Describing the phenomena of the surrounding world is an interesting task that
has long attracted the attention of scientists. However, even in seemingly simple phenomena,
complex dynamics can be revealed. In particular, leaves on the surface of various bodies of
water exhibit complex behavior. This paper addresses an idealized description of the mentioned
phenomenon. Namely, the problem of the plane-parallel motion of an unbalanced circular disk
moving in a stream of simple structure created by a point source (sink) is considered. Note
that using point sources, it is possible to approximately simulate the work of skimmers used
for cleaning swimming pools. Equations of coupled motion of the unbalanced circular disk
and the point source are derived. It is shown that in the case of a fixed-position source of
constant intensity the equations of motion of the disk are Hamiltonian. In addition, in the case
of a balanced circular disk the equations of motion are integrable. A bifurcation analysis of
the integrable case is carried out. Using a scattering map, it is shown that the equations of
motion of the unbalanced disk are nonintegrable. The nonintegrability found here can explain
the complex motion of leaves in surface streams of bodies of water.
Keywords:
ideal fluid, motion in the presence of a source, nonintegrability, scattering map, chaotic scattering
Citation:
Artemova E. M., Vetchanin E. V., The Motion of an Unbalanced Circular Disk in the Field of a Point Source, Regular and Chaotic Dynamics, 2022, vol. 27, no. 1, pp. 24-42
This paper is concerned with the motion of an aquatic robot whose body has the form of a
sharp-edged foil. The robot is propelled by rotating the internal rotor without shell deformation.
The motion of the robot is described by a finite-dimensional mathematical model
derived from physical considerations. This model takes into account the effect of added
masses and viscous friction. The parameters of the model are calculated from comparison
of experimental data and numerical solution to the equations of rigid body motion and the
Navier–Stokes equations. The proposed mathematical model is used to define controls
implementing straight-line motion, motion in a circle, and motion along a complex trajectory.
Experiments for estimation of the efficiency of the model have been conducted.
Keywords:
aquatic robot, propulsion in a fluid, periodic control action, motion planning, identification of parameters of the model
Citation:
Karavaev Y. L., Klekovkin A. V., Mamaev I. S., Tenenev V. A., Vetchanin E. V., A Simple Physical Model for Control of an Propellerless Aquatic Robot, Journal of Mechanisms and Robotics, 2022, vol. 14, no. 1, 011007, 11 pp.
The Problem of Balancing an Inverted Spherical Pendulum on an Omniwheel Platform Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, pp. 507-525
This paper addresses the problem of balancing an inverted pendulum on an omnidirectional
platform in a three-dimensional setting. Equations of motion of the platform – pendulum system
in quasi-velocities are constructed. To solve the problem of balancing the pendulum by controlling
the motion of the platform, a hybrid genetic algorithm is used. The behavior of the system is
investigated under different initial conditions taking into account a necessary stop of the platform
or the need for continuation of the motion at the end point of the trajectory. It is shown that
the solution of the problem in a two-dimensional setting is a particular case of three-dimensional
balancing.
Keywords:
balancing of an inverted pendulum, omnidirectional platform, hybrid genetic algorithm, Poincaré equations in quasi-velocities
Citation:
Shaura A. S., Tenenev V. A., Vetchanin E. V., The Problem of Balancing an Inverted Spherical Pendulum on an Omniwheel Platform, Russian Journal of Nonlinear Dynamics, 2021, vol. 17, no. 4, pp. 507-525
This paper addresses the problem of the motion of an unbalanced circular foil in the
field of a point source. Equations of joint motion of the source and the foil are constructed. It is
shown that in the case of a fixed source of constant intensity the equations of motion of the foil are
Hamiltonian. In addition, in the case of a balanced circular foil the equations of motion are integrable.
A bifurcation analysis of the integrable case is carried out. Using a scattering map, it is shown that
the equations of motion of the unbalanced foil are nonintegrable.
Citation:
Artemova E. M., Vetchanin E. V., The motion of an unbalanced circular foil in the field of a point source, arXiv:2109.13041, 2021, 21 pp.
