating planetary bodies. In the case when the axis of proper rotation is one of the principal axes of inertia, the formulas for the rates of precession and proper rotation have been simplified. For a particular case, when the shape of the core differs little from spherical, it is shown that the precession and proper rotation rates differ from the rates of empty axisymmetric rigid mantle by values of the first order of smallness. The ratio of these rates differs from the ratio for the rigid mantle by the second-order value. For a system with the axis of proper rotation deviated from a principal axis of inertia, the expression of the principal moments of inertia of the mantle through the moments of inertia of the core and one arbitrary parameter is found. The formulas for finding the angular velocities and for determining the position of the axis of proper rotation relative to the mantle are written in simple parametric form. The possibility of regular precession with the axis of proper rotation, which does not coincide with any principal axes, is studied in the case when the shape of the core differs little from the spherical one. Examples of elongated non-axisymmetric systems that allow precession with the axis of proper rotation deviated from principal axis of inertia are given. Keywords Rigid body with liquid core · Poincaré–Hough–Zhukovsky model · Precession rates of an asymmetrical system Mathematics Subject Classification 70E40 · 74F10 · 70H12
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V. Yu. Ol’shanskii [email protected] Institute of Precision Mechanics and Control, Russian Academy of Sciences, 24, ul. Rabochaya, Saratov 410028, Russia 0123456789().: V,-vol
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V. Yu. Ol’shanskii
1 Introduction The well-known Poincaré–Hough–Zhukovsky model is widely used from the time of its creation (Zhukovsky 1885; Hough 1895; Poincaré 1910) to the present due to the possibility of describing the motion of a rigid body filled with inviscid fluid using a system of ordinary differential equations. The ”rigid mantle + liquid core” system can permanently rotate around one of its principal axes of inertia. Small perturbations of such motion were considered by Hough (1895), Poincaré (1910), in particular, in connection with the problem of the Earth free nutation, and the eigenfrequencies of small nutational oscillations were found. The work (Ol’shanskii 2019) l
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