Hostname: page-component-f554764f5-44mx8 Total loading time: 0 Render date: 2025-04-22T03:08:14.341Z Has data issue: false hasContentIssue false
  • English
  • Français

Nonlinear dynamics of a buoyant pendulum submerged under surface gravity waves

Published online by Cambridge University Press:  21 April 2025

Md Shadab Hasan ,
P. Deepu Open the ORCID record for P. Deepu [Opens in a new window]  and
Kamlesh Kumar
Md Shadab Hasan
Affiliation:
Indian Institute of Technology, Patna 801103 Bihar, India
P. Deepu*
Affiliation:
Indian Institute of Technology, Patna 801103 Bihar, India
Kamlesh Kumar
Affiliation:
Indian Institute of Technology, Patna 801103 Bihar, India
*
Corresponding author: P. Deepu, [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

This study investigates the hydrodynamic interaction between a fully submerged buoyant pendulum and surface gravity waves, focusing on its primary and subharmonic resonance behaviour. The oscillatory motion of the pendulum is driven by fluid drag, with primary resonance occurring at the forcing frequency (viz. the wave frequency) and subharmonic resonance manifesting at half the forcing frequency. Both resonances exhibit nonlinear characteristics, including jump-up, jump-down phenomena and hysteresis. Furthermore, particle image velocimetry results reveal that the velocity fields of the surrounding fluid oscillate at the forcing frequency, confirming that subharmonic resonance is not induced by subharmonic excitation within the velocity field. Experimental observations are validated through both analytical and numerical methods, particularly within the primary and subharmonic resonance frequency ranges. The theoretical model describes the transverse motion of the pendulum using a nonlinear ordinary differential equation, with the method of multiple scales employed for the analytical solution. These analyses reveal the nonlinear characteristics of the system, e.g. bistable response of the primary/subharmonic resonances, and identify three distinct response regions based on the forcing frequency and amplitude. The system exhibits primary resonance regardless of the excitation strength; however, an unstable solution arises if the excitation level surpasses a specific threshold value. In contrast, subharmonic resonance is triggered only when the excitation amplitude exceeds a critical value. Furthermore, the experimental hysteresis curve confirms the theoretically predicted primary and subharmonic resonances, along with the jump-up and jump-down characteristics.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1009 , 25 April 2025 , A45
Check for updates
Copyright
© The Author(s), 2025. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

Footnotes

*

Department of Biomedical Engineering, Worcester Polytechnic Institute, MA 01609, USA

