Propulsive motion of cylindrical vibration-driven robot in a viscous fluid
V. D. Anisimov, A. G. Egorov, A. N. Nuriev, O. N. Zaitseva

References
1.	Prandtl L., “Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben”, Vorträge aus dem Gebiete der Hydro- und Aerodynamik (Innsbruck 1922), eds. T.V. Kármán, T. Levi-Civita, Springer, Berlin–Heidelberg, 1924, 18–33 (In German)
2.	Birnbaum W., “Der Schlagflügelpropeller und die kleinen Schwingungenelastisch befestigter Tragflügel”, Z. Flugtech. Motorluftschiffahrt, 15 (1924), 128–134 (In German)
3.	Theodorsen T., General Theory of Aerodynamic Instability and the Mechanism of Flutter, NACA Report 496, Natl. Advis. Comm. Aeronaut., 1935, 291–311
4.	Garrick I.E., Propulsion of a Flapping and Oscillating Airfoil, NACA Report 567, Natl. Advis. Comm. Aeronaut., 1936, 14 pp.
5.	Wagner H., “Über die Entstehung des dynamischen Auftriebes von Tragflügeln”, ZAMM, 5:1 (1925), 17–35 (In German)
6.	Glauert H., “The force and moment on an oscillating aerofoil”, Vorträge aus dem Gebiete der Aerodynamik und verwandter Gebiete (Aachen 1929), eds. A. Gilles, L. Hopf, T.v. Kármán, Springer, Berlin–Heidelberg, 1930, 88–95
7.	Küssner H.G., “Zusammenfassender Bericht über den instationären Auftrieb von Flügeln”, Luftfahrtforschung, 13:12 (1936), 410–424 (In German)
8.	Küssner H.G., Schwartz I.R., The Oscillating Wing with Aerodynamically Balanced Elevator, Technical Memorandum 991, Natl. Advis. Comm. Aeronaut., 1936, 32 pp.
9.	Keldysh M.V., Lavrent'ev M.A., “On the theory of the oscillating wing”, Technical Notes of the Central Aerohydrodynamic Institute, TsAGI, M., 1935, 48–52 (In Russian)
10.	Sedov L.I., Plane Problems of Hydrodynamics and Aerodynamics, GITTL, M.–L., 1950, 444 pp. (In Russian)
11.	Nekrasov A.I., Theory of a Wing in a Nonsteady Flow, Akad. Nauk SSSR, M.–L., 1947, 260 pp. (In Russian)
12.	Alben S., “Collective locomotion of two-dimensional lattices of flapping plates. Part 1. Numerical method, single-plate case and lattice input power”, J. Fluid Mech., 915 (2021), A20
13.	Alben S., “Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency”, J. Fluid Mech., 915, A21
14.	Alben S., Shelley M., “Coherent locomotion as an attracting state for a free flapping body”, Proc. Natl. Acad. Sci. U. S. A, 102:32 (2005), 11163–11166
15.	Isogai K., Shinmoto Y., Watanabe Y., “Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil”, AIAA J., 37:10 (1999), 1145–1151
16.	Lewin G.C., Haj-Hariri H., “Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow”, J. Fluid Mech., 492 (2003), 339–362
17.	Liu H., “A computational fluid dynamic study of hawkmoth hovering”, J. Exp. Biol., 201:4 (1998), 461–477
18.	Lua K., “On the thrust performance of a flapping two-dimensional elliptic airfoil in a forward flight”, J. Fluids Struct., 66 (2016), 91–109
19.	Maertens A.P., Triantafyllou M.S., Yue D.K.P., “Efficiency of fish propulsion”, Bioinspiration Biomimetics, 10:4 (2015), 046013
20.	Pedro G., Suleman A., Djilali N., “A numerical study of the propulsive efficiency of a flapping hydrofoil”, Int. J. Numer. Methods Fluids, 42:5 (2003), 493–526
21.	Spagnolie S.E., Moret L., Shelley M.J., Zhang J., “Surprising behaviors in flapping locomotion with passive pitching”, Phys. Fluids, 22:4 (2010), 041903
22.	Taha H.E., “Geometric nonlinear control of the lift dynamics of a pitching-plunging wing”, Proc. AIAA Scitech 2020 Forum, 2020, AIAA 2020-0824
23.	Zhang J., Liu N.-S., Lu X.-Y., “Locomotion of a passively flapping flat plate”, J. Fluid Mech., 659 (2010), 43–68
24.	Dynnikov Ya.A. On the calculation of a flapping flexible airfoil in the flow of viscous incompressible fluid, Izv. Vyssh. Uchebn. Zaved., Mashinostr., 2016, no. 4, 22–30 (In Russian)
25.	Koval' K.A., Sukhorukov A.L., Chernyshev I.A., “Verification results of the numerical method for calculating the hydrodynamic and hydroacoustic characteristics of a fin propulsor”, Fundam. Prikl. Gidrofiz., 9:4 (2016), 60–72 (In Russian)
26.	Wu X., Zhang X., Tian X., Li X., Lu W., “A review on fluid dynamics of flapping foils”, Ocean Eng., 195 (2020), 106712
27.	Schlichting H., “Berechnung ebener periodischer Grenzschichtströmungen”, Phys. Zeit., 33 (1932), 327–335 (In German)
