The Chaplygin Sleigh with Parametric Excitation: Chaotic Dynamics and Nonholonomic Acceleration

Abstract

This paper is concerned with the Chaplygin sleigh with timevarying mass distribution (parametric excitation). The focus is on the case where excitation is induced by a material point that executes periodic oscillations in a direction transverse to the plane of the knife edge of the sleigh. In this case, the problem reduces to investigating a reduced system of two first-order equations with periodic coefficients, which is similar to various nonlinear parametric oscillators. Depending on the parameters in the reduced system, one can observe different types of motion, including those accompanied by strange attractors leading to a chaotic (diffusion) trajectory of the sleigh on the plane. The problem of unbounded acceleration (an analog of Fermi acceleration) of the sleigh is examined in detail. It is shown that such an acceleration arises due to the position of the moving point relative to the line of action of the nonholonomic constraint and the center of mass of the platform. Various special cases of existence of tensor invariants are found.

Speedup of a Chaplygin top by means of rotors

“…В частности, его изучение является одной из задач управления различными роботами. Недавно был получен ряд новых результатов в связи с ускорением саней Чаплыгина (являющихся простейшим примером колёсного робота), которое достигается с помощью изменения положения центра масс или гиростатического момента [1][2][3].…”

The Chaplygin sleigh with friction moving due to periodic oscillations of an internal mass

“…Various generalizations of the problem of the Chaplygin sleigh were considered by many authors [20,21,22,23,24,25,26,27,28]. In particular, attention was given to problems of the motion of the sleigh in the presence of periodic impulse impacts [20,21], with periodic switchovers of the nonholonomic constraint to different locations [23,24], and with periodic transfers of the massive load placed on the sleigh [25,26]. Analysis was also made of the motion of the sleigh under the action of random forces which model a fluctuating continuous medium [28], when, according to the authors, the motion resembles random walks of bacterial cells with a diffusion component.…”

“…In [25], general equations are formulated which describe the motion of the Chaplygin sleigh with a given periodic transfer of several internal point masses relative to the platform. It is shown that, depending on the parameters, the system exhibits different types of motions, including those corresponding to strange attrac-tors, with which chaotic (diffusion) trajectories of the sleigh on a plane are associated.…”

Fermi-like acceleration and power-law energy growth in nonholonomic systems

“…Suppose that n material points P (i) , i = 1, .., n move on the platform in a prescribed manner. In this case the kinetic energy of the entire system can be represented in the following form [3]:…”

“…Starting with [12], strange attractors of different nature [6,30,5] are detected in nonholonomic systems. A strange attractor for the Chaplygin sleigh with a material point, which executes periodic oscillations in the direction transverse to the plane of the knife edge, is found in [1,3].…”