Theoretical and experimental investigations of the controlled motion of a roller racer
Alexander A. Kilin, Tatiana B. Ivanova, Yury L. Karavaev, and Kirill S. Yefremov
Available online 25 December 2024

Abstract
In this paper, we address the problem of the controlled motion of a roller racer on a plane. We assume that the angle between the platforms is a given periodic function of time (control function), and the no-slip conditions (nonholonomic constraint) and viscous friction forces act at the points of contact of the wheels with the plane. In this case, all trajectories of the reduced system tend asymptotically to a periodic solution. In this paper, we show that for a selected periodic control function there exists a motion of the system that is bounded (along a circle) and unbounded (along a straight line). Unbounded motion corresponds to the resonant case which takes place at zero average value of the control function. The theoretical dependence of the trajectory and the velocity of the roller racer on its parameters and the parameters of the selected control function is investigated. These dependences are confirmed experimentally.

Mathematics Subject Classification
37J60, 70E60

Keywords
roller racer, nonholonomic constraint, viscous friction, control, periodic solution

DOI
https://doi.org/10.2298/TAM241203010K

Experimental observation of stabilization of tippe top spinning on a vibrating plane
Alexander A. Kilin and Yury L. Karavaev
Available online 10 January 2025

Abstract
This paper presents an experimental study of the motion of a spherical body with a displaced center of mass on a surface undergoing oscillations in the vertical plane. The phenomenon of vibrostabilization of an unstable position in which the center of mass of the body is above the geometric center of the sphere is revealed. The influence of the frequency and amplitude of oscillations of the surface on the stabilization of the unstable position of the motion of the spherical body is analyzed.

Mathematics Subject Classification
74H45; 70-05, 70E50

Keywords
spherical top, vibrating plane, stabilization, experiment

DOI
https://doi.org/10.2298/TAM241204001K