On the motion of a heavy material point on a rotating sphere (dry friction case)

Abstract

The problem of motion of a heavy particle on a sphere uniformly rotating about a fixed axis is considered in the case of dry friction. It is assumed that the angle of inclination of the rotation axis is constant. The existence of equilibria in an absolute coordinate system and their linear stability are discussed. The equilibria in a relative coordinate system rotating with the sphere are also studied. These equilibria are generally nonisolated. The dependence of the equilibrium sets of this kind on the system parameters is also considered.MSC2010 numbers: 70F40, 70K42, 70K50 a particle on a sphere, absolute and relative equilibria, bifurcations of equilibria On the occasion of the 65th anniversary of Acad. V. V. Kozlov, a prominent scientist and teacherBased on the results of computer simulation (see [1,2]), it is possible to investigate the dynamics of systems with a large number of moving parts. However, the output of such a simulation usually does not represent any qualitative results. That is why it is reasonable to consider some simple problems such as the motion of a material particle on some surface under the action of friction force. In [3] the problem of motion of a heavy bead on a circular hoop rotating about its vertical diameter was studied. A similar problem for a circular hoop rotating about some other vertical axis was investigated in [4].In these papers the dependence of the nonisolated equilibrium sets of a bead on system parameters was considered. The fact of existence of these sets for systems with dry friction is well known (see [5,6]). The investigation of the existence and stability of nonisolated equilibria in gyro systems with friction [7, 8] laid a foundation for the development of the stability theory for systems with dry friction [9]. Methods of stability analysis of equilibria of this kind based on the general theory of systems with discontinuous right-hand sides were later developed in [10][11][12]. An original approach to studying the dependence of nonisolated equilibrium sets on the parameters of a system for two-dimensional and three-dimensional problems was suggested by A. P. Ivanov [13,14]. The same kind of bifurcations, as well as sufficient conditions for stability of equilibrium sets, were considered in [15]. Remark 1. The motion of rigid bodies on moving surfaces is generally investigated in the presence of nonholonomic constraints and under the assumption of no slippage (see, for example, [16]). The same problems for nonholonomic systems under the assumption of dry friction are still very poorly understood.

On the motion of a material point on a rotating sphere with dry friction (the case of the vertical axis)

“…Указанные соответствия и свойства исследовались, вообще говоря, для изолированных решений. В механике систем с сухим трением наблюдают как множества неизолированных относительных равновесий [2,22,23], так и множества неизолированных установившихся движений [9,10,17]. Для таких систем указанные выше соответствия, насколько известно автору, не изучались.…”