Аннотация:
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.
Ключевые слова:
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.
The work of A.V. Borisov (Introduction and Section 1) was carried out within the framework of the state assignment to the Udmurt State University 1.2404.2017/4.6. The work of E.V. Vetchanin and I. S.Mamaev (Sections 2 and 3) was carried out within the framework of the state assignment to the Izhevsk State Technical University 1.2405.2017/4.6 and was supported by the RFBR grant No 15-08-09093-a.
Поступила в редакцию: 15.05.2018 Принята в печать: 19.06.2018
Образец цитирования:
Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502
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\by Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin
\paper Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation
\jour Regul. Chaotic Dyn.
\yr 2018
\vol 23
\issue 4
\pages 480--502
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Эта публикация цитируется в следующих 14 статьяx:
А. В. Клековкин, Ю. Л. Караваев, А. А. Килин, А. В. Назаров, “Влияние хвостовых плавников на скорость водного робота, приводимого в движение внутренними подвижными массами”, Компьютерные исследования и моделирование, 16:4 (2024), 869–882
L.A. Klimina, S.A. Golovanov, M.Z. Dosaev, Y.D. Selyutskiy, A.P. Holub, “Plane-parallel motion of a trimaran capsubot controlled with an internal flywheel”, International Journal of Non-Linear Mechanics, 150 (2023), 104341
Sergey Golovanov, Liubov Klimina, Marat Dosaev, Yury Selyutskiy, Andrei Holub, 2022 16th International Conference on Stability and Oscillations of Nonlinear Control Systems (Pyatnitskiy's Conference), 2022, 1
Elizaveta M. Artemova, Yury L. Karavaev, Ivan S. Mamaev, Evgeny V. Vetchanin, “Dynamics of a Spherical Robot with Variable Moments of Inertia and a Displaced Center of Mass”, Regul. Chaotic Dyn., 25:6 (2020), 689–706
E. M. Artemova, E. V. Vetchanin, “Control of the motion of a circular cylinder in an ideal fluid using a source”, Вестн. Удмуртск. ун-та. Матем. Мех. Компьют. науки, 30:4 (2020), 604–617
E. V. Vetchanin, I. S. Mamaev, “Asymptotic behavior in the dynamics of a smooth body in an ideal fluid”, Acta Mech., 231:11 (2020), 4529–4535
A. V. Borisov, E. V. Vetchanin, I. S. Mamaev, “Motion of a smooth foil in a fluid under the action of external periodic forces. II”, Russ. J. Math. Phys., 27:1 (2020), 1–17
E. V. Vetchanin, “The Motion of a Balanced Circular Cylinder in an Ideal Fluid Under the Action of External Periodic Force and Torque”, Rus. J. Nonlin. Dyn., 15:1 (2019), 41–57
E. V. Vetchanin, E. A. Mikishanina, “Vibrational Stability of Periodic Solutions of the Liouville Equations”, Rus. J. Nonlin. Dyn., 15:3 (2019), 351–363
A. V. Borisov, E. V. Vetchanin, I. S. Mamaev, “Motion of a smooth foil in a fluid under the action of external periodic forces. I”, Russ. J. Math. Phys., 26:4 (2019), 412–427
Alexey V. Borisov, Ivan S. Mamaev, Evgeny V. Vetchanin, “Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation”, Regul. Chaotic Dyn., 23:7-8 (2018), 850–874
Ivan S. Mamaev, Evgeny V. Vetchanin, “The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor”, Regul. Chaotic Dyn., 23:7-8 (2018), 875–886
I. S. Mamaev, V. A. Tenenev, E. V. Vetchanin, “Dynamics of a Body with a Sharp Edge in a Viscous Fluid”, Nelin. Dinam., 14:4 (2018), 473–494
Kilin A.A. Pivovarova E.N., “Chaplygin TOP With a Periodic Gyrostatic Moment”, Russ. J. Math. Phys., 25:4 (2018), 509–524
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