Publications

Online First Articles

• 1.

  Kilin A. A., Karavaev Y. L.

  Experimental observation of stabilization of tippe top spinning on a vibrating plane
  Theoretical and Applied Mechanics, 2025

  Abstract          pdf  (840.04 Kb)

  
  

  Citation	Kilin A. A., Karavaev Y. L., Experimental observation of stabilization of tippe top spinning on a vibrating plane, Theoretical and Applied Mechanics, 2025, https://doi.org/10.2298/TAM241204001K
  DOI:	10.2298/TAM241204001K
  Full text:	pdf  (840.04 Kb)
• 2.

  Kilin A. A., Ivanova T. B., Karavaev Y. L., Yefremov K. S.

  Theoretical and experimental investigations of the controlled motion of a roller racer
  Theoretical and Applied Mechanics, 2024

  Abstract          pdf  (1.37 Mb)

  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). 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.

  
  

  Citation	Kilin A. A., Ivanova T. B., Karavaev Y. L., Yefremov K. S., Theoretical and experimental investigations of the controlled motion of a roller racer, Theoretical and Applied Mechanics, 2024, https://doi.org/10.2298/TAM241203010K
  DOI:	10.2298/TAM241203010K
  Full text:	pdf  (1.37 Mb)
• 3.

  Kilin A. A., Pivovarova E. N., Ivanova T. B.

  Rolling of a Homogeneous Ball on a Moving Cylinder
  Regular and Chaotic Dynamics, 2024

  Abstract          pdf  (710.83 Kb)

  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 quasiperiodic 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.

  
  

  Keywords:	homogeneous ball, nonholonomic constraint, surface of revolution, moving cylinder, unbounded drift, nonautonomous system, quadrature, integrability
  Citation:	Kilin A. A., Pivovarova E. N., Ivanova T. B., Rolling of a Homogeneous Ball on a Moving Cylinder, Regular and Chaotic Dynamics, 2024, https://doi.org/10.1134/S1560354724590027
  DOI:	10.1134/S1560354724590027
  Full text:	pdf  (710.83 Kb)
• 4.

  Kilin A. A., Gavrilova A. M., Artemova E. M.

  Dynamics of an Elliptic Foil with an Attached Vortex in an Ideal Fluid: The Integrable Case
  Regular and Chaotic Dynamics, 2024

  Abstract          pdf  (1.52 Mb)

  This paper is concerned with the plane-parallel motion of an elliptic foil with an attached vortex of constant strength in an ideal fluid. Special attention is given to the case in which the vortex lies on the continuation of one of the semiaxes of the ellipse. It is shown that in this case there exist no attracting solutions and the system is integrable by the Euler – Jacobi theorem. A complete qualitative analysis of the equations of motion is carried out for cases where the vortex lies on the continuation of the large or the small semiaxis of the ellipse. Possible types of trajectories of an elliptic foil with an attached vortex are established: quasi-periodic, unbounded (going to infinity) and periodic trajectories.

  
  

  Keywords:	ideal fluid, elliptic foil, point vortex, integrable system, bifurcation analysis
  Citation:	Kilin A. A., Gavrilova A. M., Artemova E. M., Dynamics of an Elliptic Foil with an Attached Vortex in an Ideal Fluid: The Integrable Case, Regular and Chaotic Dynamics, 2024, https://doi.org/10.1134/S1560354724590015
  DOI:	10.1134/S1560354724590015
  Full text:	pdf  (1.52 Mb)
• 5.

  Artemova E. M., Lagunov D. A., Vetchanin E. V.

  The Motion of an Elliptic Foil in the Field of a Fixed Vortex Source
  Rus. J. Nonlin. Dyn., 2024

  Abstract          pdf  (656.91 Kb)

  This paper is concerned with the motion of an elliptic foil in the field of a fixed point singularity. A complex potential of the fluid flow is constructed, and the forces and the torque which act on the foil from the fluid are obtained. It is shown that the equations of motion of the elliptic foil in the field of a fixed point vortex source can be represented as Lagrange – Euler equations. It is also shown that the system has an additional first integral due to the conservation of the angular momentum. An effective potential of the system under consideration is constructed. For the cases where the singularity is a vortex or a source, unstable relative equilibrium points corresponding to the circular motion of the foil around the singularity are found.

  
  

  Keywords:	ideal fluid, elliptic foil, point vortex, point source, Lagrangian form, Hamiltonian form
  Citation:	Artemova E. M., Lagunov D. A., Vetchanin E. V., The Motion of an Elliptic Foil in the Field of a Fixed Vortex Source, Rus. J. Nonlin. Dyn., 2024, https://doi.org/10.20537/nd241203
  DOI:	10.20537/nd241203
  Full text:	pdf  (656.91 Kb)
• 6.

  Bizyaev I. A.

  Trajectories of Light Beams in a Kerr Metric: the Influence of the Rotation of an Observer on the Shadow of a Black Hole
  Rus. J. Nonlin. Dyn., 2025

  Abstract          pdf  (5.33 Mb)

  This paper investigates the trajectories of light beams in a Kerr metric, which describes the gravitational field in the neighborhood of a rotating black hole. After reduction by cyclic coordinates, this problem reduces to analysis of a Hamiltonian system with two degrees of freedom. A bifurcation diagram is constructed and a classification is made of the types of trajectories of the system according to the values of first integrals. Relations describing the boundary of the shadow of the black hole are obtained for a stationary observer who rotates with an arbitrary angular velocity about the axis of rotation of the black hole.

  
  

  Keywords:	Kerr metric, trajectories of light beams, shadow of a black hole, bifurcation diagram, gravitational lensing
  Citation:	Bizyaev I. A., Trajectories of Light Beams in a Kerr Metric: the Influence of the Rotation of an Observer on the Shadow of a Black Hole, Rus. J. Nonlin. Dyn., 2025, https://doi.org/10.20537/nd250101
  DOI:	10.20537/nd250101
  Full text:	pdf  (5.33 Mb)