Skip to main content
SpringerLink
Log in

Motion of a system of interacting bodies in a medium with quadratic resistance

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

The rectilinear motion of a chain of identical bodies in a viscous medium with a quadratic law of resistance is considered. Neighboring bodies interact with each other. There are no restrictions on the magnitude of the interaction forces. Motions are constructed in which each of the bodies of the system shifts by the same specified distance, provided that the velocities of each of the bodies of the system coincide at the initial and final moments of time. In particular, the case where the system is at rest at the initial moment of time is considered. In the case where the velocity of the center of mass of the system at the initial moment of time is not equal to zero, a motion is constructed in which the velocity of each of the bodies is piecewise constant and the velocity of the center of mass of the system is constant. This motion is optimized under the condition that the velocity of each of the bodies is bounded. The obtained results can be used to control a locomotion system of several bodies moving in a viscous medium by means of changing its configuration.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

Data availability

The manuscript has no associated data.

References

  1. Gray, J.: Animal Locomotion. Norton, New York (1968)

    Google Scholar 

  2. Alexander, R.M.: Principles of Animal Locomotion. Princeton University Press, Princeton (2003)

    Google Scholar 

  3. Zimmermann, K., Zeidis, I., Behn, C.: Mechanics of Terrestrial Locomotion with a Focus on Nonpedal Motion Systems. Springer, Heidelberg (2010)

    Google Scholar 

  4. Steigenberger, J., Behn, C.: Worm-like Locomotion Systems: an Intermediate Theoretical Approach. Oldenbourg Wissenschaftsverlag, Munich (2012)

    Book  Google Scholar 

  5. Wagner, G.L., Lauga, E.: Crawling scallop: Friction-based locomotion with one degree of freedom. J. Theor. Biol. 324, 42–51 (2013). https://doi.org/10.1016/j.jtbi.2013.01.021

    Article  MathSciNet  Google Scholar 

  6. Chernous’ko, F.L.: The optimum rectilinear motion of a two-mass system. J. Appl. Math. Mech. 66(1), 1–7 (2002). https://doi.org/10.1016/S0021-8928(02)00002-3

    Article  MathSciNet  Google Scholar 

  7. Chernous’ko, F.L.: Analysis and optimization of the rectilinear motion of a two-body system. J. Appl. Math. Mech. 75(5), 493–500 (2011). https://doi.org/10.1016/j.jappmathmech.2011.11.001

    Article  MathSciNet  Google Scholar 

  8. Bolotnik, N., Figurina, T.: Optimal control of a two-body limbless crawler along a rough horizontal straight line. Nonlinear Dyn. 102, 1627–1642 (2020). https://doi.org/10.1007/s11071-020-05999-4

    Article  Google Scholar 

  9. Memon, A.B., Verriest, E.I., Hyun, N.P.: Graceful gait transitions for biomimetic locomotion—the worm. In: Proceedings of 53rd IEEE conference on decision and control, Los Angeles, CA, USA, 2014, 2958–2963 (2015). https://doi.org/10.1109/CDC.2014.7039844

  10. Bolotnik, N., Pivovarov, M., Zeidis, I., Zimmermann, K.: The motion of a two-body limbless locomotor along a straight line in a resistive medium. ZAMM 96(4), 429–452 (2016). https://doi.org/10.1002/zamm.201400302

    Article  MathSciNet  Google Scholar 

  11. Knyaz’kov, D.Y., Figurina, T.Y.: On the existence, uniqueness, and stability of periodic modes of motion of a locomotion system with a mobile internal mass. J. Comput. Syst. Sci. Int. 59(1), 129–137 (2020). https://doi.org/10.1134/S1064230719060108

    Article  MathSciNet  Google Scholar 

  12. Figurina, T., Knyazkov, D.: Periodic regimes of motion of capsule system on rough plane. Meccanica (2022). https://doi.org/10.1007/s11012-022-01572-y