Calculation of instability regions of the Liouville problem based on the harmonic balance method 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, pp. 1-5
This paper considers the application of the harmonic balance method for calculating the boundaries of the instability region of the Liouville problem. Explicit equations for the boundaries of the first four instability regions are given.
Keywords:
harmonic balance method, Liouville’s problem, equations with periodic coefficients
Citation:
Vetchanin E. V., Calculation of instability regions of the Liouville problem based on the harmonic balance method, 2021 International Conference ''Nonlinearity, Information and Robotics'' – IEEE, 2021, pp. 1-5
Control of the motion of a circular cylinder in an ideal fluid using a source Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, pp. 604-617
The motion of a circular cylinder in an ideal fluid in the field of a fixed source is considered. It is shown that, when the source has constant strength, the system possesses a momentum integral and an energy integral. Conditions are found under which the equations of motion reduced to the level set of the momentum integral admit an unstable fixed point. This fixed point corresponds to circular motion of the cylinder about the source. A feedback is constructed which ensures stabilization of the above-mentioned fixed point by changing the strength of the source.
Keywords:
control, ideal fluid, feedback, motion in the presence of a source
Citation:
Artemova E. M., Vetchanin E. V., Control of the motion of a circular cylinder in an ideal fluid using a source, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2020, vol. 30, no. 4, pp. 604-617
Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 689-706
The motion of a spherical robot with periodically changing moments of inertia,
internal rotors and a displaced center of mass is considered. It is shown that, under some
restrictions on the displacement of the center of mass, the system of interest features chaotic
dynamics due to separatrix splitting. A stability analysis is made of the upper equilibrium
point of the ball and of the periodic solution arising in its neighborhood, in the case of periodic
rotation of the rotors. It is shown that the lower equilibrium point can become unstable in the
case of fixed rotors and periodically changing moments of inertia.
Keywords:
nonholonomic constraint, rubber rolling, unbalanced ball, rolling on a plane
Citation:
Artemova E. M., Karavaev Y. L., Mamaev I. S., Vetchanin E. V., Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass, Regular and Chaotic Dynamics, 2020, vol. 25, no. 6, pp. 689-706
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 plane-parallel motion
of the sphere with rotation about the minor, middle and major axes of inertia. The abovementioned
motions are quasi-periodic, 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 so-called “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 period-doubling bifurcations or after a finite
number of torus-doubling bifurcations.
Mamaev I. S., Vetchanin E. V., Dynamics of Rubber Chaplygin Sphere under Periodic Control, Regular and Chaotic Dynamics, 2020, vol. 25, no. 2, pp. 215-236
Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. II Russian Journal of Mathematical Physics, 2020, vol. 27, no. 1, pp. 1-17
This paper considers the plane-parallel motion of an elliptic foil in a fluid with
a nonzero constant circulation under the action of external periodic forces and torque. The
existence of the first integral is shown for the case in which there is no external torque
and an external force acts along one of the principal axes of the foil. It is shown that, in
the general case, in the absence of friction, an extensive stochastic layer is observed for the
period advance map. When dissipation is added to the system, strange attractors can arise
from the stochastic layer.
Citation:
Borisov A. V., Vetchanin E. V., Mamaev I. S., Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. II, Russian Journal of Mathematical Physics, 2020, vol. 27, no. 1, pp. 1-17
This paper addresses a conservative system describing the motion of a smooth body in an ideal fluid under the action of an external periodic torque with nonzero mean and of an external periodic force. It is shown that, in the case where the body is circular in shape, the angular velocity of the body increases indefinitely (linearly in time), and the projection of the phase trajectory onto the plane of translational velocities is attracted to a circle. Asymptotic orbital stability (or asymptotic stability with respect to part of variables) exists in the system. It is shown numerically that, in the case of an elliptic body, the projection of the phase trajectory onto the plane of translational velocities is attracted to an annular region.
Citation:
Vetchanin E. V., Mamaev I. S., Asymptotic behavior in the dynamics of a smooth body in an ideal fluid, Acta Mechanica, 2020, vol. 231, pp. 4529-4535
Propellerless aquatic robots Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, pp. 404-411
This paper is devoted to investigations of the motion of the propellerless aquatic robots. There are two models of aquatic robots under consideration that move due to rotation of internal rotors. Mathematical models to describe the motion of the robots are proposed. Experiments with different control actions for fabricated prototypes to verify mathematical models have been conducted.
