Alexander S. Kuleshov

Mathematical Model of a Skateboard with One Degree of Freedom

In this paper, we construct a mathematical model describing the motion of a skateboard. It is assumed that the rider does not control the skateboard. Equations of motion of the model are presented in the form of Gibbs‐Appell equations. The stability of the motion of a skateboard is analyzed.

The steady rolling of a disc on a rough plane

Abstract The well-known problem of the rolling without slipping of a heavy circular disc along a horizontal plane is considered. The steady motions of a disc for which the angle between the plane of the disc and the supporting plane (the angle of nutation) is constant are investigated. The problem of the range of variation of the angle of nutation within which the given motions are stable, irrespective of the values of the constants of the two linear first integrals or, in other variables, irrespective of the angular velocities of precession and proper rotation, is investigated. It is shown that this range is wider than was established earlier in [1].

Further development of the mathematical model of a snakeboard

This paper gives the further development for the mathematical model of a derivative of a skateboard known as the snakeboard. As against to the model, proposed by Lewis et al. [1] and investigated by various methods in [1–13], our model takes into account an opportunity that platforms of a snakeboard can rotate independently from each other. This assumption has been made earlier only by Golubev [13]. Equations of motion of the model are derived in the Gibbs-Appell form. Analytical and numerical investigations of these equations are fulfilled assuming harmonic excitations of the rotor and platforms angles. The basic snakeboard gaits are analyzed and shown to result from certain resonances in the rotor and platforms angle frequencies. All the obtained theoretical results are confirmed by numerical experiments.

The steady motions of a disc on an absolutely rough plane

Abstract The branching of the steady motions of a heavy circular disc on an absolutely rough horizontal plane is investigated. The motions corresponding to critical points of the energy integral at fixed levels of two other integrals having the form of hypergeometric series are considered.

Nonlinear Dynamics of a Simplified Skateboard Model (P24)

In this paper the further investigation and development for the simplified mathematical model of a skateboard with a rider are obtained. This model was first proposed by Mont Hubbard (Hubbard 1979, Hubbard 1980). It is supposed that there is no rider’s control of the skateboard motion. To derive equations of motion of the skateboard the Gibbs-Appell method is used. The problem of integrability of the obtained equations is studied and their stability analysis is fulfilled. The effect of varying vehicle parameters on dynamics and stability of its motion is examined.

On moving Chaplygin sleigh on a convex surface

We study the problem of moving Chaplygin sleigh along an arbitrary surface. Motion equations for the sleigh are represented as Appel equations. We consider the case when the sleigh moves along a surface of rotation, in particular, along a plane, sphere, and cylinder. We show several cases when the sleigh’s motion equations can be fully integrated.

First Integrals of Equations of Motion of a Heavy Rotational Symmetric Body on a Perfectly Rough Plane

We consider the problem of the motion of a heavy dynamically symmetric rigid body bounded by a surface of rotation on a fixed perfectly rough horizontal plane. The integrability of this problem was proved by S.A. Chaplygin [1]. Chaplygin has found that the equations of motion of given mechanical system have, besides the energy integral, two first integrals, linear in generalized velocities. However, the explicit form of these integrals is known only in the case, when the moving body is a nonhomogeneous dynamically symmetric ball. In the case, when the moving body is a round disk or a hoop, the integrals, linear in the velocities, are expressed using hypergeometric series [1],[2],[3]. In the paper of Kh.M. Mushtari [4] the investigation of this problem was continued. For additional restrictions, imposed on the surface of moving body and its mass distribution, Mushtari has found two particular cases, when the motion of the body can be investigated completely. In the first case the moving rigid body is bounded by a surface formed by rotation of an arc of a parabola about the axis, passing through its focus, and in the second case, the moving body is a paraboloid of rotation. For other bodies, bounded by a surface of rotation and moving without sliding on a horizontal plane, the explicit form of additional first integrals is unknown.

Existence of liouvillian solutions in the problem of motion of a rotationally symmetric body on a sphere

The problem of rolling without sliding of a rotationally symmetric rigid body on a sphere is considered. The rolling body is assumed to be subjected to the forces, the resultant of which is directed from the center of mass G of the body to the center O of the sphere, and depends only on the distance between G and O. In this case the solution of this problem is reduced to solving the system of two first order linear differential equations over the projections ω3 and n of the angular velocity of the body onto its axis of symmetry and onto the normal to the sphere respectively. The problem of determination of the shape of the rolling body for which the equation for ω3 can be solved by separation of variables is studied.

Nonexistence of Liouvillian Solutions in the Problem of Motion of a Rotationally Symmetric Ellipsoid on a Perfectly Rough Plane

Using the Kovacic algorithm, the nonexistence of liouvillian solutions in the problem of motion of a rotationally symmetric ellipsoid on a perfectly rough horizontal plane for almost all values of parameters of the problem is proven.

Horizontal motion of a body consisting of two symmetric plates

The problem of motion of a rigid body on a fixed horizontal plane is considered. The body consists of two identical symmetric plates. These plates are connected perpendicularly to one another in such a way that their symmetry axes form a single axis being the symmetry axis of the body. All equilibrium positions of the body on the plane are found and their stability analysis is performed. The particular case of a body consisting of two identical elliptic plates is studied.