To describe the rotation of a rigid body with an ellipsoidal cavity filled with an ideal vorticated liquid, the Poincaré–Hough–Zhukovsky equations are used. It is obtained constraints (hereinafter referred to as configuration conditions) on the mass distribution and cavity dimensions of an asymmetric liquid-filled rigid body under which the rigid body can perform the regular precession. Two possible nontrivial cases are indicated when one or two components of the direct vector of the axis of proper rotation are equal to zero. It is shown that if the axis of proper rotation coincides with one of the principal axes of inertia of the system, it suffices to fulfill one configuration condition. The ratio between the periods of proper rotation and precession is found. The regular precession of a system in which the principal moments of inertia are close to each other and the cavity is close to spherical is considered. For the case when the difference between the two equatorial moments of inertia is an order of magnitude smaller than the difference between the equatorial and polar moments, the main part of the ratio between the periods coincides with the Euler period, and the correction is due to the inequality between the equatorial moments of inertia. In the case when the axis of proper rotation is orthogonal to a principal axis and is not coincident with any other principal axis, the number of configuration conditions is two. Expressions for the rates of precession and proper rotation are obtained, and the position in the body of the axis of proper rotation is indicated. Special cases are given that allow simplification of the configuration conditions.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Boué, G., Rambaux, N., Richard, A.: Rotation of a rigid satellite with a fluid component: a new light onto Titan’s obliquity. Celest. Mech. Dyn. Astron. 129(4), 449–485 (2017)
Chen, W., Shen, W.: New estimates of the inertia tensor and rotation of the triaxial nonrigid Earth. J. Geophys. Res. 115, B12419 (2010)
Dufey, J., Noyelles, B., Rambaux, N., Lemaitre, A.: Latitudinal librations of Mercury with a fluid core. Icarus 203, 1–12 (2009)
Grioli, G.: Esistenza e determinazione delle precessioni regolari dinamicamente possibili per un solido pesante asimmetrico. Ann. Mat. Pura Appl. 26(3–4), 271–281 (1947). (in Italian)
Henrard, J.: The rotation of Io with a liquid core. Celest. Mech. Dyn. Astron. 101, 1–12 (2008)
Hough, S.S.: The oscillations of a rotating ellipsoidal shell containing fluid. Philos. Trans. R. Soc. Lond. A 186, 469–506 (1895)
Karapetyan, A.V.: On stability of regular precession a symmetric rigid body with ellipsoidal cavity. Vestn. Mosk. Univ., Ser. 1: Mat., Mech. 6, 122–125 (1972)
Lamb, H.: Hydrodynamics, 6th edn. Dover Publications, New York (1945)
Levi-Civita, T., Amaldi, U.: Lezioni di Meccanica Razionale. Vol. 2, part 2. N. Zanichelli, Bologna (1928) (in Italian)
Lukovsky, Ivan, A.: Nonlinear Dynamics: Mathematical Models for Rigid Bodies with a Liquid. De Gruyter, Series: De Gruyter Studies in Mathematical Physics. 27 (2017)
Melchior, P.: Physique et dynamique planétaires, vol. 4. Vander-éditeur, Louvain (1973)
Noyelles, B., Dufey, J., Lemaitre, A.: Core–mantle interactions for Mercury. Mon. Not. R. Astron. Soc. 407, 479–496 (2010)
Noyelles, B.: Behavior of nearby synchronous rotations of a Poincaré–Hough satellite at low eccentricity. Celest. Mech. Dyn. Astron. 112, 353–383 (2012)
Ol’shanskii, Yu, V.: A new linear invariant relation of the Poincaré–Zhukovskii equations. J. Appl. Math. Mech. 76(6), 636–645 (2012)
Ol’shanskii, Yu, V.: Linear invariant relations of the Poincaré–Zhukovskii equations. J. Appl. Math. Mech. 78(1), 18–29 (2014)
Ol’shanskii, Yu, V.: Partial linear integrals of the Poincaré–Zhukovskii equations (the general case). J. Appl. Math. Mech. 81(4), 270–285 (2017)
Ol’shanskii, Yu, V.: On the regular precessions of an asymmetric liquid-filled rigid body. Mech. Solids 53(Suppl. 2), S95–S106 (2018)
Poincaré, H.: Sur la precession des corps deformables. Bull. Astron. 27, 321–356 (1910)
Touma, J., Wisdom, J.: Nonlinear core–mantle coupling. Astron. J. 122, 1030–1050 (2001)
Zhukovsky, N.E.: On motion of rigid body with cavities filled by homogenous drop-like liquid. Journal of the Russian Physico-Chemical Society, Physical part. 17(6), 81–113, 17(7), 145–199, 17(8), 231–280 (1885). In: Sobranie sochinenii (Collection of Scientific Works), 2, 31–152. Gostekhizdat, Moscow (1948)
Conflict of interest
The author certify that he has not any conflict of interest to declare.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Ol’shanskii, V.Y. New cases of regular precession of an asymmetric liquid-filled rigid body. Celest Mech Dyn Astr 131, 57 (2019). https://doi.org/10.1007/s10569-019-9929-x
- Liquid-filled rigid body
- Poincaré–Hough–Zhukovsky equations
- Regular precession of an asymmetric system
Mathematics Subject Classification