Citation:
Klekovkin A. V., Mamaev I. S., Vetchanin E. V., Tenenev V. A., Karavaev Y. L., Propellerless aquatic robots, Robots in Human Life: Proceedings of the 23rd International Conference on Climbing and Walking Robots and the Support Technologies for Mobile Machines, CLAWAR 2020, 2020, pp. 404-411
Dynamics of a spherical robot with periodically changing moments of inertia 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-5
The motion of a spherical robot with periodically changing moments of inertia and gyrostatic momentum is considered. Equations of motion are derived within the framework of the model of "rubber" rolling (without slipping and twisting). The stability of partial solutions of the system is studied numerically. It is shown that the system is nonconservative, and, as a consequence, limit cycles and strange attractors exist in the phase space of the system.
Keywords:
stability of the motion, periodic controls, nonholonomic model, rubber rolling
Citation:
Mamaev I. S., Vetchanin E. V., Dynamics of a spherical robot with periodically changing moments of inertia, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-5
Stabilization of rotations of a rigid body with a fixed point by periodic perturbations 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-5
The dynamics of a body with a fixed point is considered in the case where the moments of inertia of the system depend periodically on time. A stability of permanent rotations is estimated by a numerical approach. In a neighborhood of permanent rotations the linearization of equations of motion results in Hill's equation, and the stability is determined by the eigenvalues of a monodromy matrix. It is shown that stable rotations may be destabilized by periodically changing the moments of inertia due to parametric resonance.
Citation:
Vetchanin E. V., Stabilization of rotations of a rigid body with a fixed point by periodic perturbations, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-5
Experimental evaluation of simplified physical model for control of aquatic robot with internal rotor 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-2
This paper is devoted to investigations of the motion of an aquatic propeller-less robot. The robot motion implemented by rotation of internal rotor. A simple finite-dimensional mathematical model to describe the motion of the robot is proposed. Experiments with control actions providing the motion along a straight line and a circle have been conducted.
Citation:
Klekovkin A. V., Karavaev Y. L., Mamaev I. S., Vetchanin E. V., Tenenev V. A., Experimental evaluation of simplified physical model for control of aquatic robot with internal rotor, 2020 International Conference Nonlinearity, Information and Robotics – IEEE, 2020, pp. 1-2
The dynamics of a body with a fixed point, variable moments of inertia and internal rotors
are considered. A stability analysis of permanent rotations and periodic solutions of the system is
carried out. In some simplest cases the stability analysis is reduced to investigating the stability
of the zero solution of Hill’s equation. It is shown that by periodically changing the moments of
inertia it is possible to stabilize unstable permanent rotations of the system. In addition, stable
dynamical regimes can lose stability due to a parametric resonance. It is shown that, as the
oscillation frequency of the moments of inertia increases, the dynamics of the system becomes
close to an integrable one.
Vetchanin E. V., Mikishanina E. A., Vibrational Stability of Periodic Solutions of the Liouville Equations, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 3, pp. 351-363
The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 1, pp. 41-57
The motion of a circular cylinder in a fluid in the presence of circulation and external periodic force and torque is studied. It is shown that for a suitable choice of the frequency of external action for motion in an ideal fluid the translational velocity components of the body undergo oscillations with increasing amplitude due to resonance. During motion in a viscous fluid no resonance arises. Explicit integration of the equations of motion has shown that the unbounded propulsion of the body in a viscous fluid is impossible in the absence of external torque. In the general case, the solution of the equations is represented in the form of a multiple series.
