On some particular cases of the rotational dynamics of a rigid body around central but non-principal axis of inertia under action of dry friction in supports

The article studies the rotational dynamics of a rigid body (rotator) around the central but non-principal axis $Oz$ passing through its center of mass under the action of dry frictional torque $M_{fr} =\alpha \sqrt {\varepsilon ^2+\omega ^4} $ caused by inertia forces in the axis's supports and the drag momentum $M_R =-c|\omega|\omega $ quadratic in angular speed $\omega $. It has been shown that the dynamical equations and the equations of the kinetics of the body's rotation, which follow from the dynamical equations, are qualitatively different in general and particular cases of the inertial and dissipative parameters involved: the axial moment of inertia $J_{zz} $ and the coefficients $c$ and $\alpha ={M_{fr}}/{\sqrt {\varepsilon ^2+\omega ^4} }$ where ($\omega$ is the angular acceleration). It is found that in the particular case of the equality $J_{zz} =c=\alpha $ a physical feasible solution for the inertial rotation within the dynamics of a perfectly rigid body is absent. The paradox is resolved by the introduction of the lagged angular velocity $\omega (t-\tau)$ and acceleration $\varepsilon (t-\tau )$ as factors defining due to D'Alembert principle the supports' transversal reactions $M_{x,y} (t-\tau)$ and hence the value of $M_{fr} (t-\tau)$. The last one determines the loss rate of kinetic momentum, i.e. the ${dK_z (t)}/{dt}$ at time $t$. The rotational kinetics had a type of frictional-aerodynamic impact. Also, by numerical integration, there was shown the unusual angular kinetics $\phi (t)$ of the damping oscillations of the rotator under the action of the elastic torque $M_e =-\kappa \phi $. The kinetics was characterized by the presence of two phases: the short starting part strongly depending on initial conditions followed by the phase of almost sine wave oscillations with extremely slow damping.