On the Optimization of the Equilibrium of a Tripod on a Rough Surface

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.