Section

Mechanics

Title

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

Author(s)

Chistyakov V.V.^{a}

Affiliations

Yaroslavl State Agricultural Academy^{a}

Abstract

The article studies the rotational dynamics of a rigid body (rotator) around the central but nonprincipal 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 frictionalaerodynamic 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.

Keywords

central axis of inertia, inertia caused torques, dry friction, paradox, quadratic drag, delayed acceleration, aerodynamicfrictional impact

UDC

531.01, 531.47, 531.536

MSC

70E40,70F40, 74H15, 70K40

DOI

10.20537/vm140211

Received

1 February 2014

Language

Russian

Citation

Chistyakov V.V. On some particular cases of the rotational dynamics of a rigid body around central but nonprincipal axis of inertia under action of dry friction in supports, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 153163.

References

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