﻿Abstract
Interpretation of the motion of a heavy body around a stationary axis and dynamic (kinetic) pressures on bearing by means of the mass moment vector for the pole and the axis

Katica (Stevanović) Hedrih

This paper presents the derivation of the dynamic (kinetic) rotation equations and/or oscillations of a rigid body around a stationary axis in the Earth gravitational field. In the general case the rotation axis is not horizontal. The kinetic equations of the rigid body motion around the stationary axis are interpreted by means of the introduced vectors: $\vec{\mathfrak S}^{(A)}_n$ of the body mass linear moment for the point in the stationary (fixed) bearing $A$ and for the rotation axis oriented by unit vector $\vec n$: $\vec{\mathfrak I}^{(A)}_n$ of the body mass inertia moment and its deviation part of the vector $\vec{\mathfrak D}^{(A)}_n$ of the deviation load by the body mass inertia moment of the rotation axis for the pole in the stationary (fixed) bearing. The formed kinetic equations are used for determining scalar equations of the rigid body rotation/oscillation around the stationary rotation axis as well as for determining the bearings reaction components: that is, axial and devi-ational reaction components - resistance (pressure) of the stationary bearings and deviational (normal to the axis) reaction - the moveable bearing resistance (pressure). The first integral, that is, the integral of energy is determined from the scalar equation of the rigid body rotation/oscillation and by means of it the motion character analysis is performed by means of the phase trajectories and constant energy curves in the phase plane. The analysis of the singularity and phase trajectories in the phase plane leads to the conclusion about a possible appearence of the rigid body asymptotic stochastic behaviour in its rotation around the stationary axis in the Earth gravitational field in the case when at the initial moment the body is communicated a certain bifurcational value of the overall energy by mean of the kinetic and/or potential energy, that is, when at the initial moment the body is communicated a certain angular velocity and/or initial elongation. The time interval in which the motion is done is also taken into consideration in this case as well as the time elongation at inginitum. The appearence of the singularity triggers coupled set is pointed out on the phase portrait and at this bifurcational value of the initial kinetic parameters the sensitivity of the moment character is also pointed out within the scope of that value of the initial motion parameters. From the expressions for the bearing resistances (pressures) parts are selected corresponding to the kinetic pressures - dynamic bearing resistance from the parts that would correspond to the bearing resistances in the case of the determined system static equilibrium. These parts - the kinetic pressures are expressed by means of the vector $\vec{\mathfrak S}^{(A)}_n$ of the body mass linear moment and by the vector $\vec{\mathfrak D}^{(A)}_n$ of the deviation load by the body mass inertia moment of the rotation axis for the rotation axis and for the pole in the stationary (fixed) bearing. On the basis of the expression for the dynamic pressures it can be seen that a part of the stationary bearing reaction coming from the body dynamic properties with respect to the rotation axis its rotation around it depends on the rotator vector $\vec{\mathfrak R}$ and the absolute value of the vector $\vec{\mathfrak S}^{(A)}_n$ of the body mass linear moment for the rotation axis and for the pole in the stationary (fixed) bearing while the reaction part of both the moveable and the stationary (fixed) bearing which also comes from the rigid body moment properties which rotates for the rotation axis and for the pole in the stationary (fixed) bearing depends on the rotator vector and on the absolute value of the vector $\vec{\mathfrak D}^{(A)}_n$ of the deviation load by the body mass inertia moment for the rotation axis. Further on in the paper the rotator vector $\vec{\mathfrak R}$ behaviour is discussed as well as the change of its intensity in the body rotation or oscillation around the stationary axis in the Earth gravitational field. The numerical experiment is performed and the graphical a chemes are formed of the phase trajectories in the phase plane, of the rotator change as the elongation function and the rotator change as the versor in the plane perpendicular to the rotation axis. The diagram shows that the vector has the zero value only at the bifurcational value and that it has extreme values at the singular points; that is, it has maximum in the stable equilibrium positions, and minimums in the unstable equilibrium positions, namely, at the reciprocating points. The reciprocating points appear in oscillation and they correspond to the maximal elongations and the angular velocities are equal to zero at them. I am sure that in this paper I have given a modest contribution to this much explored topic. In my opinion this contribution is in the interpretation of the kinetic equations by means of two newly - introduced vectors $\vec{\mathfrak S}^{(A)}_n$ and $\vec{\mathfrak I}^{(A)}_n$ and in their use in interpreting the kinetic pressures as well as in the introduction of the rotator vectors.