Exact normal forces and trajectories for a rotating tripod sliding on a smooth surface

Abstract

We study the motion of a rotating tripod sliding over a smooth surface. The tripod is important and unique in that it is the only case where all aspects of the motion can be solved exactly. From the three constraint equations, we calculate the three normal forces and friction forces, which we then use to calculate the trajectory and other physical quantities of interest. We report novel results: the normal force has an interesting form when the sliding speed and the speed of rotation are almost equal; the tripod can execute serpentine trajectories; trajectories can have large lateral deflections; the translational kinetic energy can decrease and increase during a full rotation; the tripod can tip at a very late stage of its motion. The tripod trajectories are compared to trajectories of rotating, sliding cylinders that have continuous contact rings. We discuss significant implications of our results. PACS No.: 45.40.–f

Dynamics of a body sliding on a rough plane and supported at three points

“…As a tripod we consider a round solid cylinder of radius which slides on three point supports (the radius of inertia is equal, accordingly, to = √ 2 ). Let us compare the results with those obtained previously in [11] for this system and in [6] for a cylinder with a flat base (disk) under the assumption of linear pressure distribution of the body on the plane of sliding.…”

“…Remark 2.1. In [11], the moving coordinate system has been chosen such that one of its axes coincides with the linear velocity vector of the tripod. However, such a choice requires constantly recalculating the projections of the position of the supporting points onto its axes.…”

“…In [4], more general quadratures were obtained, but they are entirely useless for analysis of the dynamics. A modern investigation of the motion of a tripod was carried out by Shegelski and coauthors in [11]. This paper presents numerically constructed trajectories for different tripods.…”

On the motion of a heavy body with a circular base on a horizontal plane and riddles of curling

“…Исследованию динамики керлинга посвящено значительное число работ, включая [1][2][3][4][5][6][7][8][9]. Отметим следующие результаты.…”

Coupling between static friction force and torque for a tripod

“…Another class of systems which has been studied extensively in a series of papers by Shegelski and coworkers is that of sliding tripods with symmetrically placed legs (2π/3 radians apart from each other, see Fig. 1) [8]. One observes, for given initial conditions, trajectories which are serpentine-like (albeit with low lateral deflection), to curved trajectories with very large lateral deflections (curling).…”