Rapid rotation of a heavy gyrostat about a fixed point in a resisting medium

Abstract

iI 2. 3. LITERATURE CITED D. R. Merkin, Gyroscopic Systems [in Russian], Nauka, Moscow (1974). S. P. Sosnitskii, "One property of gyroscopic systems," Dokl. Akad. Nauk Ukr. SSR, Ser. A, No. 2, 118-120 (1979). L. L. Helms and C. R. Putnam, "Stability of incompressible systems," J. Math. Mech., 7, No. 6, 901-903 (1958). RAPID ROTATION OF A HEAVY GYROSTAT ABOUT A POINT IN A RESISTING MEDIUM L. D. Akulenko and D. D. Leshchenko FIXED UDC 531.1 We consider an asymmetric heavy rigid body with a spherical cavity filled with a high-viscosity liquid rotating rapidly about a fixed point in a weakly resisting medium. We call motions rapid when the moment of the applied forces about the fixed point is small in comparison with the instantaneous value of the kinetic energy of rotation. To investigate the motion of the body with a liquid we introduce three Cartesian coordinate systems: a stationary x i system (i = 1, 2, 3); a Yi system with the Y3 axis along the angular momentum vector G of the body and liquid (Fig. 1); a z i system whose axes coincide with the principal axes of inertia of the rigid body. The x i system is transformed into the Yi system by two rotations: by an angle k about xa, and by an angle 0 about Y2. The position of the z i axes with respect to the Yi system is determined by the Eulerian angles 0, go, and 9- Table 1 lists the cosines of the angles between axes. The equations of motion of an asymmetric body with respect to a fixed point have the following form [131: dG =L3; d6 Li . dL L~ . d'-'T ~ = '"G- ' at = Gsin-'6' (1 1 ) Lscos,--L, sinr dOdt -- G sin 0sin q~ cos q~ "Jl B + 6 " d,~ OcosO ( 1 sinZ qo cos~ ~0) L, cos,+ Lzsin, . d'-t- = C A B + Gsin0 ' (1) 6