Skip to main content
SpringerLink
Log in

Wave-induced dynamics of a particle on a thin circular plate

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

Abstract

In this paper, the dynamics of a particle placed on a thin circular plate carrying circumferential harmonic travelling wave is studied. Coulomb friction is used to model the particle–surface interaction. Distinct regions on the plate surface are identified where either of the three phases of particle motion, namely jumping, sliding and sticking, occurs. Also, the effect of wave frequency and the plate geometry on these regions is studied. Interestingly, there exists an optimum plate thickness for which the region of sliding is maximum. At certain wave frequencies, from the numerical simulations within sticking and sliding regions, it is observed that the average particle motion spirals inwards towards the plate centre. Such an average motion is observed whenever the particle is placed initially with a zero velocity relative to the plate surface. The Gedanken experiments discussed herein provide cogent explanations to all the observed average (slow) dynamics and are also found to be useful in predicting the slow dynamics of the particle a priori, that is, before the actual numerical simulations. The particle’s velocity couples the radial and tangential sliding friction components and is found to be the key physical feature that explains the observed behaviour. Also, it is observed that the plate surface excited by circumferential travelling waves can provide acoustic lubrication to a particle by reducing the limiting force required to move it relative to the surface. The methods discussed in this paper can be extended to study the dynamics of a group of particles (granular materials) and extended rigid bodies, interacting with such surface waves.

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
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16
Fig. 17

Similar content being viewed by others

References

  1. Aranson, I.S., Tsimring, L.S.: Patterns and collective behavior in granular media: theoretical concepts. Rev. Mod. Phys. 78, 641–692 (2006). https://doi.org/10.1103/RevModPhys.78.641

    Article  Google Scholar 

  2. Avirovik, D., Malladi, V.V.N.S., Priya, S., Tarazaga, P.A.: Theoretical and experimental correlation of mechanical wave formation on beams. J. Intell. Mater. Syst. Struct. 27(14), 1939–1948 (2016). https://doi.org/10.1177/1045389X15615967

    Article  Google Scholar 

  3. Blekhman, I., Dzhanelydze, G.Y.: Vibratsionnoye peremeshtchenie (Vibrational Transport). Nauka, Moscow (1964)

    Google Scholar 

  4. Blekhman, I.I.: Vibrational Mechanics: Nonlinear Dynamic Effects, General Approach, Applications. World Scientific, Farrer Road, Singapore (2000)

    Book  Google Scholar 

  5. Bucher, I., Setter, E.: A micro-scale swimmer propelled by traveling surface waves. In: Volume 7: 5th International Conference on Micro- and Nanosystems; 8th International Conference on Design and Design Education; 21st Reliability, Stress Analysis, and Failure Prevention Conference. Washington, DC, USA, pp. 101–106 (2011)

  6. Bucher, I.: Estimating the ratio between travelling and standing vibration waves under non-stationary conditions. J. Sound Vib. 270(1), 341–359 (2004). https://doi.org/10.1016/S0022-460X(03)00539-X

    Article  Google Scholar 

  7. Buguin, A., Brochard, F., de Gennes, P.G.: Motions induced by asymmetric vibrations. Eur. Phys. J. E 19(1), 31–36 (2006). https://doi.org/10.1140/epje/e2006-00013-8

    Article  Google Scholar 

  8. Daniel, S., Chaudhury, M.K., De Gennes, P.G.: Vibration-actuated drop motion on surfaces for batch microfluidic processes. Langmuir 21(9), 4240–4248 (2005)

    Article  Google Scholar 

  9. de Boer, M.P., Luck, D.L., Ashurst, W.R., Maboudian, R., Corwin, A.D., Walraven, J.A., Redmond, J.M.: High-performance surface-micromachined inchworm actuator. J. Microelectromech. Syst. 13(1), 63–74 (2004). https://doi.org/10.1109/JMEMS.2003.823236

    Article  Google Scholar 

  10. Deb Singha, T., DasGupta, A.: Theoretical and experimental study of vibration induced directed transport of particles on a rigid surface. In: 25th International Congress on Sound and Vibration (ICSV25) (2018)

  11. Denisov, G., Novilov, V., Smirnova, M.: The momentum of waves and their effect on the motion of lumped objects along one-dimensional elastic systems. J. Appl. Math. Mech. 76(2), 225–234 (2012). https://doi.org/10.1016/j.jappmathmech.2012.05.014

