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
Abstract
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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
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
Abstract
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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
Control of the Motion of a Triaxial Ellipsoid in a Fluid Using Rotors
Mathematical Notes, 2017, vol. 102, no. 4, pp. 455-464
Abstract
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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
Dynamics of Two Point Vortices in an External Compressible Shear Flow
Regular and Chaotic Dynamics, 2017, vol. 22, no. 8, pp. 893–908
Abstract
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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
Experimental investigation of the fall of helical bodies in a fluid
Russian Journal of Nonlinear Dynamics, 2017, vol. 13, no. 4, pp. 585–598
Abstract
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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
Abstract
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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
Abstract
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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
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
Abstract
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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
Abstract
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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
Abstract
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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
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
Abstract
pdf (1.23 Mb)
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
Abstract
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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
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
Abstract
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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.
Abstract
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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
Abstract
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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
Abstract
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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
A model of a screwless underwater robot
Bulletin of Udmurt University. Mathematics. Mechanics. Computer Science, 2015, vol. 25, no. 4, pp. 544-553
Abstract
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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
Abstract
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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
Abstract
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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
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
Abstract
pdf (1.71 Mb)
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
Motion control of a rigid body in viscous fluid
Computer Research and Modeling, 2013, vol. 5, no. 4, pp. 659-675
Abstract
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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
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
Abstract
pdf (15.9 Mb)
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
Abstract
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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