This paper is concerned with free and controlled motions of a spherical robot of combined type moving by displacing the center of mass and by changing the internal gyrostatic momentum. Equations of motion for the nonholonomic model are obtained and their first integrals are found. Fixed points of the reduced system are found in the absence of control actions. It is shown that they correspond to the motion of the spherical robot in a straight line and in a circle. A control algorithm for the motion of the spherical robot along an arbitrary trajectory is presented. A set of elementary maneuvers (gaits) is obtained which allow one to transfer the spherical robot from any initial point to any end point.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we investigate the dynamics of a body with a flat base (cylinder) sliding on a horizontal rough plane. For analysis we use two approaches. In one of the approaches using a friction machine we determine the dependence of friction force on the velocity of motion of cylinders. In the other approach using a high-speed camera for video filming and the method of presentation of trajectories on a phase plane for analysis of results, we investigate the qualitative and quantitative behavior of the motion of cylinders on a horizontal plane. We compare the results obtained with theoretical and experimental results found earlier. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we investigate the dynamics of a body with a flat base sliding on a horizontal and inclined rough plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. For analysis we use the descriptive function method similar to the methods used in the problems of Hamiltonian dynamics with one degree of freedom and allowing a qualitative analysis of the system to be made without explicit integration of equations of motion. In addition, we give a systematic review of the well-known experimental and theoretical results in this area.

In this paper we consider the control of a dynamically asymmetric balanced ball on a plane in the case of slipping at the contact point. Necessary conditions under which a control is possible are obtained. Specific algorithms of control along a given trajectory are constructed.

In this paper we consider the dynamics of a rigid body with a sharp edge in contact with a rough plane. The body can move so that its contact point is fixed or slips or loses contact with the support. In this paper, the dynamics of the system is considered within three mechanical models which describe different regimes of motion. The boundaries of the domain of definition of each model are given, the possibility of transitions from one regime to another and their consistency with different coefficients of friction on the horizontal and inclined surfaces are discussed.

In this paper we investigate the dynamics of a body with a flat base sliding on a inclined plane under the assumption of linear pressure distribution of the body on the plane as the simplest dynamically consistent friction model. Computer-aided analysis of the system?s dynamics on the inclined plane using phase portraits has allowed us to reveal dynamical effects that have not been found earlier.

Inšthis paper wešconsider the control ofšašdynamically asymmetric balanced ball onšašplane inšthe case ofšslipping atšthe contact point. Necessary conditions under which ašcontrol isšpossible are obtained. Specific algorithms ofšcontrol along ašgiven trajectory are constructed.

In this paper we describe an inertiameter, which is an experimental facility for determining the inertia tensor components and the position of the center of mass of compound bodies. An algorithm for determining these dynamical properties is presented. Using the algorithm obtained, the displacement of the center of mass and the tensor of inertia are determined experimentally for a spherical robot of combined type.

Inšthis paper wešconsider the dynamics ofšrigid body whose sharp edge isšinšcontact with ašrough plane. The body can move sošthat its contact point does not move oršslips oršloses touch with the support. Inšthis paper, the dynamics ofšthe system isšconsidered within three mechanical models that describe different modes ofšmotion. The boundaries ofšdefinition range ofšeach model are given, the possibility ofštransitions from one mode tošanother and their consistency with different coefficients ofšfriction onšthe horizontal and inclined surfaces isšdiscussed.

This paper investigates the possibility ofšthe motion control ofšašball with ašpendulum mechanism with non-holonomic constraints using gaitsš? the simplest motions such asšacceleration and deceleration during the motion inšašstraight line, rotation through ašgiven angle and their combination. Also, the controlled motion ofšthe system along ašstraight line with ašconstant acceleration isšconsidered. For this problem the algorithm for calculating the control torques isšgiven and itšisšshown that the resulting reduced system has the first integral ofšmotion.

The problem ofšašuniform straight cylinder (disc) sliding onšašhorizontal plane under the action ofšdry friction forces isšconsidered. The contact patch between the cylinder and the plane coincides with the base ofšthe cylinder. Wešconsider axisymmetric discs, i.e. wešassume that the base ofšthe cylinder isšsymmetric with respect tošthe axis lying inšthe plane ofšthe base. The focus isšonšthe qualitative properties ofšthe dynamics ofšdiscs whose circular base, triangular base and three points are inšcontact with ašrough plane.

In the paper we study the stability of a spherical shell rolling on a horizontal plane with Lagrange?s gyroscope inside. A linear stability analysis is made for the upper and lower position of a top. A bifurcation diagram of the system is constructed. The trajectories of the contact point for di?erent values of the integrals of motion are constructed and analyzed.

The classical problem about the motion of a heavy symmetric rigid body (top) with a fixed point on the horizontal plane is discussed. Due to the unilateral nature of the contact, detachments (jumps) are possible under certain conditions. We know two scenarios of detachment related to changing the sign of the normal reaction or the sign of the normal acceleration, and the mismatch of these conditions leads to a paradox. To determine the nature of paradoxes an example of the pendulum (rod) within the limitations of the real coefficient of friction was studied in detail. We showed that in the case of the first type of the paradox (detachment is impossible and contact is impossible) the body begins to slide on the support. In the case of the paradox of the second type (detachment is possible and contact is possible) contact is retained up to the sign change of the normal reaction, and then at the detachment the normal acceleration is non-zero.

The paper considers two two-dimensional dynamical problems for anšabsolutely rigid cylinder interacting with ašdeformable flat base (the motion ofšanšabsolutely rigid disk onšašbase which inšnon-deformed condition isšašstraight line). The base isšašsufficiently stiff viscoelastic medium that creates ašnormal pressure $p(x) = kY(x)+?\dot{Y}(x)$, where $x$šisšašcoordinate onšthe straight line, $Y(x)$ isšašnormal displacement ofšthe pointš$x$, and $k$šand $?$šare elasticity and viscosity coefficients (the Kelvin?Voigt medium). Wešare also ofšthe opinion that during deformation the base generates friction forces, which are subject tošCoulomb?s law. Wešconsider the phenomenon ofšimpact that arises during anšarbitrary fall ofšthe disk onto the straight line and investigate the disk?s motion ?along the straight line? including the stages ofšsliding and rolling.

We consider the inhomogeneous self-gravitating liquid spheroid with confocal stratification which rotates around the minor semiaxis and is in equilibrium. General relationships for pressure, angular velocity and gravitational potential of the spheroid with any density function are obtained. Special cases of piecewise constant and continuous density functions are analyzed.

Wešconsider figures ofšequilibrium and stability ofšašliquid self-gravitating elliptic cylinder. The flow within the cylinder isšassumed tošbešdew tošanšelliptic perturbation. Ašbifurcation diagram isšplotted and conditions for steady solutions tošexist are indicated.

Figures of equilibrium are considered and the stability of liquid self-gravitating elliptic cylinder with an internal flow in a class of elliptic indignations are researched. The bifurcation diagram of given system is constructed, areas of existence of the stationary solutions are specified.

The generalized model of formation of a new phase is considered. The basic stages of process of growth are gathered in a model at phase transition of the first sort. The numerical solution of the kinetic equation of Fokker–Planck is received. Dependence of the solution on parametres of system is investigated. Areas of applicability of assumptions made by Zeldovich, Lifshits and Slezov are revealed. Also it is shown, that depending on parametres of system it is possible to reserve both equilibrium distribution, and automodelling distribution of Lifshits–Slezov. At some values of parametres the equation has the oscillatory solution.