    Article  MathSciNet  MATH  Google Scholar 

  12. Derendyayev, N., Soldatov, I.: The motion of a point mass along a vibrating string. J. Appl. Math. Mech. 61(4), 681–684 (1997). https://doi.org/10.1016/S0021-8928(97)00086-5

    Article  Google Scholar 

  13. Dong, L., Chaudhury, A., Chaudhury, M.K.: Lateral vibration of a water drop and its motion on a vibrating surface. Eur. Phys. J. E 21(3), 231–242 (2006). https://doi.org/10.1140/epje/i2006-10063-7

    Article  Google Scholar 

  14. Erdész, K., Szalay, A.: Experimental study on the vibrational transport of bulk solids. Powder Technol. 55(2), 87–96 (1988). https://doi.org/10.1016/0032-5910(88)80091-3

    Article  Google Scholar 

  15. Ferretti, M., Gavrilov, S.N., Eremeyev, V.A., Luongo, A.: Nonlinear planar modeling of massive taut strings travelled by a force-driven point-mass. Nonlinear Dyn. 97(4), 2201–2218 (2019). https://doi.org/10.1007/s11071-019-05117-z

    Article  MATH  Google Scholar 

  16. Ferretti, M., Piccardo, G., dell’Isola, F., Luongo, A.: Dynamics of taut strings undergoing large changes of tension caused by a force-driven traveling mass. J. Sound Vib. 458, 320–333 (2019). https://doi.org/10.1016/j.jsv.2019.06.035

    Article  Google Scholar 

  17. Flach, S., Yevtushenko, O., Zolotaryuk, Y.: Directed current due to broken time-space symmetry. Phys. Rev. Lett. 84, 2358–2361 (2000). https://doi.org/10.1103/PhysRevLett.84.2358

    Article  Google Scholar 

  18. Fleishman, D., Asscher, Y., Urbakh, M.: Directed transport induced by asymmetric surface vibrations: making use of friction. J. Phys. Condens. Matter 19(9), 096004 (2007). https://doi.org/10.1088/0953-8984/19/9/096004

    Article  Google Scholar 

  19. Fleishman, D., Filippov, A.E., Urbakh, M.: Directed molecular transport in an oscillating symmetric channel. Phys. Rev. E 69, 011908 (2004). https://doi.org/10.1103/PhysRevE.69.011908

    Article  Google Scholar 

  20. Gabai, R., Bucher, I.: Excitation and sensing of multiple vibrating traveling waves in one-dimensional structures. J. Sound Vib. 319(1), 406–425 (2009). https://doi.org/10.1016/j.jsv.2008.06.013

    Article  Google Scholar 

  21. Gabai, R., Bucher, I.: Spatial and temporal excitation to generate traveling waves in structures. J. Appl. Mech. 77(2), 021010 (2009). https://doi.org/10.1115/1.3176999

    Article  Google Scholar 

  22. Gabai, R., Ilssar, D., Shaham, R., Cohen, N., Bucher, I.: A rotational traveling wave based levitation device—modelling, design, and control. Sens. Actuators A 255, 34–45 (2017). https://doi.org/10.1016/j.sna.2016.12.016

    Article  Google Scholar 

  23. Gavrilov, S.: Nonlinear investigation of the possibility to exceed the critical speed by a load on a string. Acta Mech. 154(1), 47–60 (2002). https://doi.org/10.1007/BF01170698

    Article  MATH  Google Scholar 

  24. Gavrilov, S.: The effective mass of a point mass moving along a string on a winkler foundation. J. Appl. Math. Mech. 70(4), 582–589 (2006). https://doi.org/10.1016/j.jappmathmech.2006.09.009

    Article  MathSciNet  Google Scholar 

  25. Gavrilov, S.N., Eremeyev, V.A., Piccardo, G., Luongo, A.: A revisitation of the paradox of discontinuous trajectory for a mass particle moving on a taut string. Nonlinear Dyn. 86(4), 2245–2260 (2016). https://doi.org/10.1007/s11071-016-3080-y

    Article  MathSciNet  MATH  Google Scholar 

  26. Golovanevskiy, V.A., Arsentyev, V.A., Blekhman, I.I., Vasilkov, V.B., Azbel, Y.I., Yakimova, K.S.: Vibration-induced phenomena in bulk granular materials. Int. J. Miner. Process. 100(3), 79–85 (2011). https://doi.org/10.1016/j.minpro.2011.05.001

    Article  Google Scholar 

  27. Goohpattader, P.S., Mettu, S., Chaudhury, M.K.: Stochastic rolling of a rigid sphere in weak adhesive contact with a soft substrate. Eur. Phys. J. E 34(11), 120 (2011). https://doi.org/10.1140/epje/i2011-11120-x