    Article  Google Scholar 

  13. Bolotnik, N., Pivovarov, M., Zeidis, I., Zimmermann, K.: On the motion of lumped-mass and distributed-mass self-propelling systems in a linear resistive environment. ZAMM 96(6), 747–757 (2016). https://doi.org/10.1002/zamm.201500091

    Article  MathSciNet  Google Scholar 

  14. Bolotnik, N., Pivovarov, M., Zeidis, I., Zimmermann, K.: The undulatory motion of a chain of particles in a resistive medium. ZAMM 91(4), 259–275 (2011). https://doi.org/10.1002/zamm.201000112

    Article  MathSciNet  Google Scholar 

  15. Karmanov, S.P., Chernousko, F.L.: An elementary model of the rowing process. Dokl. Phys. 60(3), 140–144 (2015). https://doi.org/10.1134/S1028335815030106

    Article  Google Scholar 

  16. Figurina, T., Knyazkov, D.: Periodic gaits of a locomotion system of interacting bodies. Meccanica 57(4), 1463–1476 (2022). https://doi.org/10.1007/s11012-022-01473-0

  17. Figurina, T.Y.: Optimal control of system of material points in a straight line with dry friction. J. Comput. Syst. Sci. Int. 54(5), 671–677 (2015). https://doi.org/10.1134/S1064230715050056

    Article  MathSciNet  Google Scholar 

  18. Chernous’ko, F.L.: Translational motion of a chain of bodies in a resistive medium. J. Appl. Math. Mech. 81(4), 256–261 (2017). https://doi.org/10.1016/j.jappmathmech.2017.12.002

    Article  MathSciNet  Google Scholar 

  19. Behn, C., Schale, F., Zeidis, I., Zimmermann, K., Bolotnik, N.: Dynamics and motion control of a chain of particles on a rough surface. Mech. Syst. Signal Process. 89, 3–13 (2017). https://doi.org/10.1016/j.ymssp.2016.11.001

    Article  Google Scholar 

  20. Tanaka, Y., Ito, K., Nakagaki, T., Kobayashi, R.: Mechanics of peristaltic locomotion and role of anchoring. J. R. Soc. Interface 9(67), 222–233 (2014). https://doi.org/10.1098/rsif.2011.0339

    Article  Google Scholar 

  21. Fang, H., Xu, J.: Controlled motion of a two-module vibration-driven system induced by internal acceleration-controlled masses. Arch. Appl. Mech. 82, 461–477 (2012). https://doi.org/10.1007/s00419-011-0567-3

    Article  Google Scholar 

  22. Fang, H., Xu, J.: Dynamics of a three-module vibration-driven system with non-symmetric coulomb’s dry friction. Multibody Syst. Dyn. 27, 455–485 (2012). https://doi.org/10.1007/s11044-012-9304-0

    Article  MathSciNet  Google Scholar 

  23. Fang, H., Zhao, Y., Xu, J.: Steady-state dynamics and discontinuity-induced sliding bifurcation of a multi-module piecewise-smooth vibration-driven system with dry friction. Commun. Nonlinear Sci. Numer. Simul. 114, 106704–127 (2022). https://doi.org/10.1016/j.cnsns.2022.106704

    Article  MathSciNet  Google Scholar 

  24. Tkachenko, E.A., Merkulov, D.I., Pelevina, D.A., Turkov, V.A., Vinogradova, A.S., Naletova, V.A.: Mathematical model of a mobile robot with a magnetizable material in a uniform alternating magnetic field. Meccanica 58, 357–369 (2023). https://doi.org/10.1007/s11012-022-01486-9

    Article  MathSciNet  Google Scholar 

  25. Merkulov, D.I., Pelevina, D.A., Turkov, V.A., Vinogradova, A.S., Naletova, V.A.: Mobile robots with magnetizable materials in alternating uniform inclined magnetic fields. Acta Astronaut. 181, 579–584 (2021). https://doi.org/10.1016/j.actaastro.2020.11.052

