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Dynamics of an Ellipsoid on a Rough Surface for Two Cases of Friction

Maksim S. Salimov and Vasily E. Pogreev
National Research University “Moscow Power Engineering Institute” Department of Robotics, Mechatronics, Dynamics and Strength of Machines, Moscow, Russia

Abstract—The movement of an ellipsoid with a displaced center of mass is an unusual phenomenon. When it moves on a horizontal plane relative to the vertical axis, the ellipsoid rapidly slows down its rotation. Then there are oscillations around other axes and then the rotation relative to the vertical axis is restored, but the new rotation occurs in a different direction. This phenomenon is similar to the movement of a semi-ellipsoid, the so-called “Celtic stone”. In this paper, we examine the dynamics of an axisymmetric ellipsoid on a rough surface. The contact between the body and the plane is determined by forces that take into account sliding and spinning in an associated form. The contact of the ellipsoid and the surface is considered as a small area (point) concerning the size of the body. This allows us to describe the contact with a combined two-dimensional friction model, and also to neglect the moment of dry friction forces relative to this point. The center of mass of the ellipsoid lies on the axis of its dynamic symmetry and is displaced by some distance from the geometric center of the ellipsoid. This nonuniform distribution of the ellipsoid's mass allows these unusual phenomena to appear because the ellipsoid tends to rotate in the direction where its mass is excessive. The integral expression for the friction force is replaced by Pade approximations and then the corresponding expansion coefficients are determined. The dependence of the found friction force on the sliding and rotation velocities is plotted depending on both arguments. 

Index Terms—ellipsoid, dynamics, solid, dry friction, Padé approximations, sliding, spin, rotation, horizontal surface

Cite: Maksim S. Salimov and Vasily E. Pogreev, "Dynamics of an Ellipsoid on a Rough Surface for Two Cases of Friction," International Journal of Mechanical Engineering and Robotics Research, Vol. 10, No. 8, pp.452-457, August 2021. DOI: 10.18178/ijmerr.10.8.452-457

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