Keywords:
rigid body dynamics, ideal fluid, viscous fluid, propulsion in a fluid, resonance
Citation:
Vetchanin E. V., The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 1, pp. 41-57
Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. I Russian Journal of Mathematical Physics, 2019, vol. 26, no. 4, pp. 412-428
A plane-parallel motion of a circular foil is considered in a fluid with a nonzero
constant circulation under the action of external periodic force and torque. Various integrable
cases are treated. Conditions for the existence of resonances of two types are found. In the
case of resonances of the first type, the phase trajectory of the system and the trajectory of
the foil are unbounded. In the case of resonances of the second type, the foil trajectory is
unbounded, while the phase trajectory of the system remains bounded during the motion.
Citation:
Borisov A. V., Vetchanin E. V., Mamaev I. S., Motion of a Smooth Foil in a Fluid under the Action of External Periodic Forces. I, Russian Journal of Mathematical Physics, 2019, vol. 26, no. 4, pp. 412-428
Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 850-874
This paper addresses the problem of the self-propulsion 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 speed-up 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 period-doubling bifurcations.
Keywords:
self-propulsion in a fluid, smooth body, viscous fluid, periodic oscillation of circulation, control of a rotor
Citation:
Borisov A. V., Mamaev I. S., Vetchanin E. V., Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 850-874
The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 875-886
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 one-parameter bifurcation diagram are calculated. It is shown that strange attractors appear in the system due to a cascade of period-doubling bifurcations.
Keywords:
self-propulsion, Zhukovskii foil, foil with a sharp edge, motion in a viscous fluid, controlled motion, period-doubling bifurcation
Citation:
Mamaev I. S., Vetchanin E. V., The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor, Regular and Chaotic Dynamics, 2018, vol. 23, no. 7-8, pp. 875-886
Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 480-502
This paper addresses the problem of self-propulsion 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), quasi-periodic regimes (attracting tori) and strange attractors. One-parameter bifurcation diagrams are constructed, and Neimark – Sacker bifurcations and period-doubling 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 (Fermi-like acceleration) is possible.
Keywords:
self-propulsion in a fluid, motion with speed-up, parametric excitation, viscous dissipation, circulation, period-doubling bifurcation, Neimark – Sacker bifurcation, Poincaré map, chart of dynamical regimes, chart of Lyapunov exponents, strange att
Citation:
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, Regular and Chaotic Dynamics, 2018, vol. 23, no. 4, pp. 480-502
This paper addresses the problem of plane-parallel motion of the Zhukovskii foil in a viscous
fluid. Various motion regimes of the foil are simulated on the basis of a joint numerical solution
of the equations of body motion and the Navier – Stokes equations. According to the results
of simulation of longitudinal, transverse and rotational motions, the average drag coefficients
and added masses are calculated. The values of added masses agree with the results published
previously and obtained within the framework of the model of an ideal fluid. It is shown that
between the value of circulation determined from numerical experiments, and that determined
according to the model of and ideal fluid, there is a correlation with the coefficient $\mathcal{R} = 0.722$.
Approximations for the lift force and the moment of the lift force are constructed depending
on the translational and angular velocity of motion of the foil. The equations of motion of
the Zhukovskii foil in a viscous fluid are written taking into account the found approximations and the drag coefficients. The calculation results based on the proposed mathematical model
are in qualitative agreement with the results of joint numerical solution of the equations of body
motion and the Navier – Stokes equations.
Mamaev I. S., Tenenev V. A., Vetchanin E. V., Dynamics of a Body with a Sharp Edge in a Viscous Fluid, Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 4, pp. 473-494
Identification of parameters of the model of toroidal body motion using experimental data Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 1, pp. 99-121
This paper is concerned with the motion of heavy toroidal bodies in a fluid. For experimental purposes, models of solid tori with a width of 3 cm and external diameters of 10 cm, 12 cm and 15 cm have been fabricated by the method of casting chemically solidifying polyurethane (density 1100 kg/m^{3}). Tracking of the models is performed using the underwater Motion Capture system. This system includes 4 cameras, computer and specialized software. A theoretical description of the motion is given using equations incorporating the influence of inertial forces, friction and circulating motion of a fluid through the hole. Values of the model parameters are selected by means of genetic algorithms to ensure an optimal agreement between experimental and theoretical data.
