Grigory Rozenblat

The problem of the equilibrium and motion of a tripod (a three-legged stand, or in other words, a rigid body moving with three point supports in contact with a horizontal plane) under the classical law of dry Coulomb friction has been the subject of numerous studies. This paper deals with a rigid body acted upon by a system of active forces with an arbitrary and fixed principal vector. The problem of optimizing the equilibrium of this body is posed, which consists in choosing a distribution of normal loads over the supports that guarantees the greatest resistance of the body to rotations about the vertical under conditions of dry friction and in the presence of certain constraints on an arbitrary system of external active forces applied to the body.
In this paper, optimal distributions of loads on the three supports of the body are obtained. These loads provide the corresponding maximum external vertical moment of active forces required for the static equilibrium of the body under dry friction conditions. This moment is obtained in explicit form for various situations. It is shown that, if this threshold moment is exceeded even slightly (in the nondegenerate situation where the body moves with all three points in contact with the plane), the body begins to rotate about the vertical axis passing through the center of the circle circumscribed around the supporting triangle. Furthermore, it is shown that the resulting maximum moment of active forces about the axis does not depend on the horizontal projection of the principal vector of the active forces. Degenerate situations where the body moves with only two points in contact with the plane are also discussed.

This paper deals with a formulation and a solution of problems of the dynamics of mechanical systems for which solutions that do not take into account the unilateral nature of the constraints imposed on the objects under study have been obtained before. The motive force in all the cases considered is the gravity force applied to the center of mass of each body of the mechanical system. Since unilateral constraints are imposed on all systems of bodies considered in the abovementioned problems, their correct solution requires taking into account the unilateral action of the constraint reaction forces applied to the bodies of the systems under study. A detailed analysis of the motion of the systems after zeroing out the constraint reaction forces is carried out. Results of numerical experiments are presented which are used to construct motion patterns of the systems of bodies illustrating the motions of the above-mentioned systems after they lose contact with the supporting surfaces.

This paper presents secure upper and lower estimates for solutions to the equations of rigid body motion in the Euler case (in the absence of external torques). These estimates are expressed by simple formulae in terms of elementary functions and are used for solutions that are obtained in a neighborhood of the unstable steady rotation of the body about its middle axis of inertia. The estimates obtained are applied for a rigorous explanation of the flip-over phenomenon which arises in the experiment with Dzhanibekov’s nut.

The problem of motion of an axisymmetric rigid body on a horizontal plane in the presence of gravity is considered. The body touches the plane at one point, and the plane is assumed to be perfectly smooth. In the already-known and integrable case of symmetric body the normal reaction force exerted by the plane onto the body is calculated and its sign is examined. The condition that the body remains in contact with the plane is that the reaction is positive because the constraint at the point of contact is assumed to be unilateral. In some cases a comparatively trivial analytical representation (a polynomial of degree two) for the reaction force is obtained which allows determination of the initial conditions and the body’s parameters for the body to remain in contact with the plane.

We study the motion (rolling motion) of a flat plate whose boundary is an arbitrary convex curve along a straight line. During the motion the plate is always in contact with the supporting line and subject to a dry friction. Plus, the plate is acted on by an arbitrary plane system of forces and at the point of contact only the unilateral constraint is assumed. All possible transitions from a rolling motion with slipping to a pure rolling without slipping and vice versa are classified. Necessary conditions for the plate to remain in contact with the line are obtained. The results obtained are used to study 1) the motion of a non-uniform circular disk, subject to gravity, on a rough horizontal straight line in the vertical plane and 2) the motion of a slender rod, subject to gravity, on a rough straight line.