    Article  Google Scholar 

  28. Hagedorn, P., Wallaschek, J.: Travelling wave ultrasonic motors, part i: working principle and mathematical modelling of the stator. J. Sound Vib. 155(1), 31–46 (1992). https://doi.org/10.1016/0022-460X(92)90643-C

    Article  Google Scholar 

  29. Hashimoto, Y., Koike, Y., Ueha, S.: Near-field acoustic levitation of planar specimens using flexural vibration. J. Acoust. Soc. Am. 100(4), 2057–2061 (1996)

    Article  Google Scholar 

  30. Hashimoto, Y., Koike, Y., Ueha, S.: Transporting objects without contact using flexural traveling waves. J. Acoust. Soc. Am. 103(6), 3230–3233 (1998). https://doi.org/10.1121/1.423039

    Article  Google Scholar 

  31. Havelock, T.: Some dynamical illustrations of the pressure of radiation and of adiabatic invariance. Lond. Edinb. Dublin Philos. Mag. J. Sci. 47(280), 754–771 (1924). https://doi.org/10.1080/14786442408634415

    Article  Google Scholar 

  32. Kumar, A., DasGupta, A.: Generation of harmonic waves in beams using boundary excitation. Int. J. Mech. Sci. 159, 234–245 (2019). https://doi.org/10.1016/j.ijmecsci.2019.05.021

    Article  Google Scholar 

  33. Kumar, A., DasGupta, A.: Generation of circumferential harmonic travelling waves on thin circular plates. J. Sound Vib. 478, 115343 (2020). https://doi.org/10.1016/j.jsv.2020.115343

    Article  Google Scholar 

  34. Malladi, V.V.N.S., Albakri, M., Tarazaga, P.A.: An experimental and theoretical study of two-dimensional traveling waves in plates. J. Intell. Mater. Syst. Struct. 28(13), 1803–1815 (2017). https://doi.org/10.1177/1045389X16679284

    Article  Google Scholar 

  35. Malladi, V.V.N.S., Albakri, M.I., Gugercin, S., Tarazaga, P.A.: Application of projection-based model reduction to finite-element plate models for two-dimensional traveling waves. J. Intell. Mater. Syst. Struct. 28(14), 1886–1904 (2017). https://doi.org/10.1177/1045389X16679295

    Article  Google Scholar 

  36. Malladi, V.V.N.S., Avirovik, D., Priya, S., Tarazaga, P.A.: Travelling wave phenomenon through a piezoelectric actuation on a free-free beam. In: ASME 2014 Conference on Smart Materials, Adaptive Structures and Intelligent Systems, Volume 1: Development and Characterization of Multifunctional Materials; Modeling, Simulation and Control of Adaptive Systems; Structural Health Monitoring; Keynote Presentation. American Society of Mechanical Engineers Digital Collection (2014). https://doi.org/10.1115/SMASIS2014-7529

  37. Mettu, S., Chaudhury, M.K.: Motion of liquid drops on surfaces induced by asymmetric vibration: role of contact angle hysteresis. Langmuir 27(16), 10327–10333 (2011). https://doi.org/10.1021/la201597c

    Article  Google Scholar 

  38. Mingjie, D., Koyama, D., Nakamura, K.: Noncontact ultrasonic transportation of droplet using an acoustic waveguide. In: 2012 IEEE International Ultrasonics Symposium, pp. 1990–1993 (2012). https://doi.org/10.1109/ULTSYM.2012.0498

  39. Mracek, M., Wallaschek, J.: A system for powder transport based on piezoelectrically excited ultrasonic progressive waves. Mater. Chem. Phys. 90(2), 378–380 (2005). https://doi.org/10.1016/j.matchemphys.2004.09.048

    Article  Google Scholar 

  40. Nicolai, E.: On a dynamical illustration of the pressure of radiation. Lond. Edinb. Dublin Philos. Mag. J. Sci. 49(289), 171–177 (1925). https://doi.org/10.1080/14786442508634593

    Article  MATH  Google Scholar 

  41. Pohl, D.W.: Dynamic piezoelectric translation devices. Rev. Sci. Instrum. 58(1), 54–57 (1987). https://doi.org/10.1063/1.1139566

    Article  Google Scholar 

  42. Rademacher, F.J.C., ter Borg, L.: On the theoretical and experimental conveying speed of granular bulk solids on vibratory conveyors. Forsch. Ingenieurwes. 60(10), 261–283 (1994)