Keywords:
fall through a fluid, torus, body with a hole, multiply connected body, finitedimensional model, object tracking, genetic algorithms
Citation:
Vetchanin E. V., Gladkov E. S., Identification of parameters of the model of toroidal body motion using experimental data, Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 1, pp. 99-121
The motion of a body shaped as a triaxial ellipsoid and controlled by the rotation of three internal rotors is studied. It is proved that the motion is controllable with the exception of a few particular cases. Partial solutions whose combinations enable an unbounded motion in any arbitrary direction are constructed.
Keywords:
ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gate
Citation:
Borisov A. V., Vetchanin E. V., Kilin A. A., Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors, Mathematical Notes, 2017, vol. 102, no. 4, pp. 455-464
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 pitch-fork” 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 period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.
Keywords:
point vortices, shear flow, perturbation by an acoustic wave, bifurcations, reversible pitch-fork, period doubling
Citation:
Vetchanin E. V., Mamaev I. S., Dynamics of Two Point Vortices in an External Compressible Shear Flow, Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 893–908
This paper presents a comparative analysis of computations of the motion of heavy three-bladed screws in a fluid along with experimental results. Simulation of the motion is performed using the theory of an ideal fluid and the phenomenological model of viscous friction. For experimental purposes, models of three-bladed screws with various configurations and sizes were manufactured by casting from chemically hardening polyurethane. Comparison of calculated and experimental results has shown that the mathematical models considered essentially do not reflect the processes observed in the experiments.
Keywords:
motion in a fluid, helical body, experimental investigation
Citation:
Vetchanin E. V., Klenov A. I., Experimental investigation of the fall of helical bodies in a fluid, Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 585–598
Optimal control of the motion in an ideal fluid of a screw-shaped body with internal rotors Computer Research and Modeling, 2017, vol. 9, no. 5, pp. 741-759
In this paper we consider the controlled motion of a helical body with three blades in an ideal fluid, which is executed by rotating three internal rotors. We set the problem of selecting control actions, which ensure the motion of the body near the predetermined trajectory. To determine controls that guarantee motion near the given curve, we propose methods based on the application of hybrid genetic algorithms (genetic algorithms with real encoding and with additional learning of the leader of the population by a gradient method) and artificial neural networks. The correctness of the operation of the proposed numerical methods is estimated using previously obtained differential equations, which define the law of changing the control actions for the predetermined trajectory.
In the approach based on hybrid genetic algorithms, the initial problem of minimizing the integral functional reduces to minimizing the function of many variables. The given time interval is broken up into small elements, on each of which the control actions are approximated by Lagrangian polynomials of order 2 and 3. When appropriately adjusted, the hybrid genetic algorithms reproduce a solution close to exact. However, the cost of calculation of 1 second of the physical process is about 300 seconds of processor time.
To increase the speed of calculation of control actions, we propose an algorithm based on artificial neural networks. As the input signal the neural network takes the components of the required displacement vector. The node values of the Lagrangian polynomials which approximately describe the control actions return as output signals . The neural network is taught by the well-known back-propagation method. The learning sample is generated using the approach based on hybrid genetic algorithms. The calculation of 1 second of the physical process by means of the neural network requires about 0.004 seconds of processor time, that is, 6 orders faster than the hybrid genetic algorithm. The control calculated by means of the artificial neural network differs from exact control. However, in spite of this difference, it ensures that the predetermined trajectory is followed exactly.
Keywords:
motion control, genetic algorithms, neural networks, motion in a fluid, ideal fluid
Citation:
Vetchanin E. V., Tenenev V. A., Kilin A. A., Optimal control of the motion in an ideal fluid of a screw-shaped body with internal rotors, Computer Research and Modeling, 2017, vol. 9, no. 5, pp. 741-759
Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation Around the Body Journal of Dynamical and Control Systems, 2017, vol. 23, pp. 435-458
In this paper we study the controlled motion of an arbitrary two-dimensional body in an ideal fluid with a moving internal mass and an internal rotor in the presence of constant circulation around the body. We show that by changing the position of the internal mass and by rotating the rotor, the body can be made to move to a given point, and discuss the influence of nonzero circulation on the motion control. We have found that in the presence of circulation around the body the system cannot be completely stabilized at an arbitrary point of space, but fairly simple controls can be constructed to ensure that the body moves near the given point.
