Control of the motion of a circular cylinder in an ideal fluid using a source
Udmurt State Universitya
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.
control, ideal fluid, feedback, motion in the presence of a source
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29 September 2020
Artemova E.M., Vetchanin E.V. Control of the motion of a circular cylinder in an ideal fluid using a source, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 4, pp. 604-617.
- Arnold V.I. Mathematical methods of classical mechanics, New-York-Heidelberg-Berlin: Springer, 1974.
- Belotelov V.N., Martynenko Yu.G. Control of spatial motion of an inverted pendulum mounted on a wheel pair, Mechanics of Solids, 2006, no. 6, pp. 10-28. https://www.elibrary.ru/item.asp?id=9438845
- Bizyaev I.A., Borisov A.V., Mamaev I.S. The dynamics of vortex sources in a deformation flow, Regular and Chaotic Dynamics, 2016, vol. 21, no. 3, pp. 367-376. https://doi.org/10.1134/S1560354716030084
- Bizyaev I.A., Borisov A.V., Mamaev I.S. The dynamics of three vortex sources, Regular and Chaotic Dynamics, 2014, vol. 19, no. 6, pp. 694-701. https://doi.org/10.1134/S1560354714060070
- Bogomolov V.A. The movement of an ideal fluid of constant density in the presence of sinks, Izv. AN SSSR, Mekhanika Zhidkosti i Gaza, 1976, no. 4, pp. 21-27 (in Russian).
- Borisov A.V., Kilin A.A., Mamaev I.S. Absolute and relative choreographies in the problem of the motion of point vortices in a plane, Doklady Mathematics, 2005, vol. 71, no. 1, pp. 139-144. https://www.elibrary.ru/item.asp?id=13491841
- 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. https://doi.org/10.1134/S1560354718040081
- 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. https://doi.org/10.1134/S1560354718070043
- Borisov A.V., Mamaev I.S., Bizyaev I.A. Three vortices in spaces of constant curvature: reduction, Poisson geometry, and stability, Regular and Chaotic Dynamics, 2018, vol. 23, no. 5, pp. 613-636. https://doi.org/10.1134/S1560354718050106
- Borisov A.V., Kilin A.A. Stability of Thomson's configurations of vortices on a sphere, Regular and Chaotic Dynamics, 2000, vol. 5, no. 2, pp. 189-200. https://doi.org/10.1070/RD2000v005n02ABEH000141
- Borisov A.V., Kilin A.A., Mamaev I.S. Transition to chaos in dynamics of four point vortices on a plane, Doklady Physics, 2006, vol. 51, no. 5, pp. 262-267. https://doi.org/10.1134/S1028335806050089
- Borisov A.V., Kilin A.A., Karavaev Y.L., Klekovkin A.V. Stabilization of the motion of a spherical robot using feedbacks, Applied Mathematical Modelling, 2019, vol. 69, pp. 583-592. https://doi.org/10.1016/j.apm.2019.01.008
- Borisov A.V., Mamaev I.S. On the problem of motion of vortex sources on a plane, Regular and Chaotic Dynamics, 2006, vol. 11, no. 4, pp. 455-466. https://doi.org/10.1070/RD2006v011n04ABEH000363
- Borisov A.V., Mamaev I.S., Ramodanov S.M. Motion of a circular cylinder and $n$ point vortices in a perfect fluid, Regular and Chaotic Dynamics, 2003, vol. 8, no. 4, pp. 449-462. https://doi.org/10.1070/RD2003v008n04ABEH000257
- 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. https://doi.org/10.1134/S0001434617090176
- Chaplygin S.A. On the pressure of a plane-parallel flow on obstructing bodies (to the theory of an airplane), Mat. Sb., 1911, vol. 28, pp. 120-166 (in Russian). http://mi.mathnet.ru/eng/msb6652
- Childress S., Spagnolie S.E., Tokieda T. A bug on a raft: recoil locomotion in a viscous fluid, Journal of Fluid Mechanics, 2011, vol. 669, pp. 527-556. https://doi.org/10.1017/S002211201000515X
- Fridman A.A., Polubarinova P.Ya. On moving singularities of a flat motion of an incompressible fluid, In: Geofizicheskii Sbornik, 1928, pp. 9-23 (in Russian).