    Article  Google Scholar 

  43. Rayleigh, L.: On the pressure of vibrations. Lond. Edinb. Dublin Philos. Mag. J. Sci. 3(15), 338–346 (1902). https://doi.org/10.1080/14786440209462769

    Article  MATH  Google Scholar 

  44. Reznik, D., Canny, J.: A flat rigid plate is a universal planar manipulator. In: Proceedings. 1998 IEEE International Conference on Robotics and Automation (Cat. No. 98CH36146), vol. 2, pp. 1471–1477. IEEE, Leuven, Belgium (1998)

  45. Reznik, D., Canny, J.F.: C’mon part, do the local motion! In: Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No.01CH37164), vol. 3, pp. 2235–2242. Seoul, South Korea (2001). https://doi.org/10.1109/ROBOT.2001.932955

  46. Reznik, D., Canny, J.F., Goldberg, K.Y.: Analysis of part motion on a longitudinally vibrating plate. In: Proceedings of the 1997 IEEE/RSJ International Conference on Intelligent Robot and Systems. Innovative Robotics for Real-World Applications. IROS’97, vol. 1, pp. 421–427 (1997)

  47. Ruhela, G., DasGupta, A.: Hopping on a wave: from periodic to chaotic transport. Nonlinear Dyn. (2016). https://doi.org/10.1007/s11071-016-2984-x

    Article  MathSciNet  Google Scholar 

  48. Ruhela, G., Dasgupta, A.: Motion periodicity and bifurcation of a wave excited hopping ball. Proc. R. Soc. A. 475, 20190137 (2019). https://doi.org/10.1098/rspa.2019.0137

    Article  MathSciNet  Google Scholar 

  49. Setter, E., Bucher, I.: Flexural vibration patterning using an array of actuators. J. Sound Vib. 330(6), 1121–1140 (2011). https://doi.org/10.1016/j.jsv.2010.09.027

    Article  Google Scholar 

  50. Setter, E., Bucher, I., Sas, P., Bergen, B.: Elastic travelling waves in multi-dimensional structures with application to self propulsion. In: Proceedings of International Conference on Noise and Vibration Engineering (ISMA2010), pp. 3785–3800 (2010)

  51. Sloot, E., Kruyt, N.: Theoretical and experimental study of the transport of granular materials by inclined vibratory conveyors. Powder Technol. 87(3), 203–210 (1996). https://doi.org/10.1016/0032-5910(96)03091-4

    Article  Google Scholar 

  52. Takano, T., Tomikawa, Y.: Excitation of a progressive wave in a lossy ultrasonic transmission line and an application to a powder-feeding device. Smart Mater. Struct. 7(3), 417–421 (1998). https://doi.org/10.1088/0964-1726/7/3/016

    Article  Google Scholar 

  53. Thomas, G.P., Andrade, M.A., Adamowski, J.C., Silva, E.C.: Acoustic levitation transportation of small objects using a ring-type vibrator. Phys. Proc. 70, 59–62 (2015). https://doi.org/10.1016/j.phpro.2015.08.041. (proceedings of the 2015 ICU International Congress on Ultrasonics, Metz, France)

    Article  Google Scholar 

  54. Verma, N., DasGupta, A.: Particle current on flexible surfaces excited by harmonic waves. Phys. Rev. E 88(5), 052915 (2013)

    Article  Google Scholar 

  55. Viswarupachari, C., DasGupta, A., Pratik Khastgir, S.: Vibration induced directed transport of particles. J. Vib. Acoust. 134, 5 (2012). https://doi.org/10.1115/1.4006412.051005

    Article  Google Scholar 

  56. Vose, T.H., Umbanhowar, P., Lynch, K.M.: Friction-induced velocity fields for point parts sliding on a rigid oscillated plate. Int. J. Robot. Res. 28(8), 1020–1039 (2009). https://doi.org/10.1177/0278364909340279

    Article  Google Scholar 

  57. Zhou, Q., Sariola, V., Latifi, K., Liimatainen, V.: Controlling the motion of multiple objects on a chladni plate. Nat. Commun. 7(1), 1–10 (2016)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mechanical Engineering, and Center for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Aman Kumar & Anirvan DasGupta

Authors
  1. Aman Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Anirvan DasGupta
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Aman Kumar.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Kumar, A., DasGupta, A. Wave-induced dynamics of a particle on a thin circular plate. Nonlinear Dyn 103, 293–308 (2021). https://doi.org/10.1007/s11071-020-06158-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-020-06158-5

Keywords

Search

Navigation