ВПЛИВ ДИНАМІЧНОЇ НЕСИМЕТРІЇ НА СТІЙКІСТЬ ОБЕРТАННЯ У СЕРЕДОВИЩІ З ОПОРОМ ТВЕРДОГО ТІЛА ПІД ДІЄЮ ПОСТІЙНОГО МОМЕНТУ У ІНЕРЦІАЛЬНІЙ СИСТЕМІ ВІДЛІКУ

Abstract

Under the assumption that the center of mass of an asymmetric rigid body is located on the third principal axis of inertia of a rigid body, the previously obtained conditions for the asymptotic stability of uniform rotation in a medium with resistance of a dynamically asymmetric rigid body are investigated. A rigid body rotates around a fixed point, is under the action of gravity, dissipative moment and constant moment in an inertial frame of reference. The stability conditions are presented as a system of three inequalities. The first and second inequalities have the first degree relative to the dynamic unbalance, and the third inequality has the third degree. The first and third inequalities are of the second degree with respect to the overturning or restoring moment, and the second inequality is of the first degree. The first and third inequalities are of the fourth degree with respect to the constant moment, and the second inequality is of the second degree. The third inequality is the most difficult to study. Analytical studies of the influence of dynamic unbalance, restoring and overturning moments on the conditions of asymptotic stability are carried out. Conditions for the asymptotic stability of uniform rotation in a medium with resistance to an asymmetric rigid body are obtained for sufficiently small values of dynamic unbalance. Sufficient stability conditions are written out up to the second order of smallness with respect to the constant moment and the first order of smallness with respect to the restoring and overturning moments. Instability conditions are obtained for sufficiently large dynamic unbalance. The effect of dynamic unbalance on the stability conditions for the rotation of a rigid body around the center of mass is studied. It is shown that in the absence of dissipative asymmetry, it is sufficient for asymptotic stability that the axial moment of inertia of a rigid body be greater than the double equatorial moment and that the well-known necessary stability condition for a symmetric rigid body be satisfied.