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COMMON PART

Project Number22-21-00836

Project titleDynamics and control of the motion of rigid bodies in a fluid in the presence of singularities of the fluid flow

Project LeadMamaev Ivan

AffiliationFederal State-Funded Educational Institution of Higher Education "Udmurt State University",

Implementation period | 2022 - 2023 |

Research area 01 - MATHEMATICS, INFORMATICS, AND SYSTEM SCIENCES, 01-317 - Regular and chaotic dynamics of mechanical systems

Keywordsdynamical system, rigid body, fluid, mobile robot, smooth foil, vortex

PROJECT CONTENT

Annotation

The project is concerned with investigating the problems of rigid bodies moving in a fluid, which are of much current interest for applications, using modern methods of dynamical systems theory. In particular, the project addresses two main problems. One of these problems is that of the motion of a smooth (circular or elliptic) foil in a fluid in the presence of vortices and sources. The results of theoretical and experimental investigations of the dynamics of such systems show that the motion of a body is greatly influenced by the intensive process of vortex formation which accompanies the motion of the body in the fluid. Therefore, one of the main problems of this project is to investigate the motion of bodies in the presence of singularities of the flow (vortices, sources, and sinks). The results of research in this direction are fundamental in character and will provide further insight into the dynamics of bodies in the presence of vortex structures and other singularities of the flow. The other problem is concerned with investigating the dynamics of a drop-shaped foil. This problem is closely related to the analysis of the motion and control of aquatic mobile robots. Their advantage is the absence of external moving elements such as blades and screws. The motion of such devices is achieved by periodic motion of internal masses and by rotation of rotors. The practical importance of research in this direction stems from the necessity of enhancing the efficiency of the motion of nondeformable aquatic robots while maintaining the key advantages of this modification, namely, its reliability and ease of fabrication.

Expected results

1. The mathematical model of the motion of a circular foil and an elliptic foil with periodically varying mass distribution or gyrostatic momentum in an ideal fluid in the presence of point vortices or sources. Results of search for first integrals and particular solutions to the equations of motion of the system under consideration, and their stability analysis. Poincaré maps. Results of investigation of the problem of unbounded self-propulsion and acceleration.
2. Results of experimental investigations with the available specimen of the drop-shaped amphibian robot. The mathematical model of the motion of the drop-shaped foil, which takes into account resistance (with friction models linear or quadratic in velocities). Results of comparative analysis of the constructed models for qualitative and quantitative description of experimental data.
3. Methods for solving the problem of controlling the drop-shaped robot along its trajectory, taking into account the singularities of its motion. The algorithm for controlling the motion of the drop-shaped robot by means of feedback, which ensures its motion in the neighborhood of the prescribed trajectory.
To date, there has been no systematic study of the problem of the motion of rigid bodies in a fluid in the presence of sources or sinks. Recent research has shown that new interesting results in this direction can be obtained, both those of theoretical interest and those having a potential for practical application (for example, for development of new methods of self-propulsion in a fluid).
The problem of the motion of a drop-shaped robot is of immediate interest for applications and will be valuable for the development of robotics.

REPORTS

Annotation of the results obtained in 2023

Within the framework of the model of an ideal fluid, an analysis is made of the problem of the motion of a balanced circular foil (cylinder) possessing proper circulation, in the presence of a point source. The complex potential describing the motion of the fluid in the system is presented. Using Sedov’s formulae (generalization of the Blasius-Chaplygin formulae to the case of nonstationary motion), the forces acting on the circular foil from the fluid are calculated. Equations of joint motion of the foil and the singularity are derived under the assumption that the singularity is transferred by the velocity field similarly to a point vortex. It is shown that in the absence of proper circulation the fixed sources, vortex and vortex source have a qualitatively identical force impact on the circular foil and the system reduces to that studied earlier.
Various variants of the foil and the vortex source (a fixed source, a fixed vortex) are considered. For each of the variants, a qualitative analysis of the dynamics is made and a classification of possible motions is carried out. It is shown that the qualitative difference of the dynamics of a balanced circular foil with proper circulation during motion in the field of a fixed vortex and a fixed source arises due to the presence or absence of symmetry of pressure distribution on the boundary of the foil.
An analysis is made of the problem of actuating an aqueous drop-shaped robot with a symmetric housing by periodic rotations of the inner rotor. The case of impulse controls which corresponds to an instantaneous change in the velocity of rotation of the rotor is considered. In this case, a delta-singularity arises in the equations of motion, leading to jumps in the transverse component of the translational velocity and the angular velocity. Using numerical analysis of the Poincaré map, it is shown that, depending on the form of control (symmetry or asymmetry of the control law), the system can have fixed points or limit cycles corresponding to different types of the robot’s motion. In addition, it is shown that the system can have both quasi-periodic modes of motion and strange attractors.

