On motion of a rigid body with mobile internal mass on a horizontal plane in a viscous medium

Theoretical mechanics


Panev A. S.

Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

e-mail: a.s.panev@gmail.com


We consider a movement of a solid body carrying sliding single mass point. A body of M mass is situated on a horizontal plane, and a single mass point of m mass is moving inside it over circumference of R radius, which center coincides with the center of mass of the body, counterclock-wise. The forces of dry and viscous friction act between the body and the horizontal surface. Moreover, the angular velocity of the points’ circulatory movement is constant. We assume that at the initial instant the body stays at rest, and the moving mass occupies its lowest position

This work aims is complete qualitative investigation of all possible modes of motion body without the horizontal plane liftoff.

In addition to theoretical interest, the study of mechanical systems of this type can have an applied value, for example, as the creation of vibrating robots moving due to the forces of inertia that arise when moving internal masses. The advantage of such devices is that they do not require special propulsion (wheels, tracks, legs, and so forth.) And can be performed in an enclosure, the latter fact makes them resistant to attack by the external environment and allows to apply, on solid surfaces, and in liquids. Such devices are promising for the modern space industry. In particular, they can be useful for the study of celestial objects: asteroids, planets, solar systems and their satellites.

Based on the analytical and numerical studies made in this work we obtained the following conclusions. Three possible modes of motion were found, the conditions for their existence depend on the parameters of the problem: κ is the coefficient of dry friction,ν is the coefficient of viscous friction, and 

  1. The body performs a periodical reciprocating motion with a period equal to the period of a full turn of the point along the circle. The body moves during equal intervals of time in positive and negative directions. These time intervals of movements interlace with quiescence intervals of the body on horizontal plane.

  2. The body moves with periodic velocity. With that, during one cycle the body shifts in positive direction, changing twice the direction of its movement. Time intervals of motion in negative direction are separated from those in positive direction by quiescence intervals.

  3. The body moves non-periodically. The motion is of asymptotic nature, i.e., it approaches a certain limiting periodic mode of motion. In this limiting mode, the body moves in positive direction.


periodic motion, contact surface, rigid body, mobile robots


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