References

Aashish, R., Deepu, P. & Kumar, S. 2024 Dynamics of a buoyant collapsible tube under a free surface. Exp. Therm. Fluid Sci. 154, 111168.CrossRefGoogle Scholar
Bolster, D., Hershberger, R.E. & Donnelly, R.J. 2010 Oscillating pendulum decay by emission of vortex rings. Phys. Rev. E 81 (4), 046317.CrossRefGoogle ScholarPubMed
Byrtus, M., Hajžman, M. & Zeman, V. 2013 Linearization of friction effects in vibration of two rotating blades. Appl. Comput. Mech. 7 (1), 5–22.Google Scholar
Chizfahm, A. & Jaiman, R. 2021 Data-driven stability analysis and near-wake jet control for the vortex-induced vibration of a sphere. Phys. Fluids 33 (4).CrossRefGoogle Scholar
Eshbal, L., Krakovich, A. & van Hout, R. 2012 Time resolved measurements of vortex-induced vibrations of a positively buoyant tethered sphere in uniform water flow. J. Fluid. Struct. 35, 185–199.CrossRefGoogle Scholar
Faleiros, D.E., Tuinstra, M., Sciacchitano, A. & Scarano, F. 2021 The slip velocity of nearly neutrally buoyant tracers for large-scale piv. Exp. Fluids 62 (9), 186.CrossRefGoogle Scholar
Gold, T., Reiterer, K., Worf, D., Kaiblinger, N., Khosronejad, A., Habersack, H. & Sindelar, C. 2024 Dynamics of heavy subaqueous spherical pendulums. J. Fluid Mech. 978, A19.CrossRefGoogle Scholar
Gold, T., Reiterer, K., Worf, D., Khosronejad, A., Habersack, H. & Sindelar, C. 2023 Lagrangian particle tracking velocimetry investigation of vortex shedding topology for oscillating heavy spherical pendulums underwater. J. Fluid Mech. 960, A14.CrossRefGoogle Scholar
Govardhan, R.N. & Williamson, C.H.K. 2005 Vortex-induced vibrations of a sphere. J. Fluid Mech. 531, 11–47.CrossRefGoogle Scholar
Gudmestad, O.T. & Connor, J.J. 1983 Linearization methods and the influence of current on the nonlinear hydrodynamic drag force. Appl. Ocean Res. 5 (4), 184–194.CrossRefGoogle Scholar
Hasan, M.S., Hoskoti, L., Deepu, P. & M. Sucheendran, M. 2023 Nonlinear oscillations of a flexible fiber under gravity waves. Eur. Phys. J. Special Topics 232 (6), 867–876.CrossRefGoogle Scholar
Hashiehbaf, A. & Romano, G.P. 2020 A phase averaged piv study of circular and non-circular synthetic turbulent jets issuing from sharp edge orifices. Intl J. Heat Fluid Flow 82, 108536.CrossRefGoogle Scholar
He, C., Liu, Y. & Gan, L. 2020 Instantaneous pressure determination from unsteady velocity fields using adjoint-based sequential data assimilation. Phys. Fluids 32 (3), 035101.CrossRefGoogle Scholar
Huang, G., Xia, Y., Dai, Y., Yang, C. & Wu, Y. 2021 Fluid–structure interaction in piezoelectric energy harvesting of a membrane wing. Phys. Fluids 33 (6), 063610.CrossRefGoogle Scholar
Karanfilian, S.K. & Kotas, T.J. 1978 Drag on a sphere in unsteady motion in a liquid at rest. J. Fluid Mech. 87 (1), 85–96.CrossRefGoogle Scholar
Kumar, K., Kumar, V., Deepu, P. & Ramya, P. 2021 Oscillations of a flexible filament under surface gravity waves. Phys. Rev. Fluids 6 (11), 114004.CrossRefGoogle Scholar
Kundu, P.K., Cohen, I.M. & Dowling, D.R. 2015 Fluid Mechanics. Academic Press.Google Scholar
Lin, J.-C. & Rockwell, D. 1996 Force identification by vorticity fields: techniques based on flow imaging. J. Fluid. Struct. 10 (6), 663–668.CrossRefGoogle Scholar
Mathai, V., Loeffen, L.A.W.M., Chan, T.T.K. & Wildeman, S. 2019 Dynamics of heavy and buoyant underwater pendulums. J. Fluid Mech. 862, 348–363.CrossRefGoogle Scholar
McQueen, T., Zhao, J., Sheridan, J. & Thompson, M.C. 2020 Feedback control of flow-induced vibration of a sphere. J. Fluid Mech. 889, A30.CrossRefGoogle Scholar
Mongelli, M. & Battista, N.A. 2020 A swing of beauty: pendulums, fluids, forces, and computers. Fluids 5 (2), 48.CrossRefGoogle Scholar
Mousavisani, S., Samandari, H. & Seyed-Aghazadeh, B. 2024 Experimental investigation of flow-induced vibration and flow field characteristics of a flexible triangular cylinder. J. Fluid Mech. 979, A15.CrossRefGoogle Scholar
Mullarney, J.C. & Henderson, S.M. 2010 Wave-forced motion of submerged single-stem vegetation. J. Geophys. Res.: Oceans 115 (C12), C12061.CrossRefGoogle Scholar
Nayfeh, A.H. & Mook, D.T. 2008 Nonlinear Oscillations. John Wiley & Sons.Google Scholar
Noca, F., Shiels, D. & Jeon, D. 1997 Measuring instantaneous fluid dynamic forces on bodies, using only velocity fields and their derivatives. J. Fluid. Struct. 11 (3), 345–350.CrossRefGoogle Scholar
Rajamuni, M.M., Hourigan, K. & Thompson, M.C. 2021 Vortex-induced vibration of a sphere close to or piercing a free surface. J. Fluid Mech. 929, A41.CrossRefGoogle Scholar
Rajamuni, M.M., Thompson, M.C. & Hourigan, K. 2018 Transverse flow-induced vibrations of a sphere. J. Fluid Mech. 837, 931–966.CrossRefGoogle Scholar
Rand, R.H. 2005 Lecture notes on nonlinear vibrations (version 52). Available from: http://audiophile.tam.cornell.edu/randdocs/nlvibe52.pdf.Google Scholar
Sanchez-Sanz, M., Fernandez, B. & Velazquez, A. 2009 Energy-harvesting microresonator based on the forces generated by the Kármán street around a rectangular prism. J. Microelectromech. Syst. 18 (2), 449–457.CrossRefGoogle Scholar
Sareen, A., Zhao, J., Sheridan, J., Hourigan, K. & Thompson, M.C. 2018 Vortex-induced vibrations of a sphere close to a free surface. J. Fluid Mech 846, 1023–1058.CrossRefGoogle Scholar
Strogatz, S.H. 2018 Nonlinear Dynamics and Chaos: with Applications to Physics, Biology, Chemistry, and Engineering. CRC Press.CrossRefGoogle Scholar
Thomsen, J.J., Thomsen, J.J. & Thomsen, J.J. 2003 Vibrations and Stability. vol. 2, Springer.CrossRefGoogle Scholar
Tondl, A. & Nabergoj, R. 1994 Non-periodic and chaotic vibrations in a flow induced system. Chaos, Solitons Fractals 4 (12), 2193–2202.CrossRefGoogle Scholar
Van Hout, R., Katz, A. & Greenblatt, D. 2013 Time-resolved particle image velocimetry measurements of vortex and shear layer dynamics in the near wake of a tethered sphere. Phys. Fluids 25 (7), 077102 .CrossRefGoogle Scholar
Wang, T. 2023 Pendulum-based vibration energy harvesting: mechanisms, transducer integration, and applications. Energy Convers. Manage. 276, 116469.CrossRefGoogle Scholar
Wong, K.W.L., Zhao, J., Lo Jacono, D., Thompson, M.C. & Sheridan, J. 2017 Experimental investigation of flow-induced vibration of a rotating circular cylinder. J. Fluid Mech. 829, 486–511.CrossRefGoogle Scholar
Zamanzadeh, M., Ouakad, H.M. & Azizi, S. 2020 Theoretical and experimental investigations of the primary and parametric resonances in repulsive force based mems actuators. Sensors Actuators A: Phys. 303, 111635.CrossRefGoogle Scholar
Supplementary material: File

Hasan et al. supplementary material movie 1

Movie showing response of the pendulum near primary resonance.
Download Hasan et al. supplementary material movie 1(File)
File 5.2 MB
Supplementary material: File

Hasan et al. supplementary material movie 2

Movie showing response of the pendulum near subharmonic resonance.
Download Hasan et al. supplementary material movie 2(File)
File 4.8 MB