Publications

**1.** *Vetchanin E.V., Mamaev I.S.* **Numerical analysis of a drop-shaped aquatic robot** Mathematics, - (year - 2023)

Annotation of the results obtained in 2022

Two problems of the motion of rigid bodies in a fluid have been addressed.
Equations of motion which describe the controlled motion of an unbalanced circular foil in an ideal fluid in the presence of point vortices under the action of internal mechanisms have been constructed. The equations of controlled motion of the foil have been represented in Lagrangian form with a generalized potential depending both on coordinates and on velocities.
The Hamiltonian form of the equations of motion whose Hamiltonian explicitly depends on the coordinates defining the position and orientation of the foil has been represented. First integrals of the equations of motion corresponding to the laws of conservation of momentum and angular momentum have been found.
Equations of motion have been represented which are reduced by symmetry fields corresponding to the law of conservation (first integrals) of linear momentum and angular momentum, with a Hamiltonian independent of the coordinates defining the position and orientation of the foil, and a noncanonical Poisson bracket. An additional first integral of the reduced system (a Casimir function of the Poisson bracket) which depends functionally on the first integrals of linear momentum and angular momentum has been found.
Using a computer analysis, the periodic perturbation of the integrable case which corresponds to the joint motion of the foil in the presence of one vortex with zero total circulation and zero values of the components of the linear momentum of the system has been investigated. Using a period advanced map (stroboscopic Poincaré map), it was shown that under periodic changes in the moment of inertia of the system the onset of chaotic regimes of motion is possible. It was shown that chaos arises according to the scenario of separatrix splitting.
A mathematical model of the motion of a foil with a sharp edge in a viscous fluid has been constructed. When the model of motion was constructed, it was assumed that inside the shell there is a heavy rotor that produces gyrostatic momentum and sets the system in motion. For this object, the equations of motion were represented in the form of Kirchhoff equations for the motion of a rigid body in an ideal fluid, which were supplemented with viscous resistance terms. Within the framework of this investigation, two resistance models – a linear model and a model quadratic in velocities - were considered.
Experimental investigations of the motion of a drop-shaped robot have been conducted. For experimental research, a prototype of a drop-shaped floating underwater robot was used which is set in motion by rotating a rotor that is placed inside the shell. The rotor is not connected with the environment and produces angular momentum, thus making the motion possible. For the experiments, controls implementing the motion along a straight line and along the arc of a circle were constructed.
The experiments have shown that this prototype of an underwater robot, which is controlled by rotating one rotor, can move in the fluid in a plane parallel to the surface of the fluid, and that, by combining the motion by rotating the rotor with submersion/emersion by means of buoyancy modules (intake of fluid into and discharge of fluid from the robot’s trunk), one can implement motion in the volume of the fluid. By comparing the simulation of the motion with experiment it was shown that the model adequately describes the motion for the considered values of the control parameters.

Publications

**1.** *Vetchanin E.V., Mamaev I.S.* **Периодическое возмущение движения неуравновешенного кругового профиля в присутствии точечных вихрей в идеальной жидкости** Вестник Удмуртского университета. Математика. Механика. Компьютерные науки, - (year - 2022)