- Gebhard B., Ortega R. Stability of periodic solutions of the $N$-vortex problem in general domains, Regular and Chaotic Dynamics, 2019, vol. 24, no. 6, pp. 649-670. https://doi.org/10.1134/S1560354719060054
- Gonchar V.Yu., Ostapchuk P.N., Tur A.V., Yanovsky V.V. Dynamics and stochasticity in a reversible system describing interaction of point vortices with a potential wave, Physics Letters A, 1991, vol. 152, pp. 287-292. https://doi.org/10.1016/0375-9601(91)90107-J
- Heughan D.M. An experimental study of a symmetrical aerofoil with a rear suction slot and a retractable flap, The Aeronautical Journal, 1953, vol. 57, no. 514, pp. 627-645. https://doi.org/10.1017/S0368393100126719
- Ivanova T.B., Kilin A.A., Pivovarova E.N. Controlled motion of a spherical robot with feedback. I, Journal of Dynamical and Control Systems, 2018, vol. 24, no. 3, pp. 497-510. https://doi.org/10.1007/s10883-017-9387-2
- Ivanova T.B., Kilin A.A., Pivovarova E.N. Controlled motion of a spherical robot with feedback. II, Journal of Dynamical and Control Systems, 2019, vol. 25, no. 1, pp. 1-16. https://doi.org/10.1007/s10883-017-9390-7
- Ivlev V.I., Misyurin S.Y., Nosova N.Y. Servopneumatic actuator of a robot with compensation for the mutual influence of movements of the degrees of mobility, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2018, vol. 28, issue 2, pp. 231-239. https://doi.org/10.20537/vm180209
- Kalman R.E., Falb P.L., Arbib M.A. Topics in mathematical system theory, New York: McGraw-Hill, 1969.
- Kanso E., Marsden J.E., Rowley C.W., Melli-Huber J. Locomotion of articulated bodies in a perfect fluid, Journal of Nonlinear Science, 2005, vol. 15, pp. 255-289. https://doi.org/10.1007/s00332-004-0650-9
- Kasper W. Aircraft wing with vortex generation, U.S. Patent No. 3 831 885 (27 Aug 1974). https://patents.google.com/patent/US3831885A/en
- Kelly S.D., Xiong H. Self-propulsion of a free hydrofoil with localized discrete vortex shedding: analytical modeling and simulation, Theoretical and Computational Fluid Dynamics, 2010, vol. 24, pp. 45-50. https://doi.org/10.1007/s00162-009-0174-x
- Kelly S.D., Hukkeri R.B. Mechanics, dynamics, and control of a single-input aquatic vehicle with variable coefficient of lift, IEEE Transactions on Robotics, 2006, vol. 22, no. 6, pp. 1254-1264. https://doi.org/10.1109/TRO.2006.882934
- 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. https://doi.org/10.20537/nd1504001
- Kiselev L.V., Medvedev A.V. Comparative analysis and optimization of the dynamic properties of autonomous underwater robots of various designs and configurations, Podvodnye Issledovaniya i Robototekhnika, 2012, no. 1, vol. 13, pp. 24-35 (in Russian). http://jmtp.febras.ru/journal/1-13-2012/24-35.pdf
- Kiselev O.M. Stable feedback control of a fast wheeled robot, Russian Journal of Nonlinear Dynamics, 2018, vol. 14, no. 3, pp. 409-417. https://doi.org/10.20537/nd180310
- Klenov A.I., Kilin A.A. Influence of vortex structures on the controlled motion of an above-water screwless robot, Regular and Chaotic Dynamics, 2016, vol. 21, no. 7-8, pp. 927-938. https://doi.org/10.1134/S1560354716070145
- Kochin N.E., Kibel I.A., Roze N.V. Theoretical hydrodynamics, New York: Wiley, 1964.
- Koiller J., Castilho C., Rodrigues A.R. Vortex pairs on the triaxial ellipsoid: axis equilibria stability, Regular and Chaotic Dynamics, 2019, vol. 24, no. 1, pp. 61-79. https://doi.org/10.1134/S1560354719010039
- Kozlov V.V., Ramodanov S.M. The motion of a variable body in an ideal fluid, Journal of Applied Mathematics and Mechanics, 2001, vol. 65, no. 4, pp. 579-587. https://doi.org/10.1016/S0021-8928(01)00063-6
- Kozlov V.V., Onishchenko D.A. The motion in a perfect fluid of a body containing a moving point mass, Journal of Applied Mathematics and Mechanics, 2003, vol. 67, no. 4, pp. 553-564. https://doi.org/10.1016/S0021-8928(03)90058-X
- Kutta W.M. Auftriebskräfte in strömenden Flüssigkeiten, Illustr. Aeronaut. Mitteilungen, 1902, vol. 6, pp. 133-135.
