On the equilibria and uniform rotations of a dumbbell-shaped body on a rough horizontal plane with two contact points

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Abstract

A problem of motion of a dumbbell-shaped body on a horizontal rough plane is considered. It is assumed that the dumbbell is a weightless inextensible rod, with masses being concentrated at two points of it, and there is dry friction between these points and the plane. It is also assumed that a constant force acts perpendicular to the rod on some fixed point on it. The conditions under which the rod is at rest, as well as the conditions under which the rod uniformly rotates around one of its points of support, are determined. The relationship between the magnitude of the angular velocity of uniform rotation and the force providing such a rotation is revealed. Bifurcation diagrams are constructed and analyzed.

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About the authors

A. A. Burov

FRC CSC RAS

Author for correspondence.
Email: jtm@narod.ru
Russian Federation, Moscow

V. I. Nikonov

FRC CSC RAS

Email: nikon_v@list.ru
Russian Federation, Moscow

E. S. Shalimova

FRC CSC RAS; Lomonosov Moscow State University

Email: ekateryna-shalimova@yandex.ru
Russian Federation, Moscow; Moscow

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Supplementary files

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2. Fig. 1. A rod on a rough horizontal plane under the action of external forces.

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3. 2. The equilibrium of the rod on a rough horizontal plane under the action of external forces.

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4. 3. The set of points on the plane (c, f ), where inequalities (3.5) are fulfilled, at 0 ≤ K < 1 (a), K = 1 (b), K > 1 (c).

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5. 4. Distribution of forces in the case of uniform rotation of the rod around point A.

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6. Fig. 5. Distribution of forces in the case of uniform rotation of the rod around point B.

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7. Fig. 6. Distribution of forces in the case of uniform rotation of the rod around the O point.

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8. 7. The case of mA = mB, K = 1.

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9. 8. The case of mA = mB, K = 2.

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10. Fig. 9. The case of mA = mB, K = 1/2.

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