28.	Holtsmark J.,, Johnsen I., Sikkeland T., Skavlem S., “Boundary layer flow near a cylindrical obstacle in an oscillating, incompressible fluid”, J. Acoust. Soc. Am., 26:1 (1954), 26–39
29.	Riley N., “Oscillatory viscous flows. Review and extension”, IMA J. Appl. Math., 3:4 (1967), 419–434
30.	Riley N., “The steady streaming induced by a vibrating cylinder”, J. Fluid Mech., 68:4 (1975), 801–812
31.	Nuriev A.N., Egorov A.G., “Asymptotic investigation of hydrodynamic forces acting on an oscillating cylinder at finite streaming Reynolds numbers”, Lobachevskii J. Math., 40:6 (2019), 794–801
32.	Nuriev A.N., Egorov A.G., Kamalutdinov A.M., “Hydrodynamic forces acting on the elliptic cylinder performing high-frequency low-amplitude multi-harmonic oscillations in a viscous fluid”, J. Fluid Mech., 913 (2021), A40
33.	Riley N., Watson E., “Eccentric oscillations of a circular cylinder in a viscous fluid”, Mathematika, 40:2 (1993), 187–202
34.	Nuriev A.N., Egorov A.G., “Asymptotic theory of a flapping wing of a circular cross-section”, J. Fluid Mech., 941 (2022), A23
35.	Nuriev A.N., Egorov A.G., Zaitseva O.N., Kamalutdinov A.M., “Asymptotic study of the aerohydrodynamics of a flapping cylindrical wing in the high-frequency approximation”, Lobachevskii J. Math., 43:8 (2022), 2250–2256
36.	Egorov A.G., Nuriev A.N., “Cruising speed of a cylindrical wing performing small translational-rotational oscillations”, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 164, no. 2–3, 2022, 170–180 (In Russian)
37.	Egorov A.G., Nuriev A.N., Anisimov V.D., Zaitseva O.N., “Propulsive motion of an oscillating cylinder in a viscous fluid”, Phys. Fluids, 36:2 (2024), 021908
38.	Chernous'ko F.L., “On the motion of a body containing a movable internal mass”, Dokl. Phys., 50:11 (2005), 593–597
39.	Chernous'ko F.L., “Analysis and optimization of the motion of a body controlled by means of a movable internal mass”, J. Appl. Math. Mech., 70:6 (2006), 819–842
40.	Bolotnik N.N., Figurina T.Yu., Chernous'ko F.L., “Optimal control of the rectilinear motion of a twobody system in a resistive medium”, J. Appl. Math. Mech., 76:1 (2012), 1–14
41.	Bolotnik N., Pivovarov M., Zeidis I., Zimmermann K., “The undulatory motion of a chain of particles in a resistive medium in the case of a smooth excitation mode”, ZAMM, 93:12 (2013), 895–913
42.	Yegorov A.G., Zakharova O.S., “The energy-optimal motion of a vibration-driven robot in a resistive medium”, J. Appl. Math. Mech., 74:4 (2010), 443–451
43.	Egorov A.G., Zakharova O.S., “The optimal quasistationary motion of a vibration-driven robot in a viscous medium”, Russ. Math., 56:2 (2012), 50–55
44.	Du Z., Fang H., Zhan X., Xu J., “Experiments on vibration-driven stick-slip locomotion: A sliding bifurcation perspective”, Mech. Syst. Signal Process, 105 (2018), 261–275
45.	Diao B., Zhang X., Fang H., Xu J., “Optimal control of the multi-module vibration-driven locomotion robot”, J. Sound Vib., 527 (2022), 116867
46.	Egorov A.G., Zakharova O.S., “The energy-optimal motion of a vibration-driven robot in a medium with a inherited law of resistance”, J. Comput. Syst. Sci. Int., 54:3 (2015), 495–503
47.	Vetchanin E.V., Mamaev I.S., Tenenev V.A., “The self-propulsion of a body with moving internal masses in a viscous fluid”, Regular Chaotic Dyn., 18:1–2 (2013), 100–117
48.	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 Chaotic Dyn., 23:7–8 (2018), 850–874
49.	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 Chaotic Dyn., 25:6 (2020), 689–706
50.	Egorov A.G., Nuriev A.N., Anisimov V.D., “Optimization of the movement of a cylindrical vibration-driven robot in a viscous fluid, induced by pendulum oscillations of the internal mass”, Lobachevskii J. Math., 444:10 (2023), 4438–4447
51.	Purcell E.M., “Life at low Reynolds number”, Am. J. Phys., 45:1 (1977), 3–11
52.	Becker L.E., Koehler S.A., Stone H.A., “On self-propulsion of micro-machines at low Reynolds number: Purcell's three-link swimmer”, J. Fluid Mech., 490 (2003), 15–35
53.	Sánchez-Rodríguez J., Raufaste C., Argentina M., “Scaling the tail beat frequency and swimming speed in underwater undulatory swimming”, Nat. Commun., 14:1 (2023), 5569