Evgeny Vetchanin
Publications:
Mamaev I. S., Vetchanin E. V.
Dynamics of Rubber Chaplygin Sphere under Periodic Control
2020, vol. 25, no. 2, pp. 215236
Abstract
This paper examines the motion of a balanced spherical robot under the action of
periodically changing moments of inertia and gyrostatic momentum. The system of equations
of motion is constructed using the model of the rolling of a rubber body (without slipping and
twisting) and is nonconservative. It is shown that in the absence of gyrostatic momentum the
equations of motion admit three invariant submanifolds corresponding to planeparallel motion
of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned
motions are quasiperiodic, and for the numerical estimate of their stability charts
of the largest Lyapunov exponent and charts of stability are plotted versus the frequency and
amplitude of the moments of inertia. It is shown that rotations about the minor and major axes
of inertia can become unstable at sufficiently small amplitudes of the moments of inertia. In
this case, the socalled “Arnol’d tongues” arise in the stability chart. Stabilization of the middle
unstable axis of inertia turns out to be possible at sufficiently large amplitudes of the moments
of inertia, when the middle axis of inertia becomes the minor axis for a part of a period. It
is shown that the nonconservativeness of the system manifests itself in the occurrence of limit
cycles, attracting tori and strange attractors in phase space. Numerical calculations show that
strange attractors may arise through a cascade of perioddoubling bifurcations or after a finite
number of torusdoubling bifurcations.

Borisov A. V., Mamaev I. S., Vetchanin E. V.
Selfpropulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation
2018, vol. 23, no. 78, pp. 850874
Abstract
This paper addresses the problem of the selfpropulsion of a smooth body in a fluid by periodic oscillations of the internal rotor and circulation. In the case of zero dissipation and constant circulation, it is shown using methods of KAM theory that the kinetic energy of the system is a bounded function of time. In the case of constant nonzero circulation, the trajectories of the center of mass of the system lie in a bounded region of the plane. The method of expansion by a small parameter is used to approximately construct a solution corresponding to directed motion of a circular foil in the presence of dissipation and variable circulation.
Analysis of this approximate solution has shown that a speedup is possible in the system in the presence of variable circulation and in the absence of resistance to translational motion. It is shown that, in the case of an elliptic foil, directed motion is also possible. To explore the dynamics of the system in the general case, bifurcation diagrams, a chart of dynamical regimes
and a chart of the largest Lyapunov exponent are plotted. It is shown that the transition to chaos occurs through a cascade of perioddoubling bifurcations.

Mamaev I. S., Vetchanin E. V.
The Selfpropulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor
2018, vol. 23, no. 78, pp. 875886
Abstract
This paper addresses the problem of controlled motion of the Zhukovskii foil in a viscous fluid due to a periodically oscillating rotor. Equations of motion including the added mass effect, viscous friction and lift force due to circulation are derived. It is shown that only limit cycles corresponding to the direct motion or motion near a circle appear in the system at the standard parameter values. The chart of dynamical regimes, the chart of the largest Lyapunov exponent and a oneparameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of perioddoubling bifurcations.

Borisov A. V., Mamaev I. S., Vetchanin E. V.
Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation
2018, vol. 23, no. 4, pp. 480502
Abstract
This paper addresses the problem of selfpropulsion of a smooth profile in a medium with viscous dissipation and circulation by means of parametric excitation generated by oscillations of the moving internal mass. For the case of zero dissipation, using methods of KAM theory, it is shown that the kinetic energy of the system is a bounded function of time, and in the case of nonzero circulation the trajectories of the profile lie in a bounded region of the space. In the general case, using charts of dynamical regimes and charts of Lyapunov exponents, it is shown that the system can exhibit limit cycles (in particular, multistability), quasiperiodic regimes (attracting tori) and strange attractors. Oneparameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and perioddoubling bifurcations are found. To analyze the efficiency of displacement of the profile depending on the circulation and parameters defining the motion of the internal mass, charts of values of displacement for a fixed number of periods are plotted. A hypothesis is formulated that, when nonzero circulation arises, the trajectories of the profile are compact. Using computer calculations, it is shown that in the case of anisotropic dissipation an unbounded growth of the kinetic energy of the system (Fermilike acceleration) is possible.

Vetchanin E. V., Mamaev I. S.
Dynamics of Two Point Vortices in an External Compressible Shear Flow
2017, vol. 22, no. 8, pp. 893–908
Abstract
This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincar´e map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitchfork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of perioddoubling bifurcations is a typical scenario of transition to chaos in the system under consideration.

Vetchanin E. V., Kilin A. A., Mamaev I. S.
Control of the Motion of a Helical Body in a Fluid Using Rotors
2016, vol. 21, no. 78, pp. 874884
Abstract
This paper is concerned with the motion of a helical body in an ideal fluid, which is controlled by rotating three internal rotors. It is proved that the motion of the body is always controllable by means of three rotors with noncoplanar axes of rotation. A condition whose satisfaction prevents controllability by means of two rotors is found. Control actions that allow the implementation of unbounded motion in an arbitrary direction are constructed. Conditions under which the motion of the body along an arbitrary smooth curve can be implemented by rotating the rotors are presented. For the optimal control problem, equations of subRiemannian geodesics on $SE(3)$ are obtained.

Vetchanin E. V., Mamaev I. S., Tenenev V. A.
The Selfpropulsion of a Body with Moving Internal Masses in a Viscous Fluid
2013, vol. 18, no. 12, pp. 100117
Abstract
An investigation of the characteristics of motion of a rigid body with variable internal mass distribution in a viscous fluid is carried out on the basis of a joint numerical solution of the Navier–Stokes equations and equations of motion for a rigid body. A nonstationary threedimensional solution to the problem is found. The motion of a sphere and a dropshaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of selfpropulsion of a body in an arbitrary given direction is shown.
