The motion of the 2D hydrodynamic Chaplygin sleigh in the presence of circulation

Abstract

We consider the motion of a planar rigid body in a potential flow with circulation and subject to a certain nonholonomic constraint. This model is related to the design of underwater vehicles.The equations of motion admit a reduction to a 2-dimensional nonlinear system, which is integrated explicitly. We show that the reduced system comprises both asymptotic and periodic dynamics separated by a critical value of the energy, and give a complete classification of types of the motion. Then we describe the whole variety of the trajectories of the body on the plane.

Control of locomotion systems and dynamics in relative periodic orbits

“…Reconstruction for SE(2)-invariant systems appear in various works, see particularly [22,21], and the results in Section 4.1 are known, except for the introduction of the frequencies of unbounded motions. 4.1.…”

“…Hence (see also [22,21]): We now consider the frequencies of motions in the relative periodic orbit. As discussed in Section 2.4, they are the frequencies produced by reconstructing the gait with the action of S 1 , namely, with the first equation (15).…”

“…These behaviours are revealed in numerical integrations and have been investigated in a number of works, particularly oriented towards infinite dimensional systems and having many applications (see e.g. [2,22,23,21,20,14] and references therein). In the case of quasiperiodic motions, the origin of these behaviours is clarified by the reconstruction procedure of Krupa and Field: one frequency is due to the periodicity of the gait in shape space, while the remaining k are produced by the action of a compact abelian subgroup of G, isomorphic to a torus.…”

Nonholonomic LL systems on central extensions and the hydrodynamic Chaplygin sleigh with circulation

“…and with arbitrary circulation constants α, β. The particular expressions for α, β and I in the case that the body is of elliptic shape are given in [7].…”

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

“…The authors of [60] assert that the Kutta-Zhukovsky condition is equivalent to a nonholonomic constraint, which is, generally speaking, incorrect from the viewpoint of physical principles of mechanics. By the way, a nonholonomic model is also used in [23,24,27] to describe the motion of a plate in a fluid. It should be noted that, when it comes to describing the motion of a rigid body in an ideal fluid, the equations with nonintegrable constraints arise within the framework of vakonomic mechanics.…”