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Propulsive Motion of Cylindrical Vibration-Driven Robot in a Viscous Fluid

https://doi.org/10.26907/2541-7746.2024.3.277-296

Abstract

The propulsive motion of a multimass system, vibration-driven robot (VR), in a viscous incompressible fluid was studied. The VR consisted of a round cylindrical body submerged in the fluid and an internal mass (IM) performing small-amplitude pendulum-like oscillations inside the body. Using the method of asymptotic expansions, the combined mechanical and hydrodynamic problems that describe the self-propulsion of the system in the fluid were solved. The hydrodynamic problem was formulated on the basis of the complete non-stationary Navier–Stokes equation. An analytical solution was derived to describe the cruising regime of the VR motion in the fluid. The non-stationary hydrodynamic influence on the VR was determined. The efficiency of the propulsive system’s motion was assessed.

About the Authors

V. D. Anisimov
Kazan Federal University
Russian Federation

Kazan, 420008



A. G. Egorov
Kazan Federal University
Russian Federation

Kazan, 420008



A. N. Nuriev
Kazan Federal University
Russian Federation

Kazan, 420008



O. N. Zaitseva
Kazan Federal University
Russian Federation

Kazan, 420008



References

1. Prandtl L. Über die Entstehung von Wirbeln in der idealen Flüssigkeit, mit Anwendung auf die Tragflügeltheorie und andere Aufgaben. In: Vorträge aus dem Gebiete der Hydround Aerodynamik (Innsbruck 1922). Kàrmàn T.V., Levi-Civita T. (Hrsg.). Berlin, Heidelberg, Springer, 1924. S. 18–33. https://doi.org/10.1007/978-3-662-00280-3. (In German)

2. Birnbaum W. Der Schlagflügelpropeller und die kleinen Schwingungen elastisch befestigter Tragflügel. Z. Flugtech. Motorluftschiffahrt, 1924, Bd. 15, S. 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, pp. 291–311.

4. Garrick I.E. Propulsion of a Flapping and Oscillating Airfoil. NACA Report 567. Natl. Advis. Comm. Aeronaut., 1936, pp. 1–14.

5. Wagner H. Über die Entstehung des dynamischen Auftriebes von Tragflügeln. ZAMM, 1925, Bd. 5, H. 1, S. 17–35. https://doi.org/10.1002/zamm.19250050103. (In German)

6. Glauert H. The force and moment on an oscillating aerofoil. In: Gilles A., Hopf L., Kàrmàn Th.v. (Hrsg.) Vorträge aus dem Gebiete der Aerodynamik und verwandter Gebiete: Aachen 1929. Berlin, Heidelberg, Springer, 1930, S. 88–95. https://doi.org/10.1007/978-3-662-33791-2_16.

7. Küssner H.G. Zusammenfassender Bericht über den instationären Auftrieb von Flügeln. Luftfahrtforschung, 1936, Bd. 13, H. 12, S. 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 p.

9. Keldysh M.V., Lavrent’ev M.A. On the theory of the oscillating wing. In: Tekhn. Zametki TsaGI [Technical Notes of the Central Aerohydrodynamic Institute]. Moscow, TsAGI, 1935, pp. 48–52. (In Russian)

10. Sedov L.I. Ploskie zadachi gidrodinamiki i aerodinamiki [Plane Problems of Hydrodynamics and Aerodynamics]. Moscow, Leningrad, GITTL, 1950. 444 p. (In Russian)

11. Nekrasov A.I. Teoriya kryla v nestatsionarnom potoke [Theory of a Wing in a Nonsteady Flow]. Moscow, Leningrad, Akad. Nauk SSSR, 1947. 260 p. (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., 2021, vol. 915, art. A20. https://doi.org/10.1017/jfm.2021.6

13. Alben S. Collective locomotion of two-dimensional lattices of flapping plates. Part 2. Lattice flows and propulsive efficiency. J. Fluid Mech., 2021, vol. 915, art. A21. https://doi.org/10.1017/jfm.2021.43.

14. Alben S., Shelley M. Coherent locomotion as an attracting state for a free flapping body. Proc. Natl. Acad. Sci. U. S. A., 2005, vol. 102, no. 32, pp. 11163–11166. https://doi.org/10.1073/pnas.0505064102.

15. Isogai K., Shinmoto Y., Watanabe Y. Effects of dynamic stall on propulsive efficiency and thrust of flapping airfoil. AIAA J., 1999, vol. 37, no. 10, pp. 1145–1151. https://doi.org/10.2514/2.589.

16. Lewin G.C., Haj-Hariri H. Modelling thrust generation of a two-dimensional heaving airfoil in a viscous flow. J. Fluid Mech., 2003, vol. 492, pp. 339–362. https://doi.org/10.1017/S0022112003005743.

17. Liu H. A computational fluid dynamic study of hawkmoth hovering. J. Exp. Biol., 1998, vol. 201, no. 4, pp. 461–477. https://doi.org/10.1242/jeb.201.4.461.

18. Lua K., Dash S.M., Lim T.T., Yeo K.S. On the thrust performance of a flapping twodimensional elliptic airfoil in a forward flight. J. Fluids Struct., 201, vol. 66, pp. 91–109. https://doi.org/10.1016/j.jfluidstructs.2016.07.012.

19. Maertens A.P., Triantafyllou M.S., Yue D.K.P. Efficiency of fish propulsion. Bioinspiration Biomimetics, 2015, vol. 10, no. 4, art. 046013. https://doi.org/10.1088/1748-3190/10/4/046013.

20. Pedro G., Suleman A., Djilali N. A numerical study of the propulsive efficiency of a flapping hydrofoil. Int. J. Numer. Methods Fluids, 2003, vol. 42, no. 5, pp. 493–526. https://doi.org/10.1002/fld.525.