Keywords:
ideal fuid, controllability, Kirchhoff equations, circulation around the body
Citation:
Vetchanin E. V., Kilin A. A., Control of Body Motion in an Ideal Fluid Using the Internal Mass and the Rotor in the Presence of Circulation Around the Body, Journal of Dynamical and Control Systems, 2017, vol. 23, pp. 435-458
The motion controlled by the rotation of three internal rotors of a body with helical symmetry in an ideal liquid is considered. The problem is to select controls that ensure the displacement of the body with minimum effort. The optimality of particular solutions found earlier is studied.
Citation:
Vetchanin E. V., Mamaev I. S., Optimal control of the motion of a helical body in a liquid using rotors, Russian Journal of Mathematical Physics, 2017, vol. 24, no. 3, pp. 399-411
Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body Doklady Physics, 2016, vol. 466, no. 3, pp. 293-297
In this paper, we study the free and controlled motion of an arbitrary two-dimensional body with a moving internal material point through an ideal fluid in presence of constant circulation around the body. We perform bifurcation analysis of free motion (with fixed internal mass). We show that by changing the position of the internal mass the body can be made to move to a specified point. There are a number of control problems associated with the nonzero drift of the body in the case of fixed internal mass.
Citation:
Vetchanin E. V., Kilin A. A., Free and controlled motion of a body with moving internal mass though a fluid in the presence of circulation around the body, Doklady Physics, 2016, vol. 466, no. 3, pp. 293-297
Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 302-332
We consider the controlled motion in an ideal incompressible fluid of a rigid body
with moving internal masses and an internal rotor in the presence of circulation of the fluid
velocity around the body. The controllability of motion (according to the Rashevskii–Chow
theorem) is proved for various combinations of control elements. In the case of zero circulation,
we construct explicit controls (gaits) that ensure rotation and rectilinear (on average) motion.
In the case of nonzero circulation, we examine the problem of stabilizing the body (compensating
the drift) at the end point of the trajectory. We show that the drift can be compensated for if
the body is inside a circular domain whose size is defined by the geometry of the body and the
value of circulation.
Citation:
Vetchanin E. V., Kilin A. A., Controlled Motion of a Rigid Body with Internal Mechanisms in an Ideal Incompressible Fluid, Proceedings of the Steklov Institute of Mathematics, 2016, vol. 295, pp. 302-332
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 sub-Riemannian geodesics on $SE(3)$ are obtained.
Keywords:
ideal fluid, motion of a helical body, Kirchhoff equations, control of rotors, gaits, optimal control
Citation:
Vetchanin E. V., Kilin A. A., Mamaev I. S., Control of the Motion of a Helical Body in a Fluid Using Rotors, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 874-884
Control of the motion of an unbalanced heavy ellipsoid in an ideal fluid using rotors Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 663–674
This paper is concerned with the motion of an unbalanced heavy three-axial ellipsoid in an ideal fluid controlled by rotation of three internal rotors. It is proved that the motion of the body considered is controlled with respect to configuration variables except for some special cases. An explicit control that makes it possible to implement unbounded motion in an arbitrary direction has been calculated. Directions for which control actions are bounded functions of time have been determined.
Keywords:
ideal fluid, motion of a rigid body, Kirchhoff equations, control by rotors, gaits
Citation:
Vetchanin E. V., Kilin A. A., Control of the motion of an unbalanced heavy ellipsoid in an ideal fluid using rotors, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 4, pp. 663–674
This paper is concerned with the process of the free fall of a three-bladed screw in a fluid. The investigation is performed within the framework of theories of an ideal fluid and a viscous fluid. For the case of an ideal fluid the stability of uniformly accelerated rotations (the Steklov solutions) is studied. A phenomenological model of viscous forces and torques is derived for investigation of the motion in a viscous fluid. A chart of Lyapunov exponents and bifucation diagrams are computed. It is shown that, depending on the system parameters, quasiperiodic and chaotic regimes of motion are possible. Transition to chaos occurs through cascade of period-doubling bifurcations.
