. 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). An 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.

This paper addresses the problem of a homogeneous ball rolling on the inner surface of a circular cylinder in a field of gravity parallel to its axis. It is assumed that the ball rolls without slipping on the surface of the cylinder, and that the cylinder executes plane-parallel motions in a circle perpendicular to its symmetry axis. The integrability of the problem by quadratures is proved. It is shown that in this problem the trajectories of the ball are quasi-periodic in the general case, and that an unbounded elevation of the ball is impossible. However, in contrast to a fixed (or rotating) cylinder, there exist resonances at which the ball moves on average downward with constant acceleration.

This paper treats the problem of a spherical robot with an axisymmetric pendulum drive rolling without slipping on a vibrating plane. The main purpose of the paper is to investigate the stabilization of the upper vertical rotations of the pendulum using feedback (additional control action). For the chosen type of feedback, regions of asymptotic stability of the upper vertical rotations of the pendulum are constructed and possible bifurcations are analyzed. Special attention is also given to the question of the stability of periodic solutions arising as the vertical rotations lose stability.

This paper addresses the problem of a sphere with axisymmetric mass distribution rolling on a horizontal plane. It is assumed that there is no slipping of the sphere as it rolls in the direction of the projection of the symmetry axis onto the supporting plane. It is also assumed that, in the direction perpendicular to the above-mentioned one, the sphere can slip relative to the plane. Examples of realization of the above-mentioned nonholonomic constraint are given. Equations of motion are obtained and their first integrals are found. It is shown that the system under consideration admits a redundant set of first integrals, which makes it possible to perform reduction to a system with one degree of freedom.

The problems of constructing exact solutions for the Navier–Stokes equations are related not only to the nonlinearity of the hydrodynamic equations, but also to their vector nature. In this paper, we propose a method for systematic constructing exact solutions to the linearized Navier–Stokes equations in the case of viscous flows by the eigenfunction expansion of Hermitian operators. As an example, the corresponding basis for the flow in the inner region of the circle is constructed. For clarity, expressions and graphs of current functions for individual modes are given. The obtained solutions can be used as test solutions for computer modeling of viscous fluid flows.

We investigate the model of controlled motion of a pendulum-actuated spherical robot on a horizontal plane, taking rolling resistance into account. We derive equations of motion and obtain partial steady-state solutions. We present algorithms for designing elementary maneuvers (gaits) which ensure transition between two steady motions of the system. These gaits correspond to the acceleration along a straight line and to the turn through a given angle.

The paper presents the model of rolling resistance and the application of this model for the control of a pendulum actuated spherical robot on a horizontal plane. Control actions are derived in the form of maneuvers (gaits) which ensure the transition between two steady motions of the system. The experiments confirming the applicability of the model of viscous rolling friction and a method for determining coefficients of rolling resistance from experimental data are presented.

In this paper, we investigate the motion of a homogeneous heavy ball rolling without slipping on the surface of a rotating cylinder in two settings: a setting without dissipation and a setting with rolling friction torque which is proportional to the angular velocity of the ball. In both cases we assume that there exists a non-holonomic constraint that corresponds to the condition that there be no slipping at the point of contact. In the first case, the system of five differential equations on the level set of first integrals is reduced to quadratures. To define possible types of motion, we carry out a bifurcation analysis of the reduced system. In the case with friction we show that all trajectories shift on average downward and the ball falls.

This paper is concerned with the problem of a dynamically symmetric heavy ball rolling without slipping on a cone which rotates uniformly about its symmetry axis. The equations of motion of the system are obtained, partial periodic solutions are found, and their stability is analyzed.

This paper is concerned with the rolling of a homogeneous ball with slipping on a uniformly rotating horizontal plane.We take into account viscous friction forces arising when there is slipping at the contact point.It is shown that, as the coefficient of viscosity tends to infinity, the solution of the generalized problem on each fixed time interval tends to a solution of the corresponding nonholonomic problem.

This paper investigates the rolling without slipping of a homogeneous heavy ball on the surface of a rotating cone in two settings: without dissipation in a nonholonomic setting and with rolling friction torque which is proportional to the angular velocity of the ball. In the nonholonomic setting, the resulting system of five differential equations on the level set of first integrals is reduced to quadratures. A bifurcation analysis of the above system is carried out to determine the possible types of motion. In the second case, it is shown that there are not only trajectories emanating from the lower point of the cone (its vertex), but also trajectories to the vertex of the cone (fall). An analysis of the dependence of the type of terminal motion of the center of mass of the ball on initial conditions is carried out.

This paper is concerned with the study of the rolling without slipping of a dynamically symmetric (in particular, homogeneous) heavy ball on a cone which rotates uniformly about its symmetry axis. The equations of motion of the system are obtained, partial periodic solutions are found and their stability is analyzed.

This paper is concerned with a model of the controlled motion of a spherical robot with an axisymmetric pendulum actuator on an inclined plane. First integrals of motion and partial solutions are presented and their stability is analyzed. It is shown that the steady solutions exist only at an inclination angle less than some critical value and only for constant control action.

In this work we consider the controlled motion of a pendulum spherical robot on an inclined plane. The algorithm for determining the control actions for the motion along an arbitrary trajectory and examples of numerical simulation of the controlled motion are given.

In this work we investigate the motion of a homogeneous ball rolling without slipping on uniformly rotating horizontal and inclined planes under the action of a constant external force supplemented with the moment of rolling friction, which depends linearly on the angular velocity of the ball. We systematise well-known results and supplement them with the stability analysis of partial solutions of the system. We also perform an experimental investigation whose results support the adequacy of the rolling friction model used. Comparison of numerical and experimental results has shown a good qualitative agreement.

In this paper, we develop a model of a controlled spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum, with a feedback that stabilizes given partial solutions for a free system at the final stage of motion. According to the proposed approach, feedback depends on phase variables (current position, velocities) and does not depend on the specific type of trajectory. We present integrals of motion and partial solutions, analyze their stability, and give examples of computer simulations of motion with feedback that demonstrate the efficiency of the proposed model.

The dynamics of a Painlevé–Appell system consisting of two point masses joined by a weightless rigid rod is studied within two mechanical models, which describe different motion regimes. One of the masses can slide or can be supported at rest on a rough straight line. The boundaries of the region of definition of each of the models are presented, and the transitions between them are analysed for various friction coefficients.

The experimental stand is described and the results of investigation of the motion of a brake shoe are presented. In the noncritical region, the friction coefficient is determined experimentally. It is shown that its value corresponds to the condition of uniqueness of the solution for construction of this brake shoe. The dynamics observed in the paradoxical-motion region is described.