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## Archive of Issues

Russia Yaroslavl
Year
2014
Issue
2
Pages
153-163
 Section Mechanics Title 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 Author(-s) Chistyakov V.V.a Affiliations Yaroslavl State Agricultural Academya Abstract 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. Keywords central axis of inertia, inertia caused torques, dry friction, paradox, quadratic drag, delayed acceleration, aerodynamic-frictional 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 non-principal axis of inertia under action of dry friction in supports, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2014, issue 2, pp. 153-163. References Lur'e A.I. Analiticheskaya mekhanika (Analytical mechanics), Moscow: Nauka, 1961. Smirnov Yu.P. On dynamical equations of the systems with dry friction, Sbornik nauchno-metodicheskikh statei po teoreticheskoi mekhanike, Moscow: Vysshaya shkola, 1981, vol. 11, pp. 184-188 (in Russian). Kozlov V.V. Lagrangian mechanics and dry friction, Nelineinaya Dinamika, 2010, vol. 6, no. 4, pp. 855-868 (in Russian). Astapov I.S. On rotational stability of Celtic stone, Vestnik Moskovskogo Universiteta. Seriya I. Matematika, Mekhanika, 1980, no. 2, pp. 97-100 (in Russian). Markeev A.P. Teoreticheskaya mekhanika (Theoretical mechanics), Moscow: CheRo, 1999, 572 p. Painleve P. Le cons sur le frottement, Paris: Hermann, 1895. Translated under the title Lektsii o trenii, Moscow: Gostekhizdat, 1954, 316 p. Somov O.I. On the acceleration of various orders of magnitude in relative motion, Zap. Imp. Akad. Nauk, 1866, vol. 9, pp. 121-132 (in Russian). Korn G.A., Korn T.M. Mathematical handbook, McGrow-Hill Book Company, 1968. Full text