21. Spagnolie S.E., Moret L., Shelley M.J., Zhang J. Surprising behaviors in flapping locomotion with passive pitching. Phys. Fluids, 2010, vol. 22, no. 4, art. 041903. https://doi.org/10.1063/1.3383215.

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. https://doi.org/10.2514/6.2020-0824.

23. Zhang J., Liu N.-S., Lu X.-Y. Locomotion of a passively flapping flat plate. J. Fluid Mech., 2010, vol. 659, pp. 43–68. https://doi.org/10.1017/S0022112010002387.

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, pp. 22–30. https://10.18698/0536-1044-2016-4-22-30. (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., 2016, vol. 9, no. 4, pp. 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., 2020, vol. 195, art. 106712. https://doi.org/10.1016/j.oceaneng.2019.106712.

27. Schlichting H. Berechnung ebener periodischer Grenzschichtstr¨omungen. Phys. Zeit, 1932, Bd. 33, S. 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., 1954, vol. 26, no. 1, pp. 26–39. https://doi.org/10.1121/1.1907285.

29. Riley N. Oscillatory viscous flows. Review and extension. IMA J. Appl. Math., 1967, vol. 3, no. 4, pp. 419–434. https://doi.org/10.1093/imamat/3.4.419.

30. Riley N. The steady streaming induced by a vibrating cylinder. J. Fluid Mech., 1975, vol. 68, no. 4, pp. 801–812. https://doi.org/10.1017/S0022112075001243.

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., 2019, vol. 40, no. 6, pp. 794–801. https://doi.org/10.1134/S1995080219060180.

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., 2021, vol. 913, art. A40. https://doi.org/10.1017/jfm.2020.1180.

33. Riley N., Watson E. Eccentric oscillations of a circular cylinder in a viscous fluid. Mathematika, 1993, vol. 40, no. 2, pp. 187–202. https://doi.org/10.1112/S0025579300006975.

34. Nuriev A.N., Egorov A.G. Asymptotic theory of a flapping wing of a circular cross-section. J. Fluid Mech., 2022, vol. 941, art. A23. https://doi.org/10.1017/jfm.2022.287.

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., 2022, vol. 43, no. 8, pp. 2250–2256. https://doi.org/10.1134/S1995080222110233.

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, 2022, vol. 164, nos. 2–3, pp. 170–180. https://doi.org/10.26907/2541-7746.2022.2-3.170-180. (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, 2024, vol. 36, no. 2, art. 021908. https://doi.org/10.1063/5.0189346.

38. Chernous’ko F.L. On the motion of a body containing a movable internal mass. Dokl. Phys., 2005, vol. 50, no. 11, pp. 593–597. https://doi.org/10.1134/1.2137795.

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., 2006, vol. 70, no. 6, pp. 819–842. https://doi.org/10.1016/j.jappmathmech.2007.01.003.

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., 2012, vol. 76, no. 1, pp. 1–14. https://doi.org/10.1016/j.jappmathmech.2012.03.001.

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, 2013, vol. 93, no. 12, pp. 895–913. https://doi.org/10.1002/zamm.201200124.

42. Yegorov A.G., Zakharova O.S. The energy-optimal motion of a vibration-driven robot in a resistive medium. J. Appl. Math. Mech., 2010, vol. 74, no. 4, pp. 443–451. https://doi.org/10.1016/j.jappmathmech.2010.09.010.

43. Egorov A.G., Zakharova O.S. The optimal quasistationary motion of a vibration-driven robot in a viscous medium. Russ. Math., 2012, vol. 56, no. 2, pp. 50–55. https://doi.org/10.3103/S1066369X12020065.

44. Du Z., Fang H., Zhan X., Xu J. Experiments on vibration-driven stick-slip locomotion: A sliding bifurcation perspective. Mech. Syst. Signal Process., 2018, vol. 105, pp. 261–275. https://doi.org/10.1016/j.ymssp.2017.12.001.

45. Diao B., Zhang X., Fang H., Xu J. Optimal control of the multi-module vibration-driven locomotion robot. J. Sound Vib. 2022, vol. 527, art. 116867. https://doi.org/10.1016/j.jsv.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., 2015, vol. 54, no. 3, pp. 495–503. https://doi.org/10.1134/S1064230715030065.

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., 2013, vol. 18, nos. 1–2. https://doi.org/10.1134/S1560354713010073.

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., 2018, vol. 23, nos. 7–8, pp. 850–874. https://doi.org/10.1134/S1560354718070043.

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., 2020, vol. 25, no. 6, pp. 689–706. https://doi.org/10.1134/S156035472006012X.

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., 2023, vol. 444, no. 10, pp. 4438–4447. https://doi.org/10.1134/S1995080223100104.

51. Purcell E.M. Life at low Reynolds number. Am. J. Phys., 1977, vol. 45, no. 1, pp. 3–11. https://doi.org/10.1119/1.10903.

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., 2003, vol. 490, pp. 15–35. https://doi.org/10.1017/S0022112003005184.

53. Sànchez-Rodríguez J., Raufaste C., Argentina M. Scaling the tail beat frequency and swimming speed in underwater undulatory swimming. Nat. Commun., 2023, vol. 14, no. 1, art. 5569. https://doi.org/10.1038/s41467-023-41368-6.


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Anisimov V.D., Egorov A.G., Nuriev A.N., Zaitseva O.N. Propulsive Motion of Cylindrical Vibration-Driven Robot in a Viscous Fluid. Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki. 2024;166(3):277-296. (In Russ.) https://doi.org/10.26907/2541-7746.2024.3.277-296

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