Keywords:
ideal fluid, viscous fluid, motion of a rigid body, dynamical system, stability of motion, bifurcations, chart of Lyapunov exponents
Citation:
Tenenev V. A., Vetchanin E. V., Ilaletdinov L. F., Chaotic dynamics in the problem of the fall of a screw-shaped body in a fluid, Russian Journal of Nonlinear Dynamics, 2016, vol. 12, no. 1, pp. 99-120
Bifurcations and chaos in the dynamics of two point vortices in an acoustic wave International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 4, 1650063, 13 pp.
In this paper, we consider a system governing the motion of two point vortices in a flow excited by an external acoustic forcing. It is known that the system of two vortices is integrable in the absence of acoustic forcing. However, the addition of the acoustic forcing makes the system much more complex, and the system becomes nonintegrable and loses the phase volume preservation property. The objective of our research is to study chaotic dynamics and typical bifurcations. Numerical analysis has shown that the reversible pitchfork bifurcation is typical. Also, we show that the existence of strange attractors is not characteristic for the system under consideration.
Keywords:
reversible pitchfork, point vortices, acoustic forcing, chaos
Citation:
Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the dynamics of two point vortices in an acoustic wave, International Journal of Bifurcation and Chaos, 2016, vol. 26, no. 4, 1650063, 13 pp.
Experimental determination of the added masses by method of towing Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 568-582
This paper is concerned with the experimental determination of the added masses of bodies completely or partially immersed in a fluid. The paper presents an experimental setup, a technique of the experiment and an underlying mathematical model. The method of determining the added masses is based on the towing of the body with a given propelling force. It is known (from theory) that the concept of an added mass arises under the assumption concerning the potentiality of flow over the body. In this context, the authors have performed PIV visualization of flows generated by the towed body, and defined a part of the trajectory for which the flow can be considered as potential. For verification of the technique, a number of experiments have been performed to determine the added masses of a spheroid. The measurement results are in agreement with the known reference data. The added masses of a screwless freeboard robot have been defined using the developed technique.
Keywords:
added mass, movement on a free surface, hydrodynamic resistance, method of towing
Citation:
Klenov A. I., Vetchanin E. V., Kilin A. A., Experimental determination of the added masses by method of towing, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 568-582
Optical measurement of a fluid velocity field around a falling plate Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 554-567
The paper is devoted to the experimental verification of the Andersen–Pesavento–Wang model describing the falling of a heavy plate through a resisting medium. As a main research method the authors have used video filming of a falling plate with PIV measurement of the velocity of surrounding fluid flows. The trajectories of plates and streamlines were determined and oscillation frequencies were estimated using experimental results. A number of experiments for plates of various densities and sizes were performed. The trajectories of plates made of plastic are qualitatively similar to the trajectories predicted by the Andersen–Pesavento–Wang model. However, measured and computed frequencies of oscillations differ significantly. For a plate made of high carbon steel the results of experiments are quantitatively and qualitatively in disagreement with computational results.
Keywords:
PIV — Particle Image Velocimetry, Maxwell problem, model of Andersen–Pesavento–Wang
Citation:
Vetchanin E. V., Klenov A. I., Optical measurement of a fluid velocity field around a falling plate, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 554-567
The paper is devoted to the development of a model of an underwater robot actuated by inner rotors. This design has no moving elements interacting with an environment, which minimizes a negative impact on it, and increases noiselessness of the robot motion in a liquid. Despite numerous discussions on the possibility and efficiency of motion by means of internal masses' movement, a large number of works published in recent years confirms a relevance of the research. The paper presents an overview of works aimed at studying the motion by moving internal masses. A design of a screwless underwater robot that moves by the rotation of inner rotors to conduct theoretical and experimental investigations is proposed. In the context of theoretical research a robot model is considered as a hollow ellipsoid with three rotors located inside so that the axes of their rotation are mutually orthogonal. For the proposed model of a screwless underwater robot equations of motion in the form of classical Kirchhoff equations are obtained.