    Article  Google Scholar 

  26. Bolotnik, N., Schorr, P., Zeidis, I., Zimmermann, K.: Periodic locomotion of a two-body crawling system along a straight line on a rough inclined plane. ZAMM 98, 1930–1946 (2018). https://doi.org/10.1002/zamm.201800107

    Article  MathSciNet  Google Scholar 

  27. Bolotnik, N., Figurina, T.: Controllabilty of a two-body crawling system on an inclined plane. Meccanica 58, 321–336 (2023). https://doi.org/10.1007/s11012-021-01466-5

    Article  MathSciNet  Google Scholar 

  28. Bardin, B.S., Panev, A.S.: On the motion of a body with a moving internal mass on a rough horizontal plane. Rus. J. Nonlin. Dyn. 14(4), 519–542 (2018). https://doi.org/10.20537/nd180407

    Article  MathSciNet  Google Scholar 

  29. Golitsyna, M.V.: Periodic regime of motion of a vibratory robot under a control constraint. Mech. Solids 53, 49–59 (2018). https://doi.org/10.3103/S002565441803007X

    Article  Google Scholar 

  30. Golitsyna, M.V., Samsonov, V.A.: Estimating the domain of admissible parameters of a control system of a vibratory robot. J. Comput. Syst. Sci. Int. 57, 255–272 (2018). https://doi.org/10.1134/S1064230718020089

  31. Zarychta, S., Balcerzak, M., Denysenko, V., Stefanski, A., Dabrowski, A., Lenci, S.: Optimization of the closed-loop controller of a discontinuous capsule drive using a neural network. Meccanica 58, 537–553 (2023). https://doi.org/10.1007/s11012-023-01639-4

    Article  Google Scholar 

  32. Liu, P., Yu, H., Cang, S.: On the dynamics of a vibro-driven capsule system. Arch. Appl. Mech. 88, 2199–2219 (2018). https://doi.org/10.1007/s00419-018-1444-0

    Article  Google Scholar 

  33. Zhu, J., Liao, M., Zheng, Y., Qi, S., Li, Z., Zeng, Z.: Multi-objective optimisation based on reliability analysis of a self-propelled capsule system. Meccanica 58, 397–419 (2023). https://doi.org/10.1007/s11012-022-01519-3

    Article  MathSciNet  Google Scholar 

  34. Fang, H., Wang, K.W.: Piezoelectric vibration-driven locomotion systems: exploiting resonance and bistable dynamics. J. Sound Vib. 391, 153–169 (2017). https://doi.org/10.1016/j.jsv.2016.12.009

    Article  Google Scholar 

  35. Xu, J., Fang, H.: Improving performance: recent progress on vibration-driven locomotion systems. Nonlinear Dyn. 98, 2651–2669 (2019). https://doi.org/10.1007/s11071-019-04982-y

    Article  Google Scholar 

  36. Duong, T., Van, C.N., Ho, K., La, N., Ngo, Q., Nguyen, K., Hoang, T., Chu, N., Nguyen, V.: Dynamic response of vibro-impact capsule moving on the inclined track and stochastic slope. Meccanica 58, 421–439 (2023). https://doi.org/10.1007/s11012-022-01521-9

    Article  MathSciNet  Google Scholar 

  37. Nunuparov, A., Becker, F., Bolotnik, N., Zeidis, I., Zimmermann, K.: Dynamics and motion control of a capsule robot with an opposing spring. Arch. Appl. Mech. 89, 2193–2208 (2019). https://doi.org/10.1007/s00419-019-01571-8

    Article  Google Scholar 

  38. Ivanov, A.P.: Analysis of an impact-driven capsule robot. Int. J. Nonlinear Mech. 119, 103257 (2020). https://doi.org/10.1016/j.ijnonlinmec.2019.103257