- Kurakin L.G., Ostrovskaya I.V. On the stability of Thomson’s vortex $N$-gon and a vortex tripole/quadrupole in geostrophic models of Bessel vortices and in a two-layer rotating fluid: a review, Russian Journal of Nonlinear Dynamics, 2019, vol. 15, no. 4, pp. 533-542. https://doi.org/10.20537/nd190412
- Lighthill M.J. On the squirming motion of nearly spherical deformable bodies through liquids at very small Reynolds numbers, Communications on Pure and Applied Mathematics, 1952, vol. 5, no. 2, pp. 109-118. https://doi.org/10.1002/cpa.3160050201
- Lin X., Guo S. Development of a spherical underwater robot equipped with multiple vectored water-jet-based thrusters, Journal of Intelligent and Robotic Systems, 2012, vol. 67, no. 3-4, pp. 307-321. https://doi.org/10.1007/s10846-012-9651-3
- 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. https://doi.org/10.1134/S1560354718070055
- Mason R.J. Fluid locomotion and trajectory planning for shape-changing robots, PhD Dissertation, Pasadena, California: California Institute of Technology, 2003, 264 p. https://doi.org/10.7907/MFM1-0866
- Martynenko Yu.G., Formalskiy A.M. The theory of the control of a monocycle, Journal of Applied Mathematics and Mechanics, 2005, vol. 69, no. 4, pp. 516-528. https://doi.org/10.1016/j.jappmathmech.2005.07.003
- Mazumdar A., Asada H.H. Pulse width modulation of water jet propulsion systems using high-speed Coanda-effect valves, Journal of Dynamic Systems, Measurement, and Control, 2013, vol. 135, no. 5, 051019, 11 pp. https://doi.org/10.1115/1.4024365
- Michelin S., Llewellyn Smith S.G. An unsteady point vortex method for coupled fluid-solid problems, Theoretical and Computational Fluid Dynamics, 2009, vol. 23, no. 2, pp. 127-153. https://doi.org/10.1007/s00162-009-0096-7
- Milne-Thomson L.M. Theoretical hydrodynamics, Macmillan and Co. Ltd, 1962.
- Nelson R., Protas B., Sakajo T. Linear feedback stabilization of point-vortex equilibria near a Kasper wing, Journal of Fluid Mechanics, 2017, vol. 827, pp. 121-154. https://doi.org/10.1017/jfm.2017.484
- Novikov A.E., Novikov E.A. Vortex-sink dynamics, Physical Review E, 1996, vol. 54, no. 4, pp. 3681-3686. https://doi.org/10.1103/PhysRevE.54.3681
- Olver P.J. Application of Lie groups to differential equations, Grad. Texts in Math., vol. 107, Springer, 2000.
- Ramodanov S.M. Motion of a circular cylinder and $N$ point vortices in a perfect fluid, Regular and Chaotic Dynamics, 2002, vol. 7, no. 3, pp. 291-298. https://doi.org/10.1070/RD2002v007n03ABEH000211
- Ramodanov S.M., Tenenev V.A. Motion of a body with variable distribution of mass in a boundless viscous liquid, Russian Journal of Nonlinear Dynamics, 2011, vol. 7, no. 3, pp. 635-647 (in Russian). https://doi.org/10.20537/nd1103016
- Ramodanov S.M., Tenenev V.A., Treschev D.V. Self-propulsion of a body with rigid surface and variable coefficient of lift in a perfect fluid, Regular and Chaotic Dynamics, 2012, vol. 17, no. 6, pp. 547-558. https://doi.org/10.1134/S1560354712060068
- Rust I.C., Asada H.H. The eyeball ROV: design and control of a spherical underwater vehicle steered by an internal eccentric mass, 2011 IEEE International Conference on Robotics and Automation, IEEE, 2011, pp. 5855-5862. https://doi.org/10.1109/ICRA.2011.5979835
- Sedov L.I. Ploskie zadachi gidrodinamiki i aerodinamiki (Two-dimensional problems in hydro- and aeromechanics), Moscow: Gostekhizdat, 1950.
- Shashikanth B.N., Marsden J.E., Burdick J.W., Kelly S.D. The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with $N$ point vortices, Physics of Fluids, 2002, vol. 14, no. 3, pp. 1214-1227. https://doi.org/10.1063/1.1445183
- Smith C.B. A solution for the lift and drag of airfoils with air inlets and suction slots, Journal of the Aeronautical Sciences, 1949, vol. 16, no. 10, pp. 581-589. https://doi.org/10.2514/8.11863
- Llewellyn Smith S.G. How do singularities move in potential flow? Physica D: Nonlinear Phenomena, 2011, vol. 240, no. 20, pp. 1644-1651. https://doi.org/10.1016/j.physd.2011.06.010
- 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. https://doi.org/10.1007/s10883-016-9345-4
- 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. https://doi.org/10.1134/S0081543816080186
- 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 (in Russian). https://doi.org/10.20537/nd1604009
- 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. https://doi.org/10.1142/S0218127416500632
- 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. https://doi.org/10.1134/S1560354717080019