The Chaplygin sleigh with parametric excitation: chaotic dynamics and nonholonomic acceleration

TL;DR

This study investigates the chaotic dynamics and potential for unbounded acceleration in a Chaplygin sleigh with time-varying mass distribution, revealing complex behaviors due to parametric excitation and nonholonomic constraints.

Contribution

It introduces a reduced model for the sleigh with parametric excitation, analyzing chaotic behavior and conditions for unbounded acceleration, which are novel insights into nonholonomic systems.

Findings

Chaotic trajectories with strange attractors observed.

Unbounded acceleration linked to excitation parameters and geometry.

Existence of tensor invariants in special cases.

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.