    Article  Google Scholar 

  39. Nguyen, K.T., La, N.T., Ho, K.T., Ngo, Q.H., Chu, N.H., Nguyen, V.D.: The effect of friction on the vibro-impact locomotion system: modeling and dynamic response. Meccanica 56, 2121–2137 (2021). https://doi.org/10.1007/s11012-021-01348-w

    Article  MathSciNet  Google Scholar 

  40. Xue, J., Zhang, S., Xu, J.: Coordinated optimization of locomotion velocity and energy consumption in vibration-driven system. Meccanica 58, 371–385 (2023). https://doi.org/10.1007/s11012-022-01488-7

    Article  MathSciNet  Google Scholar 

  41. 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, 495–503 (2015). https://doi.org/10.1134/S1064230715030065

    Article  MathSciNet  Google Scholar 

  42. Egorov, A.G., Zakharova, O.S.: The energy-optimal motion of a vibration-driven robot in a resistive medium. J. Appl. Math. Mech. 74, 443–451 (2010). https://doi.org/10.1016/j.jappmathmech.2010.09.010

    Article  MathSciNet  Google Scholar 

  43. Tahmasian, S., Jafaryzad, A., Bulzoni, N.L., Staples, A.E.: Dynamic analysis and design optimization of a drag-based vibratory swimmer. Fluids 5, 38 (2020). https://doi.org/10.3390/fluids5010038

    Article  Google Scholar 

  44. Tahmasian, S.: Dynamic analysis and optimal control of drag-based vibratory systems using averaging. Nonlinear Dyn. 104, 2201–2217 (2021). https://doi.org/10.1007/s11071-021-06440-0

  45. Knyazkov, D., Figurina, T.: Periodic regimes of motion of a body with a moving internal mass. In: Proc. of 2019 24th International conference on methods and models in automation and robotics (MMAR), Miedzyzdroje, Poland, 26–29 August 2019, pp. 331–336 (2019). https://doi.org/10.1109/mmar.2019.8864630

  46. Figurina, T., Glazkov, T.: Optimization of the rectilinear motion of a capsule system along a rough plane. Z. Angew. Math. Mech. 101, 202000111 (2021). https://doi.org/10.1002/zamm.202000111

    Article  MathSciNet  Google Scholar 

  47. Kamamichi, N., Furuta, K.: Locomotion analysis of self-propelled board by inclined internal mass motion with slider-crank mechanism. Meccanica 58, 473–492 (2023). https://doi.org/10.1007/s11012-022-01538-0

    Article  MathSciNet  Google Scholar 

  48. Liu, Y., Islam, S., Pavlovskaya, E., Wiercigroch, M.: Optimization of the vibro-impact capsule system. J. Mech. Eng. 62, 430–439 (2016). https://doi.org/10.5545/sv-jme.2016.3754

    Article  Google Scholar 

  49. Liu, Y., Jiang, H., Pavlovskaia, E., Wiercigroch, M.: Experimental investigation of the vibro-impact capsule system. Proc IUTAM 22, 237–243 (2017). https://doi.org/10.1016/j.piutam.2017.08.029

    Article  Google Scholar 

  50. Tian, J., Afebu, K.O., Wang, Z., Liu, Y., Prasad, S.: Dynamic analysis of a soft capsule robot self-propelling in the small intestine via finite element method. Nonlinear Dyn. 111, 9777–9798 (2023). https://doi.org/10.1007/s11071-023-08376-z

    Article  Google Scholar 

  51. Yan, Y., Zhang, B., Liu, Y., Prasad, S.: Dynamics of a vibro-impact self-propelled capsule encountering a circular fold in the small intestine. Meccanica 58, 451–472 (2023). https://doi.org/10.1007/s11012-022-01528-2