Keywords:
mobile robot, screwless underwater robot, movement in ideal fluid
Citation:
Vetchanin E. V., Karavaev Y. L., Kalinkin A. A., Klekovkin A. V., Pivovarova E. N., A model of a screwless underwater robot, Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
The contol of the motion through an ideal fluid of a rigid body by means of two moving masses Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 633–645
In this paper we consider the problem of motion of a rigid body in an ideal fluid with two material points moving along circular trajectories. The controllability of this system on the zero level set of first integrals is shown. Elementary “gaits” are presented which allow the realization of the body’s motion from one point to another. The existence of obstacles to a controlled motion of the body along an arbitrary trajectory is pointed out.
Keywords:
ideal fluid, Kirchhoff equations, controllability of gaits
Citation:
Kilin A. A., Vetchanin E. V., The contol of the motion through an ideal fluid of a rigid body by means of two moving masses, Russian Journal of Nonlinear Dynamics, 2015, vol. 11, no. 4, pp. 633–645
Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
This paper is concerned with the dynamics of two point vortices of the same intensity which are affected by an acoustic wave. Typical bifurcations of fixed points have been identified by constructing charts of dynamical regimes, and bifurcation diagrams have been plotted.
Keywords:
point vortices, nonintegrability, bifurcations, chart of dynamical regimes
Citation:
Vetchanin E. V., Kazakov A. O., Bifurcations and chaos in the problem of the motion of two point vortices in an acoustic wave, Russian Journal of Nonlinear Dynamics, 2014, vol. 10, no. 3, pp. 329-343
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 three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid in a gravitational field, which is caused by the motion of internal material points, is explored. The possibility of self-propulsion of a body in an arbitrary given direction is shown.
Keywords:
finite-volume numerical method, Navier–Stokes equations, variable internal mass distribution, motion control
Citation:
Vetchanin E. V., Mamaev I. S., Tenenev V. A., The Self-propulsion of a Body with Moving Internal Masses in a Viscous Fluid, Regular and Chaotic Dynamics, 2013, vol. 18, no. 1-2, pp. 100-117
We consider the optimal motion control problem for a mobile device with an external rigid shell moving along a prescribed trajectory in a viscous fluid. The mobile robot under consideration possesses the property of self-locomotion. Self-locomotion is implemented due to back-and-forth motion of an internal material point. The optimal motion control is based on the Sugeno fuzzy inference system. An approach based on constructing decision trees using the genetic algorithm for structural and parametric synthesis has been proposed to obtain the base of fuzzy rules.
Vetchanin E. V., Tenenev V. A., Shaura A. S., Motion control of a rigid body in viscous fluid, Computer Research and Modeling, 2013, vol. 5, no. 4, pp. 659-675
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. A non-stationary three-dimensional solution to the problem is found. The motion of a sphere and a drop-shaped body in a viscous fluid, which is caused by the motion of internal material points, in a gravitational field is explored. The possibility of motion of a body in an arbitrary given direction is shown.
Keywords:
finite-volume numerical method, Navier-Stokes equations, variable internal mass distribution, motion control
Citation:
Vetchanin E. V., Mamaev I. S., Tenenev V. A., The motion of a body with variable mass geometry in a viscous fluid, Russian Journal of Nonlinear Dynamics, 2012, vol. 8, no. 4, pp. 815-836
Motion control simulating in a viscous liquid of a body with variable geometry of weights Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371-381
Statement of a problem of management of movement of a body in a viscous liquid is given. Movement bodies it is induced by moving of internal material points. On a basis the numerical decision of the equations of movement of a body and the hydrodynamic equations approximating dependencies for viscous forces are received. With application approximations the problem of optimum control of body movement dares on the set trajectory with application of hybrid genetic algorithm. Possibility of the directed movement of a body under action is established back and forth motion of an internal point. Optimum control movement direction it is carried out by motion of other internal point on circular trajectory with variable speed
Keywords:
optimum control, the equations of movement, Navier–Stokes equations, numerical methods, fuzzy decision trees, genetic algorithm
Citation:
Vetchanin E. V., Tenenev V. A., Motion control simulating in a viscous liquid of a body with variable geometry of weights, Computer Research and Modeling, 2011, vol. 3, no. 4, pp. 371-381