On periodic motions of the body with movable internal mass over a horizontal surface

Theoretical mechanics


Bardin B. S.1*, Panev A. S.2**

1. ,
2. Moscow Aviation Institute (National Research University), 4, Volokolamskoe shosse, Moscow, А-80, GSP-3, 125993, Russia

*e-mail: bardin@yandex.ru
**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 relatively to it over circumference of R radius, which center coincides with the centre of mass of the body. Moreover, the angular velocity of the points’ circulatory movement is constant. Frictional forces between the body and horizontal plane are specified by Coulomb friction model. We assume that at the initial instant the body stays at rest, and the moving mass occupies its lowest position.

Thus stated the problem has the two parameters: k — Coulomb friction factor and . With certain values of these parameters, the relative motion of the point can cause motions of the body over the horizontal plane. This work aims at complete qualitative investigation of periodic motions of the body without the horizontal plane liftoff.

The study of the motion of the abovementioned mechanical system presents not only theoretical interest, but also can be applied to the development of mobile devices (vibrating robots) moving due to the displacement of internal mass. 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. The set of values of parameters k and μ wherein the movement of the body is possible is broken down into three areas: I, II and III. The movement of the body in the abovementioned areas bears qualitatively different character. If the values of the parameters belong to region I, then 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. With parameters from area II, the body moves with alternate 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. But if parameter values belong to region III, there are no quiescence intervals, and body motion in general will not be periodic. The movement is of asymptotic nature, i.e., it approaches a certain limiting periodic mode of movement. In this limiting mode, the body moves in positive direction.

The work is carried out at the Moscow Aviation Institute (National Research University) at the cost of the grant of the Russian Scientific Foundation (project № 14-21-00068).


periodic motion, friction, rigid body, mobile robots


  1. Chernous’ko F.L. Prikladnaya matematika i mekhanika, 2006, vol. 70, no.6, pp. 915-946.

  2. Chernous’ko F.L. Prikladnaya matematika i mekhanika, 2008, vol. 72, no.2, pp. 203-215.

  3. Bolotnik N.N., Chernous’ko F.L. Trudy Instituta matematiki i mekhaniki UrO RAN, 2010, vol. 16, no.5, pp. 213–222.

  4. Bolotnik N.N., Zeidis I.M., Tsimmermann K., Yatsun S.F. Izvestiya RAN. Teoriya i sistemy upravleniya., 2006, vol. 70, no. 5, pp. 157-167.


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