    Article  MathSciNet  Google Scholar 

  52. Tian, J., Liu, Y., Prasad, S.: Exploring the dynamics of a vibro-impact capsule moving on the small intestine using finite element analysis. In: Lacarbonara, W., Balachandran, B., Leamy, M.J., Ma, J., Tenreiro Machado, J.A., Stepan, G. (eds.) Advances in Nonlinear Dynamics, pp. 127–136. Springer, Cham (2022)

    Chapter  Google Scholar 

  53. Liao, M., Zhang, J., Liu, Y., Zhu, D.: Speed optimisation and reliability analysis of a self-propelled capsule robot moving in an uncertain frictional environment. Int. J. Mech. Sci. 221, 107156 (2022). https://doi.org/10.1016/j.ijmecsci.2022.107156

    Article  Google Scholar 

  54. Zhang, J., Liu, Y., Zhu, D., Prasad, S., Liu, C.: Simulation and experimental studies of a vibro-impact capsule system driven by an external magnetic field. Nonlinear Dyn. 109, 1501–1516 (2022). https://doi.org/10.1007/s11071-022-07539-8

    Article  Google Scholar 

  55. Liu, Y., Chavez, J.P., Zhang, J., Tian, J., Guo, B., Prasad, S.: The vibro-impact capsule system in millimetre scale: numerical optimisation and experimental verification. Meccanica 55, 1885–1902 (2020). https://doi.org/10.1007/s11012-020-01237-8

    Article  MathSciNet  Google Scholar 

  56. Guo, B., Liu, Y., Prasad, S.: Modelling of capsule-intestine contact for a self-propelled capsule robot via experimental and numerical investigation. Nonlinear Dyn. 98, 3155–3167 (2019). https://doi.org/10.1007/s11071-019-05061-y

  57. Guo, B., Ley, E., Tian, J., Zhang, J., Liu, Y., Prasad, S.: Experimental and numerical studies of intestinal frictions for propulsive force optimisation of a vibro-impact capsule system. Nonlinear Dyn. 101, 65–83 (2020). https://doi.org/10.1007/s11071-020-05767-4

    Article  Google Scholar 

  58. Gidoni, P., DeSimone, A.: Stasis domains and slip surfaces in the locomotion of a bio-inspired two-segment crawler. Meccanica 52, 587–601 (2017). https://doi.org/10.1007/s11012-016-0408-0

    Article  MathSciNet  Google Scholar 

  59. Zhao, Y., Fang, H., Xu, J.: Dynamics and phase coordination of multi-module vibration-driven locomotion robots with linear or nonlinear connections. Meccanica 58, 509–535 (2023). https://doi.org/10.1007/s11012-022-01623-4

  60. Figurina, T.: On the periodic motion of a two-body system upward along an inclined straight line with dry friction. MATHMOD 2018 extended abstract volume, ARGESIM Report 55, 13–14 (2018). https://doi.org/10.11128/arep.55.a55151

Download references

Acknowledgements

We appreciate Prof. Bolotnik N.N. for reading the manuscript and useful comments.

Funding

The study is supported by Russian Foundation for Basic Research (RFBR) and German Research Foundation (DFG), grant No. 21-51-12004, and by the state program No. 123021700055-6.

Author information

Authors and Affiliations

  1. Laboratory of Control of Mechanical Systems, Ishlinsky Institute for Problems in Mechanics RAS, Prospekt Vernadskogo 101-1, Moscow, Russia, 119526

    Tatiana Figurina & Dmitri Knyazkov

Authors
  1. Tatiana Figurina
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dmitri Knyazkov
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

TF and DK contributed equally to this work.

Corresponding author

Correspondence to Dmitri Knyazkov.

Ethics declarations

Conflict of interest

The authors have no relevant financial or non-financial interests to disclose.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Figurina, T., Knyazkov, D. Motion of a system of interacting bodies in a medium with quadratic resistance. Nonlinear Dyn 112, 273–288 (2024). https://doi.org/10.1007/s11071-023-09048-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09048-8

Keywords

Search

Navigation